equivariant sheaves on flag varieties, dg modules and

Equivariant Sheaves on Flag

Varieties, DG Modules

and Formality

Dissertation zur Erlangung des Doktorgrades

der Fakultat fur Mathematik und Physik

der Albert-Ludwigs-Universitat

Freiburg im Breisgau

vorgelegt von

Olaf M. Schnurer

Dezember 2007

Dekan: Prof. Dr. Jorg Flum

1. Gutachter: Prof. Dr. Wolfgang Soergel

2. Gutachter: Prof. Dr. Valery Lunts

Datum der Promotion: 10. Marz 2008

Meinen Eltern

Contents

Chapter 1. Introduction 3Danksagung 8

Chapter 2. Formality of Derived Categories 112.1. Differential Graded Modules 112.2. Differential Graded Graded Algebras and Formality 132.3. Subcategories of Triangulated Categories 162.4. Derived Categories and DG Modules 162.5. Sheaves and Perverse Sheaves 192.6. Mixed Hodge Structures 212.7. Mixed Hodge Modules 222.8. Construction of Epimorphisms from Projective Objects 252.9. Existence of Enough Perverse-Projective Hodge Sheaves 272.10. Comparison of Mixed Hodge Structures 322.11. Local-to-Global Spectral Sequence 342.12. Purity 382.13. Formality of some DG Algebras 392.14. Formality of Cell-Stratified Varieties 412.15. Formality and Intersection Cohomology Sheaves 422.16. Formality of Partial Flag Varieties 44

Chapter 3. Formality and Closed Embeddings 473.1. The Goal of the Chapter 473.2. Formality and Normally Nonsingular Inclusions 503.3. Derived Category of (Perverse) Sheaves 513.4. Closed Embeddings 523.5. Tensor Product with a DG Bimodule 533.6. Passage from Geometry to DG Modules 543.7. DG Bimodules and Transformations 563.8. DGG Bimodules 583.9. Triangulated Functors on Objects 583.10. Restriction of Projective Objects 603.11. Passage to Cohomology Algebras 623.12. Passage to Extension Algebras 64

Chapter 4. t-Structures, Koszul Duality and Limits 674.1. t-Structures 674.2. Minimal Homotopically Projective Modules 72

1

2 CONTENTS

4.3. Heart 744.4. Koszul-Duality 774.5. Inverse Limits of Categories 804.6. Inverse Limits of Categories of DG Modules 83

Chapter 5. Formality of Equivariant Derived Categories 875.1. Acyclic Maps 875.2. Equivariant Derived Categories of Topological Spaces 885.3. Equivariant Derived Categories of Varieties 915.4. Equivariant Intersection Cohomology Complexes 935.5. Better Approximations and Formality 955.6. Existence and Examples of ABC-Approximations 965.7. Formality of Equivariant Flag Varieties 102

Bibliography 103

CHAPTER 1

Introduction

Let G be a complex connected reductive affine algebraic group andB ⊂ P ⊂ G a Borel and a parabolic subgroup. The main result ofthis thesis is an algebraic description of the B-equivariant (bounded,constructible) derived category Db

B,c(X) (see [BL94]) of sheaves of realvector spaces on the partial flag variety X := G/P .

Let S be the stratification ofX intoB-orbits and ICB(S) ∈ DbB,c(X)

the equivariant intersection cohomology complex of the closure of thestratum S ∈ S. The (ICB(S))S∈S are the simple equivariant perversesheaves onX. Let ICB(S) be their direct sum and E = Ext(ICB(S)) itsgraded algebra of self-extensions in Db

B,c(X). We consider E as a differ-ential graded (dg) algebra with differential d = 0. Let dgDer(E) be thederived category of (right) dg E-modules (see [Kel94]) and dgPer(E)the smallest strict full triangulated subcategory of dgDer(E) contain-ing E and closed under forming direct summands. We give alternativedescriptions of dgPer(E) below.

Theorem 1 (cf. Theorem 107). There is an equivalence of trian-gulated categories

DbB,c(X) ∼= dgPer(Ext(ICB(S))).

Similar equivalences between equivariant derived categories andcategories of dg modules over the extension algebra of the simple equi-variant perverse sheaves are known for a connected Lie group acting ona point ([BL94, 12.7.2]), for a torus acting on an affine or projectivenormal toric variety ([Lun95]), and for a complex semisimple adjointgroup acting on a smooth complete symmetric variety ([Gui05]). Thekey point in the proof of these equivalences is the formality of some dgalgebra whose cohomology is the extension algebra.

Conjecturally ([Lun95, 0.1.3], [Soe01, 4]), the analog of Theorem 1should hold for the equivariant derived category of a complex reductivegroup acting on a projective variety with a finite number of orbits.(Theorem 1 is a special case of this conjecture since Db

G,c(G×BX) and

DbB,c(X) are equivalent by the induction equivalence.)

Let Db(X) be the (bounded) derived category of sheaves of realvector spaces on X = G/P , and Db(X,S) the full subcategory ofS-constructible objects. Let IC(S) be the direct sum of the (non-equivariant) simple S-constructible perverse sheaves on X, and F =

3

4 1. INTRODUCTION

Ext(IC(S)) its graded algebra of self-extensions in Db(X). The cate-gory dgPer(F) is defined similarly as dgPer(E) above. In the course ofthe proof of Theorem 1 we obtain the following non-equivariant analog.

Theorem 2 (cf. Theorem 43). There is an equivalence of triangu-lated categories

Db(X,S) ∼= dgPer(Ext(IC(S))).

The category PervB(X) of equivariant perverse sheaves on X isthe heart of the perverse t-structure on Db

B,c(X), and similarly for

Perv(X,S) ⊂ Db(X,S). The corresponding t-structure on dgPer(E)and dgPer(F) can be defined for a more general class of dg algebras;we explain this below. It turns out that the heart of such a t-structure isequivalent to a full abelian subcategory dgFlag of the abelian categoryof dg modules. The equivalences in Theorems 1 and 2 are in fact t-exactand induce equivalences

PervB(X) ∼= dgFlag(Ext(ICB(S))),

Perv(X,S) ∼= dgFlag(Ext(IC(S))),

i. e. algebraic descriptions of the categories of (equivariant) perversesheaves. The simple object ICB(S) is mapped to eSE (where eS ∈ E isthe projector from ICB(S) onto the direct summand ICB(S)), which isan indecomposable projective dg E-module. This seems to be part of aKoszul duality (cf. [BGS96, 1.2.6]). We have not yet tried to describethe standard t-structures on Db

B,c(X) and Db(X,S) algebraically.

The forgetful functor For : DbB,c(X) → Db(X,S) induces a sur-

jective morphism E → F of dg algebras and an extension of scalars

functor (?L⊗E F) : dgPer(E) → dgPer(F). These two functors provide

a connection between the equivalences in Theorems 1 and 2, i. e. thereis a commutative (up to natural isomorphism) square (see Remark 109)

DbB,c(X)

∼ //

For��

dgPer(Ext(ICB(S)))

(?L⊗EF)

��Db(X,S)

∼ // dgPer(Ext(IC(S))).

Assume for this paragraph that B = P and that we work withsheaves of complex vector spaces. The analogously defined extensionalgebras EC and FC are isomorphic to morphism spaces of Soergel’sbimodules (see [Soe01, Soe92, Soe90]). (The analogous descriptionalso works for real coefficients.) These bimodules are isomorphic to the(B-equivariant) intersection cohomologies of Schubert varieties and canbe described using the moment graph picture (see [BM01]). If T ⊂ Bis a maximal torus, the moment graph of the T -variety X = G/Bwith the stratification S is constructed from the following datum: T

1. INTRODUCTION 5

acting on the one-skeleton (the union of the fixed points and the one-dimensional orbits), and the Bruhat order on the fixed points. If theanalogs of Theorems 1 and 2 hold for sheaves with complex coeffi-cients, this implies that the corresponding (equivariant) derived cate-gories only depend on this datum.

Let us now comment on the purely algebraic result concerning t-structures mentioned above. Let A = (A =

⊕i≥0A

i, d) be a positively

graded dg algebra with A0 a semisimple ring and d(A0) = 0 (i. e. A0 is adg subalgebra). Let (Lx)x∈W be the finite collection of non-isomorphicsimple (right) A0-modules, and dgPrae(A) the smallest strict full trian-gulated subcategory of the derived category dgDer(A) of dg A-modules

that contains all Lx := Lx ⊗A0 A (where Lx is concentrated in degreezero). Let dgMod(A) be the abelian category of dg A-modules, anddgFlag(A) the full subcategory of dgMod(A) consisting of objects that

have an Lx-flag, i. e. a finite filtration with subquotients in {Lx}x∈W .

Theorem 3 (cf. Theorem 65, Propositions 71 and 73).The category dgFlag(A) is a full abelian subcategory of dgMod(A),and there is a t-structure on dgPrae(A) so that the obvious functordgFlag(A) → dgPrae(A) induces an equivalence between dgFlag(A)and the heart of this t-structure.

The construction of this t-structure was motivated by a similarmethod in [BL94, 11.4], where A is a polynomial ring with generatorsin degree two. A related construction can be found in [BL06].

The existence of a bounded t-structure on dgPrae(A) implies thatdgPrae(A) is Karoubian (see [LC07]). Thus dgPrae(A) ⊂ dgDer(A) isclosed under taking direct summands, hence dgPrae(A) = dgPer(A),where dgPer(A) is the smallest strict full triangulated subcategory ofdgDer(A) containing A and closed under taking direct summands. Us-ing Ravenel-Neeman’s characterization of compact objects ([Rav84,Nee92]), Keller shows in [Kel98, 8] that the objects of dgPer(A) areprecisely the compact objects of dgDer(A).

This reasoning applies in particular to the dg algebras E and F de-fined above and gives alternative descriptions of dgPer(E) and dgPer(F).Likewise, the categories of dg modules appearing in the main equiva-lences of [Lun95] and [Gui05] are in fact of the form dgPer.

We prove that if the differential of A is zero and its underlyinggraded ring is a Koszul ring satisfying some finiteness conditions, thenthe heart of dgPer(A) is governed by the Koszul dual ring.

Let us describe in more detail our approach to prove Theorem 1. Atseveral points we could replace dgPrae by dgPer, but we find it morenatural to write dgPrae. We use notation from subsequent chapterswithout further explanation.

6 1. INTRODUCTION

Step 1 (see Chapter 2). Let X be a complex variety with a strati-fication T into cells (i. e. T ∼= CdT for each T ∈ T ). Under some purityassumptions explained below we will establish an equivalence

(1) Db(X, T ) ∼= dgPrae(Ext(IC(T )))

of triangulated categories, of which Theorem 2 is a special case. Theproof works as follows. Since T is a cell-stratification, there is anequivalence

Db(Perv(X, T ))∼−→ Db(X, T ).

There are enough projective objects in Perv(X, T ), so we find projec-tive resolutions PT → IC(T ) of finite length

. . .→ P−2T → P−1

T → P 0T → IC(T )→ 0.

Let P → IC(T ) be the direct sum of these resolutions and B = End (P )the dg algebra of endomorphisms of P . The functor Hom (P, ?) inducesan equivalence

Db(Perv(X, T ))∼−→ dgPraeB({eTB}T∈T ).

Note that the cohomology of B is isomorphic to Ext(IC(T )). Thus weobtain equivalence (1) if B is formal. In order to prove formality, weneed to choose the resolutions PT → IC(T ) more carefully.

Each IC(T ) is the underlying perverse sheaf of a mixed Hodge

module IC(T ) that is pure of weight dT . We construct resolutions

PT → IC(T ) in the category of mixed Hodge modules so that theunderlying resolutions PT → IC(T ) are projective resolutions as con-sidered above. From these resolutions we get a dg algebra of mixedHodge structures with underlying dg algebra B = End (P ). If each

IC(T ) is T -pure of weight dT (i. e. all restrictions to strata in T arepure of weight dT ), this additional structure on B enables us to con-struct a dg subalgebra Sub(B) of B and quasi-isomorphisms

B ←↩ Sub(B) � H(B)

of dg algebras, establishing the formality of B.We will need the following slightly more general statement than

equivalence (1), with essentially the same proof.

Theorem 4 (cf. Theorem 35). Let (X,S) be a stratified complexvariety with irreducible and simply connected strata. Let T be a cell-

stratification refining S. If IC(S) is T -pure of weight dS for each S ∈S, there is a triangulated equivalence

Db(X,S) ∼= dgPrae(Ext(IC(S))).

Step 2 (see Chapter 3). Let (X,S) and (Y, T ) be stratified complexvarieties with irreducible and simply connected strata. Let i : Y →X be a closed embedding so that S → T , S 7→ S ∩ Y , is bijective

1. INTRODUCTION 7

and i|S∩Y : S ∩ Y → S is a normally nonsingular inclusion of a fixedcodimension c for all S ∈ S. Then

(2) [−c]i∗(IC(S))∼−→ IC(S ∩ Y )

for all S ∈ S. If both stratifications S and T have compatible refine-ments by cell-stratifications satisfying the purity conditions of Theorem4, we obtain the vertical equivalences in diagram

(3) Db(X,S)[−c]i∗

//

∼��

Db(Y, T )

∼��

dgPrae(Ext(IC(S)))?

L⊗

Ext(IC(S))Ext(IC(T ))

// dgPrae(Ext(IC(T ))).

The extension of scalars functor in the lower row is induced by theisomorphisms (2). This diagram is commutative (up to natural iso-morphism). Unfortunately the proof is rather technical.

Step 3 (see Chapter 5). Let X = G/P be a partial flag varietywith stratification S into B-orbits as before. We construct a sequence

E0f0−→ E1

f1−→ E2 → . . .→ Enfn−→ En+1 → . . .

of resolutions pn : En → X of X satisfying several nice properties. Forexample, each pn is smooth and n-acyclic (in the classical topology), thequotient morphisms qn : En → En := B\En are Zariski locally trivialfiber bundles, and each Sn := {qn(p−1

n (S)) | S ∈ S} is a stratification

of En. The induced morphisms fn : En → En+1 define functors f∗n :

Db(En+1,Sn+1)→ Db(En,Sn), and we obtain a sequence of categorieswhose inverse limit is equivalent to the category we want to describe,

DbB,c(X) ∼= lim←−D

b(En,Sn).

Moreover, the morphisms fn satisfy the assumptions of Step 2 (in par-ticular, the stratifications Sn admit refinements where the purity con-ditions hold), and the obtained commutative diagrams of the form (3)provide an equivalence

lim←−Db(En,Sn)

∼−→ lim←− dgPrae(Ext(IC(Sn))).

Finally, the obvious morphisms E = Ext(ICB(S)) → Ext(IC(Sn)) ofdg algebras induce an equivalence

dgPrae(Ext(ICB(S)))∼−→ lim←− dgPrae(Ext(IC(Sn))).

Replacing dgPrae by dgPer finishes the sketch of proof of Theorem 1.

8 1. INTRODUCTION

The methods used to prove Theorem 1 will be developed in abroader context and may be applied to other situations. For exam-ple, we obtain analogs of this theorem for the following varieties withgroup actions:

• B as above acting on a Schubert variety in G/P ,• GLn or a parabolic subgroup of GLn acting on a point,• a connected solvable affine algebraic group acting on a point.

These are examples of groups acting on varieties where so called ABC-approximations (sequences of resolutions as considered in Step 3 above)exist. We set up the defining conditions for an ABC-approximation inan ad hoc manner. These conditions enable us to prove Theorem 1.We believe that there are variations suitable for other situations.

It would be nice to know whether the analog of Theorem 1 is truefor Db

Q,c(G/P ) if Q is a parabolic subgroup of G containing B. It is aconsequence of Theorem 43 that the non-equivariant version holds, i. e.we can replace the stratification S in Theorem 2 by the stratificationinto Q-orbits.

Let us finally mention a relation to the formality of compact Kahlermanifolds. If a nonsingular irreducible complex variety X is simplyconnected and has a cell-stratification, we can apply Theorem 4 to thetrivial stratification and obtain an equivalence

Db(X, {X}) ∼= dgPer(H(X; R)),

where H(X; R) is the real cohomology ring of X. This equivalence isalso true for a compact Kahler manifold X, since such a manifold is al-ways formal, i. e. its de Rham algebra is formal ([DGMS75, Huy05]).

This thesis is organized as follows: Chapters 2, 3 and 5 containessentially the results explained above in Steps 1, 2 and 3 respectively.Chapter 4 is independent of the previous chapters (except for somebasic results on dg modules). It contains the proof of Theorem 3,the interlude on Koszul rings, and some results on inverse limits ofcategories (of dg modules) used in Step 3.

Danksagung

Wolfgang Soergel hat mir das Thema dieser Arbeit vorgeschlagenund mich in vorzuglicher Weise betreut. Seine Begeisterung und seineIdeen haben entscheidenden Anteil am Entstehen dieser Arbeit. Ichdanke ihm, dass ich in einem so spannenden Gebiet forschen durfteund darf.

Ich danke Peter Fiebig, Geordie Williamson, Anne Balthasar, Mar-tin Entrup und Richard Vale fur anregende und hilfreiche mathema-tische Diskussionen, fur die Bereitschaft, ihr Wissen mit mir zu teilen,und fur die nette Atmosphare in der Arbeitsgruppe. Matthias Ansorge,Frank Ettwein, Scott Marsden, Hannes Junginger, Lars Diening, Ste-fan Suhr, Bijan Afshordel, Stephan Wehrheim, Bernhard Link und Olaf

DANKSAGUNG 9

Munsonius haben mir das Leben am Institut und außerhalb angenehmgemacht. Ich danke auch fur Hilfe am Computer.

Wahrend der Anfertigung dieser Arbeit durfte ich an vielen Kon-ferenzen und Schulen teilnehmen, u. a. in Utrecht, Oberwolfach, Lu-miny, Reisensburg, Wuppertal und Gottingen. Diese Abwechslungenzum Uni-Alltag habe ich sehr genossen.

Ich danke fur die finanzielle Unterstutzung, die ich als Promo-tionsstipendiat nach dem Landesgraduiertenforderungsgesetz erhaltenhabe.

Ich danke allen, die mich in meiner Freiburger Zeit als Freundebegleitet haben; stellvertretend seien Christine, Sebastian und Arnegenannt.

Meine Eltern haben mir viele Freiraume gewahrt und mich in jegli-cher Hinsicht unterstutzt. Ihnen, meinem Bruder Oliver und Luzie giltmein herzlicher Dank.

CHAPTER 2

Formality of Derived Categories

2.1. Differential Graded Modules

We review the language of differential graded (dg) modules over adg algebra (see [Kel94, Kel98, BL94, Huy05]).

Let k be a commutative ring and A = (A =⊕

i∈ZAi, d) a differ-

ential graded k-algebra (= dg algebra). (The k-algebra structure isdefined by a ring homomorphism σ : k → A0 with central image, i. e.aσ(x) = σ(x)a for all a ∈ A, x ∈ k.) A dg (right) module over A willalso be called an A-module or a dg module if there is no doubt aboutthe dg algebra. We often write M for a dg module (M,dM).

We consider the category dgMod(A) of dg modules, the homotopycategory dgHot(A) and the derived category dgDer(A) of dg modules.In [Kel94], these categories are denoted by C(A), H(A) and D(A)respectively. We often omit A from the notation.

The category dgMod is equipped with an autoequivalence {1}. IfM is a dg module, its shift {1}M is defined by

({1}M)j = M j+1, d{1}M = −dMand has the same A-module structure as M . (We do not have to twistthe A-module structure since we work with right modules.) If f : M →N is a morphism of A-modules, with components f j : M j → N j, then{1}f is defined by ({1}f)j = f j+1. This shift functor induces shiftfunctors {1} on dgHot and dgDer. We use the same notation for theshift functor on the category of graded modules over a graded ring. Wedefine {n} = {1}n for n ∈ Z.

The homotopy category dgHot(A) with the shift functor {1} andthe distinguished triangles isomorphic to standard triangles (= map-ping cones of morphisms) is a triangulated category (see [BL94, 10.3]).Any short exact sequence 0 → K → M → N → 0 of A-modules thatis A-split (= it splits as a sequence of graded A-modules) can be com-pleted to a distinguished triangle K →M → N → {1}K in dgHot(A).

The category dgDer(A) inherits the triangulation from dgHot(A).If R is a (graded) ring, we denote the category of (graded) right

R-modules by Mod(R) (by gMod(R), repectively). If M and N are A-modules, we define the complex Hom (M,N) of k-modules as follows:

Hom j(M,N) := HomgMod(A)(M, {j}N)

= HomgMod(A)({−j}M,N);

11

12 2. FORMALITY OF DERIVED CATEGORIES

if f ∈ Hom j(M,N), then

df = dN ◦ f − (−1)jf ◦ dM = dN ◦ f − f ◦ d{−j}M .

Note that

(4) HomdgHot(M,N) = H0(Hom (M,N))

We call a dg module P homotopically projective ([Kel98]), if itsatisfies one of the following equivalent conditions ([BL94, 10.12.2.2]):

(a) HomdgHot(P, ?) = HomdgDer(P, ?).(b) For each acyclic dg module M (that is H(M) = 0), we have

HomdgHot(P,M) = 0.(c) For each acyclic dg module M , the complex Hom (P,M) is

acyclic.

In [Kel94, 3.1] such a module is said to have property (P), in [BL94,10.12.2] the term K-projective is used. Since evaluation at 1 ∈ A0 is anisomorphism of complexes

(5) Hom (A,M)∼−→M,

A and each direct summand of A is homotopically projective. (Anobject Y is a direct summand of an object X if there is an object Zand an isomorphism X

∼−→ Y ⊕ Z.)Let dgHotproj(A) be the full subcategory of dgHot(A) consisting

of homotopically projective dg modules (this is the category Hp in[Kel94, 3.1]). It is a triangulated subcategory of dgHot(A) and closedunder taking direct summands. The quotient functor

q : dgHot(A)→ dgDer(A)

induces a triangulated equivalence ([Kel94, 3.1, 4.1])

(6) dgHotproj(A)∼−→ dgDer(A).

The category dgPer(A) (dgPer(A) respectively) is defined to bethe smallest full triangulated subcategory of dgHot(A) (of dgDer(A)respectively) that contains A and is closed under taking direct sum-mands. The category dgPer(A) of perfect dg modules is a subcategoryof dgHotproj and denoted by per(A) in [Kel98, 8.1.4, 8.2.6]. Equiva-lence (6) restricts to an equivalence

(7) dgPer(A)∼−→ dgPer(A).

If A → B is a morphism of dg algebras (dga-morphism), we havethe extension of scalars functor ([BL94, 10.11])

(8) prodBA = (?⊗A B) : dgMod(A)→ dgMod(B).

It descends to a triangulated functor

prodBA = (?⊗A B) : dgHot(A)→ dgHot(B)

2.2. DIFFERENTIAL GRADED GRADED ALGEBRAS AND FORMALITY 13

and has the left derived functor

prodBA = (?L⊗A B) : dgDer(A)→ dgDer(B).

The right adjoint functors of prodBA are given by the obvious restric-tion of scalars functors resAB .

Each dga-morphism f : A → B induces on cohomology a dga-morphism H(f) : H(A)→ H(B). If H(f) is an isomorphism, f is calleda dga-quasi-isomorphism. Two dg algebrasA and B are equivalentif there is a sequence

A ← C1 → C2 ← . . .→ . . .← Cn → B

of dga-quasi-isomorphisms. A dg algebra A is formal if it is equivalentto a dg algebra with differential d = 0. In this case, A is equivalent toH(A).

Theorem 5. If A → B is a dga-quasi-isomorphism, the extensionand restriction functors prodBA and resAB are mutually inverse equiva-lences between dgDer(A) and dgDer(B). They induce equivalences be-tween dgPer(A) and dgPer(B).

Proof. See [Kel98, 8.1.4, 8.2.6] and [BL94, 10.12.5.1] (or [Kel94,6.1]). �

2.2. Differential Graded Graded Algebras and Formality

We show that some dg algebras with an extra grading are formal.Let k be a commutative ring and R a differential graded graded (dgg)algebra, i. e. a Z2-graded associative k-algebra R =

⊕i,j∈ZR

ij endowedwith a k-linear differential d : R → R that is homogeneous of degree(1, 0) and satisfies the Leibniz rule

d(ab) = (da)b+ (−1)iadb

for all a ∈ Rij, b ∈ Rkl.A dgg module M = (M,d) over R is a Z2-graded right R-module

M =⊕

i,j∈ZMij with a k-linear differential d : M →M of degree (1, 0)

satisfying

d(ma) = (dm)a+ (−1)imda

for all m ∈M ij, a ∈ Rkl. We often write M instead of M or (M,d).Morphisms of dgg modules are morphisms of the underlying Z2-

graded R-modules of degree (0, 0) that commute with the differentials.We denote the category of dgg modules over R (or for short R-

modules or dgg modules) by dggMod(R).Let R be a dgg algebra as above and define Ri :=

⊕j∈ZR

ij. This

yields a dg algebra R = (R =⊕

i∈ZRi, d). Similarly, every dgg module

yields a dg module over R, and this defines a functor dggMod(R) →dgMod(R).


The cohomology of a dgg module over a dgg algebra R is a dggmodule over the the dgg algebra H(R). Morphisms of dgg algebras(dgga-morphisms) are algebra homomorphisms that are morphismsof dgg modules. The meanings of dgga-quasi-isomorphism, equiv-alent and formal are the obvious generalizations from dg algebras.

A Z2-graded k-module M =⊕

M ij is pure of weight w, if M ij 6=0 implies j = i + w. A dgg module or algebra is pure of weight w ifthe underlying bigraded module is pure of weight w. Every pure dggalgebra R of weight w 6= 0 is the zero algebra, since 1 ∈ R00.

Let M be in dggMod(R). We define a bigraded k-module Γ(M) by

(9) Γ(M)ij =

M ij if i < j,

ker(dij : M ij →M i+1,j) if i = j,

0 if i > j.

Here is an illustration of the obvious inclusion Γ(M) ↪→M (the differ-entials go to the right and the cocycle modules of the complexes M∗j

are denoted by Z(M)ij):∣∣∣∣ M01 → Z(M)11

Z(M)00 → 0

∣∣∣∣ ↪→∣∣∣∣M01 → M11

M00 → M10

∣∣∣∣The differential of M restricts to a differential of Γ(M). The multipli-cation on R restricts to a multiplication on Γ(R) and Γ(R) becomes adgg algebra. Similarly, Γ(M) is a dgg module over Γ(R). In fact, weobtain a functor

(10) Γ : dggMod(R)→ dggMod(Γ(R)).

Proposition 6. If the cohomology H(R) of a dgg algebra R is pureof some weight, R is formal. More precisely,

R ←↩ Γ(R) � H(R)

are dgga-quasi-isomorphisms. Here Γ(R) ↪→ R is the obvious inclusionand Γ(R) � H(R) the componentwise projection.

Proof. Let us assume that H(R) is pure of weight zero. (If H(R)is pure of weight 6= 0, then H(R) = 0. Hence H(R) is also pure ofweight zero.)

We include the following picture illustrating the morphisms R ←↩Γ(R) � H(R). The differentials go to the right, the cocycle and co-homology modules of the complexes R∗j are denoted by Zij and Hij

2.2. DIFFERENTIAL GRADED GRADED ALGEBRAS AND FORMALITY 15

respectively.∣∣∣∣∣∣R02 → R12 → R22

R01 → R11 → R21

R00 → R10 → R20

∣∣∣∣∣∣∣∣∣∣∣∣R02 → R12 → Z22

R01 → Z11 → 0Z00 → 0 → 0

∣∣∣∣∣∣? _oo

��∣∣∣∣∣∣0 → 0 → H22

0 → H11 → 0H00 → 0 → 0

∣∣∣∣∣∣Let R be an arbitrary dgg algebra. The proposition results from

the following evident statements.

(a) The dgga-inclusion Γ(R) ↪→ R induces on cohomology an iso-morphism in degrees (i, j) with i ≤ j (above the diagonal inour picture).

(b) If H(R) vanishes in degrees (i, j) with i < j (the cohomologylives below the diagonal), componentwise projection Γ(R) →H(R) is a well-defined dgga-morphism and induces on coho-mology an isomorphism in degrees (i, i) (on the diagonal).

�

We generalize Proposition 6 slightly to dgg algebras that look likematrices.

Let R be a dgg algebra and {eα}α∈I a finite set of orthogonal idem-potent elements of R00 satisfying 1 =

∑α∈I eα and d(eα) = 0 for

all α ∈ I. We get a direct sum decomposition R =⊕

Rαβ whereRαβ := eαReβ for α, β ∈ I. The differential of R induces differentialson each component Rαβ. In particular, we can consider the cohomolo-gies H(Rαβ).

Proposition 7. Let R and {eα}α∈I be as above. If there are inte-gers (nα)α∈I such that each H(Rαβ) is pure of weight nα − nβ, then Ris formal. More precisely, there are a dgg subalgebra S of R containingall {eα}α∈I and quasi-isomorphisms

R ←↩ S � H(R)

of dgg algebras.

Proof. Define S =⊕

Sijαβ ⊂ R by

Sijαβ =

Rijαβ if i+ nα − nβ < j,

ker(dijαβ : Rijαβ → Ri+1,j

αβ ) if i+ nα − nβ = j,

0 if i+ nα − nβ > j.


With the induced multiplication and differential, S becomes a dggsubalgebra S of R. The inclusion S ↪→ R and the obvious projec-tion S � H(R) are easily seen to be quasi-isomorphisms of dgg alge-bras. �

2.3. Subcategories of Triangulated Categories

Let T be a triangulated category, with shift X 7→ [1]X. If M is aset of objects of T , we denote by

PraeVerd(M) = PraeVerd(M, T )

the smallest strict (= closed under isomorphisms) full triangulated sub-category of T that contains all objects of M , and by

Verd(M) = Verd(M, T )

the smallest full triangulated subcategory of T that contains all ob-jects of M and is closed under taking direct summands. In particular,Verd(M) is closed under isomorphisms.

We call PraeVerd(M) and Verd(M) the Prae-Verdier and Verdiersubcategory generated by M . If X is an object of T , we abbreviateVerd({X}, T ) by Verd(X, T ), and similarly for PraeVerd.

Alternatively, we may characterize these categories by their classof objects: The class of objects of PraeVerd(M) is the smallest classthat contains M , is closed under taking isomorphic objects, under theshift, and that contains the third object of a distinguished triangle Σif it contains two objects of Σ. The class of objects of Verd(M) is thesmallest class that satisfies these conditions and is closed under takingdirect summands.

Lemma 8. Let F : T → T ′ be a triangulated functor between tri-angulated categories, and let M be a set of objects of T . If F inducesisomorphisms

HomT (X, [i]Y )∼−→ HomT ′(F (X), [i]F (Y )),

for all X, Y in M and all i ∈ Z, it induces a triangulated equivalence

PraeVerd(M)∼−→ PraeVerd(F (M)),

where F (M) = {F (X) | X ∈M}.

Proof. This follows by a standard devissage argument. �

2.4. Derived Categories and DG Modules

Let A be an abelian category. We denote by Ket(A), Hot(A) andDer(A) the category of (cochain) complexes in A, the homotopy cat-egory of complexes in A and the derived category of A respectively.We also write D instead of Der. We use upper indices +, −, b forthe bounded below, bounded above and bounded subcategories respec-tively. The canonical functor Hot(A) → Der(A) is usually denoted

2.4. DERIVED CATEGORIES AND DG MODULES 17

by q. If A is a complex, its i-th cohomology is Hi(A). We have theshift functor A 7→ [1]A. We often consider A as a full subcategoryof Der(A), consisting of complexes (with cohomology) concentrated indegree 0. If I is a subset of Z, we write DerI(A) for the full subcategoryof Der(A) with objects whose cohomology vanishes in degrees outsideI. For objects A and B in the derived category of A, we define

ExtnA(A,B) := HomnA(A,B)

:= HomnDer(A)(A,B)

:= HomDer(A)(A, [n]B),

ExtA(A,B) :=⊕n∈Z

ExtnA(A,B)(11)

and the extension algebra

Ext(A) := ExtA(A,A).(12)

All functors between additive categories we consider are additive.If M , N are complexes in A, let Hom (M,N) or HomA(M,N)

denote the complex of abelian groups with n-th component

(13) Hom n(M,N) =∏i+j=n

HomA(M−i, N j)

and differential

df = d ◦ f − (−1)nf ◦ dfor each homogeneous f of degree n. The n-th cohomology of thiscomplex is

(14) Hn(Hom (M,N)) = HomHot(A)(M, [n]N).

With the obvious composition, Hom (M,M) becomes a dg algebrathat we denote by End (M) or EndA(M). The functor

Hom (M, ?) : Ket(A)→ dgMod( End (M)),

induces a triangulated functor

Hom (M, ?) : Hot(A)→ dgHot( End (M)).

Recall the categories dgPer(R) and dgPer(R) for a dg algebra Rfrom section 2.1. By definition, we have

dgPer(R) = Verd(R, dgHot(R)),

dgPer(R) = Verd(R, dgDer(R)).

If M is a set of R-modules, we define

dgPraeR(M) := PraeVerd(M, dgHot(R)),(15)

dgPraeR(M) := PraeVerd(M, dgDer(R)).(16)


If all elements of M are homotopically projective, equivalence (6) re-stricts to a triangulated equivalence

(17) dgPraeR(M)∼−→ dgPraeR(M).

Proposition 9. Let A be an abelian category, and {Pα}α∈I a finiteset of complexes in A such that the canonical maps

(18) HomHot(A)(Pα, [n]Pβ)→ HomD(A)(Pα, [n]Pβ)

are isomorphisms for all n ∈ Z and all α, β ∈ I. (For example, all Pαcould be bounded above complexes of projective objects of A.) DefineP =

⊕Pα and R = End (P ). Let eα ∈ R0 be the projector from P

onto its direct summand Pα. Then the functor Hom (P, ?) induces atriangulated equivalence

(19) PraeVerd({Pα}α∈I ,D(A))∼−→ dgPraeR({eαR}α∈I).

Proof. Consider the diagram

(20) PraeVerd({Pα}α∈I ,Hot(A))

q

��

Hom (P,?)// dgPraeR({eαR}α∈I)

q

��PraeVerd({Pα}α∈I ,D(A)) dgPraeR({eαR}α∈I)

with obvious vertical functors. We claim that all arrows are equiva-lences. For the arrow on the left this follows from (18) and Lemma 8.Since all eαR are direct summands of the homotopically projective dgmodule R (see (5)), the arrow on the right is equivalence (17).

Since HomHot(A)(Pα, [n]Pβ) and HomdgHot(R)(eαR, [n]eβR) are bothnaturally identified with Hn(eβReα) and these identifications are com-patible with the maps

Hom (P, ?) : HomHot(A)(Pα, [n]Pβ)→ HomdgHot(R)(eαR, eβR[n]),

Lemma 8 proves our claim. �

Remark 10. (a) Let P be a complex in an abelian category Awith endomorphism complexR = End (P ). If the composition

(21) Hot(A)Hom (P,?)−−−−−−→ dgHot(R)

q−→ dgDer(R)

vanishes on acyclic complexes, it factors through q : Hot(A)→Der(A) to a triangulated functor

(22) Hom (P, ?) : Der(A)→ dgDer(R).

This is the case, for example, if P is a bounded above complexof projective objects of A: If A is a acyclic, we have

Hn(Hom (P,A)) = HomHot(P, [n]A)

= HomDer(P, [n]A) = 0.

So Hom (P,A) is acyclic and hence isomorphic to the zeroobject in dgDer.

2.5. SHEAVES AND PERVERSE SHEAVES 19

(b) Keep the assumptions of Proposition 9 and assume that thecomposition (21) vanishes on acyclic complexes. Then therestriction of (22) yields directly equivalence (19).

2.5. Sheaves and Perverse Sheaves

All varieties we consider are assumed to be complex (algebraic) va-rieties. Let X be a variety. We denote by Sh(X) the abelian cat-egory of sheaves of real vector spaces with respect to the classicaltopology on X, by D+(X) = Der+(Sh(X)) its bounded below andby Db(X) = Derb(Sh(X)) its bounded derived category. Let Db

c (X)be the full triangulated subcategory of Db(X), consisting of complexeswith algebraically constructible cohomology ([BBD82, 2.2.1]).

The functors tensor product⊗ and sheaf homomorphisms Hom areexact and left exact respectively, and there derived and right derivedfunctors are denoted by ⊗ and Hom respectively. Any morphismf : X → Y of complex varieties gives rise to adjoint pairs (f ∗, f∗) and(f!, f

!) relating Db(X) and Db(Y ). These functors would classicallybe written ⊗, RHom , f−1, Rf∗, Rf! and f ! respectively. Let pt bethe final object in the category of varieties, and pt the constant sheafwith stalk R on pt. Let c = cX : X → pt be the constant map andX := c∗(pt) the constant sheaf on X. Verdier duality is defined by

D = DX = Hom (?, c!(pt)). We have Df∗ = f!D, Df ∗ = f !D, and

D2 = id on Dbc (X).

An algebraic stratification of X is a finite partition S of X into non-empty locally closed subvarieties, called strata, such that the closure ofeach stratum is a union of strata. If S ∈ S is a stratum, we denote by lSthe inclusion of S in X. From now on, if we speak about stratifications,we always mean algebraic Whitney stratifications. In particular, allstrata are nonsingular. We assume in the following that all strataare irreducible varieties. The (complex) dimension of a stratum S isdenoted by dS. A cell-stratification is a stratification such that eachstratum S is isomorphic to an affine linear space, so S ∼= CdS . If S ∈ Sis a stratum of a cell-stratification, the inclusion lS : S → X is an affinemorphism of varieties, since S is an open affine subset of its closure (theintersection of open affine subsets of a variety is open affine) and theclosed embedding S ⊂ X is an affine morphism.

A sheaf F ∈ Sh(X) is called smooth (along a stratification S)or S-constructible, if l∗S(F ) is a local system on S, for all S ∈ S. LetSh(X,S) ⊂ Sh(X) be the full subcategory of such sheaves. An objectF of D+(X) is called smooth (along S) or S-constructible, if allHi(F ) are in Sh(X,S).

Proposition 11. Let (X,S) be a stratified variety. The full subcat-egory D+(X,S) ⊂ D+(X) of S-constructible objects is a triangulated


subcategory and closed under taking direct summands. Similarly forDb.

Proof. Let us show first that D+(X,S) is a triangulated subcat-egory of D+(X). We have to show that the cone of a morphism of twosmooth objects is smooth. If we restrict to any stratum S ∈ S andtake the long exact sequence of cohomology sheaves, we have to show:If M1 → M2 → E → M4 → M5 is an exact sequence in Sh(S) andall Mi are local systems on S, then E is a local system on S. But thecategory of local systems on S is a full abelian subcategory of Sh(S)and closed under extensions.

The last statement implies also that D+(X,S) is closed under tak-ing direct summands: A direct summand of a local system M on S isthe kernel of an endomorphism of M and hence a local system. �

Middle perversity defines perverse t-structures on Db(X,S) andDb

c (X), see [BBD82, 2.1, 2.2]. Their hearts Perv(X,S) and Perv(X)are the categories of smooth perverse sheaves and of perverse sheavesrespectively. We have Perv(X,S) = Perv(X) ∩ Db(X,S).

Proposition 12. If (X,S) is a stratified variety, then

Db(X,S) = Verd(Perv(X,S),Db(X))

= PraeVerd(Perv(X,S),Db(X)).

Proof. It is clear that PraeVerd ⊂ Verd. We have Verd ⊂ Db(X,S)by Proposition 11. Since any object of Db(X,S) has perverse cohomol-ogy in finitely many degrees only ([BBD82, 2.1.2.1]), perverse trunca-tion shows the inclusion Db(X,S) ⊂ PraeVerd. �

There is a natural triangulated equivalence of categories (see [Beı87,BBD82])

(23) real = realX : Db(Perv(X))∼−→ Db

c (X).

We denote this functor often by A 7→ A := real(A). In particular,

(24) real : ExtnPerv(X)(A,B)∼−→ ExtnSh(X)(A,B)

is an isomorphism for all A, B ∈ Db(Perv(X)). The correspondingstatement for sheaves that are smooth along a fixed stratification S isusually false. However, if S is a cell-stratification, it is true, as we willstate below, see Theorem 13.

Now assume that S is a cell-stratification of X. Let S ∈ S and recallS ∼= CdS . Denote by CS = [dS]S the constant perverse sheaf on S andconsider the standard and costandard objects ∆S = lS!CS and ∇S =lS∗CS. Since lS is affine, both ∆S and∇S belong to Perv(X,S). Let ICSbe the image of the canonical morphism ∆S → ∇S. (This object wasdenoted IC(S) in the introduction.) The (ICS)S∈S are precisely the

2.6. MIXED HODGE STRUCTURES 21

simple objects of Perv(X,S). There is a natural triangulated functor(cf. [BBD82, 3.1])

(25) real = realX,S : Db(Perv(X,S))→ Db(X,S).

We denote this functor also by A 7→ A. Its construction will be ex-plained in section 3.3.

Theorem 13. Let (X,S) be a cell-stratified variety. Then the cat-egory Perv(X,S) is artinian and has enough projective and injectiveobjects. Each projective object has a finite filtration with standard sub-quotients. Each object has a projective resolution of finite length. Thefunctor realX,S (see (25)) is an equivalence of categories. In particular,

(26) real : ExtnPerv(X,S)(A,B)∼−→ ExtnSh(X)(A,B)

is an isomorphism for all A, B ∈ Db(Perv(X,S)).

Proof. This follows from [BGS96, 3.2, 3.3].We sketch the proof that real is an equivalence. The restriction of

real to Perv(X,S) is the identity. Both Db(Perv(X,S)) and Db(X,S)are generated, as triangulated categories, by the standard objects ∆S,for S ∈ S. The same statement holds for the costandard objects ∇S.Hence it is enough to prove that

real : ExtnPerv(X,S)(∆S,∇T )∼−→ ExtnSh(X)(∆S,∇T )

is an isomorphism for all strata S, T and all n ∈ Z. The right hand sideis 6= 0 if and only if S = T and n = 0, since we have the adjunction(l∗T , lT∗) and S is a cell. That the same statement holds for the lefthand side follows from [BGS96, 3.2]. �

2.6. Mixed Hodge Structures

The following definitions and results about mixed Hodge structuresare taken from [Del71, Del94]; see also [DMOS82] for some defini-tions.

A (real) mixed Hodge structure M consists of

(a) a real vector space MR of finite dimension,(b) a finite increasing filtration W on MR, called weight filtration,

and(c) a finite decreasing filtration F on the complexification MC =

C⊗R MR, called Hodge filtration,

such that the filtration WC, obtained by extension of scalars, the filtra-tion F and its complex conjugate filtration F form a system of threeopposed filtrations on MC, i. e. grpF grq

FgrWCn (MC) = 0 if n 6= p + q. A

morphism f : M → N of mixed Hodge structures is an R-linear mapfR : MR → NR that is compatible with the weight filtrations and whosecomplexification fC is compatible with the Hodge filtration.


A mixed Hodge structure M has weights ≤ n (resp. ≥ n), ifgrWj,R(M) := grWj (MR) = WjMR/Wj−1MR = 0 for j > n (resp. j < n).It is pure of weight n, if it is of weight ≤ n and of weight ≥ n.

Let R(n) be the Tate structure of weight −2n. It is a pure Hodgestructure of weight −2n, with R(n)R = (2πi)nR ⊂ C = R(n)C.

The category MHS of mixed Hodge structures is a rigid abelian R-linear tensor category. It admits the fiber functor “underlying vectorspace” ω0 : MHS → R- mod to the category of finite dimensional realvector spaces and is hence neutral tannakian. A mixed Hodge structureM is polarizable, if each graded piece grWn (M) is a polarizable Hodgestructure ([Del71, 2.1.16]). The polarizable mixed Hodge structuresare a rigid tensor subcategory of MHS.

The functor grWR : MHS → R- gmod, M 7→⊕

n∈Z grWn,R(M), isan exact faithful R-linear tensor functor from the category of mixedHodge structures to the category of finite dimensional graded vectorspaces over R. We denote the composition of grWR with the functor“underlying vector space” η : R- gmod→ R- mod by ωW . This functorωW : MHS→ R- mod is a fiber functor and there is an isomorphism offiber functors ([Del94, p. 513])

(27) a : ω0∼−→ ωW .

2.7. Mixed Hodge Modules

We denote by MHM(X) the abelian category of mixed Hodge mod-ules (over R) on a complex variety X (see [Sai94, Sai89, BGS96]).Instead of mixed Hodge module we also say Hodge sheaf. There isa faithful and exact functor rat : MHM(X) → Perv(X). It inducesa triangulated functor rat : Db(MHM(X)) → Db(Perv(X)). Objectsand morphisms in MHM(X) or in Db(MHM(X)) are often denoted bya letter with a tilde, and omission of the tilde means application of rat,

e. g. M 7→M := rat(M).There are functors Hom , ⊗ and Verdier duality D. For f : X → Y

a morphism of complex varieties, we have functors f ∗, f∗, f!, f! relat-

ing Db(MHM(X)) and Db(MHM(Y )). We have the usual adjunctions(f ∗, f∗) and (f!, f

!) between these functors, and Df∗ = f!D, Df ∗ = f !Dand D2 = id. All these functors “commute” with the composition

(28) v := real ◦ rat : Db(MHM(X))→ Db(Perv(X))∼−→ Db

c (X),

where real is the equivalence (23). We have v ◦ Hi = pHi ◦ v, wherepHi : Db(X) → Perv(X) ⊂ Db(X) is the i-th perverse cohomologyfunctor.

The Hodge sheaves on the point pt are the polarizable mixed Hodgestructures ([Sai89, 1.4]). Each Tate structure R(n) is in MHM(pt).

Each Hodge sheaf M ∈ MHM(X) has a finite increasing filtrationW in MHM(X) called weight filtration. This filtration is functorial,

2.7. MIXED HODGE MODULES 23

and M 7→ grWn (M) is an exact functor ([Sai89, 1.5]). A Hodge sheafM has weights ≤ n (resp. ≥ n), if grWj (M) = 0 for j > n (resp. j < n).More generally, a complex of Hodge sheaves M has weights ≤ n (resp.≥ n), if each Hi(M) has weights ≤ n + i (resp. ≥ n + i). It is calledpure of weight n, if it has weights ≤ n and ≥ n.

We give some properties of mixed Hodge modules.

(P1) If M ∈ Db(MHM(X)) is of weight ≤ w (resp. ≥ w), so aref!M , f ∗M (resp. f∗M , f !M) ([Sai89, 1.7]).

(P2) M is of weight ≤ w if and only if DM is of weight ≥ −w.(P3) For any M ∈ MHM(X), every grWn (M) is a semisimple object

of MHM(X) ([Sai89, 1.9]).(P4) If M ∈ Db(MHM(X)) is pure of weight n, we have a non-

canonical isomorphism M ∼=⊕

j∈Z[−j] Hj(M) ([Sai89, 1.11]).

In the following, f : X → Y is a morphism of complex varieties, M ,N , A, B, C, D are objects of Db(MHM(X)) or Db(MHM(Y )), andc : X → pt is the constant map.

(P5) We have f ∗(A⊗B) = f ∗A⊗ f ∗B.(P6) The Tate twist M(n) of M is defined by M(n) = M⊗c∗(R(n))

([Sai89, 1.15]), satisfies M(0) = M and commutes with allfunctors f ∗, f !, f∗, f!.

(P7) If M is of weight ≤ w, then M(n) is of weight ≤ w − 2n,and [n]M is of weight ≤ w + n. The same statement with ≤replaced by ≥.

(P8) If j : U ↪→ X is an open embedding and i : X − U ↪→ X theclosed embedding of its complement, then j! = j∗, i∗ = i!, andthere are distinguished triangles

j!j∗M →M → i∗i

∗M → [1]j!j∗M

i∗i!M →M → j∗j

!M → [1]i∗i!M

functorial in M ∈ Db(MHM(X)). These distinguished trian-gles are compatible with those for the underlying complexesin Db

c (X) ([Sai89, 4.3.3, 4.4.1]).(P9) There is a functorial isomorphism ([Sai90, 2.9])

Hom(A⊗B,C) = Hom(A,Hom (B,C)),

where Hom = HomDb(MHM(X)). In particular, there is an ad-junction (?⊗B,Hom (B, ?)).

(P10) The adjunction from (P9) yields an adjunction morphism

Hom (M,N)⊗M → N,

called evaluation. Using this adjunction morphism twice andthe adjunction (P9), we get the composition morphism

Hom (B,C)⊗Hom (A,B)→Hom (A,C).


Similarly, using the symmetry of the tensor product, we getthe morphism

Hom (A,B)⊗Hom (C,D)→Hom (A⊗ C,B ⊗D).

(P11) We have ([Sai90, 2.9.3])

Hom (A,B) = D(A⊗ DB).

(P12) From (P11) and (P5) we get

f !Hom (A,B) = Df ∗(A⊗ DB)

= D(f ∗A⊗ Df !B)

= Hom (f ∗A, f !B).

(P13) If f is smooth of relative (complex) dimension n, we have([Soe89, 1.4])

[2n]f ∗(M)(n) = f !(M).

Let (X,S) be a stratified variety and MHM(X,S) the full abeliansubcategory of MHM(X) consisting of Hodge sheaves M satisfyingrat(M) ∈ Perv(X,S). We denote by Db(MHM(X),S) the full subcat-egory of Db(MHM(X)) consisting of complexes M satisfying v(M) ∈Db(X,S) (or, equivalently Hi(M) ∈ MHM(X,S), for all i ∈ Z). Ob-jects of MHM(X,S) and Db(MHM(X),S) are called smooth (alongS).

Proposition 14. Let S = Cn for some n ∈ N, and c : S → pt bethe constant map.

(a) If E ∈ MHM(pt) is pure of weight w, then c∗(E) is pure ofweight w. (Note that [n]c∗(E) is in MHM(S, {S}) and pure ofweight w + n).

(b) If M ∈ MHM(S, {S}) is a pure Hodge sheaf of weight w andsmooth along the trivial stratification, there is a pure Hodgestructure E ∈ MHM(pt) of weight w − n such that M ∼=[n]c∗(E).

Proof. Properties (P1), (P7) and (P13) imply (a). Let us prove(b). By [Sai89, 2.2 Theorem], M corresponds to a polarizable variationV of Hodge structure of weight w − n on S = Cn. The fiber V0 ofV at 0 ∈ Cn is a polarizable Hodge structure of weight w − n. Wedenote its constant extension to Cn by V0. Obviously, there is an

isomorphism V∼−→ V0 of the underlying local systems that respects

the Hodge filtration at 0 ∈ Cn. By the Rigidity Theorem ([Sch73,7.24], see also [CMSP03, 13.1.9, 13.1.10]), this isomorphism is anisomorphism of polarizable variations of Hodge structures of weightw − n. We obtain M ∼= [n]c∗(V0), where we now consider V0 as apolarizable Hodge structure of weight w − n on pt, in particular as anelement of MHM(pt). �

2.8. CONSTRUCTION OF EPIMORPHISMS FROM PROJECTIVE OBJECTS 25

If Y is a variety, we define Y = c∗(R(0)) ∈ Db(MHM(Y )), so

v(Y ) = Y . Let X be an irreducible variety of dimension dX and j :U → X the inclusion of a nonsingular affine open dense subset. Theintersection complexes of X are defined by ([Sai89, 1.13])

IC(X) := im(j!([dX ]U)→ j∗([dX ]U)) ∈ Perv(X) and

IC(X) := im(j!([dx]U)→ j∗([dx]U)) ∈ MHM(X).

This definition does not depend on the choice of U , IC(X) is simple

and pure of weight dX := dimCX and satisfies rat(IC(X)) = IC(X).If lS : S → X is the inclusion of the closure of a stratum S in a

stratified variety (X,S), we denote lS∗(IC(S)) by ICS, and similarly

for ICS. (This coincides with the previous definition of ICS for a cell-

stratified variety.) These objects are smooth, ICS is simple and pure

of weight dS, and we have rat(ICS) = ICS.

In the introduction, we wrote IC(S) and IC(S) instead of ICS and

ICS. We will use this notation later on again.

2.8. Construction of Epimorphisms from Projective Objects

In this section we describe an algorithm for constructing an epi-morphism from a projective object onto a given object. This algorithmwill be used in section 2.9 in order to show that there are enoughperverse-projective mixed Hodge modules.

Let A be an artinian k-category, where k is a field. We write Hom,End, Ext, ⊗ instead of HomA, EndA, ExtA, ⊗k, respectively. We makethe following assumptions:

(E1) End(L) = k for all simple objects L in A.(E2) There are enough projective objects in A.

Note that (E1) implies that Hom(M,N) is finite dimensional, for allM , N ∈ A. Then (E2) shows that Ext1(M,N) is finite dimensional,for all M , N ∈ A.

The following algorithm keeps extending simple objects to a givenobject until this is no longer possible. In doing this, only non-trivialextensions are used. We will prove in Proposition 15 that our algo-rithm stops after finitely many steps and that the object we get backis projective.

Step 1: Take an object A ∈ A as input datum.Step 2: Set i = 0 and A0 = A.Step 3: While there is a simple object L ∈ A with Ext1(Ai, L) 6= 0

Step 3.1: Take a simple object L ∈ A with E := Ext1(Ai, L) 6= 0.


Step 3.2: The element id ∈ E∗ ⊗ E = Ext1(Ai, E∗ ⊗ L) gives rise1

to an extension E∗ ⊗ L ↪→ Ai+1 � Ai.Step 3.3: Increase i by 1.

Step 4: DefineQ = Ai and return the epimorphismQ = Ai � A0 = A.

Proposition 15. Given any A ∈ A, the above algorithm termi-nates after finitely many steps. It returns an epimorphism Q → Afrom a projective object Q in A.

Proof. We denote the length of an object X ∈ A by λ(X). As-sume that our algorithm does not stop. Then it constructs a se-quence . . . � A2 � A1 � A0 of objects and epimorphisms withλ(A0) < λ(A1) < λ(A2) < . . . .

By (E2), there are a projective object P and an epimorphism π0 :P → A0 = A. Since A1 → A0 is epimorphic, there is a morphismπ1 : P → A1 lifting π0 : P → A0. Proceeding in this manner weobtain liftings πi : P → Ai of πi−1 for all i > 0. Now Lemma 16below shows that all these πi are epimorphisms. In particular, we getthe contradiction λ(Ai) ≤ λ(P ) for all i. So our algorithm stops. Theobject Q that is returned is projective, since Ext1(Q,L) = 0 for allsimple objects L ∈ A. �

Lemma 16. Let π : P � A be an epimorphism from an object P(not necessarily projective) onto A. Let L be a simple object, E =Ext1(A,L) and

(29) 0→ E∗ ⊗ L i−→Mc−→ A→ 0

the extension defined by id ∈ E∗⊗E = Ext1(A,E∗⊗L). Let π : P →Mbe a morphism such that

(30) Peπ~~}}}}}}}}

π��

M c// // A

commutes. Then π is an epimorphism.

Proof. We have to show that

(∗)N The map π∗ : Hom(M,N)→ Hom(P,N) is injective.

holds for all objects N .

1Perhaps we should explain what we mean by the object E∗ ⊗ L. Let Nat bethe following category: Its objects are the natural numbers N; if m, n are objectsof Nat, we define HomNat(m,n) to be the set of n×m-matrices over k; compositionof morphisms is matrix multiplication. We fix an equivalence of categories φ :k- mod ∼−→ Nat between the category of finite dimensional vector spaces over k andNat. There is an obvious functor (?⊗?) : Nat×A → A, (n,M) 7→ n⊗M := M⊕n. IfV is a finite dimensional vector space and M is in A, we define V ⊗M := φ(V )⊗M .

2.9. EXISTENCE OF ENOUGH PERVERSE-PROJECTIVE HODGE SHEAVES27

If 0 → N ′ → N → N ′′ → 0 is an exact sequence and (∗)N ′ and(∗)N ′′ hold, we get a commutative diagram with exact rows

0 // Hom(M,N ′) //� _

��

Hom(M,N) //

��

Hom(M,N ′′)� _

��0 // Hom(P,N ′) // Hom(P,N) // Hom(P,N ′′)

and see that (∗)N is satisfied. So it is enough to prove (∗)N for simpleobjects N .

Let N be simple. By applying Hom(?, N) to the exact sequence(29), we obtain the exact sequence

0→ Hom(A,N)c∗−→ Hom(M,N)

i∗−→ Hom(E∗ ⊗ L,N)δ−→ Ext1(A,N).

Our claim is that c∗ is bijective. For N 6∼= L this is clear, sinceHom(E∗ ⊗ L,N) vanishes. If N = L there is an obvious map cansuch that

Ecan

xxppppppppppppid

$$JJJJJJJJJJ

. . . // Hom(E∗ ⊗ L,L)δ // Ext1(A,L) // . . .

commutes. Since Hom(L,L) = k by (E1), can and hence δ are isomor-phisms. This shows that c∗ is an isomorphism if N = L or N ∼= L.

We now apply Hom(?, N) to diagram (30) and get a commutativediagram

Hom(P,N)

Hom(M,N)

eπ∗ 77nnnnnnnnnnnnHom(A,N)

c∗∼oo

?�π∗

OO

Since c∗ is bijective and π∗ is injective, π∗ is injective and (∗)N istrue. �

2.9. Existence of Enough Perverse-Projective Hodge Sheaves

Let (X,S) be a cell-stratified complex variety. A smooth Hodge

sheaf P ∈ MHM(X,S) is called perverse-projective, if the under-

lying perverse sheaf P = rat(P ) is a projective object of Perv(X,S).A complex in MHM(X,S) is called perverse-projective, if all itscomponents are perverse-projective.

Proposition 17. If (X,S) is a cell-stratified complex variety, thereare enough perverse-projective objects in MHM(X,S), i. e. for every

smooth Hodge sheaf A ∈ MHM(X,S), there is a perverse-projective

smooth Hodge sheaf P ∈ MHM(X,S) and an epimorphism P � A.

Proof. Postponed to the end of this section. �


Corollary and Definition 18. Every smooth Hodge sheaf A ∈MHM(X,S) has a perverse-projective resolution P → A, i. e.

there is a perverse-projective complex P = (P n, dn) in MHM(X,S)

with P n = 0 for n > 0 and a quasi-isomorphism

(31) P =

��

(. . . // P−2d−2

//

��

P−1d−1

//

��

P 0 //

��

0 //

��

. . . )

A = (. . . // 0 // 0 // A // 0 // . . . )

in Ket(MHM(X,S)). Moreover, we can assume that this resolution is

of finite length, i. e. P n = 0 for n� 0.

Proof. Let P 0 � A be an epimorphism from a perverse-projective

object, and let K0 be its kernel. Then we find an epimorphism P−1 �K0 with P−1 perverse-projective. Proceeding in this way, we construct

a perverse-projective resolution of A. Since rat(A) has a projective

resolution of finite length in Perv(X,S) (Theorem 13), we see that Ki

is perverse-projective for i� 0. �

If M , N ∈ MHM(X) are Hodge sheaves on X with underlying per-verse sheaves M and N , there is a (polarizable) mixed Hodge structureon all ExtiPerv(X)(M,N), defined as follows. Let c : X → pt be theconstant map and M = real(M), N = real(N). On a point, perversecohomology and ordinary cohomology coincide, and we get

v(Hi c∗Hom (M, N)) = Hi c∗Hom (M,N)

= HiRHomSh(X)(M,N)

= ExtiSh(X)(M,N).

Thus, Hi c∗Hom (M, N) defines a natural mixed Hodge structure onExtiSh(X)(M,N). Using isomorphism (24), we transfer this structure

to ExtiPerv(X)(M,N) and denote the obtained mixed Hodge structure

by ExtiPerv(X)(M, N), as it depends on M and N . We also write

HomPerv(X)(M, N) instead of Ext0Perv(X)(M, N).

If M , N are smooth along our cell-stratification, the analogous argu-ment (using isomorphism (26) instead of (24)) equips ExtiPerv(X,S)(M,N)

with a mixed Hodge structure ExtiPerv(X,S)(M, N), and

(32) ExtiPerv(X,S)(M, N) = ExtiPerv(X)(M, N)

holds for all i ∈ Z.

Remark 19. Our construction defines bifunctors

ExtiPerv(X)(?, ?) : MHM(X)◦ ×MHM(X)→ MHS .


The usual long exact Ext-sequences in both variables underlie exact

sequences of mixed Hodge structures. Furthermore, if A, B and Care Hodge sheaves, composition defines a morphism of mixed Hodgestructures

(33) ExtiPerv(X)(B, C)⊗ ExtjPerv(X)(A, B)→ Exti+jPerv(X)(A, C).

This can be seen as follows. If F , G ∈ Db(MHM(X)) are complexes ofHodge sheaves, the tensor product of the adjunction morphisms c∗c∗ →id for F and G is the morphism c∗(c∗F ⊗ c∗G)→ F ⊗G (using (P5)).Using the adjunction (c∗, c∗) again, we get a morphism c∗F ⊗ c∗G →c∗(F ⊗ G). We compose this morphism for F = Hom (B, C) and

G = Hom (A, B) with c∗ of the composition morphism Hom (B, C)⊗Hom (A, B)→Hom (A, C) (cf. (P10)) and get a morphism

c∗Hom (B, C)⊗ c∗Hom (A, B)→ c∗Hom (A, C).

Now we take the (i+ j)-th cohomology and use the obvious morphismof mixed Hodge structures

Hi c∗Hom (B, C)⊗ Hj c∗Hom (A, B)

→ Hi+j(c∗Hom (B, C)⊗ c∗Hom (A, B)

)in order to get morphism (33).

The analogous remarks are valid for ExtiPerv(X,S).

Lemma 20. Let F : A → B be an exact faithful functor betweenabelian categories, F : D(A) → D(B) its derived functor, and f amorphism in D(A). Then f is an isomorphism if and only if F (f) is anisomorphism. In particular, the same statement holds for morphismsf in A.

Proof. The “only if” part is trivial. Let f : X → Y be a morphism

in D(A). Embed f in a distinguished triangle Xf−→ Y → Z → [1]X.

Assume that F (f) is an isomorphism. Then F (Z) is isomorphic to thezero object, so all Hi(F (Z)) = F (Hi(Z)) vanish. Since F is faithful, allHi(Z) vanish, so Z ∼= 0 and f is an isomorphism. �

Lemma 21. Let A, B ∈ MHM(X) be Hodge sheaves on X, let M ∈MHM(pt) be a polarizable mixed Hodge structure, and let c : X → ptbe the constant map. Then there are natural isomorphisms

M ⊗ c∗A∼−→ c∗(c

∗M ⊗ A)(34)

c∗M ⊗Hom (A,B)∼−→Hom (A, c∗M ⊗B).(35)

in Db(MHM(pt)) and Db(MHM(X)) respectively.


Proof. The morphism (34) is the image of the identity morphismof c∗M ⊗ A under the chain of obvious morphisms

Hom(c∗M ⊗ A, c∗M ⊗ A)→ Hom(c∗M ⊗ c∗c∗A, c∗M ⊗ A)

= Hom(c∗(M ⊗ c∗A), c∗M ⊗ A)

= Hom(M ⊗ c∗A, c∗(c∗M ⊗ A)),

where Hom = HomDb(MHM). Since v(M) is a finite dimensional vectorspace, v(34) is an isomorphism, and Lemma 20, applied to rat, andequivalence (23) for X = pt, where it is a triviality, show that (34) isan isomorphism.

The morphism (35) comes from the identifications ((P6), (P11))

c∗M = D(c∗R(0)⊗ Dc∗M) = Hom (c∗R(0), c∗M)

and the morphism ((P10), (P6))

Hom (c∗R(0), c∗M)⊗Hom (A,B)→Hom (c∗R(0)⊗ A, c∗M ⊗B)

= Hom (A, c∗M ⊗B).

All these morphisms are mapped to isomorphisms by the functor v, so(35) is an isomorphism by Lemma 20 and equivalence (23). �

Lemma 22. Let A, B ∈ MHM(X,S), M ∈ MHM(pt), and let

c : X → pt be the constant map. Then c∗M ⊗ B ∈ MHM(X,S), andthere is a natural isomorphism

M ⊗ ExtiPerv(X,S)(A, B)∼−→ ExtiPerv(X,S)(A, c

∗M ⊗ B)

of (polarizable) mixed Hodge structures, for all i ∈ Z.

Proof. The first statement is obvious. Isomorphisms (34) and(35) yield isomorphisms

M ⊗ c∗Hom (A, B)∼−→ c∗(c

∗M ⊗Hom (A, B))∼−→ c∗Hom (A, c∗M ⊗ B)

We take the i-th cohomology, use the exactness of the functor (M⊗?),and obtain

M ⊗ Hi c∗Hom (A, B)∼−→ Hi c∗Hom (A, c∗M ⊗ B).

�

Proof of Proposition 17. If M , N ∈ MHM(X) are Hodgesheaves, there is a short exact sequence (see [Sai90])

0→ H1MHM(pt)(HomPerv(X)(M, N))→ Ext1

MHM(X)(M, N)

→H0MHM(pt)(Ext1

Perv(X)(M, N))→ 0,(36)


where HiMHM(pt) is the absolute Hodge cohomology functor: For a mixed

Hodge structure A ∈ MHM(pt) one has

HiMHM(pt)(A) := ExtiMHM(pt)(R(0), A).

The categories MHM(X,S) and Perv(X,S) are closed under exten-sions in Db(MHM(X) and Db(X) ([BBD82, 1.3.6, 3.1.17]). Thus, for

smooth M , N ∈ MHM(X,S), there is a short exact sequence

0→ H1MHM(pt)(HomPerv(X,S)(M, N))→ Ext1

MHM(X,S)(M, N)

→H0MHM(pt)(Ext1

Perv(X,S)(M, N))→ 0,(37)

Let M , N ∈ MHM(X,S) and consider the polarizable mixed Hodge

structure E = Ext1Perv(X,S)(M, N). The map can : R(0) → E∗ ⊗ E,

1 7→ id eE, is a morphism of polarizable mixed Hodge structures, i. e. an

element can ∈ H0MHM(pt)(E

∗ ⊗ E). Lemma 22 yields an isomorphism

(38) E∗⊗E = E∗⊗Ext1Perv(X,S)(M, N)

∼−→ Ext1Perv(X,S)(M, c∗(E∗)⊗N)

of polarizable mixed Hodge structures. The exact sequence (37) showsthat there is an extension of smooth Hodge sheaves

c∗(E∗)⊗ N ↪→ K � M

such that the underlying extension of perverse sheaves is given by theelement idE ∈ E∗ ⊗ E

∼−→ Ext1Perv(X,S)(M, c∗(E∗)⊗N).

We now use the following algorithm in order to prove our proposi-tion.

Step 1: Take an object A ∈ MHM(X,S) as input datum.

Step 2: Set i = 0 and A0 = A.

Step 3: While there is a stratum S ∈ S with Ext1Perv(X,S)(Ai, ICS) 6= 0

Step 3.1: Take a stratum S ∈ S with E = Ext1Perv(X,S)(Ai, ICS) 6=

0.Step 3.2: Choose, as explained above, an extension c∗(E∗)⊗ICS ↪→

Ai+1 � Ai of smooth Hodge sheaves such that the under-lying extension of perverse sheaves is given by id ∈ E∗⊗E.

Step 3.3: Increase i by 1.

Step 4: Define P = Ai and return the epimorphism P = Ai � A0 = Aof smooth Hodge sheaves.

The underlying algorithm is the algorithm from section 2.8 forA = Perv(X,S). This choice of A is justified by Theorem 13. Thus,Proposition 15 shows that our algorithm terminates and returns an

epimorphism P � A from a perverse-projective smooth Hodge sheaf

P . �


2.10. Comparison of Mixed Hodge Structures

Let A, B ∈ MHM(X,S) be smooth Hodge sheaves on a cell-stratified variety (X,S), with underlying smooth perverse sheaves Aand B. As explained in section 2.9, there is a (polarizable) mixed

Hodge structure ExtiPerv(X,S)(A, B) on ExtiPerv(X,S)(A,B).

Now assume that P → A and Q→ B are perverse-projective resolu-tions of finite length (cf. Corollary and Definition 18), with underlyingprojective resolutions P → A and Q→ B. We apply HomPerv(X,S)(?, ?)

to P and Q and get a double complex of (polarizable) mixed Hodge

structures (see Remark 19) with (i, j)-component HomPerv(X,S)(P−i, Qj).

We denote its simple complex by Hom Perv(X,S)(P , Q) or simply by

Hom (P , Q). (We use this notation also for arbitrary complexes P and

Q in MHM(X,S).) The underlying complex of real vector spaces is thecomplex Hom (P,Q) = Hom Perv(X,S)(P,Q) from section 2.4 (slightlygeneralized to an R-linear abelian category). The n-th cohomology

Hn(Hom Perv(X,S)(P , Q)) is a mixed Hodge structure, and its underly-ing vector space is

Hn(Hom Perv(X,S)(P,Q)) = HomHot(Perv(X,S))(P, [n]Q)(39)

= HomD(Perv(X,S))(P, [n]Q)

= ExtnPerv(X,S)(P,Q)∼−→ ExtnPerv(X,S)(P,B)∼← ExtnPerv(X,S)(A,B).

Proposition 23. The (polarizable) mixed Hodge structures

ExtnPerv(X,S)(A, B) and Hn(Hom Perv(X,S)(P , Q))

with underlying vector space ExtnPerv(X,S)(A,B) coincide.

Proof. We write Hom, Hom and Ext instead of HomPerv(X,S),Hom Perv(X,S) and ExtPerv(X,S) respectively, and show the existence ofisomorphisms of mixed Hodge structures

(40) Extn(A, B)∼← Hn(Hom (P , B))

∼← Hn(Hom (P , Q)).

Let us construct the isomorphism on the left in (40). We decompose

P → A into short exact sequences as follows. For i ≤ 0, let Ki be the

image of the differential P i−1 → P i, and define K1 = A. For eachi ≤ 0, we get a short exact sequence

0→ Ki → P i → Ki+1 → 0.

2.10. COMPARISON OF MIXED HODGE STRUCTURES 33

The long exact Ext-sequence in the first variable gives an exact se-quence of mixed Hodge modules (see Remark 19)

0→ Hom(Ki+1, B)→ Hom(P i, B))→ Hom(Ki, B)(41)

→ Ext1(Ki+1, B)→ 0

and isomorphisms

(42) Extj(Ki, B)∼−→ Extj+1(Ki+1, B)

for all j ≥ 1.

The 0-th cohomology of the complex C := Hom (P , B) is

H0(C) = ker(

Hom(P 0, B))→ Hom(K0, B)) ∼−→ Hom(A, B))

by (41) for i = 0. For m ≥ 0 we have

Hm+1(C) = cok(

Hom(P−m, B))→ Hom(K−m, B))

∼−→ Ext1(K−m+1, B)∼−→ Extm+1(A, B)

by (41) and repeated use of (42).The existence of the isomorphism on the right in (40) will be a con-

sequence of the following more general claim: The perverse-projective

resolution Q→ B induces a quasi-isomorphism

Hom (P , Q)→ Hom (P , B).

This follows from Lemma 24 below. �

Lemma 24. Let P , Q and R be complexes in MHM(X,S), and

assume that P is perverse-projective and bounded above. Then any

quasi-isomorphism f : Q → R in Ket(MHM(X,S)) induces a quasi-isomorphism

Hom (P , Q)→ Hom (P , R).

Proof. It suffices to prove that rat(f) : Q→ R in Ket(Perv(X,S))induces a quasi-isomorphism Hom (P,Q) → Hom (P,R). But thisfollows from Remark 10. �

Remark 25. Assume that P → A, Q → B and R → C are

perverse-projective resolutions of finite length. Then Hom (P , Q) and

Hom (Q, R) are complexes of (polarizable) mixed Hodge structures.The obvious composition map

Hom (Q, R)⊗ Hom (P , Q)→ Hom (P , R)

is a morphism of complexes of mixed Hodge structures. It induces amorphism of mixed Hodge structures

Hi(Hom (Q, R))⊗ Hj(Hom (P , Q))→ Hi+j(Hom (P , R)).


Under the identifications from (the proof of) Proposition 23, this mor-phism corresponds to the morphism (33) in Remark 19. The reasonfor this fact is that both morphisms are just the composition of mor-phisms in the derived category of perverse sheaves, if we forget aboutthe mixed Hodge structures.

2.11. Local-to-Global Spectral Sequence

If X is a complex variety and M a complex of sheaves (of Hodgesheaves respectively) onX, the hypercohomology H(M) := H(X;M) :=H(c∗M) is a complex (with differential zero) of vector spaces (of mixedHodge structures respectively). Here c : X → pt is the constant map.If l : Y ↪→ X is a locally closed subvariety, the local hypercohomologyof M along Y is HY (M) := H(l!M) = H(l∗l

!M).Consider now a complex variety X filtered by closed subvarieties

X = X0 ⊃ X1 ⊃ · · · ⊃ Xr = ∅. If M ∈ Db(X) is a complex ofsheaves, there is a local-to-global spectral sequence with E1-term Ep,q

1 =Hp+qXp−Xp+1

(M) converging to Ep,q∞ = grp(Hp+q(M)). As described in

[BGS96, 3.4], it can be constructed by use of an injective resolutionof M .

Even though there are not enough injective Hodge sheaves, we shallconstruct a similar spectral sequence of mixed (polarizable) Hodgestructures, if M is a complex of Hodge sheaves on our filtered vari-ety X. In order to do so, we need the following technical proposition.(When we formulated and proved this proposition, we were not awareof the similar statement given in [BBD82, 3.1.2.7] without proof.)

Proposition 26. Let A be an abelian category and

(43) A0 A1a1oo . . .a2oo Ar−1

ar−1oo Ar = 0aroo

a finite sequence of objects and morphisms in Db(A). Then there is abounded complex K in A with a finite filtration F by subcomplexes, K =F 0K ⊃ F 1K ⊃ · · · ⊃ F r−1K ⊃ F rK = 0, and quasi-isomorphismsup : F pK → Ap in Ket(A) such that the diagram

(44) A0 A1a1oo A2

a2oo . . .a3oo Ar−1ar−1oo Ar

aroo

F 0K

u0

OO

F 1Kk1oo

u1

OO

F 2Kk2oo

u2

OO

. . .k3oo F r−1Kkr−1oo

ur−1

OO

F rKkroo

ur

OO

commutes in Db(A). Here kp : F pK ↪→ F p+1K denotes the inclusion.

2.11. LOCAL-TO-GLOBAL SPECTRAL SEQUENCE 35

Proof. We explain how the following diagram is constructed.

(45) A0 A1a1oo A2

a2oo A3a3oo A4

a4oo . . .a5oo

B0

r0

B1

b1ccHHHHHHHHH

r1

OO

B2

b2ccHHHHHHHHH

r2

OO

B3

b3ccHHHHHHHHH

r3

OO

B4

b4ccHHHHHHHHH

r4

OO

. . .

bbEEEEEEEEEE

C0

s0

C1c1oo

s1

C2c2oo

s2

OO

C3c3oo

s3

OO

C4c4oo

s4

OO

. . .oo

F 0K

t0

OO

F 1Kk1oo

t1

OO

F 2Kk2oo

t2

OO

F 3Kk3oo

t3

OO

F 4Kk4oo

t4

OO

. . .oo

The first row is the given one. We define C0 = B0 = A0 and s0 =r0 = idA0 . The derived category Db(A) is the localization of Hotb(A)by the quasi-isomorphisms. So each morphism ap can be representedby a “roof”

Ap−1 bp←− Bp rp−→ Ap

in Hotb(A), where rp is a quasi-isomorphism. We can and do chooseBr = 0. This explains the second row of our diagram. By definition,we put C1 = B1, s1 = idB1 and c1 = b1. The rest of the third rowis constructed as follows. By the axioms of a multiplicative system([KS94, 1.6.1]), the diagram given by the morphisms r1s1 and b2 canbe completed in Hotb(A) by an object C2, a morphism c2 and a quasi-isomorphism s2 such that b2s2 = r1s1c2. Now we complete the diagramgiven by r2s2 and b3 by C3, c3 and a quasi-isomorphism s3, and proceedin this manner. Again we can and do choose Cr = 0.

By choosing representatives we may assume that all cr are mor-phisms in Ketb(A). We describe the construction of the fourth rowonly in the case r = 3, but the general procedure should then be obvi-ous. The morphism cp fits in the commutative diagram

Cp−1 Cpcpoo

Cp−1 ⊕M(Cp)

( id 0 0 )

OO

Cp

(cpid0

)oo

id

OO

in Ket(A), where M(Cp) = Cp ⊕ Cp[1] is the mapping cone of idCp .Note that M(Cp) is an acyclic object, so both vertical maps are quasi-isomorphisms. Furthermore, the lower horizontal map is a monomor-phism. We add the identity of M(C2) to the lower horizontal map ofthis diagram for p = 1, combine it with the above diagram for p = 2


and get the two left squares in the commutative diagram

C0 C1c1oo C2

c2oo C3 = 0c3oo

C0 ⊕M(C1)⊕M(C2)

OO

C1 ⊕M(C2)oo

OO

C2oo

OO

C3 = 0oo

OO

with obvious maps. Note that all morphisms in the lower row aremonomorphisms of complexes. We define this diagram to be the lowertwo rows in diagram (45) (still in the case r = 3). It remains to defineup = rpsptp (to be more precise, we have to choose representatives ofrp and sp in Ketb(A) first). �

Now let M ∈ Db(MHM(X)) be a complex of Hodge sheaves on X,where X is a complex variety, filtered by closed subvarieties X = X0 ⊃X1 ⊃ · · · ⊃ Xr = ∅. If we let ip : Xp ↪→ X denote the inclusion, theadjunctions (ip∗ = ip!, i

!p) yield a sequence in Db(MHM(X))

(46)

M = i0∗i!0M i1∗i

!1M

oo . . .oo ir−1∗i!r−1M

oo ir∗i!rM = 0.oo

We apply c∗ to this sequence, where c : X → pt. Proposition 26 showsthat there is a diagram(47)

c∗M c∗i1∗i!1M

oo c∗i2∗i!2M

oo . . .oo 0oo

K = F 0K

u0

OO

F 1Kk1oo

u1

OO

F 2Kk2oo

u2

OO

. . .k3oo F rK = 0,kroo

ur

OO

where the lower horizontal row is a finite filtration on a complex K inMHS, the vertical maps are quasi-isomorphisms in Ketb(MHS), and thediagram commutes in Db(MHS). (We could write MHM(pt) instead ofMHS.)

By [Lan02, Proposition XX.9.3], there exists a spectral sequence(Er, dr)r≥0 with

Ep,q0 = grpF (Kp+q) = F pKp+q/F p+1Kp+q,

Ep,q1 = Hp+q(grpF (K)),

Ep,q∞ = grp(Hp+q(K)),

where H(K) is filtered by the images of the obvious maps H(F pK)→H(K). This filtration corresponds under the isomorphism H(u0) :

H(K)∼−→ H(c∗M) = H(M) to the filtration by the images of the obvious

maps H(c∗ip∗i!pM) = HXp(M)→ H(M).

We want to identify the E1-term Ep,q1 of our spectral sequence with

Hp+qXp−Xp+1

(M). If j1 : X0 −X1 ↪→ X0 is the open embedding, property

2.11. LOCAL-TO-GLOBAL SPECTRAL SEQUENCE 37

(P8) yields a distinguished triangle

i1∗i!1M →M → j1∗j

∗1M

[1]−→ .

The decomposition of Xp in the closed subvariety Xp+1 and its opencomplement Xp−Xp+1 gives a similar distinguished triangle with mid-dle term i!pM . We apply ip∗ to it and get the distinguished triangle

(48) ip+1∗i!p+1M → ip∗i

!pM → jp+1∗j

!p+1M

[1]−→

where jp : Xp−1 − Xp ↪→ X is the locally closed embedding. On theother hand, the filtration on K gives rise to distinguished triangles

F p+1Kkp+1−−→ F pK → F pK/F p+1K

[1]−→

in Db(MHS) and the maps (up+1, up) can be completed to a morphismof distinguished triangles

F p+1Kkp+1 //

up+1

��

F pK //

up

��

F pK/F p+1K[1]

//

up/p+1

��c∗ip+1∗i

!p+1M // c∗ip∗i

!pM // c∗jp+1∗j

!p+1M

[1]//

in Db(MHS), where the lower distinguished triangle is c∗(48). Sinceup+1 and up are quasi-isomorphisms, up/p+1 is an isomorphism. Takingthe (p+ q)-th cohomology yields an isomorphism

Ep,q1 = Hp+q(grpF (K))

Hp+q(up/p+1)

∼// Hp+q(c∗jp+1∗j

!p+1M) = Hp+q

Xp−Xp+1(M).

Under this identification, the differential d1 : Ep,q1 → Ep+1,q

1 gets inden-tified with the composition

(49) Hp+qXp−Xp+1

(M) −→ Hp+1+qXp+1

(M) −→ Hp+1+qXp+1−Xp+2

(M),

where both morphisms come from triangles of the type (48). Let usstate what we have proved.

Proposition 27. Let X be a complex variety, filtered by closedsubvarieties X = X0 ⊃ X1 ⊃ · · · ⊃ Xr = ∅, and M ∈ Db(MHM(X))a complex of Hodge sheaves on X. Then there is a spectral sequence(Er, dr)r≥0 of (polarizable) mixed Hodge structures with E1-term Ep,q

1 =Hp+qXp−Xp+1

(M) and differential d1 given by the composition (49) that

converges to Ep,q∞ = grp(Hp+q(M)), where H(M) is filtered by the images

of the obvious maps HXp(M)→ H(M).


2.12. Purity

Let (X,S) be a stratified complex variety, M ∈ MHM(X) and

w ∈ Z. We say that M is S-∗-pure of weight w, if, for all strata

S ∈ S, the restrictions l∗SM are pure of weight w. It is S-!-pure of

weight w, if all restrictions l!SM are pure of weight w, and S-pure ofweight w, if it is S-∗-pure and S-!-pure of weight w.

Theorem 28. Let (X,S) be a cell-stratified complex variety, M ,

N ∈ MHM(X,S) smooth Hodge sheaves, and m, n ∈ Z. If M is S-

∗-pure of weight m and N is S-!-pure of weight n, the complex (withdifferential zero) of (polarizable) mixed Hodge structures

ExtPerv(X,S)(M, N) =⊕i∈N

ExtiPerv(X,S)(M, N)

is pure of weight n−m.

Proof. Recall that the mixed Hodge structure ExtiPerv(X,S)(M, N)

comes from Hi(c∗Hom (M, N)) = Hi(Hom (M, N)). We define Xp tobe the union of all strata whose codimension in X is greater or equal top, Xp =

⋃S ∈ S, dS + p ≤ dimC X

S. This defines a filtration of X by closed

subvarieties, X = X0 ⊃ X1 ⊃ · · · ⊃ Xr = ∅, where r = dimCX + 1.Proposition 27 shows that there is a spectral sequence of mixed Hodge

structures with E1-term Ep,q1 = Hp+q

Xp−Xp+1(Hom (M, N)) converging to

Ep,q∞ = grp(Hp+q(Hom (M, N))). Lemma 29 below shows that Ep,q

1 isa pure Hodge structure of weight p+ q+n−m. There are no non-zeromorphisms between pure Hodge structures of different weights, henceour spectral sequence degenerates at the E1-term, i. e. E1 = E2 = . . . =

E∞. Furthermore, Hp+q(Hom (M, N)) is pure of weight p+ q+n−m,since it has a finite filtration with successive subquotients that are pureand of the same weight (it is in fact isomorphic to the direct sum ofthese subquotients, see (P3)). �

Lemma 29. Under the assumptions of Theorem 28 and with the

notation introduced in its proof, Ep,q1 = Hp+q

Xp−Xp+1(Hom (M, N)) is a

pure Hodge structure of weight p+ q + n−m.

Proof. The decomposition Xp − Xp+1 =⋃S ∈ S, dS + p = dimC X

Sinto strata of codimension p is the decomposition of Xp − Xp+1 intoconnected components. Therefore, we have

Hp+qXp−Xp+1

(Hom (M, N)) =⊕

S ∈ S, dS + p = dimC X

Hp+q(l!SHom (M, N)),

where lS : S ↪→ X is the inclusion of the stratum S ∈ S. For eachS ∈ S, we have by property (P12)

l!SHom (M, N) = D(l∗SM ⊗ Dl!SN).

2.13. FORMALITY OF SOME DG ALGEBRAS 39

The restrictions l∗SM and l!SN are pure, so (P4) yields isomorphisms

l∗SM∼=⊕

i∈Z[−i] Hi(l∗SM) and l!SN∼=⊕

j∈Z[−j] Hj(l!SN).

Fix i, j ∈ Z. Since Hi(l∗SM) is in MHM(S, {S}) and pure of weightm + i, Proposition 14 shows that there is A′ ∈ MHM(pt) pure of

weight m + i − dS such that Hi(l∗SM) ∼= [dS]c∗A′. Then A := [dS −i]A′ ∈ Db(MHM(pt)) is pure of weight m and we have [−i] Hi(l∗SM) ∼=c∗(A). If we proceed similarly and use (P7) and (P13), we find B ∈Db(MHM(pt)) pure of weight n such that [−j] Hj(l!SN) ∼= c!(B). Using

(P12), (P13) and the adjunction isomorphism id∼−→ c∗c

∗, we obtain

Hp+q(D(c∗(A)⊗ D(c!B))

)= Hp+q

(c∗c

!Hom (A,B))

= Hp+q([2dS]c∗c

∗Hom (A,B)(dS))

∼← Hp+q([2dS]Hom (A,B)(dS)

),

and this is pure of weight p+ q + n−m by (P2), (P7) and (P11). �

2.13. Formality of some DG Algebras

Let X be a complex variety with a cell-stratification S, and M ∈MHM(X,S) a smooth Hodge sheaf. By Corollary 18, there is a perverse-

projective resolution P → M of finite length, with underlying projec-tive resolution P →M . As in section 2.10, we consider the complex

A := End (P ) := Hom Perv(X,S)(P , P ).

of (polarizable) mixed Hodge structures. Remark 25 shows that the

multiplication (= composition map) A ⊗ A → A is a morphism ofcomplexes of mixed Hodge structures. Note that a complex of mixedHodge structures is the same as an object of the tensor category dgMHS

of differential graded mixed Hodge structures. So A is a (unital) ringobject in dgMHS (a “dg algebra of mixed Hodge structures”).

The exact faithful R-linear tensor functors “underlying vector space”ω0, “associated graded vector space” grWR , “underlying vector space”η and “underlying vector space of the associated graded vector space”ωW = η ◦ grWR (see section 2.6) induce tensor functors (denoted by thesame symbol):

(50) dgMHS

ω0

��

ωW

''OOOOOOOOOOOO

grWR // dggMod(R)

η

��dgMod(R) dgMod(R)

Here we consider R as a dg R-algebra concentrated in degree 0 and asa dgg R-algebra concentrated in degree (0, 0) (see sections 2.1 and 2.2).More elementary, dgMod(R) and dggMod(R) are the categories of dg


real vector spaces and dg graded real vector spaces respectively. Theisomorphism a from (27) induces an isomorphism

(51) a : ω0∼−→ ωW

between the induced functors.Then A = ω0(A) is the dg algebra End (P ) (see sections 2.4 and

2.10). Its cohomology is the extension algebra ExtPerv(X,S)(P ) and iso-morphic to ExtPerv(X,S)(M) (see (12) and (39)).

Theorem 30. Let (X,S), P → M , A and A be as above, and w an

integer. If M is S-pure of weight w, then A is formal. More precisely,there are a dg subalgebra Sub(A) of A and dga-quasi-isomorphismsA←↩ Sub(A) � H(A).

Proof. Consider the dgg algebra R := grWR (A). Its graded com-

ponents are Rij = grWj,R(Ai). By Proposition 23 and Theorem 28, the

complexes (with differential zero) of mixed Hodge structures H(A) and

ExtPerv(X,S)(M, M) are isomorphic and pure of weight 0. So

grWj,R(Hi(A)) = Hi(grWj,R(A)) = Hi(R∗j) = (H(R))ij

vanishes for i 6= j, and H(R) is pure of weight zero. Proposition 6shows the existence of dgga-quasi-isomorphisms

R←↩ Γ(R) � H(R).

We apply η to this sequence and obtain the dga-quasi-isomorphisms inthe first row in diagram

(52) ωW (A) = R Γ(R)? _oo // // H(R) = ωW (H(A))

ω0(A) = A

∼ a eAOO

Sub(A)? _oo

∼

OO

H(A) = ω0(H(A)),

∼ aH( eA)

OO

where R := η(R), Γ(R) := η(Γ(R)) and the dg subalgebra Sub(A) ⊂ Ais defined as the pull-back, i. e. Sub(A) := a−1(Γ(R)). All vertical(horizontal) morphisms in this diagram are dga-(quasi-)isomorphisms.

�

We generalize Theorem 30 slightly as follows. Let (X,S) be as

above, I a finite set and Pα → Mα perverse-projective resolutions of

finite length of smooth Hodge sheaves Mα (α ∈ I). Let P be the direct

sum of the (Pα)α∈I , A = End (P ), A = ω0(A) = End (P ) the dg algebra

underlying the “dg algebra of mixed Hodge structures” A, and eα ∈ A0

the projector from P onto the direct summand Pα. The cohomol-ogy H(A) of A is isomorphic to the extension algebra ExtPerv(X,S)(M),where M is the direct sum of the underlying perverse sheaves (Mα)α∈I .

2.14. FORMALITY OF CELL-STRATIFIED VARIETIES 41

Theorem 31. Let X, S, I, Pα → Mα, A, A, eα be as above. Let

wα ∈ Z (α ∈ I) be integers. If Mα is S-pure of weight wα, for all α ∈ I,then A is formal. More precisely, there are a dg subalgebra Sub(A) of Acontaining all (eα)α∈I and quasi-isomorphisms A ←↩ Sub(A) � H(A)of dg algebras.

Proof. The proof is very similar to that of Theorem 30. Mainly,we use Proposition 7 instead of Proposition 6. �

2.14. Formality of Cell-Stratified Varieties

Theorem 32. Let X be a complex variety with a cell-stratification

S, (Mα)α∈I a finite number of smooth Hodge sheaves Mα ∈ MHM(X,S)

with direct sum M =⊕

Mα. Denote by

Ext(M) := ExtSh(X)(M)

the extension algebra of M = v(M), a dg algebra with differential d = 0.Let eα ∈ Ext0(M) = EndSh(X)(M) be the projector from M onto the

direct summand Mα = v(Mα). If there are integers (wα)α∈I , such that

Mα is S-pure of weight wα, for all α ∈ I, there is an equivalence oftriangulated categories

(53) PraeVerd({Mα}α∈I ,Db(X)) ∼= dgPraeExt(M)({eα Ext(M)}α∈I).

Remarks 33. (a) Under the equivalence we construct in theproof, the objects Mα and eα Ext(M) correspond. We do notemphasize similar obvious correspondences in the following.

(b) In equivalence (53) we can write dgPrae instead of dgPrae, byequivalence (17).

(c) We can replace Ext(M) in Theorem 32 by ExtPerv(X,S)(M) orExtPerv(X)(M) (using (24) and (26)). From equivalences (23)and (25) we deduce that we can replace the left hand side of(53) by

PraeVerd({Mα}α∈I ,Db(Perv(X,S)))

or

PraeVerd({Mα}α∈I ,Db(Perv(X)).

Proof. Let Pα → Mα be perverse-projective resolutions of finitelength (Corollary 18) and Pα → Mα the underlying projective reso-

lutions in Perv(X,S). Let P =⊕

Pα and P =⊕

Pα. As in the

second part of section 2.13, we define A = End (P ) and A = ω0(A) =End (P ). Theorem 31 yields a dg subalgebra Sub(A) of A and dga-quasi-isomorphisms

(54) A←↩ Sub(A) � H(A).


We claim that

PraeVerd({Mα},Db(X))

realX,S←−−−− PraeVerd({Mα},Db(Perv(X,S)))(55)

Hom (P,?)−−−−−−→ dgPraeA({eαA})(56)

?L⊗

Sub(A)A

←−−−−− dgPraeSub(A)({eα Sub(A)})(57)

?L⊗

Sub(A)H(A)

−−−−−−−→ dgPraeH(A)({eα H(A)})(58)

?L⊗

H(A)Ext(M)

−−−−−−−→ dgPraeExt(M)({eαExt(M)}).(59)

is a sequence of triangulated equivalences. By Theorem 13, (55) is anequivalence. The isomorphisms Pα →Mα in Db(Perv(X,S)), Proposi-tion 9 and Remark 10 show that (56) is an equivalence. (The eα hereare in fact representatives of the eα in the Theorem, but we do notcare about this too much.) The dga-quasi-isomorphisms (54) induceequivalences (see Theorem 5) between the respective categories dgPerand restrict to the equivalences (57) and (58). The last equivalence issimilarly induced by the dga-isomorphism (cf. (39))

H(A) = H( End (P ))= Ext(P )P∼−→M−−−−→∼

Ext(M)real−−→ Ext(M).

�

Remark 34. We use the notation

Form eP→fM : PraeVerd({Mα},Db(X))∼−→ dgPraeExt(M)({eαExt(M)}).

for the “Formality equivalence” (53) constructed in the proof of The-orem 32 to indicate that it mainly depends on the perverse-projective

resolutions Pα → Mα.

2.15. Formality and Intersection Cohomology Sheaves

Let (X, T ) be a cell-stratified complex variety. If ICT is T -pureof weight dT , for all T ∈ I, Theorem 32 and Proposition 12 yield anequivalence

(60) Db(X, T ) ∼= dgPraeE({eTE}T∈T ),

where E = Ext(IC(T )) is the extension algebra of the direct sum IC(T )of the (ICT )T∈T , and eT is the projector from this direct sum onto ICT .In order to shorten the notation we simply write dgPrae(Ext(IC(T )))for the category on the right in (60) and use similar notation in the

2.15. FORMALITY AND INTERSECTION COHOMOLOGY SHEAVES 43

following. We introduce this notation more precisely in (99) in Section4.1. So equivalence (60) becomes

Db(X, T ) ∼= dgPrae(Ext(IC(T ))).

If T ′ is a subset of T and all ICT ′ are T -pure of weight dT ′ for allT ′ ∈ T ′, we get by Theorem 32 an equivalence

(61) PraeVerd({ICT ′}T ′∈T ′ ,Db(X)) ∼= dgPrae(Ext(IC(T ′))).

Theorem 35. Let (X,S) be a stratified variety with simply con-

nected strata. Let T be a cell-stratification refining S. If ICS is T -pureof weight dS for all S ∈ S, there is a triangulated equivalence

(62) Db(X,S) ∼= dgPrae(Ext(IC(S))).

Proof. Since each S ∈ S is irreducible, it contains a (unique)dense stratum T (S) ∈ T ; then ICS = ICT (S). We apply equivalence(61) to the set T ′ of these dense strata. Then we use Proposition 12 andthe fact that the (ICS)S∈S are the simple objects of Perv(X,S). �

Remark 36. It will turn out that we can replace dgPrae in (62) bydgPer (see Corollary 66). We will introduce in Section 4.1, Theorem65, a t-structure on dgPrae(E), where E = Ext(IC(S)). If we considerthe perverse t-structure on Db(X,S), equivalence (62) will be t-exact,since it maps ICS to eSE (eS the obvious projector). In particular, itrestricts to an equivalence between the hearts. Since the heart of thet-structure on dgPrae(E) will be described in Section 4.3 as the fullabelian subcategory dgFlag(E) of dgMod(E), we obtain an equivalence

Perv(X,S) ∼= dgFlag(E).

Remark 37. We use the notation

(63) FormTeP→fIC(S): Db(X,S)

∼−→ dgPrae(Ext(IC(S)))

for equivalence (62) to indicate its dependence on the refinement T and

the perverse-projective resolutions PS → ICS (cf. Remark 34).

Remark 38. In Theorem 35, it is sufficient to require that each ICSis T -∗-pure of weight dS. This follows from the following observation. IfS is a stratum of a stratified variety and l the inclusion of a subvariety,

we have D(ICS) ∼= ICS(dS) and obtain

l!(ICS) = D(l∗(D(ICS))) ∼= D(l∗(ICS(dS)))

= D(l∗(ICS))(−dS).

So if l∗(ICS) is pure of weight dS, then l!(ICS) is pure of weight −dS +2dS = dS.


2.16. Formality of Partial Flag Varieties

Let G ⊃ B be a complex connected reductive affine algebraic groupand a Borel subgroup. Let P , Q be parabolic subgroups of G containingB. The Bruhat decomposition of the flag variety G/B into B-orbitsis a cell-stratification. (It is a Whitney stratifications; as Peter Fiebigtold me, he learned from Tom Braden that this follows from [Kal05].)More generally, the B-orbits on the partial flag variety G/P form acell-stratification, and the Q-orbits on G/P form a stratification. (Thisshould also follow from [Kal05].)

Proposition 39. The Q-orbits in G/P are simply connected.

Proof. Let Y ⊂ G/P be a Q-orbit and T ⊂ B ⊂ G a maximaltorus. Then Y = QwP/P for some w in the normalizer of T in G.The stabilizer S of wP/P in Q is Q ∩ wPw−1 and is connected as theintersection of two parabolic subgroups of a connected reductive group([Bor91, 14.22]). Since the orbit map Q → Y , q 7→ qnP/P , is a (atleast) topological S-principal fiber bundle, we have the exact sequenceof homotopy groups

π1(S)→ π1(Q)→ π1(Y )→ π0(S)→ π0(Q)→ π0(Y )→ 1.

We know that S and Q are connected and hence have to prove thatπ1(S) → π1(Q) is surjective. Since T ⊂ S it is sufficient to show thatπ1(T ) → π1(Q) is surjective. If LQ is a Levi subgroup of Q, thenT ⊂ LQ and π1(Q) = π1(LQ). Hence our claim is a consequence of thefollowing Lemma 40. �

Lemma 40. If T is a maximal torus in a connected reductive groupL, then π1(T )→ π1(L) is surjective.

Proof. Let A ⊂ L be a Borel subgroup containing T . Thenπ1(T ) = π1(A) and the long exact homotopy sequence for the A-principal fiber bundle L→ L/A yields an exact sequence

π1(A)→ π1(L)→ π1(L/A)→ π0(A).

Since A is connected and fundamental groups of topological groups areabelian, π1(L/A) is abelian and vanishes since the flag variety L/A hasonly even cohomology (Hurewicz isomorphism and Poincare duality).So π1(A)→ π1(L) is surjective. �

Theorem 41 ([Soe89], [KL80]). Let F ∈ Db(MHM(G/B)) bepure and smooth along the stratification by Bruhat cells. Let l : Y →G/B be the inclusion of a P -orbit. Then l∗(F ) and l!(F ) are pure aswell, of the same weight as F .

Proof. See [Soe89, Parabolic Purity Theorem]. There, G is as-sumed to be semisimple, but we may replace G by (G,G) etc. �

2.16. FORMALITY OF PARTIAL FLAG VARIETIES 45

Corollary 42. Let F ∈ Db(MHM(G/P )) be pure and smoothalong the stratification by B-orbits. Let l : Y → G/P be the inclusionof a B-orbit. Then l∗(F ) and l!(F ) are pure as well, of the same weightas F .

Proof. Let π : G/B → G/P be the obvious projection and Z ⊂π−1(Y ) the unique Bruhat cell such that π induces an isomorphism

Z∼−→ Y :

Z //

∼ π

��

π−1(Y ) //

π

��

G/B

π

��

Y Yl // G/P.

Since π is smooth of relative dimension n = dimC(P/B), π∗(F ) =[−2n]π!(F )(−n) and π!(F ) are pure of the same weight as F (use (P1),(P7) and (P13)). Obviously, π∗(F ) is smooth along the stratificationby Bruhat cells, and so is π!(F ). Hence we deduce from Theorem 41that l∗(F ) and l!(F ) are pure of the same weight as F . �

Theorem 43. If Q is the stratification of G/P into Q-orbits, thereis an equivalence of triangulated categories

(64) Db(G/P,Q) ∼= dgPrae(Ext(IC(Q))).

Remark 36 applies to Theorem 43 mutatis mutandis.

Proof. The strata of Q are simply connected by Proposition 39.

The cell-stratification T of G/P into B-orbits refines Q, and every ICQis T -pure of weight dQ, for Q ∈ Q (Corollary 42). Hence we can applyTheorem 35. �

CHAPTER 3

Formality and Closed Embeddings

We formulate in Section 3.1 the goal of this Chapter and explain inSection 3.2 the main application. The proof of the goal is divided intoseveral parts and given in the following sections.

3.1. The Goal of the Chapter

Let (X,S) and (Y, T ) be cell-stratified complex varieties, and i :Y → X a closed embeding such that i(T ) := {i(T ) | T ∈ T } ⊂ S.We say for short that i : (Y, T ) → (X,S) is a closed embedding ofcell-stratified varieties. (If S and T are merely stratifications, theterm closed embedding of stratified varieties is defined similarly.)

Assume that we are in the setting of Theorem 32 on X and on Y .

More precisely, let (Mα)α∈I and (Nα)α∈I be finite collections of smoothHodge sheaves on X and on Y . Assume that there are integers (wα)α∈I(resp. (vα)α∈I) such that Mα (resp. Nα) is S-pure (resp. T -pure) ofweight wα (resp. vα), for all α ∈ I.

Let µ be an integer (in our applications, µ will be the negativecomplex codimension of the inclusion i : Y → X, and we will havewα + µ = vα for all α ∈ I). Suppose that there are isomorphisms

(65) σα : [µ]i∗(Mα)∼−→ Nα

in Db(MHM(Y )), for all α ∈ I. Let σ : [µ]i∗(M)∼−→ N be the direct

sum of these isomorphisms, where M =⊕

Mα and N =⊕

Nα.

Let πα : Pα → Mα and ρα : Qα → Nα be perverse-projective

resolutions and π : P → M and ρ : Q → N their direct sum. Thevertical equivalences in(66)

PraeVerd({Mα},Db(X))[µ]i∗

//

Form eP→fM ∼��

PraeVerd({Nα},Db(Y ))

Form eQ→ eN ∼��

dgPraeExt(M)({eα Ext(M)})?

L⊗

Ext(M)Ext(N)

//

∼/7ffffffffffffffffffffff

ffffffffffffffffffffffdgPraeExt(N)({eα Ext(N)})

come from Theorem 32, cf. Remark 34. The horizontal functors donot depend on the choice of perverse-projective resolutions. The upperarrow is the restriction of [µ]i∗ : Db(X) → Db(Y ), the lower one is

47

48 3. FORMALITY AND CLOSED EMBEDDINGS

the restriction of the extension of scalars functor coming from the dga-morphism

(67) Ext(M)[µ]i∗−−→ Ext([µ]i∗(M))

σ−→∼

Ext(N).

Here the letter σ above the arrow indicates that the arrow is con-structed using the morphism σ : [µ]i∗(M)

∼−→ N . We will sometimesuse similar notation in the following. The goal of this chapter is toprove that there is a natural isomorphism as indicated in (66). (Wesay natural transformation instead of morphism of functors and natu-ral isomorphism instead of isomorphism of functors. We consider onlynatural transformations of triangulated functors that are compatiblewith the shifts.)

Theorem 44. Keep the above assumptions. Then diagram (66)commutes up to natural isomorphism, i. e. there is a natural isomor-phism

(?L⊗

Ext(M)Ext(N)) ◦ Form eP→fM ∼−→ Form eQ→ eN ◦ [µ]i∗.

Proof. Let A = End (P ) and B = End (Q). We define A andSub(A) as in the proof of Theorem 32, and B and Sub(B) accordingly.We only prove the theorem in the case that I is a singleton, the gen-eral case being an obvious generalization. We expand diagram (66)according to the sequence of equivalences (55)-(59) to the followingdiagram.

3.1. THE GOAL OF THE CHAPTER 49

(68)

PraeVerd(M,Db(X))[µ]i∗

// PraeVerd(N,Db(Y ))

PraeVerd(M,Db(Perv(X,S)))[µ]Lpi∗

//

realX,S ∼

OO

Hom (P,?) ∼

��

PraeVerd(N,Db(Perv(Y, T )))

realY,T∼

OO

Proposition 48

∼

dl QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

Hom (Q,?)∼

��dgPraeA(A)

∼Corollary 50

2:mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm ?L⊗AHom (Q,[µ]pi∗(P ))

// dgPraeB(B)

dgPraeSub(A)(Sub(A))

?L⊗

Sub(A)A ∼

OO

?L⊗

Sub(A)Sub(Hom (Q,[µ]pi∗(P )))

//

?L⊗

Sub(A)H(A) ∼

��

dgPraeSub(B)(Sub(B))

?L⊗

Sub(B)B∼

OO

?L⊗

Sub(B)H(B)∼

��

∼Section 3.11

dl QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

∼Section 3.11

rz mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

dgPraeH(A)(H(A))?

L⊗

H(A)H(Hom (Q,[µ]pi∗(P )))

//

?L⊗

H(A)Ext(M) ∼

��

dgPraeH(B)(H(B))

?L⊗

H(B)Ext(N)∼

��

∼Section 3.12

rz mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

dgPraeExt(M)(Ext(M))?

L⊗

Ext(M)Ext(N)

// dgPraeExt(N)(Ext(N))

The horizontal functors in this diagram will be explained in the restof this chapter. It will result from the specified Proposition, Corollaryand Sections that all squares commute up to natural isomorphism, asindicated by the diagonal arrows. Since all vertical arrows are equiva-lences, this proves the theorem. �


3.2. Formality and Normally Nonsingular Inclusions

If f : Y → X is a closed embedding of irreducible varieties and, inthe classical topology, a normally nonsingular inclusion of (complex)codimension c (cf. [GM88, I.1.11]), we have a canonical isomorphism([GM83, 5.4.1], and [BBD82, 0] for the different normalization)

[−c]f ∗(IC(X))∼−→ IC(Y ) in Perv(Y ).(69)

It comes from a canonical isomorphism

[−c]f ∗(IC(X))∼−→ IC(Y ) in MHM(Y ).(70)

Let i : Y → X be a closed embedding of varieties and assume thatwe have stratifications S of X and T of Y (with irreducible strata).Suppose that S → T , S 7→ i−1(S) = S∩Y , is bijective and that i|S∩Y :S ∩ Y → S is a normally nonsingular inclusion of a fixed codimensionc, for all S ∈ S. The isomorphisms (69) and (70) induce isomorphisms

[−c]i∗(ICS)∼−→ ICS∩Y in Perv(Y, T ) and(71)

[−c]i∗(ICS)∼−→ ICS∩Y in MHM(Y, T ).(72)

Theorem 45. Let i : Y → X and S, T be as above. Assume that

(a) all strata in S and T are simply connected,(b) there are cell-stratifications S ′ and T ′ refining S and T such

that• ICS is S ′-pure of weight dS, for all S ∈ S,

• ICT is T ′-pure of weight dT , for all T ∈ T , and• i : (Y, T ′)→ (X,S ′) is a closed embedding of cell-stratified

varieties.

Let PS → ICS and QT → ICT be perverse-projective resolutions (smoothalong the cell-stratifications), for S ∈ S and T ∈ T . Then diagram

(73) Db(X,S)[−c]i∗

//

∼FormS′eP→fIC(S)

��

Db(Y, T )

∼FormT′eQ→fIC(T )

��dgPrae(Ext(IC(S)))

?L⊗

Ext(IC(S))Ext(IC(T ))

// dgPrae(Ext(IC(T )))

is commutative (up to natural isomorphism). The vertical functors aregiven by Theorem 35 (cf. Remark 37), the extension of scalars functoris induced by the isomorphisms (71).

Remark 46. Remark 36 applies to the vertical equivalences in di-agram (73). If we equip all categories in this diagram with t-structuresas described there, all functors will be t-exact.

For the extension of scalars functor this is obvious. The t-exactnessof [−c]i∗ can be proved as follows: Since all strata in S are simply

3.3. DERIVED CATEGORY OF (PERVERSE) SHEAVES 51

connected, the (ICS)S∈S are the simple objects of Perv(X,S). Everyobject of Perv(X,S) has finite length. Now the isomorphisms (71) andthe long exact perverse cohomology sequence show the t-exactness.

Proof. This is a consequence of Theorem 44 and the results ofSection 2.15. �

3.3. Derived Category of (Perverse) Sheaves

Let (X,S) be a stratified variety. We recall from [BBD82, 3.1]how the functor

real = realX,S : Db(Perv(X,S))→ Db(X,S)

(cf. (25)) is defined. We specialize the notation from [BBD82, 3.1.5]down to our needs. Let A = Sh(X) be the category of sheaves ofreal vector spaces on X with respect to the classical topology andD = D+(X,S) the full triangulated subcategory of D+A = D+(X)consisting of S-constructible objects. Let (D≤0,D≥0) be the perverset-structure on D for the middle perversity p. Its heart C is Perv(X,S).Then Db, the union of the D[a,b], is precisely Db(X,S) (cf. Proposition12).

We consider the filtered derived category D+FA = D+F (Sh(X)).Its objects are finitely filtered (we use decreasing filtrations) complexesin Sh(X). Let DF be the full subcategory consisting of objects (K,F )such that all griF (K) are in D. Similarly, DbF is defined by the con-dition that all griF (K) are in Db. We denote these categories in ourspecial case by D+F (X,S) and DbF (X,S).

If (K,F ) is an object of DbF , the distinguished triangles

gri+1F (K)→ F iK/F i+2K → griF (K)

[1]−→provide morphims griF (K)→ [1] gri+1

F (K). If we defineKi := [i] griF (K),these morphisms define differentials in the complex G(K) := (· · · →Ki → Ki+1 → · · · ). In fact, this defines a functor G : DbF → Ketb(D).

Let DbFbete be the full subcategory of DbF consisting of objects(K,F ) with all [i] griF (K) in C. The functor G induces an equivalence(see [BBD82, 3.1.8])

G : DbFbete∼−→ Ketb(C).

Let C(Inj) be the intersection of C and Ketb(Inj(A)), where Inj(A)is the category of injective objects of A. Since A has enough injectives,the inclusion C(Inj) ⊂ C is an equivalence of categories. Let A be aninverse, and denote by the same letter the induced equivalence

A : Ketb(C) ∼−→ Ketb(C(Inj))

that is inverse to Ketb(C(Inj)) ⊂ Ketb(C). In the proof of [BBD82,3.1.8], a functor

H : Ketb(C(Inj))→ DbFbete


is explicitly (with some choices) constructed, such that H ◦A is inverseto G.

Let ω : D+FA → D+A be the ”forgetting the filtration” functor.By [BBD82, 3.1.10], the realization functor

real := ω ◦H ◦ A : Ketb(C)→ Db.

factors to a triangulated functor

real : Db(C)→ Db.

This functor depends on the choices of A and of H. In our case thisfunctor is

(74) real = realX,S : Db(Perv(X,S))→ Db(X,S).

Let AX = AX,S and HX = HX,S be the equivalences A and H involvedin the definition of realX,S .

3.4. Closed Embeddings

Let i : (Y, T )→ (X,S) be a closed embedding of stratified varieties.Since i∗ is perverse t-exact [BBD82, 1.3.17, 1.4.16],

pi∗ : Perv(Y, T )→ Perv(X,S)

is exact and induces the functor pi∗ in diagram

(75) Db(Perv(Y, T ))realY,T //

pi∗��

Db(Y, T )

i∗��

Db(Perv(X,S))realX,S // Db(X,S).

Proposition 47. Keep the above assumptions. Then diagram (75)commutes up to natural isomorphism.

Proof. We even claim that the equivalences AX and HX can bechosen so that diagram (75) commutes. Since

DbFbete(Y, T )ω //

i∗��

Db(Y, T )

i∗��

DbFbete(X,S)ω // Db(X,S)

obviously commutes, it is enough to prove that(76)

Ketb(Perv(Y, T ))AY //

pi∗��

Ketb(Perv(Y, T , InjSh))HY //

pi∗��

DbFbete(Y, T )

i∗��

Ketb(Perv(X,S))AX // Ketb(Perv(X,S, InjSh))

HX // DbFbete(X,S)

3.5. TENSOR PRODUCT WITH A DG BIMODULE 53

is commutative, if we choose AX and HX wisely. The arrow in themiddle is well defined since pi∗ alias i∗ maps InjSh(Y ) to InjSh(X).Since the vertical functors pi∗ and i∗ are fully-faithful embeddings, it ispossible to choose AX and HX in such a way that (76) is commutative(cf. the definition of H in [BBD82, 3.1.8]). �

Let i : (Y, T ) ↪→ (X,S) be a closed embedding of cell-stratifiedvarieties. Theorem 13 shows that the left derived functor

Lpi∗ : Db(Perv(X,S))→ Db(Perv(Y, T ))

of the right exact functor pi∗ : Perv(X,S)→ Perv(Y, T ) exists.

Proposition 48. Let i : (Y, T ) ↪→ (X,S) be a closed embedding ofcell-stratified varieties. Then there exists a natural isomorphism

τ : realY,T ◦Lpi∗∼−→ i∗ ◦ realX,S

as indicated in the diagram

Db(Perv(X,S)) ∼realX,S //

Lpi∗

��

Db(X,S)

i∗

��Db(Perv(Y, T )) ∼

realY,T //

τ

∼

19jjjjjjjjjjjjjjj

jjjjjjjjjjjjjjjDb(Y, T ).

Proof. We have adjunctions (Lpi∗, pi∗) and (i∗, i∗). Since bothrealization functors are equivalences of categories (by Theorem 13), itfollows from Proposition 47 that real−1 ◦ i∗ ◦ real and Lpi∗ are both leftadjoint to pi∗, hence there is a natural isomorphism Lpi∗

∼−→ real−1 ◦ i∗◦real. Composition with real yields the isomorphism τ . �

3.5. Tensor Product with a DG Bimodule

By [Kel98, 8.1.1, 8.1.2, 8.2.5], there is a fully faithful functor p :dgDer→ dgHot that is left adjoint to the quotient functor q : dgHot→dgDer and has image in dgHotproj. If we consider p as a functordgDer→ dgHotproj, it is quasi-inverse to the equivalence (6).

Let A and B be dg algebras and X a dg A-B-bimodule (with Aacting on the left and B on the right). This yields a triangulatedfunctor

(?⊗A X) : dgHot(A)→ dgHot(B).

Its left derived functor is the pair ((?L⊗A X), σ) (we use the definition

of derived functors from [Del77, C.D.II.2.1.2, p. 301]), where

(?L⊗A X) := q ◦ (?⊗A X) ◦ p : dgDer(A)→ dgDer(B)

and σ is the natural transformation

(77) σ : (?L⊗A X) ◦ q → q ◦ (?⊗A X)

coming from the adjunction (p, q).


This pair ((?L⊗A X), σ) is a final coapproximation to

dgHot(A)?⊗AX−−−→ dgHot(B)

q−→ dgDer(B).

3.6. Passage from Geometry to DG Modules

Let I : A → B be a right exact functor between abelian categories.We denote the induced functor Hotb(A)→ Hotb(B) by the same sym-bol. Assume that each object of A has a projective resolution of finitelength. Then I has a left derived functor LI : Derb(A) → Derb(B).Let P and Q be bounded complexes of projective objects in A and Brespectively. Then Hom (Q, I(P )) is obviously a dg End (P )- End (Q)-bimodule. It induces (see section 3.5) the lower horizontal arrow indiagram

(78) Derb(A)LI //

Hom (P,?)

��

Derb(B)

Hom (Q,?)

��dgDer( End (P ))

?L⊗

End (P )Hom (Q,I(P ))

//

π

/7gggggggggggggggggggggggg

ggggggggggggggggggggggggdgDer( End (Q)).

The vertical functors are restrictions of the functors (22) explained inRemark 10 (a). We construct now a natural transformation

π : (?L⊗

End (P )Hom (Q, I(P ))) ◦ Hom (P, ?)→ Hom (Q, ?) ◦ LI

as indicated in the diagram. Since Hom (Q, ?) ◦ LI is the left derivedfunctor of

Hom (Q, ?) ◦ I : Hotb(A)→ dgHot( End (Q)),

it is enough, by the universal property of left derived functors, to con-struct a natural transformation

π : (?L⊗

End (P )Hom (Q, I(P ))) ◦ Hom (P, ?) ◦ q → q ◦ Hom (Q, ?) ◦ I

as indicated by the diagram

Hotb(A)I //

q

��

Hotb(B)

Hom (Q,?)

��Derb(A)

Hom (P,?)

��

dgHot( End (Q))

q

��dgDer( End (P ))

?L⊗

End (P )Hom (Q,I(P ))

//

bπ3;ooooooooooooooooooooooooooooooo

ooooooooooooooooooooooooooooooodgDer( End (Q)).

3.6. PASSAGE FROM GEOMETRY TO DG MODULES 55

Thus we define π to be the composition

Hom (P, q(?))L⊗

End (P )Hom (Q, I(P ))

= q(Hom (P, ?))L⊗

End (P )Hom (Q, I(P ))(79)

σ from (77)−−−−−−→ q(Hom (P, ?) ⊗


)I and composition−−−−−−−−−−→ q

(Hom (Q, I(?))

).

Proposition 49. Assume in addition to the above assumptionsthat [µ]I(P ) ∼= Q in Derb(B) for some integer µ. Then diagram (78)induces the diagram

(80) PraeVerd(P,Derb(A))[µ]LI

//

Hom (P,?)��

PraeVerd(Q,Derb(B))

Hom (Q,?)��

dgPrae End (P )( End (P ))?

L⊗

End (P )Hom (Q,[µ]I(P ))

//

eπ∼

/7gggggggggggggggggggg

ggggggggggggggggggggdgPrae End (Q)( End (Q))

and π is a natural isomorphism.

Proof. We have LI(P ) ∼= I(P ) in Derb(B). Thus, after replacingthe horizontal functors in diagram 78 by their composition with theshift [µ], this diagram restricts to(81)

PraeVerd(P,Derb(A))[µ]LI

//

Hom (P,?)��

PraeVerd([µ]I(P ),Derb(B))

Hom (Q,?)��

dgPrae End (P )( End (P ))?

L⊗

End (P )Hom (Q,[µ]I(P ))

//

eπ .6fffffffffffffffffffffff

fffffffffffffffffffffffdgPrae End (Q)(Hom (Q, [µ]I(P )))

Since [µ]I(P ) ∼= Q in Derb(B) we get Hom (Q, [µ]I(P )) ∼= End (Q) indgDer( End (Q)). Hence we can rename (81) to (80).

In order to show that π is a natural isomorphism, it is sufficient tocheck that πP or eqivalently πP is an isomorphism. Since πP is obtainedby plugging in P in (79) (P is a complex of projective objects), thisfollows from the isomorphism

Hom (P, q(P ))L⊗


= q(Hom (P, P )) ⊗


)∼−→ q(Hom (Q, I(P ))).

�


Corollary 50. Under the assumptions of section 3.1, the secondsquare in diagram (68) commutes up to natural isomorphism.

Proof. Take as A the category Perv(X,S) (using Theorem 13),as I : A → B the right exact functor

pi∗ : Perv(X,S)→ Perv(Y, T )

and as P and Q the complexes denoted by the same symbols in sec-

tion 3.1. By assumption, we have an isomorphism [µ]i∗(M)∼−→ N

in Db(MHM(Y )). Hence [µ]i∗(M)∼−→ N in Db(Y, T ) or equivalently

[µ]Lpi∗(M)∼−→ N in Db(Perv(Y, T )), by Proposition 48 and Theorem

13. But N is isomorphic to Q and Lpi∗(M) is isomorphic to pi∗(P )since P →M is a projective resolution in Perv(X,S). �

3.7. DG Bimodules and Transformations

Lemma 51. Let B → A be a morphism of dg algebras. If M isa homotopically projective dg B-module, then M ⊗

BA is homotopically

projective.

Proof. We have an adjunction (?⊗BA, resBA) relating the categories

dgHot(B) and dgHot(A). Let N be an acyclic dg A-module. ThenresBA N is acyclic, and we get

HomdgHot(A)(M ⊗BA,N) = HomdgHot(B)(M, resBA N) = 0

since M is homotopically projective. Hence M ⊗BA is homotopically

projective. �

Assume that we are given morphisms φ : B → A and ψ : S → Rof dg algebras. Let M be a dg A-R-bimodule. By restriction of scalarswe may view M as a dg B-S-module. Let χ : N →M be a morphismof dg B-S-bimodules. We denote this situation as follows.

(82)�� B y N x S

(φ, χ, ψ)−−−−−→�� A y M x R

We get the following diagram

(83) dgDer(B)?

L⊗BA

//

?L⊗BN

��

dgDer(A)

?L⊗AM

��dgDer(S)

?L⊗SR

//

θ4<pppppppppp

ppppppppppdgDer(R)

and construct now the indicated natural transformation

θ : (?L⊗SR) ◦ (?

L⊗BN)→ (?

L⊗AM) ◦ (?

L⊗BA).

3.7. DG BIMODULES AND TRANSFORMATIONS 57

From Lemma 51 we see that the obvious transformation

(?L⊗AM) ◦ (?

L⊗BA)

∼−→ (?L⊗BM)

is an isomorphism. So it is sufficient to define a transformation

θ : (?L⊗SR) ◦ (?

L⊗BN)→ (?

L⊗BM)

But there is an obvious natural transformation

(?L⊗SR) ◦ (?

L⊗BN) ◦ q σ from (77)−−−−−−→ (?

L⊗SR) ◦ q ◦ (?⊗

BN)(84)

σ from (77)−−−−−−→ q ◦ (?⊗SR) ◦ (?⊗

BN)

χ−−−−→ q ◦ (?⊗BM)

that induces, by the universal property of left derived functors, thetransformation we want.

Proposition 52. Keep the assumptions from above. Let n ∈ N bean element such that the maps f : S → N , s 7→ ns and g : R → M ,r 7→ χ(n)r are quasi-isomorphisms of dg modules (so n ∈ Z(N)0 :=N0 ∩ ker dN). Then diagram (83) restricts to

(85) dgPraeB(B)?

L⊗BA

//

?L⊗BN

��

dgPraeA(A)

?L⊗AM

��dgPraeS(S)

?L⊗SR

//

θ|∼

2:nnnnnnnnnnn

nnnnnnnnnnndgPraeR(R)

and θ| is a natural isomorphism.

Proof. Since S ∼= N and R ∼= M in dgDer, diagram (83) restrictsto (85).

If N is a dg B-module, then θN is obtained by plugging in p(N) in(84) (up to an isomorphism coming from the adjunction isomorphism

N∼−→ q(p(N)).In order to show that θ| is a natural isomorphism, it is enough to

check that θB is an isomorphism. Since B is homotopically projective,we may assume p(B) = B. Then θB is given by(

(BL⊗BN)

L⊗SR) ∼−→ (q(N)

L⊗SR)

→ q(N ⊗

SR)

(86)

χ−→ q(M)


We may asume that the adjunction morphism p(q(N)) → N is givenby f : S → N . Then the composition of the last two maps in (86) isidentified with the isomorphism q(g) : q(R)→ q(M) in dgDer(R). �

3.8. DGG Bimodules

In section 2.2 we considered dgg modules over a dgg algebra R. Wedefined a functor Γ : dggMod(R)→ dggMod(Γ(R)) (see (9), (10)) andused it to show that dgg algebras with pure cohomology are formal.

The construction of Γ is easily extended to bimodules. If A and Bare dgg algebras and M is a dgg A-B-bimodule, then Γ(M) becomesa dgg Γ(A)-Γ(B)-bimodule. We get the following situation similar to(82).

(87)�� Γ(A) y Γ(M) x Γ(B) ⊂

�� A y M x B

Here the inclusion Γ(M) ⊂ M is a morphism of dgg Γ(A)-Γ(B)-bimodules. The cohomology H(M) is a dgg H(A)-H(B)-bimodule. As-sume now that the cohomologies of A, B and M vanish in degrees(i, j) with i < j. Then componentwise projection defines the verticalmorphisms of dgg algebras and dgg bimodules in the following diagram

(88)�� Γ(A) y Γ(M) x Γ(B) −→

�� H(A) y H(M) x H(B)

We would like to apply Proposition 52 to the situations sketched in(87) and (88), i. e. we need an element m ∈ Γ(M) inducing quasi-isomorphisms Γ(B)→ Γ(M), B →M and H(B)→ H(M).

Lemma 53. Let B be a dgg algebra, M a dgg B-module and f :B → M a quasi-isomorphism of (right) dgg B-modules. Then m :=f(1) ∈ Γ(M)00 and the multiplication maps

(m·) : B →M, b 7→ mb,

(m·) : Γ(B)→ Γ(M), b 7→ mb,

([m]·) : H(B)→ H(M), c 7→ [m]c,

are quasi-isomorphisms of dgg modules (over B, Γ(B) and H(B)),where we denote by [m] the class of m in H(M).

Proof. Since 1 ∈ B00∩ker dB, we have m = f(1) ∈M00∩ker dM =Γ(M)00. The functor Γ is a “truncation functor” and maps quasi-isomorphisms to quasi-isomorphisms, so Γ(f) is a quasi-isomorphism.But f = (m·), Γ(f) = (m·) and H(f) = ([m]·). �

3.9. Triangulated Functors on Objects

Let A be an abelian category. The stupid truncation functors

σ≤i, σ≥i : Ket(A)→ Ket(A),

3.9. TRIANGULATED FUNCTORS ON OBJECTS 59

for i ∈ Z, are defined as follows. If

P = (. . . → P i−1 di−1

−−→ P i di

−→ P i+1 → . . . )

is a complex in A, then

σ≤i(P ) = (. . . → P i−1 di−1

−−→ P i → 0 → . . . ),

and

σ≥i(P ) = (. . . → 0 → P i di

−→ P i+1 → . . . ).

If f : P → Q is a morphism in Ket(A), σ≤i(f) and σ≥i(f) are definedin the obvious way. There are obvious transformations id → σ≤i andσ≥i → id.

Proposition 54. Let A, B be abelian categories and F : Derb(A)→Derb(B) a triangulated functor. Let

. . .→ 0→ P−nd−n

−−→ P−n+1 → . . .→ P−1 d−1

−−→ P 0 p−→M → 0→ . . .

be a bounded exact complex in A (a resolution of M). Assume thatF (P i) is an object of B, for all i = −n, . . . , 0, where we identify B with

the heart of the trivial t-structure on Derb(B). Let F (P ) be the complex

. . .→ 0→ F (P−n)F (d−n)−−−−→ F (P−n+1)→ . . .

F (d−1)−−−−→ F (P 0)→ 0→ . . .

in B. Then F (M) and F (P ) are isomorphic in Derb(B).

Remark 55. In the following proof, we show the existence of an

isomorphism F (P )∼−→ F (M). But in general it depends on many

choices.

Proof. We write Hom instead of HomDer(B). The transformation

σ≥0 → id yields an inclusion s : F (P 0) = σ≥0(F (P )) → F (P ) inKetb(B). By induction on n, we prove the following more precise state-

ment: There is an isomorphism α ∈ Hom(F (P ), F (M)) such that

F (P 0)s // F (P )

α∼��

F (P 0)F (p)

// F (M)

commutes in Der(B) .For n = 0, this is obvious. Assume that n ≥ 1. Consider the

morphism f : [−1]σ≤−1(F (P ))→ σ≥0(F (P )) = F (P 0) in Ket(B), given

by F (d−1) in degree zero. Its mapping cone is F (P ), and we get adistinguished triangle

(89) [−1]σ≤−1(F (P ))f−→ F (P 0)

s−→ F (P )[1]−→


in Derb(B). Similarly, we get a distinguished triangle

(90) [−2]σ≤−2(F (P ))g−→ F (P−1)

t−→ [−1]σ≤−1(F (P ))[1]−→

in Derb(B), where t is the obvious inclusion, defined similarly as sabove, and g is a morphism in Ket(A), given by F (d−2) in degree zero.

We factorize d−1 : P−1 → P 0 as

P−1 a−→ Kb−→ P 0,

where K = ker p = im d−1. By induction, applied to the exact complex

(. . .→ 0→ P−nd−n

−−→ P−n+1 → . . .→ P−2 d−2

−−→ P−1 a−→ K → 0→ . . . ),

there is an isomorphism β ∈ Hom([−1]σ≤−1F (P ), F (K)) such that

F (P−1)t // [−1]σ≤−1(F (P ))

β∼��

F (P−1)F (a)

// F (K)

commutes in Der(B).Consider now the diagram

(91) [−1]σ≤−1(F (P ))f //

β∼��

F (P 0)s // F (P )

[1]//

F (K)F (b)

// F (P 0)F (p)

// F (M)[1]

//

Both rows are distinguished triangles, the upper one is (89), and the

lower one comes from the short exact sequence Kb−→ P 0 p−→ M . We

claim that this diagram is commutative. If this is the case, we can com-plete the partial morphism (β, id) of distinguished triangles in (91) by

some morphism α ∈ Hom(F (P ), F (M)) to a morphism of distinguishedtriangles, and any such α is an isomorphism.

So let us show that f = F (b) ◦ β. If we apply Hom(?, F (P 0)) to

(90) and use Hom([−1]σ≤−2(F (P )), F (P 0)) = 0, we get an injection

(? ◦ t) : Hom([−1]σ≤−1(F (P )), F (P 0)) ↪→ Hom(F (P−1), F (P 0)).

Hence it is enough to check the equality f ◦ t = F (b) ◦ β ◦ t. ButF (b) ◦ β ◦ t = F (b) ◦ F (a) = F (d−1) = f ◦ t. �

3.10. Restriction of Projective Objects

Let i : (Y, T ) → (X,S) be a closed embedding of cell-stratifiedvarieties.

3.10. RESTRICTION OF PROJECTIVE OBJECTS 61

Lemma 56. If V is an object of Perv(X,S) with a finite filtrationwith standard subquotients, then i∗(V ) is in Perv(Y, T ), so i∗(V ) =pi∗(V ). If V is a projective object of Perv(X,S), then the restrictioni∗(V ) is a projective object of Perv(Y, T ).

Proof. For standard objects ∆S = lS!([dS]S), we have

i∗(∆S) =

{∆S if S ∈ T ,

0 otherwise,

hence i∗(∆S) is in Perv(Y, T ). If W ′ → W → W ′′ is a short exact se-quence in Perv(X,S), and i∗(W ′) and i∗(W ′′) are in Perv(Y, T ), thenso is i∗(W ): Our short exact sequence comes from a distinguished trian-gle (W ′,W,W ′′). We apply i∗ to this triangle and use that Perv(Y, T )is stable by extensions (see [BBD82, 1.3.6]). An obvious inductionon the length of the filtration with standard subquotients finishes theproof of the first statement.

If V ∈ Perv(X,S) is projective, it has a finite filtration with stan-dard subquotients by Theorem 13, so i∗(V ) = pi∗(V ) by the first part.Since pi∗ is left adjoint to the exact functor pi∗, it maps projectiveobjects to projective objects. �

Corollary 57. If V ∈ MHM(X,S) is perverse-projective, then

i∗(V ) is in MHM(Y, T ) and perverse-projective, where we considerMHM(Y, T ) as the heart of the trivial t-structure on Db(MHM(Y ), T ).

Proof. Lemma 56 shows that v(i∗(V )) = i∗(v(V )) is projective

in Perv(Y, T ). This implies that i∗(V ) is in MHM(Y, T ), since rat :MHM(X)→ Perv(X) is exact and faithful. �

Remark 58. If V in MHM(X,S) is perverse-projective, we define

pi∗(V ) := i∗(V ).

Corollary 57 and Proposition 48 prove that rat(pi∗(V )) ∼= pi∗(rat(V )).This justifies our definition.

Recall that we defined in section 2.9 a mixed Hodge structure

ExtiPerv(X)(M, N) on ExtiPerv(X)(M,N), for objects M , N of MHM(X).

The same definition also works for objects M , N of Db(MHM(X)).Consider the obvious composition in Db(MHM(X)) provided by sev-eral adjunction morphisms

Hom (M, N)→ i∗i∗Hom (M, N)→ i∗Hom (i∗(M), i∗(N)).

We take hypercohomology and obtain morphisms of (polarizable) mixedHodge structures

ExtjPerv(X)(M, N)→ ExtjPerv(Y )(i∗(M), i∗(N)).

These morphisms are natural in M and N .


In particular, if P and Q are smooth perverse-projective Hodge

sheaves, then i∗(P ) and i∗(Q) are smooth perverse-projective Hodgesheaves and we get a morphism

(92) HomPerv(X,S)(P , Q)→ HomPerv(Y,T )(pi∗(P ), pi∗(Q))

of mixed Hodge structures (see Corollary 57 and Remark 58).

3.11. Passage to Cohomology Algebras

We combine our results in order to prove that the third and fourthsquare in diagram (68) commute up to natural isomorphism.

Assume that we are in the setting described in section 3.1 (with Ia singleton). So we are given a perverse-projective resolution of finite

length P → M (cf. (31)). Let

pi∗(P ) := (. . .→ . . .→ pi∗(P−1)→ pi∗(P 0)→ 0→ . . . )

be the complex obtained by applying pi∗ to P (cf. Corollary 57 and

Remark 58). We may and will assume that this complex pi∗(P ) is acomplex in MHM(Y, T ). The underlying complex of smooth projectiveperverse sheaves is denoted by pi∗(P ).

The definition of the complex A = End (P ) in section 2.10 and thecomments around (92) at the end of section 3.10 show that there is amorphism of dg algebras “of mixed Hodge structures”

(93) A = End (P )→ End (pi∗(P )) = End ([µ]pi∗(P )).

Recall that B = End (Q). Hence the dg End ([µ]pi∗(P )- End (Q)-bimodule

V := Hom (Q, [µ]pi∗(P ))

becomes a dg A-B-bimodule. Note that A, B and V are complexesof mixed Hodge structures, and the differentials, multiplications andoperations are morphisms of mixed Hodge structures.

We apply the tensor functors ω0, grWR , and ωW = η ◦ grWR from

section 2.13 (cf. diagram 50) to A, B and the A-B-bimodule V andcall the obtained dg(g) algebras and bimodules as shown here:

(94)�� A y V x B

_ω0

��

�ωW

((PPPPPPPPPPPP

� grWR //

�� R y W x S_η

�� A y V x Ba

∼//�� R y W x S .

The isomorphism of dg algebras and bimodules indicated by the lowerhorizontal arrow comes from the natural isomorphism (51).

Proposition 59. There is a quasi-isomorphism f : S → W of

(right) dgg S-modules.

3.11. PASSAGE TO COHOMOLOGY ALGEBRAS 63

Proof. If A is an abelian category, we have the (intelligent) trun-cation functors

τ≤i, τ≥i : Ket(A)→ Ket(A),

for i ∈ Z, as defined, for example, in [KS94, 1.3].By Proposition 54 (using Corollary 57) and assumption (65) we

have isomorphisms

[µ]pi∗(P )∼−→ [µ]i∗(M)

∼−→ N

in Db(MHM(Y )). Hence Hj([µ]pi∗(P )) vanishes for j 6= 0 and we get asequence

Q→ N∼← H0([µ]pi∗(P )← τ≤0([µ]pi∗(P ))→ [µ]pi∗(P )

of quasi-isomorphisms in Ketb(MHM(X,S)). We apply Hom (Q, ?)to this sequence and obtain, using Lemma 24, a sequence of quasi-

isomorphisms of B-modules connecting

B = Hom (Q, Q) and V = Hom (Q, [µ]pi∗(P )).

Hence H(S) and H(W ) are isomorphic as dgg H(S)-modules. Choose

w ∈ Z(W )00 such that H(S) → H(W ), s 7→ [w]s is an isomorphism.Then

f : S → W , s 7→ ws,

is a quasi-isomorphism of dgg S-modules. �

Lemma 53 shows that multiplication by w := f(1) ∈ Γ(W )00 defines

quasi-isomorphisms S → W , Γ(S)→ Γ(W ) and H(S)→ H(W ) of dggmodules.

We apply η to the morphisms of dgg algebras and bimodules�� R y W x S ⊃��

��Γ(R) y Γ(W ) x Γ(S)

and denote the resulting situation by

(95)�� R y W x S ⊃

�� Γ(R) y Γ(W ) x Γ(S) .

Multiplication by w ∈ Z(Γ(W ))0 still defines quasi-isomorphisms S →W and Γ(S) → Γ(W ) of dg modules, hence we can apply Proposition52 and obtain a natural isomorphism

(96) dgPraeΓ(R)(Γ(R))?

L⊗

Γ(R)R

//

?L⊗

Γ(R)Γ(W )

��

dgPraeR(R)

?L⊗RW

��dgPraeΓ(S)(Γ(S))

?L⊗

Γ(S)S

//

∼

2:lllllllllllll

llllllllllllldgPraeS(S).


Since the cohomologies H(R), H(S) and hence H(W ) are pure ofweight zero (as shown in the proof of Theorem 30), componentwiseprojection defines well-defined morphisms of dgg algebras and modules�

��Γ(R) y Γ(W ) x Γ(S) →

��

��H(R) y H(W ) x H(S)

with underlying morphisms of dg algebras and modules�� Γ(R) y Γ(W ) x Γ(S) →�� H(R) y H(W ) x H(S) .

Multiplication by w ∈ Z(Γ(W ))0 and its class [w] ∈ H(W )0 definesquasi-isomorphisms Γ(S) → Γ(W ) and H(S) → H(W ) of dg modules,so application of Proposition 52 yields a natural isomorphism

(97) dgPraeΓ(R)(Γ(R))?

L⊗

Γ(R)H(R)

//

?L⊗

Γ(R)Γ(W )

��

dgPraeH(R)(H(R))

?L⊗

H(R)H(W )

��dgPraeΓ(S)(Γ(S))

?L⊗

Γ(S)H(S)

//

∼

19kkkkkkkkkkkkkk

kkkkkkkkkkkkkkdgPraeH(S)(H(S))

Let �� A y V x B ⊃�� Sub(A) y Sub(V ) x Sub(B)

be the inverse image of (95) under the isomorphism a in (94) (cf. dia-gram (52) in the proof of Theorem 30). Then diagrams (96) and (97)get transformed in the third and forth square in diagram (68).

3.12. Passage to Extension Algebras

We prove now that the fifth square in diagram (68) commutes upto natural isomorphism. The setting is as in section 3.11. Recall that

M = rat(M) and M = real(M) = v(M) and similarly for N .We define in the following isomorphisms of dg algebras and bimod-

ules (with all differentials equal to zero)

(98)�� H(A) y H(V ) x H(B)

(φ, χ, ψ)�� Ext(M) y Ext(N) x Ext(N)

(real, real, real)�� Ext(M) y Ext(N) x Ext(N) ,

where we abbreviate ExtPerv(X,S) and ExtPerv(Y,T ) in the middle row byExt. The right module structures on these bimodules and the isomor-phisms real (cf. (26)) are the obvious ones. For the definiton of the left

3.12. PASSAGE TO EXTENSION ALGEBRAS 65

module structures and the morphisms φ, χ and ψ we have to recall andestablish several isomorphisms. (The left module structures on H(V )and Ext(M) were already defined, but we repeat the definition.)

By assumption (see (65)) there is an isomorphism

σ : [µ]i∗(M)∼−→ N

in Db(MHM(Y )). Define σ := v(σ) and let τ be the natural isomor-phism from Proposition 48. We obtain isomorphisms

[µ]Lpi∗(M)[µ]τM−−−→ [µ]i∗(M)

σ−→ N.

The equivalence real : Db(Perv(X,S))→ Db(X,S) shows that there isa unique isomorphism

λ : [µ]Lpi∗(M)∼−→ N

in Db(Perv(X,S)) such that real(λ) = σ ◦ [µ]τM .

We denote the perverse-projective resolutions P → M and Q→ Nby π and ρ, and their underlying projective resolutions as π : P → Mand ρ : Q → N . Since π is a projective resolution, we may assumethat Lpi∗(P ) and Lpi∗(M) are identical to pi∗(P ) and that Lpi∗(π) isthe identity. Hence we may consider λ also as an isomorphism

λ : [µ]pi∗(P )∼−→ N.

Instead of⊕

HomnHot(Perv(X,S)) and

⊕Homn

Der(Perv(X,S)) we writeHomHot and HomDer, and similarly for (Y, T ). The above isomorphismsgive rise to dga-morphisms

H(A) = HomHot(P )[µ]pi∗−−−→ HomHot([µ]pi∗(P )),

Ext(M)[µ]Lpi∗−−−−→ Ext([µ]Lpi∗(M)

λ−→∼

Ext(N),

Ext(M)[µ]i∗−−→ Ext([µ]i∗(M))

σ−→∼

Ext(N).

The first morphism comes from (93), the last one coincides with (67).These morphisms and the obvious left multiplications define the leftoperations shown in (98). Note that

H(V ) = HomHot(Q, [µ]pi∗P ).

The morphism φ is the composition

H(A) = HomHot(P )= HomDer(P ) = Ext(P )π−→∼

Ext(M),

ψ is defined analogously, and χ is given by

H(V ) = HomHot(Q, [µ]pi∗P )= HomDer(Q, [µ]pi∗P )

= Ext(Q, [µ]pi∗(P ))ρ,λ−→∼

Ext(N).


Lemma 60. The morphisms

χ : H(V )→ Ext(N) and real : Ext(N)→ Ext(N)

are isomorphisms of dg bimodules.

Proof. They are obviously bijective. An easy calculation showsthat they are morphisms of dg bimodules. �

All morphisms in (98) are isomorphisms of dg algebras and dg bi-modules. We apply Proposition 52 to the situation (98) and the elementn ∈ H(V ) corresponding to id ∈ Ext(N) and obtain the commutativity(up to natural isomorphism) of the fifth square in diagram (68).

CHAPTER 4

t-Structures, Koszul Duality and Limits

In this chapter we further develop the theory of dg modules (cf.Section 2.1). Sections 4.2 and 4.4 are included for completeness onlyand will not be used in the rest of this thesis.

4.1. t-Structures

Let A = (A =⊕

i≥0Ai, d) be a positively graded dg algebra (over

a commutative ring). Assume that A0 is a semisimple ring containedin the kernel of the differential, d(A0) = 0. Then A0 has (up to isomor-phism) only a finite number of non-isomorphic simple (right) modules(Lx)x∈W (e. g. [Lan02]), and A0 is a dg subalgebra A0 of A. If Mis an A0-module, extension of scalars (? ⊗A0 A) (see (8)) defines anA-module

M := M ⊗A0 A.We want to define a t-structure (see [BBD82, 1.3]) on the category(see (15))

(99) dgPrae := dgPrae(A) := dgPraeA({Lx}x∈W ),

where we view the A0-modules Lx as A0-modules concentrated in de-gree zero. If A is a polynomial ring in finitely many generators ofdegree two, a similar construction can be found in [BL94, 11.4].

Remark 61. With basically the same arguments, we can also define

a t-structure on dgPraeA({Lx}x∈W ′), where W ′ is a subset of W .

We consider the following full subcategories of dgHot(A):

• dgFilt(A). Its objects are dg modules M admitting a finitefiltration 0 = F0(M) ⊂ F1(M) ⊂ · · · ⊂ Fn(M) = M by dgsubmodules with subquotients

Fi(M)/Fi−1(M) ∼= {li}Lxiin dgMod(A)

for suitable l1 ≥ l2 ≥ · · · ≥ ln and xi ∈ W . We call such afiltration an L-filtration.• dgGrproj(A). Its objects are dg modules (M,d) such that the

underlying graded A-module M is isomorphic to a direct sum

of shifted Lx, that is

(100) M ∼= {l1}Lx1 ⊕ {l2}Lx2 ⊕ · · · ⊕ {ls}Lxs in gMod(A)

for suitable li ∈ Z, xi ∈ W .

67

68 4. T-STRUCTURES, KOSZUL DUALITY AND LIMITS

As a right module over itself, A0 is isomorphic to a finite direct sumof simple modules Lx. Hence A is isomorphic to a finite direct sum of

modules Lx. In particular, each Lx is in dgPer and homotopically pro-

jective (see (5)). As a graded A-module, each Lx is projective. So any

L-filtration of a dg module yields several A-split short exact sequencesin dgMod that become distinguished triangles in dgHot. Hence everyobject of dgFilt lies in dgPrae and in dgGrproj. We have the inclusions

dgFilt ⊂ dgPrae ⊂ dgPer ⊂ dgHotproj ⊂ dgHot

and

dgFilt ⊂ dgGrproj ⊂ dgHot .

There is a unique dga-morphism A → A0 inducing the identity onA0 ⊂ A. It gives rise to an extension of scalars functor (? ⊗A A0). IfM is an A-module, we define

M =M⊗A A0 =M/MA+,

where A+ is the kernel of A → A0. (IfM is homotopically projective,then it is homotopically flat (:= K-flat, see [BL94, 10.12.4.4]), hence

M is isomorphic to ML⊗A A0 in dgDer.)

Lemma 62. Let x, y ∈ W . If f : Lx → Ly is a non-zero morphismin gMod(A) or in dgMod(A), it is an isomorphism and x = y.

Proof. It is sufficient to prove the statement for gMod(A). Letf be non-zero. Application of (? ⊗A A0) yields a non-zero morphismf : Lx → Ly of simple modules. So f is an isomorphism and x = y.

We apply (? ⊗A0 A) and obtain an isomorphism f : Lx → Ly. But fand f agree. �

If M is in dgGrproj, the number of summands occuring in anydecomposition (100) is the length λ(M) of the A0-module M. Inparticular, if M is in dgFilt, this length coincides with the length of

any L-filtration of M.If M is a graded A-module, we let

M≤i :=∑j≤i

M jA

be the graded submodule of M that is generated by the degree ≤ iparts. This defines an increasing filtration . . . ⊂ M≤i ⊂ M≤i+1 ⊂ . . .of M . If M is the underlying graded module of a dg module in dgFilt,

this filtration is coarser than any L-filtration. Define M<i := M≤i−1.

Proposition 63. Every object of dgPrae is isomorphic to an ob-ject of dgFilt. In other words: The inclusion dgFilt ⊂ dgPrae is anequivalence of categories.

4.1. T-STRUCTURES 69

Proof. All generators Lx of dgPrae are in dgFilt, and dgFilt isclosed under the shift {1}. So it is sufficient to prove:

Claim: Let f : X → Y be a morphism in dgMod of objects X, Yof dgFilt. Then the cone C(f) is isomorphic to an object of dgFilt.

We will prove this by induction on the length λ(X). We may assumethat λ(X) > 0.

Recall that the cone C(f) is given by the graded (right) A-moduleY ⊕ {1}X with differential dC(f) =

[dY f0 −dX

].

The case λ(X) = 1: By use of the shift {1} we may assume

that X = Lx for some x ∈ W . Choose an L-filtration (Fi(Y ))ni=0 of

Y and abbreviate Fi = Fi(Y ). Assume that Fi/Fi−1∼= {li}Lxi

, forl1 ≥ l2 ≥ · · · ≥ ln and xi ∈ W .

If the image of f is contained in Y<0, let s ∈ {0, . . . , n} be maximalwith Fs = Y<0. Then

Gi =

{Fi ⊕ 0 if i ≤ s,

Fi−1 ⊕ {1}Lx if i > s.

defines an L-filtration (Gi)n+1i=0 of the cone C(f) = Y ⊕{1}Lx, i. e. C(f)

is in dgFilt.Now assume im f 6⊂ Y<0. Let t ∈ {0, n} be minimal with im f ⊂ Ft.

Then t ≥ 1 and lt = 0, since X is generated by its degree zero part.The composition

(101) X = Lxf−→ Ft � Ft/Ft−1

∼= Lxt

is non-zero and hence an isomorphism by Lemma 62. Hence f induces

an isomorphism f : Lx∼−→ im f , and its mapping cone V = (im f) ⊕

{1}Lx is an acyclic dg submodule of C(f).

The inverse of im f∼−→ Ft/Ft−1 splits the short exact sequence

(Ft−1, Ft, Ft/Ft−1). This implies that, for 1 ≤ i ≤ n,

(102) Fi ∩ (Fi−1 + im f) =

{Fi−1 if i 6= t,

Fi if i = t.

The filtration (Fi + V ) of C(f) induces a filtration of C(f)/V withsuccessive subquotients

Fi + V

Fi−1 + V

∼← FiFi ∩ (Fi−1 + V )

∼← FiFi ∩ (Fi−1 + im f)

Using (102), we obtain an L-filtration of C(f)/V , hence C(f)/V is indgFilt.

The short exact sequence (V,C(f), C(f)/V ) of A-modules and theacyclicity of V show that C(f) → C(f)/V is a quasi-isomorphism.Since both C(f) and C(f)/V are homotopically projective, it is anisomorphism in dgHot. Hence C(f) is isomorphic to an object of dgFilt.


Reduction to the case λ(X) = 1: After having written down thefollowing proof, I realized that I merely reproved [BBD82, 1.3.10].

Let U := F1(X) be the first step of an L-filtration of X, u : U → Xthe inclusion and p : X → X/U the projection. Then the short exact

sequence 0 → Uu−→ X

p−→ X/U → 0 is A-split and hence defines adistinguished triangle (U,X,X/U) in dgHot. We apply the octahedralaxiom to the maps u and f and get the dotted arrows in the followingcommutative diagram (that I only include because it looks so nice).

{1}U{1}u

��?????

{1}X{1}p

��?????

{1}X/U

X/U##

��C(f ◦ u)

��

??��

C(f);;

??��

X

p ??��

f

��??????

Y;;

??��U

u??��

f◦u <<

The four paths with the bended arrows are distinguished triangles.By the length 1 case we may replace C(f ◦ u) by an isomorphic

object C(f ◦ u) of dgFilt. So C(f) is isomorphic to the cone of a

morphism X/U → C(f ◦ u) and hence, by induction, isomorphic to anobject of dgFilt. �

If M =⊕

i∈ZMi is a graded A0-module and of finite length as an

ordinary A0-module, we define its graded length

λ(M, v) :=∑i∈Z

λ(M i)vi ∈ N[v, v−1].

Define

dgPrae≤0 := {M ∈ dgPrae(A) | λ(H(ML⊗A A0), v) ∈ N[v−1]},

(103)

dgPrae≥0 := {M ∈ dgPrae(A) | λ(H(ML⊗A A0), v) ∈ N[v]}.

We could also write M = M⊗A A0 instead of ML⊗A A0, since each

object of dgPrae is homotopically projective, hence homotopically flat.

Remark 64. It is instructive to consider an object M of dgFilt.Then the differential ofM vanishes, and the polynomials λ(M, v) andλ(H(M), v) coincide. So M lies in dgPrae≤0 (in dgPrae≥0) if andonly if it is generated in degrees ≤ 0 (in degrees ≥ 0, respectively), asa graded A-module.

4.1. T-STRUCTURES 71

Theorem 65. Let A = (A =⊕

i≥0Ai, d) be a positively graded dg

algebra with A0 a semisimple ring and d(A0) = 0. Then (103) definesa t-structure on dgPrae(A).

Proof. Define

dgPrae≤n := {−n} dgPrae≤0,

dgPrae≥n := {−n} dgPrae≥0 .

We have to check the three defining properties ([BBD82, 1.3.1]).

(a) HomdgHot(X, Y ) = 0 for X ∈ dgPrae≤0 and Y ∈ dgPrae≥1:By Proposition 63 we may assume that X, Y are in dgFilt.But then any morphism of the underlying graded A-modulesis zero.

(b) dgPrae≤0 ⊂ dgPrae≤1 und dgPrae≥1 ⊂ dgPrae≥0: Obvious.(c) If X is in dgPrae there is a distinguished triangle (A,X,B)

with A in dgPrae≤0 and B in dgPrae≥1: We may assume thatX is in dgFilt. The graded A-submodule X≤0 of X is an A-

submodule, since it coincides with some step in any L-filtrationof X. The short exact sequence X≤0 → X → X/X≤0 is A-splitand hence defines a distinguished triangle (X≤0, X,X/X≤0)in dgHot. All terms of this triangle are in dgFilt, X≤0 is indgPrae≤0 and X/X≤0 in dgPrae≥1.

�

The t-structure (dgPrae≤0, dgPrae≥0) on dgPrae yields truncationfunctors τ≤n and τ≥n ([BBD82, 1.3.3]). For every X in dgPrae,

τ≤0X → X → τ≥1X → {1}τ≤0X

is, up to unique isomorphism, the only distinguished triangle (A,X,B)with A in dgPrae≤0 and B in dgPrae≥1. IfM is in dgFilt, this triangleis given by

M≤0 →M→M/M≤0 → {1}M≤0,

(we did not specify the last morphism) as follows from the proof ofTheorem 65. Note that all terms of this triangle are in dgFilt.

The derived category of dg modules over a dg algebra has infinitedirect sums and these are given by direct sums of dg modules. A dgmodule M is called compact if HomdgDer(M, ?) commutes with infinitedirect sums ([Kel98]).

Corollary 66. Let A = (A =⊕

i≥0Ai, d) be a positively graded

dg algebra with A0 a semisimple ring and d(A0) = 0. Then

(a) dgPrae(A) ⊂ dgHot(A) is closed under taking direct sum-mands, i. e. dgPrae(A) = dgPer(A): The objects of dgPraeare precisely the perfect dg modules.

(b) We have dgPrae(A) = dgPer(A), and the objects of this cate-gory are precisely the compact dg modules.


Proof. Proof of (a): We recall some results and definitions from[LC07]. A t-structure (T ≤0, T ≥0) on a triangulated category T isbounded if any object X of T is contained in T ≥m ∩ T ≤n for someintegers m ≤ n. An idempotent morphism e : X → X of an additivecategory splits if there are an object Y and morphisms u : X → Yand v : Y → X satisfying vu = e and uv = idY . It strongly splitsif e and idX −e split. In this case, we get a direct sum compositionf : Y ⊕ Z ∼−→ X such that f−1 ◦ e ◦ f =

[1 00 0

]. An additive category

is Karoubian if any idempotent morphism splits. The main resultof [LC07] is: A triangulated category with a bounded t-structure isKaroubian. Moreover, an idempotent morphism in a triangulated cat-egory splits if and only if it strongly splits.

Our t-structure on dgPrae is obviously bounded, so dgPrae isKaroubian and every idempotent morphism strongly splits. Since di-rect summands are kernels of certain idempotent morphisms and dgPraeis a strict subcategory of dgHot, it is now easy to see that dgPrae ⊂dgHot is closed under taking direct summands.

Proof of (b): Equivalence (7) restricts to a triangulated equivalence

dgPrae(A)∼−→ dgPrae(A) := dgPraeA({Lx}x∈W )

(see (16)). Since dgPrae(A) = dgPer(A), we obtain dgPrae(A) =dgPer(A).

As proved in [Kel98, 8.2.6], Ravenel-Neeman’s characterization ofcompact objects ([Rav84, Nee92]) shows that the objects of dgPer(A)are precisely the compact objects of dgDer(A). �

Remark 67. For A as above, we have seen in the proof of Corollary66 that equivalence (7) takes the form

(104) dgPrae(A) = dgPer(A)∼−→ dgPrae(A) = dgPer(A).

If we replace dgPrae in (103) by dgPer, this defines a t-structure ondgPrae(A) = dgPer(A), and (104) is t-exact.

In the rest of this chapter, we will prove several results for categoriesdgPrae. Equivalence (104) shows that all the “categorical” resultsremain true, if we replace dgPrae by dgPer, dgPrae or dgPer.

The t-structure on dgPer(A) can of course be transferred to anyequivalent category. For example, ifA → B is a dga-quasi-isomorphism,we obtain a t-structure on dgPer(B), by Theorem 5.

4.2. Minimal Homotopically Projective Modules

The purpose of this section is to relate our constructions to [BL94].Motivated by [BL94, 11.4], we define what a minimal homotopicallyprojective dg module is. It will turn out that the objects of dgFilt areprecisely the minimal homotopically projective objects in dgPrae.

As before, let A = (A =⊕

i≥0Ai, d) be a positively graded dg

algebra with A0 a semisimple ring and d(A0) = 0.

4.2. MINIMAL HOMOTOPICALLY PROJECTIVE MODULES 73

An object P of dgHotproj is called minimal homotopically pro-jective if it lies in dgGrproj and if λ(P) is minimal in its isomorphismclass in dgHotproj ∩ dgGrproj, that is, if for every isomorphic objectN of dgHotproj ∩ dgGrproj we have λ(P) ≤ λ(N ).

Note that we always have λ(P) ≥ λ(H(P)). Isomorphic objects P ,N in dgHotproj are homotopy equivalent, and the same holds for Pand N . In particular λ(H(P)) = λ(H(N )).

We prove an analog of [BL94, 11.4.7].

Proposition 68. Let P = (P, dP ) be in dgPrae∩ dgGrproj. Thenthe following statements are equivalent.

(a) P is in dgFilt.(b) for all i, P≤i is an A-submodule of P, that is dP (P≤i) ⊂ P≤i;(c) dP (P ) ⊂ PA+;(d) P = P ⊗A A0 = P/PA+ has differential d = 0;(e) λ(P) = λ(H(P));(f) P is minimal homotopically projective.

Proof. • (a) ⇒ (b): Each P≤i appears in any L-filtration.• (b) ⇔ (c): Use a decomposition of P of the form (100).• (c) ⇔ (d) ⇔ (e): Obvious.• (e) ⇒ (f): If N is in dgHotproj∩ dgGrproj and isomorphic toP , we have

λ(P) = λ(H(P)) = λ(H(N )) ≤ λ(N ).

• (f) ⇒ (e): By Proposition 63, there is an object M of dgFiltisomorphic to P in dgHot. We have already proved (a)⇒ (e),hence

λ(M) = λ(H(M)) = λ(H(P)) ≤ λ(P).

Since P is minimal, we have equality.• (f) ⇒ (a): Postponed.

�

Proposition 69. Let P, Q be minimal homotopically projectiveA-modules in dgPrae. If P and Q are quasi-isomorphic in dgHot,they are isomorphic in dgMod.

Proof. Observation: If a graded A-module M with M i = 0 fori� 0 satisfies MA+ = M , then M = 0.

Let P = (P, dP ) and Q = (Q, dQ) be quasi-isomorphic in dgHot.Since P and Q are in dgHotproj, they are isomorphic in dgHot. Letf : P → Q be a morphism in dgMod that becomes an isomorphism indgHot. We have already proved the implication (f) ⇒ (d) of Proposi-tion 68, so the differentials on P and Q vanish, and the morphisms indgMod(A0) and in dgHot(A0) between these objects coincide. Hencef : P → Q is an isomorphim in dgMod(A0). Since the functor


dgMod(A) → Mod(A0), (M,dM) 7→ M := M ⊗A A0 = M/MA+, isright exact, we find cok f = 0, and our observation implies cok f = 0.

From the short exact sequence 0→ ker f → P f−→ Q → 0 we obtain anexact sequence

TorA1 (Q,A0)→ ker f → Pf−→∼Q→ 0.

Since Q is in dgGrproj, it is projective and hence flat as an A-module.Hence TorA1 (Q,A0) = 0 and ker f = ker(f) = 0. Our observationimplies ker f = 0. So f is an isomorphism. �

Proof of Proposition 68. (f) ⇒ (a): Let P be a minimal ho-motopically projective object in dgPrae. By Proposition 63, there isan object M of dgFilt isomorphic to P in dgHot. We have alreadyseen that M is minimal homotopically projective. So Proposition 69shows that M and P are isomorphic in dgMod. �

4.3. Heart

We keep the above assumptions: A = (A =⊕

i≥0Ai, d) is a posi-

tively graded dg algebra with A0 a semisimple ring and d(A0) = 0.Let C = dgPrae≤0 ∩ dgPrae≥0 be the heart of our t-structure on

dgPrae.Let dgFilt0 be the full subcategory of dgFilt consisting of objectsM

with λ(M, v) ∈ N (cf. Remark 64). The underlying graded A-modulo

of each such object is a finite direct sum of some Lx (without shifts).From the above discussion it is clear that dgFilt0 is contained in C

and that each object of C is isomorphic to an object of dgFilt0. Thisshows

Proposition 70. The inclusion dgFilt0 ⊂ C is an equivalence ofcategories. In particular, dgFilt0 is abelian.

Let dgFlag be the full subcategory of dgMod with the same objectsas dgFilt0. We will show in Propositions 71 and 73 that dgFlag is a fullabelian subcategory of dgMod and that the obvious functor dgFlag→dgFilt0 is an equivalence. Hence the abelian structure on dgFilt0 is themost obvious one.

Proposition 71. The category dgFlag is abelian and the inclusiondgFlag ⊂ dgMod is exact. In short, dgFlag is a full abelian subcategoryof dgMod.

We will need the following remark in the proof.

Remark 72. The inclusion A0 ⊂ A defines the extension of scalarsfunctor

prodAA0 = (?⊗A0 A) : Mod(A0)→ gMod(A),

4.3. HEART 75

where we view A0-modules as graded A0-modules concentrated in de-gree zero. Since prodAA0 is exact and fully faithful (cf. proof of Lemma62), its essential image B is closed under taking kernels, cokernels, im-ages and pull-backs. In particular, if U and V are graded submodulesof M with U , V and M in B, then U ∩ V and U + V are in B.

Proof of Proposition 71. Let f : M → N be a morphism indgFlag. Let K = (K, d) and C be the kernel and cokernel of f in theabelian category dgMod. It is sufficient to prove that K and C belongto dgFlag.

Since the underlying graded A-module of an object of dgFlag is inthe essential image B of prodAA0 , the kernel K lies in B by Remark 72.

Let I ⊂ N be the image of f in dgMod. Since M/K ∼−→ I and

C ∼−→ N /I, it is sufficient to prove:Claim: If U = (U, dU) ⊂ M is an A-submodule of M ∈ dgFlag

with U in B, then U and the quotient M/U in dgMod are objects ofdgFlag.

Proof of the claim: Let (Fi(M)) be an L-filtration of M with

subquotients Fi/Fi−1∼= Lxi

, and (Fi(U)) and (Fi(M/U)) the inducedfiltrations of U andM/U . The underlying graded modules of all stepsof all these filtrations are in B: For Fi(M) this is obvious, and forFi(U) and Fi(M/U) this is a consequence of Remark 72. In the shortexact sequence of finitely filtered objects

0→ (U , Fi(U))υ−→ (M, Fi(M))

π−→ (M/U , Fi(M/U))→ 0

all morphisms are strictly compatible with the filtrations. Thus, by[Del71, 1.1.11],

0→ Gri(U)Gri(υ)−−−→ Gri(M)

Gri(π)−−−→ Gri(M/U)→ 0

is exact for all i. All underlying graded modules are in B (Remark

72). Since Gri(M) ∼= Lxiis simple in B, Gri(υ) is either zero or an

isomorphism, and Gri(π) is either an isomorphism or zero. We deducethat U and M/U are in dgFlag. �

Proposition 73. The obvious functor dgFlag → C is an equiva-lence of categories. Any object in C has finite length, and the simple

objects in C are (up to isomorphism) the {Lx}x∈W .

Proof. Let M, N be in dgFlag. Since both M and N are gen-erated in degree zero, any homotopy M → {−1}N is zero. So thecanonical map

HomdgMod(M,N )→ HomdgHot(M,N )

is an isomorphism, and the obvious functor dgFlag → dgFilt0 is anequivalence. In combination with Proposition 70, this shows the firststatement.


We prove the remaining stetements for dgFlag. If M 6= 0 is in

dgFlag, the first step of any L-filtration yields a monomorphism Ly ↪→M. If M is simple, this must be an isomorphism. Any non-zero

subobject of Lx has a subobject isomorphic to some Ly. So Lemma 62

(or Remark 72) shows that the (Lx)x∈W are (up to isomorphism) thesimple objects of dgFlag and pairwise non-isomorphic. Each object of

dgFlag has finite length, since it has an L-filtration. �

Remark 74. It is also possible to define a t-structure starting withits potential heart, see [BBD82, 1.3.13].

We would have had to show:

(a) dgFlag∼−→ dgFilt0 is an abelian category (see Proposition 71).

(b) Define C to be the smallest strict (closed under isomorphisms)full subcategory of dgHot, containing dgFilt0. Then C is ad-missible (easy for dgFilt0, short exact sequences are A-split)and stable by extensions (easy for dgFilt0, the middle term in-

herits an obvious L-filtration from such filtrations of the borderterms).

(c) The smallest strict full triangulated subcategory of dgHot con-taining all the {n}C, for n ∈ Z, is dgPrae (easy).

This looks a bit simpler. In particular, we do not have to proveProposition 63. The resulting t-structure coincides with ours.

Our method has the advantage that the truncation functors arevery explicit.

Proposition 75. An object M in the heart C is injective if andonly if H1(M) = 0.

Proof. We give some more equivalent conditions, and some ex-planations in brackets.

(a) M is injective.(b) HomC(·,M) is exact.(c) Ext1

C(·,M) = 0.

(d) Ext1C(Lx,M) = 0 (the Lx are the simple objects in the heart,

and every object of C has finite length).(e) Ext1

C(A,M) = 0.(f) Every short exact sequence 0 → M → N → A → 0 in C

splits.(g) IfM→N → A c−→ {1}M is a distinguished triangle in dgHot,

then c = 0.(h) HomdgHot(A, {1}M) = 0(i) H1(M) = 0 (see (4) and (5)).

�

4.4. KOSZUL-DUALITY 77

4.4. Koszul-Duality

We thank Peter Fiebig for drawing our attention to the case thatthe underlying graded ring of a dg algebra is Koszul.

In addition to the assumptions on our dg algebra in the previous sec-tions, we assume now that its differential vanishes and that the underly-ing graded ring is Koszul. More precisely, letA = (A =

⊕i≥0A

i, d = 0)

be a positively graded dg algebra with A0 a semisimple ring and dif-ferential zero, and assume that A is a Koszul ring (cf. [BGS96]). Thelast condition means that there is a resolution P � A0,

. . .d−3−−→ P−2

d−2−−→ P−1d−1−−→ P0 � A0,

of the graded (right) A-module A0 = A/A+ such that each P−i is aprojective object in gMod(A) and generated in degree i, P−i = P i

−iA.(We use lower indices in the complex P since we use upper indices forthe graded parts of our graded modules.) Such a resolution is uniqueup to unique isomorphism. Note that multiplication

(105) P i−i ⊗A0 A→ P−i

is an isomorphism (it is surjective, splits, and any splitting is surjective,since it is an isomorphism in degree i).

Since the differential of A vanishes, we can view each P−i as a dgA-module with differential zero. The maps d−i are then morphisms ofdg modules.

On the graded A-module

K := K(A) :=⊕n∈N

{n}P−n = P0 ⊕ {1}P−1 ⊕ . . .

we define a differential dK by

(106) dK(p0, p−1, p−2, . . . )

:= ((−1)−1d−1(p−1), (−1)−2d−2(p−2), (−1)−3d−3(p−3), . . . ).

This defines an object K(A) = (K, dK) of dgMod(A). It can be seen asan iterated mapping cone of the morphisms (−1)−id−i of dg modules.(We chose these signs from the proof of Theorem 77. If we omit all thesigns (−1)−i in (106), we obtain an isomorphic dg module.)

We say that P is of finite length if P−j = 0 for j � 0, and thatP has finitely generated components if each P−i is a finitely generatedA-module.

Proposition 76. Let A be as above. Assume that A is a Koszulring with a resolution P � A0 as above of finite length with finitelygenerated components. Then K(A) is in dgFilt0 and an injective objectof the heart C.

Proof. Since P has finite length, the filtration 0 ⊂ P0 ⊂ P0 ⊕{1}P−1 ⊂ · · · ⊂ K by A-submodules is finite. Since all subquotients


{i}P−i have differential zero and are finitely generated by their de-

gree zero part, they have an L-filtration (use (105) and note that the

differential of each Lx vanishes). Hence K(A) is in dgFilt0 ⊂ C.Since P � A0 is a resolution, the cohomology H(K(A)) is A0, sit-

ting in degree zero. Hence K(A) is an injective object of C by Propo-sition 75. �

If A is a positively graded ring, we can form the graded ring E(A) :=ExtMod(A)(A

0, A0) (see (11)) of self-extensions of the right A-moduleA0 = A/A+ (we consider self-extensions in Mod(A), not in gMod(A)).If A is a right finite (all Ai are finitely generated as right A0-modules)Koszul ring, then E(A) is a right finite Koszul ring and E(E(A)) = Acanonically (see [BGS96, Proposition 1.2.5]). In this case E(A) iscalled the Koszul dual of A.

Theorem 77. Let A = (A =⊕

i≥0Ai, d = 0) be a positively graded

dg algebra with A0 a semisimple ring and differential zero. If A is aKoszul ring with a resolution P � A0 as above of finite length withfinitely generated components, then

(a) The functor K := HomC(·,K(A)) : C → (EndC(K(A))- Mod)op

induces an equivalence

K : C ∼−→ (EndC(K(A))- mod)op

between C and the opposite category of the category of finitelygenerated (left) EndC(K(A))-modules.

(b) EndC(K(A)) is isomorphic to E(A) = ExtA(A0, A0).

Proof. Proof of (a): Since A = P0 is an A-submodule of K(A),

each Lx is contained in K(A). So every simple object of the artiniancategory C is contained in the injective object K(A). Hence K(A) is aprojective generator of the opposite category Cop, and a standard result(see e. g. [BGS96, 1.1]) proves our claim.

Proof of (b): Let v : gMod(A) → Mod(A) be the “forgetting thegrading” functor. We denote the induced functor Ket(gMod(A)) →Ket(Mod(A)) by the same letter.

If M and N are in gMod(A),⊕n∈Z

HomgMod(A)(M, {n}N)→ HomMod(A)(v(M), v(N))(107)

f = (fn) 7→∑

fn

is bijective if M is finitely generated as an A-module.Since v(P ) → v(A0) is a projective resolution, Ei(A) is the i-th

cohomology of the complex R := Hom Mod(A)(v(P ), v(P )) (see (13) forthe definition of this complex). Its i-th component is (we use the shift

4.4. KOSZUL-DUALITY 79

([1]P )m = Pm+1 as if we were using upper indices in the complex P )

Ri =∏s∈Z

HomMod(A)(v(Ps−i), v(Ps))

=⊕s∈Z

HomMod(A)(v(Ps−i), v(Ps)) since v(P ) has finite length,

∼←⊕s,n∈Z

HomgMod(A)(Ps−i, {n}Ps) by (107),

=⊕n∈Z

∏s∈Z

HomgMod(A)(Ps−i, {n}Ps) since P has finite length,

=⊕n∈Z

Hom igMod(A)(P, {n}P ).

So if we define Rn = Hom gMod(A)(P, {n}P ) for n ∈ Z, we get anisomorphism of complexes ⊕

n∈Z

Rn∼−→ R.

Cohomology commutes with (arbitrary) direct sums, hence⊕n∈Z

H i(Rn)∼−→ H i(R) = Ei(A).

By [BGS96, Proposition 2.1.3], H i(Rn) = ExtigMod(A)(A0, {n}A0) van-ishes if i 6= −n. Furthermore,

Rin =

∏s∈Z

HomgMod(A)(P−s, {n}P−s+i)

=⊕s∈Z

HomgMod(A)(P−s, {n}P−s+i)

vanishes if i < −n, since P−s = P s−sA and ({n}P−s+i)s = P s+n

−s+i. Thismeans that

Ei(A) = Hi(R−i) = ker(Ri−i → Ri+1

−i ).

Recall that

Ri−i =

⊕s∈Z

HomgMod(A)(P−s, {−i}P−s+i)

=⊕s∈Z

HomgMod(A)({s}P−s, {s− i}P−(s−i)).

An element f = (f i)i∈Z = (f is)i,s∈Z of⊕

i∈ZRi−i, where

f is ∈ HomgMod(A)({s}P−s, {s− i}P−s+i),lies in E(A) if and only if

(108) d−s+i ◦ f is − (−1)if is−1 ◦ d−s = 0


for all i, s ∈ Z. Any f = (f is) as above obviously defines an endomor-phism of the graded A-module K(A). It commutes with the differentialdK and hence defines an endomorphism of the A-module K(A) if andonly if (108) holds. So E(A) and EndC(K(A)) are isomorphic. �

In the proof of Theorem 77, we described a natural isomorphismE(A)

∼−→ EndC(K(A)). Hence EndC(K(A)) is graded. If we think ofEndC(K(A)) as a matrix algebra, it consists of upper triangular matri-ces and the grading is the diagonal grading.

4.5. Inverse Limits of Categories

We introduce the notion of inverse limit of a sequence of categories.It will be used in Section 4.6, where we consider inverse limits of dgcategories, and in Section 5.2, where we obtain a description of theequivariant derived category as an inverse limit.

Let

C0F0←− C1

F1←− C2 ← . . .← CnFn←− Cn+1 ← . . .

or in short ((Cn), (Fn)) be a sequence of categories (and functors). Wecall the following category the inverse limit of this sequence and denoteit by lim←−Cn. (I do not care whether this definition agrees with anydefinition of inverse limit in category theory.)

• Objects are sequences

((Mn)n∈N, (φn)n∈N)

of objects Mn in Cn and isomorphisms

φn : Fn(Mn+1)∼−→Mn.

• Morphisms α : ((Mn), (φn)) → ((Nn), (ψn)) are sequences(αn)n∈N of morphisms αn : Mn → Nn such that

Fn(Mn+1)Fn(αn+1)

//

∼φn

��

Fn(Nn+1)

∼ψn

��Mn

αn // Nn

commutes, for all n ∈ N.

Lemma 78. Let N ∈ N and assume that Fn : Cn+1 → Cn is anequivalence for all n ≥ N . Then the obvious projection functor prN :lim←−Cn → CN is an equivalence.

Proof. Obvious. �

4.5. INVERSE LIMITS OF CATEGORIES 81

A morphism of sequences ((Cn), (Fn)) and ((Dn), (Gn)) of categoriesis a sequence ν = (νn) of functors νn : Cn → Dn such that

Cnνn

��

Cn+1Fnoo

νn+1

��Dn Dn+1

Gnoo

commutes (up to natural isomorphism), for each n ∈ N. Any suchmorphism ν obviously defines a functor lim←− νn : lim←−Cn → lim←−Dn.

In the following, we describe a setting in which this functor is anequivalence. Let (I,≤) be a directed (i. e. for all I, J ∈ I there isK ∈ I with I ≤ K, J ≤ K) partially ordered set. (In our applicationsthis will be the set of segments in Z, partially ordered by inclusion.) AnI-filtered category is a category C together with strict full subcategories(CI)I∈I such that CI ⊂ CJ for I ≤ J . We say that C is the union ofthe CI if any (finite set of) object(s) of C is contained in some CI . Amorphism C → D of I-filtered categories (I-filtered morphism) is afunctor F : C → D inducing functors F I : CI → DI for all I ∈ I.

If ((Cn), (Fn)) is a sequence of I-filtered categories, the inverse limitlim←−Cn is filtered by the lim←−C

In.

We will use the following conditions on a sequence of I-filteredcategories ((Cn), (Fn)).

(F1) For each I ∈ I there is NI ∈ N such that, for all n ≥ NI ,F In : CIn+1 → CIn is an equivalence.

(F2) lim←−Cn is the union of the lim←−CIn.

Any morphism (νn) : ((Cn), (Fn)) → ((Dn), (Gn)) of sequences ofI-filtered categories induces an I-filtered morphism

lim←− νn : lim←−Cn → lim←−Dn.

Proposition 79. Let (νn) : ((Cn), (Fn))→ ((Dn), (Gn)) be a mor-phism of sequences of I-filtered categories and assume that both se-quences satisfy condition (F2) and that ((Cn), (Fn)) satisfies condition(F1). If for all I ∈ I there is N ∈ N such that, for all n ≥ N ,νIn : CIn → DIn is an equivalence, then lim←− νn : lim←−Cn → lim←−Dn is anequivalence and ((Dn), (Gn)) also satisfies condition (F1).

Proof. It follows from the assumptions that ((Dn), (Gn)) also sat-isfies condition (F1). By condition (F2) it is sufficient to show thateach lim←− ν

In is an equivalence. Using condition (F1) and Lemma 78 we

can show equivalently that νIN is an equivalence, for N � 0. But thisis an assumption. �

Corollary 80. Let ((Cn), (Fn)) satisfy the conditions (F1)-(F2).Assume that C∞ is an I-filtered category and the union of the CI∞,and that there are compatible I-morphisms νn : C∞ → Cn restricting


to functors νIn : CI∞ → CIn. Suppose that for each fixed I, νIn is anequivalence for n � 0. Then the obvious functor C∞ → lim←−Cn is anequivalence.

Proof. C∞ is equivalent to the inverse limit of the constant se-

quence C∞id←− C∞

id←− . . . . �

If ((Cn), (Fn)) is a sequence of categories and all Cn and Fn areadditive, then the inverse limit lim←−Cn is obviously additive. In thefollowing we describe a setting in which a similar statement holds fortriangulated categories.

Let T0F0←− T1

F1←− . . .Fn−1←−−− Tn

Fn←− . . . be a sequence of triangulatedcategories and triangulated functors. The shift functors of the variousTn induce an obvious shift functor [1] on lim←−Tn. Assume that each Tn isan I-filtered category and that all functors Fn are I-filtered morphisms(the T In are not assumed to be stable under the shift).

Proposition 81. Let ((Tn), (Fn)) as above satisfy conditions (F1)and (F2) and assume that each T In is closed under extensions in Tn.Then there is a unique class D of triangles in lim←−Tn (considered as anadditive category with shift functor [1]) such that (lim←−Tn,D) is a trian-gulated category and all projections pri : lim←−Tn → Ti are triangulated(i ∈ N).

A triangle Σ is in D if and only if all pri(Σ) are distinguished(i ∈ N).

Proof. (Cf. [BL94, 2.5.2].) We denote by E the class of trian-gles Σ in lim←−Tn such that all pri(Σ) are distinguished. We prove that(lim←−Tn, [1], E) is a triangulated category (it is obvious then that all priare triangulated). In all axioms of a triangulated category ([Ver96]),only a finite set {Xi} of objects is involved, and the existence of someobjects and morphisms is asserted. So we may check these axioms ina suitable full subcategory lim←−T

In of lim←−Tn containing all [k]Xi, for

k = −1, 0, 1. But this subcategory is equivalent to T INIby Lemma

78. (The condition that T INIis closed under extensions is used for

constructing a distinguished triangle with a given base.)If a class D of triangles satisfies the conditions of the proposition,

then obviously D ⊂ E . If Σ : Xf−→ Y → Z → [1]X is in E , there is

a triangle Σ′ : Xf−→ Y → Z ′ → [1]X in D. All objects are in some

lim←−TIn . Since Σ and Σ′ become isomorphic under prNI

: lim←−TIn∼−→ T INI

,they are isomorphic in lim←−Tn and hence Σ ∈ D. �

Remark 82. We omit the formulations and proofs of the obviousgeneralizations of Proposition 79 and Corollary 80 to I-filtered trian-gulated categories.

4.6. INVERSE LIMITS OF CATEGORIES OF DG MODULES 83

4.6. Inverse Limits of Categories of DG Modules

Let A∞ be a positively graded dg algebra with differential zeroand A0

∞ isomorphic to a finite product of division rings. If A∞ is theinverse limit of a sequence of dg algebras (An)n∈N of the same typethat “stabilizes in each degree” (see (S3)), we show that dgPrae(A∞)is the inverse limit of the categories dgPrae(Ai). We first study thespecial case where only two dg algebras are involved, and generalizeafterwards.

4.6.1. Special Case. Let A = (A =⊕

i≥0Ai, d = 0) be a posi-

tively graded dg algebra with differential zero and A0 =∏

x∈W exA0 a

finite product1 of division rings. Then the exA0 are up to isomorphism

the simple A0-modules, so

dgPrae(A) = dgPraeA({exA}x∈W ).

Let B be a dg algebra of the same type and φ : A → B a dga-morphism.We assume that φ0 : A0 → B0 is an isomorphism. Hence B0 =

∏exB

0,where we write ex instead of φ(ex).

Since exA⊗A B∼−→ exB, the extension of scalars functor induces a

triangulated functor

prodBA = (?⊗A B) : dgPrae(A)→ dgPrae(B).

By considering objects of dgFilt(A), it is easily seen that this functoris t-exact with respect to the t-structures introduced in Section 4.1.

By a segment we mean in the following a non-empty bounded in-terval in Z. If I = [a, b] is a segment (a, b ∈ Z, a ≤ b), we define|I| = b − a and dgPraeI = dgPrae≥a ∩ dgPrae≤b. If I is the posetof all segments in Z, partially ordered by inclusion, then dgPrae is anI-filtered category and the union of the dgPraeI . The functor prodBAis a morphism of I-filtered categories (as defined in Section 4.5).

Lemma 83. Let X be in dgPrae(A). If I is any segment, then

X in dgPraeI(A) ⇔ prodBA(X ) in dgPraeI(B).

Proof. This is a direct consequence of the definition of the t-structure (103) (cf. 64) and the assumption that φ0 : A0 → B0 isan isomorphism. �

1If R is a semisimple ring, there is a unique complete set {ex}x∈W of centralorthogonal idempotents in R such that each exR is a simple ring (see e. g. [Lan02]).

If a ring R is isomorphic to a finite product of division rings, then R is semisim-ple and each exR is a division ring. So R has a canonical inner decomposition∏exR = R as a finite product of division rings, and it is reasonable to say directly

that R =∏exR is a finite product of division rings.


Proposition 84. Let φ : A → B be a morphism of dg algebras asabove. If I is a segment and φ an isomorphism up to degree |I| + 2(i. e. φi : Ai → Bi is an isomorphism for all i ≤ |I|+ 2), then

prodBA : dgPraeI(A)→ dgPraeI(B)

is an equivalence of categories.

Proof. These statements are true since morphisms, homotopiesand differentials are defined and can be defined in low degrees. But letus give the details. We may assume that I = [0, b] for some b ∈ N.

prodBA is fully faithful: Let X , Y be in dgPraeI(A). We have toshow that

(109) prodBA : HomdgHot(A)(X ,Y)∼−→ HomdgHot(B)(φ(X ), φ(Y))

is an isomorphism, where we abbreviate φ(X ) = prodBA(X ) and φ(Y) =prodBA(Y). We may assume that X and Y are in dgFilt (Proposition63) and that

X = {l1}ev1A⊕ {l2}ev2A⊕ · · · ⊕ {ls}evsA,(110)

Y = {m1}ew1A⊕ {m2}ew2A⊕ · · · ⊕ {mt}ewtA

as graded A-modules, with vi, wi ∈ W , 0 ≤ −li ≤ b and 0 ≤ −mi ≤ b.Both φ(X ) and φ(Y) are given by the right hand side of (110), if wereplace A by B there.

We have

HomgMod(A)({l}evA, {m}ewA) = ({m}ewAev)−l = ewAm−lev

for v, w ∈ W , l, m ∈ Z. Since the differentials dX : X → {1}X,dY : Y → {1}Y are morphisms of graded A-modules (the differentialof A is zero), they are given by matrices x and y with entries in A.Similarly, morphisms f ∈ HomdgMod(X ,Y) are matrices satisfying yf =fx, and homotopies h : X → {−1}Y are matrices. Each entry of thesematrices is homogeneous, more precisely, we have xij ∈ evi

Ali+1−ljevj,

yij ∈ ewiAmi+1−mjewj

, fij ∈ ewiAmi−ljevj

, and hij ∈ ewiAmi−1−ljevj

.The differential of φ(X ) is given by the matrix φ(x). The functorprodBA : dgMod(A)→ dgMod(B) maps the matrix f = (fij) to φ(f) =(φ(fij)), and similarly for homotopies.

Surjectivity of (109): Let f be in HomdgMod(φ(X ), φ(Y)). Since all

entries of f are of degree ≤ b ≤ |I| + 2, there is a unique matrix f

such that φ(f) = f and deg(fij) = deg(fij) for all i, j (and fij = 0 if

fij = 0). This f defines an element of HomdgMod(X ,Y) if and only ifthe matrix equation yf = fx holds. All summands in every entry ofthis equation have degree ≤ b + 1 ≤ |I| + 2, and φ is an isomorphism

up to degree |I|+ 2. So it is enough to show that φ(y)f = fφ(x). But

this is true by assumption on f .

4.6. INVERSE LIMITS OF CATEGORIES OF DG MODULES 85

Injectivity of (109): Assume that f in HomdgMod(X ,Y) is mappedto φ(f) = 0 in HomdgHot(B)(φ(X ), φ(Y)). Then there is a homotopy

h : φ(X) → {−1}φ(Y ) alias a matrix with entries in B, such that

φ(f) = hφ(x)+φ(y)h. Since all entries of h are homogeneous of degree

≤ b − 1 ≤ |I| + 2, there is a unique matrix h with φ(h) = h and

deg(hij) = deg(hij). This matrix h defines a homotopy between f and0, because φ is an isomorphism up to degree |I|+ 2.

prodBA is surjective on isomorphism classes of objects in

dgPraeI(B): Let X be an object of dgPraeI(B). We may assume that

X is in dgFilt and has, as a graded B-module, the form

X = {l1}ev1B ⊕ {l2}ev2B ⊕ · · · ⊕ {ls}evsB,

with 0 ≤ −l1 ≤ −l2 ≤ · · · ≤ −ls ≤ b. The differential d eX is a matrix x,with all entries in B of degree ≤ b + 1 ≤ |I| + 2. Let x be the uniquematrix with entries in A such that φ(x) = x and deg(xij) = deg(xij)for all i, j. Define

X = {l1}ev1A⊕ {l2}ev2A⊕ · · · ⊕ {ls}evsA.

The matrix x defines a differential on X if and only if x2 = 0. In thismatrix equation, all summands have degree ≤ b + 2 ≤ |I| + 2. Butφ(x2) = x2 = 0 holds, and φ is an isomorphism in degrees ≤ |I| + 2.Hence (X, x) is in dgFilt and the A-module we are searching for. �

4.6.2. General Case. Let

(111) A0φ0←− A1

φ1←− . . .φn−1←−−− An

φn←− . . .

be a sequence of dg algebras and dga-morphisms. Assume that

(S1) Each An = (An =⊕

i≥0Ain, d = 0) is a positively graded dg

algebra with differential zero.(S2) A0

0 =∏

x∈W exA00 is a finite product of division rings.

(S3) There is an increasing sequence 0 < r0 ≤ r1 ≤ . . . of positiveintegers (rn) with rn →∞ for n→∞, such that each φn is anisomorphism up to degree rn. (In particular, all φ0

n : A0n+1 →

A0n are isomorphisms.)

The morphisms φn induce extension of scalars functors

φ∗n := prodAnAn+1

: dgPrae(An+1)→ dgPrae(An),

so we obtain a sequence

(112) dgPrae(A0)φ∗0←− dgPrae(A1)

φ∗1←− . . .φ∗n−1←−−− dgPrae(An)

φ∗n←− . . .

of categories or even of I-filtered categories, where I is the poset ofsegments in Z.


Proposition 85. Under the above assumptions, lim←− dgPrae(An)has a natural structure of triangulated category with the following classof distinguished triangles: A triangle Σ is distinguished if and only ifall pri(Σ) are distinguished (i ∈ N).

Proof. We want to deduce this from Proposition 81. It followsfrom (S3) and Propositon 84 that our sequence (112) satisfies condition(F1). Condition (F2) is fulfilled by Lemma 83. It is clear that eachdgPraeI is closed under extensions in dgPrae. �

Let A∞ be a dg algebra with dga-morphism νn : A∞ → An (n ∈ N)such that νn = φn ◦ νn+1 for all n ∈ N. Assume that there is anincreasing sequence 0 < r0 ≤ r1 ≤ . . . of positive integers (rn) withrn →∞ for n→∞, such that each νn is an isomorphism up to degreern. Equivalently, we could say that A∞ is the inverse limit of oursequence (111) of dg algebras, i. e. A∞ = lim←−An.

Proposition 86. Under the above assumptions, the obvious func-tor

dgPrae(A∞)∼−→ lim←− dgPrae(An)

is an equivalence of triangulated categories.

Proof. This is a consequence of Corollary 80 and Remark 82. �

CHAPTER 5

Formality of Equivariant Derived Categories

5.1. Acyclic Maps

If Y is a topological space, we denote by Sh(Y ) the category ofsheaves of real vector spaces on Y and byD+(Y ) andDb(Y ) its boundedbelow and bounded derived category.

A continuous map f : X → Y is pre-∞-acyclic, if, for all sheavesB ∈ Sh(Y ), the adjunction morphism B → f∗f

∗B in D+(Y ) is anisomorphism. Let n ∈ N. The map f is called pre-n-acyclic, ifB → τ≤nf∗f

∗B is an isomorphism for all B in Sh(Y ). Here τ≤n isthe truncation functor (see e. g. [KS94, 1.3]). Let n ∈ N ∪ {∞}. We

call f n-acyclic, if, for any base change Y → Y , the induced map

f : X ×Y Y → Y is pre-n-acyclic. A topological space X is called(pre-)n-acyclic if the constant map X → pt is (pre-)n-acyclic.

Let f : X → Y be n-acyclic and F = f−1(y) the fiber over a pointy ∈ Y . Then the sheaf cohomology Hi(F ;F ) of the constant sheaf isR in degree i = 0 and vanishes in degrees 1 ≤ i ≤ n. In particular, Fis connected and not empty, so f is surjective.

Lemma 87. The composition of n-acyclic maps is n-acyclic, forn ∈ N ∪ {∞}.

Proof. Easy exercise with truncation functors. �

Let P be a property of topological spaces. We say that a topologicalspace X is open-locally P if every neighborhood of every point x ∈ Xcontains an open neighborhood of x satisfying property P.

We believe that the following modification of [BL94, Criterion1.9.4] is true.

Proposition 88. Let f : X → Y be a locally trivial fiber bundleand n ∈ N ∪ {∞}. Assume that all fibers are pre-n-acyclic and open-locally contractible. Then f is n-acyclic.

This proposition should also hold if we replace contractible by pre-n-acyclic. We hope to write down the proof in the future.

Proof. See [Soe07, XV.3.7.2]. �

Corollary 89. Every pre-n-acyclic and open-locally contractiblespace is n-acyclic.

87

88 5. FORMALITY OF EQUIVARIANT DERIVED CATEGORIES

5.2. Equivariant Derived Categories of Topological Spaces

Following [BL94], we introduce the equivariant derived category.Let G be a topological group. A G-space is a topological space Xwith a continuous G-action G × X → X. A G-map of G-spaces is acontinuous G-equivariant map. A G-space is trivial if it has the formG × Y for some topological space Y with G-action g.(h, y) = (gh, y).A G-space X is free if it has a covering by open G-stable subspacesG-isomorphic to trivial spaces. Then the quotient map X → G\X is alocally trivial principal fiber bundle (pfb) (with structure group G).

A resolution of a G-space X is a G-map from a free G-spaceto X. Morphisms of resolutions are G-maps over X. Let p : P →X be a resolution of X and q : P → G\P the quotient map. Thecategory Db

G(X,P ) is defined as follows (with the obvious compositionof morphisms):

• Objects M of DbG(X,P ) are triples (MX ,M, µ) where MX ∈

Db(X), M in Db(G\P ) and

µ : p∗(MX)∼−→ q∗(M)

is an isomorphism in Db(P ).• Morphisms α : M → N (where M = (MX ,M, µ) and N =

(NX , N, ν)) are pairs (αX , α) where α : MX → NX and α :M → N are morphisms in Db(X) and Db(G\P ) respectivelysuch that

p∗(MX)

µ∼��

p∗(αX)// p∗(NX)

ν∼��

q∗(M)q∗(α)

// q∗(N)

commutes.

We have two obvious functors: The forgetful functor For : DbG(X,P )→

Db(X), M 7→ MX , and the functor γ : DbG(X,P ) → Db(G\P ), M 7→

M . If I is a segment in Z, we let DIG(X,P ) be the full subcategoryof Db

G(X,P ) consisting of objects M with For(M) in DI(X). If p issurjective, this is equivalent to the condition γ(M) ∈ DI(G\P ).

A morphism of resolutions ν : P → R of X induces an obviousfunctor

(113) ν∗ : DbG(X,R)→ Db

G(X,P ).

A resolution p : P → X is n-acyclic if the continuous map p isn-acyclic. Note that any n-acyclic map is surjective.

Lemma 90 ([BL94]). Let p : P → X be an n-acyclic resolution ofa G-space X, where n ∈ N ∪ {∞}. If I is a segment with n > |I|, thefunctor

γ : DIG(X,P )→ DI(G\P )

5.2. EQUIVARIANT DERIVED CATEGORIES OF TOPOLOGICAL SPACES 89

is fully faithful, and its essential image DI(G\P |p) is closed underextensions and taking direct summands in Db(G\P ).

Proof. This is [BL94, 2.3.2] (but with n > |I|). �

Proposition 91. Let ν : P → R be a morphism of n-acyclic res-olutions p : P → X and r : R → X, where n ∈ N ∪ {∞}. If Iis a segment with n > |I|, the functor ν∗ from (113) restricts to anequivalence

ν∗ : DIG(X,R)→ DIG(X,P ).

Proof. Let S = P ×X R and consider the cartesian diagram

SπR //

πP

��

R

r

��P

p // X

By [BL94, 2.2.2] (but with n > I),

DIG(X,P )π∗P−→ DIG(X,S)

π∗R←− DIG(X,R)

are equivalences of categories. Let (idP , ν) : P → S = P ×X R bethe unique morphism of resolutions with πP ◦ (idP , ν) = idP and πR ◦(idP , ν) = ν. Then (idP , ν)∗ is inverse to π∗P and ν∗ = (idP , ν)∗ ◦ π∗R isan equivalence on DIG. �

If P → X and R→ X are∞-acyclic resolutions there is an equiva-lence Db

G(X,P )∼−→ Db

G(X,R) that is defined up to a canonical naturalisomorphism (as can be seen from the first part of the proof of Proposi-tion 91). The G-equivariant derived category Db

G(X) of X is “defined”to be Db

G(X,P ), for P → X an∞-acyclic resolution ([BL94, 2.7.2]). Itis a triangulated category (cf. [BL94, 2.5.2]). (By Lemma 90 we have

DbG(X,P )

∼−→ Db(G\P |p) and the category on the right is a triangu-lated subcategory of Db(G\P ); this structure of triangulated categorycoincides with the structure defined in [BL94, 2.5.2].)

For our purposes, the following description of the equivariant de-rived category as an inverse limit will be useful. Any sequence

P0f0−→ P1

f1−→ P2 → . . .→ Pnfn−→ Pn+1 → . . .

of morphisms of resolutions pn : Pn → X gives rise to a sequence ofcategories and functors

(114) DbG(X,P0)

f∗0←− DbG(X,P1)

f∗1←− . . .f∗n−1←−− Db

G(X,Pn)f∗n←− . . .

and we can consider the inverse limit lim←−DbG(X,Pn) of these categories

as defined in Section 4.5. In fact, our sequence is a sequence of I-filtered categories (where I is the poset of segments in Z), if we filterDbG(X,Pn) by the subcategories DIG(X,Pn).


We define the forgetful functor For to be the composition

For : lim←−DbG(X,Pn)

pr0−−→ DbG(X,P0)

For−−→ Db(X)

and denote by γi the composition

lim←−DbG(X,Pn)

pri−→ DbG(X,Pi)

γ−→ Db(G\Pi).

An object M of lim←−DbG(X,Pn) is in lim←−D

IG(X,Pn) if and only if For(M)

is in DI(X). Hence lim←−DbG(X,Pn) is the union of the lim←−D

IG(X,Pn),

so our sequence (114) satisfies condition (F2).

Proposition 92. Assume that pn : Pn → X is n-acyclic, for eachn ∈ N. Then lim←−D

bG(X,Pn) carries a natural structure of triangulated

category with the following class of distinguished triangles: A triangleΣ is distinguished if and only if all γi(Σ) are distinguished (i ∈ N).

Proof. (Essentially, we reproduce the argument of [BL94, 2.5.2].)Our sequence (114) satisfies conditions (F2) (as remarked above) and(F1) (by Proposition 91).

We cannot directly apply Proposition 81, because the categoriesDbG(X,Pn) are not triangulated. But we may adapt this proposition

and its proof to our slightly more general situation using Lemma 90.The main modifications are replacing pri by γi and DIG(X,PNI

) byDI(G\PNI

|pNI), where we may assume that NI > |I|. �

Proposition 93. Let P = (P0f0−→ P1

f1−→ . . . ) and Q = (Q0 →Q1 → . . . ) be sequences of resolutions of X and assume that Pn and Qn

are n-acyclic, for each n ∈ N. Then lim←−DbG(X,Pn) and lim←−D

bG(X,Qn)

are equivalent as triangulated categories.If Q∞ → X is an∞-acyclic resolution, the categories lim←−D

bG(X,Pn)

and DbG(X,Q∞) (alias Db

G(X)) are equivalent as triangulated cate-gories.

Proof. We may assume without loss of generality that there is amorphism ν = (νn)n∈N : P → Q between our two sequences of resolu-tions (otherwise consider the sequence of n-acyclic (Lemma 87) reso-lutions Pn ×X Qn). It induces compatible functors ν∗n : Db

G(X,Qn) →DbG(X,Pn). The restrictions ν∗n : DIG(X,Qn) → DIG(X,Pn) are equiva-

lences if n > |I| (Proposition 91), so we get an equivalence

lim←−DbG(X,Qn)

∼−→ lim←−DbG(X,Pn)

by Proposition 79. It follows from Proposition 92 that this equivalencerespects distinguished triangles.

For the remaining statement use the constant sequence Q∞id−→

Q∞id−→ . . . of resolutions.

If we identifyDbG(X,Q∞) = Db

G(X), it is obvious that the structuresof triangulated category given by Proposition 92 and [BL94, 2.5.2]coincide. �

5.3. EQUIVARIANT DERIVED CATEGORIES OF VARIETIES 91

5.3. Equivariant Derived Categories of Varieties

Let G be an affine algebraic group (defined over C, as all varieties inthe following). The definitions of a G-variety and of a G-morphism be-tween G-varieties are the obvious generalizations from the topologicalcategory.

A Zariski principal fiber bundle (Zpfb) with structure groupG (or G-Zpfb) is a surjective G-morphism q : E → B between G-varieties with the following property. For every point in B, there is aZariski open neighborhood U in B and a G-isomorphism τ : G×U ∼−→q−1(U) (here the G-action on G×U is given by g.(h, u) = (gh, u)) suchthat the diagram

(115) q−1(U)

q

��

G× Uτ

∼oo

prU

��U U

commutes, where prU is the projection G×U → U . A map τ as aboveis called a local trivialization over U . By abuse of notation we oftensay that E is a Zpfb.

A morphism f : E → E ′ of G-Zpfbs is a G-morphism f : E → E ′.Each such f induces a unique morphism f : B → B′ such that q′ ◦ f =f ◦ q. We often write f instead of f .

Let X be a G-variety. A Zariski resolution (Z-resolution) of

X is a datum (Bq←− E

p−→ X) consisting of a Zpfb q : E → B togetherwith a G-morphism p : E → X. We often omit q : E → B from thenotation and say that p : E → X is a Zariski resolution or even that Eis a Zariski resolution of X. Morphisms E → E ′ of Zariski resolutionsof X are G-morphisms over X.

Let n ∈ N ∪ {∞}. A variety (morphism of varieties) is called n-acyclic, if it is n-acyclic with respect to the classical topology. A

Zariski resolution (Bq←− E

p−→ X) is n-acyclic if p is n-acyclic.

Let E0f0−→ E1 → . . . → En

fn−→ En+1 → . . . be a sequence of Z-resolutions of a G-variety X. If pn : En → X is n-acyclic, Propositions92 and 93 provide a triangulated equivalence

(116) DbG(X) ∼= lim←−D

bG(X,En).

Let (X,S) be a stratified G-variety (we do not assume that thestrata are G-stable). We denote the full subcategory of Db

G(X) con-sisting of objects M with For(M) ∈ Db(X,S) by Db

G(X,S). Similarly,we define Db

G(X,E,S) ⊂ DbG(X,E), if E is a Zariski resolution of X.

If I is a segment, we use the obvious definitions for DIG(X,S) etc..Equivalence (116) restricts to a triangulated equivalence

(117) DbG(X,S) ∼= lim←−D

bG(X,En,S).


If X is a G-variety, we denote by DbG,c(X) the full subcategory of

DbG(X) consisting of objects F such that For(F ) is constructible for

some stratification of X.

Proposition 94. Let (X,S) be a stratified G-variety. If each stra-tum is a G-orbit, then Db

G(X,S) = DbG,c(X).

Proof. Let M be in DbG,c(X) and l : S → X the inclusion of

an orbit. We have to prove that Hi(l∗(For(M))) = For(Hi(l∗(M))) is alocal system on S. But A := Hi(l∗(M)) is in the heart of the t-categoryDbG,c(S) and hence a G-equivariant constructible sheaf on S ([BL94,

2.5.3]).If H ⊂ G is the stabilizer of a point of S, the corresponding orbit

map G/H → S is a diffeomorphism. Since G → G/H is topologicalpfb, the functor F 7→ G ×H F establishes an equivalence between H-equivariant sheaves on pt and G-equivariant sheaves on G/H = G×Hpt(here we identify sheaves with their etale spaces). It follows that For(A)is a local system on S. �

A G-stratification of a G-variety is a stratification by G-stablestrata. A G-stratified variety is a G-variety X with a G-stratification.

Let (X,S) be a G-stratified variety. An approximation (E, f) of(X,S) is a sequence

E0f0−→ E1 → . . .→ En

fn−→ En+1 → . . .

of Zariski resolutions (Bnqn←− En

pn−→ X) of X. An approximation(E, f) is called an A-approximation if the following conditions hold.

(A1) Each (Bnqn←− En

pn−→ X) is n-acyclic.(A2) Sn := {qn(p−1

n (S)) | S ∈ S} is a stratification of Bn, for alln ∈ N. (In particular, each stratum of Sn is irreducible.)

(A3) Every stratum in Sn is simply connected.

Any A-approximation E0f0−→ E1

f1−→ . . . yields a sequence B0f0−→

B1f1−→ . . . of varieties, and a sequence ((Db(Bn,Sn)), (f ∗n)) of triangu-

lated categories.

Proposition 95. Let (E, f) = (. . . → Enfn−→ En+1 → . . . ) be

an A-approximation of a G-stratified variety (X,S). Then there is acanonical triangulated equivalence

(118) lim←−DbG(X,En,S)

∼−→ lim←−Db(Bn,Sn)

Proof. The functor DbG(X,En) → Db(Bn), (MX ,M, µ) 7→ M ,

restricts to a functor νn : DbG(X,En,S) → Db(Bn,Sn), since qn is

locally trivial. The inverse limit lim←− νn is the functor in (118).

All categories DbG(X,En,S) are filtered by the DIG(X,En,S), and

are the union of these subcategories. Similarly for Db(Bn,Sn). For

5.4. EQUIVARIANT INTERSECTION COHOMOLOGY COMPLEXES 93

n > |I|, restriction

(119) f ∗n : DIG(X,En+1,S)→ DIG(X,En,S)

is an equivalence (Proposition 91).We claim that

(120) νIn : DIG(X,En,S)→ DI(Bn,Sn)

is an equivalence for n > |I|. By Lemma 90 (and Proposition 11), νInis fully faithful and its essential image is closed under extensions inDb(Bn,Sn). Let S ∈ S and Sn := qn(p−1

n (S)). The inclusions lS andlSn and proper base change give rise to an object “extension by zero

of the constant sheaf on S” in D[0,0]G (X,En) that is mapped to lSn!Sn

under νn : DbG(X,En,S) → Db(Bn,Sn). It follows from Lemma 97

below that νIn is surjective on isomorphism classes of objects.Proposition 79 shows that (118) is an equivalence and that the

filtered category Db(Bn,Sn) satisfies condition (F1). Proposition 81equips lim←−D

b(Bn,Sn) with the structure of a triangulated category,and it is obvious that (118) is a triangulated equivalence. �

Corollary 96. Let (E, f) be an A-approximation, I a segmentand N ∈ N. If N > |I|, the obvious functor lim←−D

I(Bn,Sn)→ DI(BN)is fully faithful.

Proof. It follows from the proof of the Proposition 95 (cf. (119),(120)) that

f ∗n : DI(Bn+1,Sn+1)→ DI(Bn,Sn)

is an equivalence for n > |I|. Lemma 78 shows that

prN : lim←−DI(Bn,Sn)→ DI(BN ,SN)

is an equivalence for N > |I|. �

Lemma 97. Let (X,S) be a stratified variety with simply connectedstrata and I a segment in Z. Then every A ∈ DI(X,S) is an iteratedextension of objects lS!S[−i], for S ∈ S and i ∈ I.

Proof. The shift [1] and the truncation functors for the trivial t-structure allow us to assume that A ∈ Sh(X,S). If j is the inclusionof an open stratum U and i the inclusion of its closed complement, weget a distinguished triangle (j!j

∗(A), A, i∗i∗(A)). Since j∗(A) is a finite

direct sum of constant sheaves U , an induction on the number of stratafinishes the proof. �

5.4. Equivariant Intersection Cohomology Complexes

Let G be an affine algebraic group of complex dimension dG and(X,S) a G-stratified variety. On Db

G(X,S), there is the perverse t-structure with heart the category PervG(X,S) of equivariant perversesheaves (smooth along S) (see [BL94, 5]). If L is a G-equivariant local


system on S, we have the equivariant intersection cohomology complexICG(S,L) = lS!∗([dS]L) in PervG(X,S). We are mainly interested inthe case of the constant G-equivariant local system SG on S and defineICG(S) := ICG(S, SG). We will describe this object precisely usingthe following type of approximation.

An AB-approximation is an A-approximation (E, f) of (X,S)such that the following conditions are satisfied.

(B1) Each morphism pn : En → X is smooth of relative complexdimension dpn .

(B2) Each fn : Bn → Bn+1 is a closed embedding of varieties.(B3) For all n ∈ N and S ∈ S, fn : Sn → Sn+1 is a normally non-

singular inclusion of codimension cn, where Sn := qn(p−1n (S)).

Let (E, f) be an AB-approximation of (X,S). If we consider ICG(S)as an object of lim←−D

b(Bn,Sn), using equivalences (117) and (118), wehave (cf. [BL94, 5.1], using B1)

prn(ICG(S)) = [dG − dpn ]IC(Sn) ∈ Db(Bn,Sn).

In order to avoid to many shifts, we replace lim←−Db(Bn,Sn) by an equiv-

alent category as follows. Consider the following morphism of sequencesof triangulated categories

(121) Db(B0,S0)

[dp0−dG]

��

Db(B1,S1)f∗0oo

[dp1−dG]

��

Db(B2,S2)f∗1oo

[dp2−dG]

��

. . .f∗2oo

Db(B0,S0) Db(B1,S1)[−c0]f∗0oo Db(B2,S2)

[−c1]f∗1oo . . . .[−c2]f∗2oo

If we denote the inverse limit of the second row by lim←−Db[Bn,Sn], this

morphism induces a triangulated equivalence

(122) lim←−Db(Bn,Sn)

∼−→ lim←−Db[Bn,Sn].

Conditions A2, B2 and B3 show that each fn : Bn → Bn+1 meetsthe assumptions made before (71) and (72) in Section 3.2. So we obtainisomorphisms

ιS,n : [−cn]f ∗n(IC(Sn+1))∼−→ IC(Sn) in Perv(Bn,Sn) and(123)

ιS,n : [−cn]f ∗n(IC(Sn+1))∼−→ IC(Sn) in MHM(Bn,Sn).

As an object of lim←−Db[Bn,Sn], the equivariant intersection complex

ICG(S) is

ICG(S) = ((IC(Sn)), (ιS,n)) ∈ lim←−Db[Bn,Sn].

Note that ICG(S) = ((IC(Sn)), (ιS,n)) is a natural “Hodge lift” ofICG(S). Remark 46 shows that all functors [−cn]f ∗n in (121) are t-exact with respect to the perverse t-structures.

5.5. BETTER APPROXIMATIONS AND FORMALITY 95

5.5. Better Approximations and Formality

Let (E, f) be an approximation of a G-stratified variety (X,S) andassume that we are given stratifications Tn of Bn, for each n ∈ N. Thetriple (E, f, T = (Tn)) is called an ABC-approximation if (E, f) isan AB-approximation and the following conditions hold.

(C1) Each Tn is a cell-stratification of Bn that is finer than thestratification Sn.

(C2) Each fn : (Bn, Tn) → (Bn+1, Tn+1) is a closed embedding of(cell-)stratified varieties.

(C3) The Hodge sheaf IC(Sn) is Tn-pure of weight dSn , for all n ∈ Nand S ∈ S. (By Remark 38 we can equivalently require Tn-∗-purity of weight dSn .)

In Section 5.6 we will provide some methods for constructing ABC-approximations and give examples ofG-stratified varieties having ABC-approximations.

For M , N in DbG(X), define Extn(M,N) := HomDb

G(X)(M, [n]N)

and Ext(M,N) :=⊕

n∈Z ExtnG(M,N). The (equivariant) extensionalgebra of M is Ext(M) := Ext(M,M).

Theorem 98. Let G be an affine algebraic group and (X,S) aG-stratified variety. If (X,S) has an ABC-approximation, there is atriangulated equivalence

(124) DbG(X,S) ∼= dgPrae(Ext(ICG(S))),

where ICG(S) is the direct sum of the (ICG(S))S∈S . This equivalence ist-exact with respect to the perverse t-structure on Db

G(X,S) and the t-structure on dgPrae(Ext(ICG(S))) introduced in Section 4.1, Theorem65. By restriction to the heart, it induces an equivalence

(125) PervG(X,S) ∼= dgFlag(Ext(ICG(S)))

of abelian categories (cf. Section 4.3). In these statements, we canreplace dgPrae by dgPer.

Remark 99. Probably, the existence of an ABC-approximationimposes some conditions on the G-stratified variety (X,S) (e. g. eachequivariant local system on a stratum may be trivial). We did notpursue this.

Proof. Obviously, E∞ := Ext(ICG(S)) is a positively graded dgalgebra with differential zero and has as degree zero part the product of|S|-many copies of R. Hence dgPrae(E∞) and the equivalent categorydgPrae(E∞) are well defined.

Let (E, f, T ) be an ABC-approximation of (X,S). By (117), (118)and (122) we have equivalences(126)

DbG(X,S) ∼= lim←−D

bG(X,En,S)

∼−→ lim←−Db(Bn,Sn)

∼−→ lim←−Db[Bn,Sn].


of triangulated categories.Properties A3, C1 and C3 allow to apply Theorem 35 (cf. Remark

37), and we obtain equivalences

(127) Formn := FormTnePn→fIC(Sn): Db(Bn,Sn)

∼−→ dgPrae(En)

of triangulated categories, where we have fixed perverse-projective res-

olutions Pn,Sn → IC(Sn) and where En := Ext(IC(Sn)). The isomor-phisms (123) induce dga-morphisms φn : En+1 → En. Properties A2,A3, B2, B3, C1, C2, C3 and Theorem 45 yield the following com-mutative (up to natural isomorphism) diagram with triangulated andt-exact functors.

(128) Db(Bn+1,Sn+1)[−cn]f∗n //

Formn+1 ∼��

Db(Bn,Sn)

Formn ∼��

dgPrae(En+1)?

L⊗En+1

En

// dgPrae(En)

Hence the sequence (Formn)n∈N defines a morphism between se-quences of triangulated categories. Its inverse limit establishes an equi-valence

(129) lim←−Db[Bn,Sn]

∼−→ lim←− dgPrae(En).

Using (126), there are obvious dga-morphisms νn : E∞ → En suchthat νn = φn ◦ νn+1. Let J be a segment such that ICG(S) ∈ DJG(X).We deduce from Corollary 96 that νn is an isomorphism up to degreen− |J | − 1. So our sequence of dg algebras ((En), (φn)) satisfies (if weforget the first |J |+ 2 members and renumerate, which is harmless forthe following) the conditions (S1)-(S3) considered in Section 4.6.2, andE∞ is the inverse limit of this sequence. Proposition 85 (and Remark67) shows that lim←− dgPrae(En) carries a natural structure of triangu-lated category. Since all functors Formn are triangulated, it followsfrom Proposition 92 that equivalence (129) is triangulated. Finally,Proposition 86 provides a triangulated equivalence

dgPrae(E∞)∼−→ lim←− dgPrae(En).

This establishes (124). Since ICG(S) is mapped to eSE∞, equivalence(124) is t-exact, and we obtain (125) by use of Proposition 73. Finally,by Corollary 66, we can replace dgPrae by dgPer. �

5.6. Existence and Examples of ABC-Approximations

Let G be an affine algebraic group. An ABCD-approximationof a G-stratified variety (X,S) is an ABC-approximation (E, f, T ) of(X,S) satisfying the following condition.

5.6. EXISTENCE AND EXAMPLES OF ABC-APPROXIMATIONS 97

(D) For each n ∈ N, the Zpfb qn : En → Bn can be trivializedaround each stratum T ∈ Tn (this means that there is anopen subvariety U of Bn containing T such that qn has a localtrivialization over U).

An ABCD-approximation for G is an ABCD-approximation ofthe G-stratified variety (pt, {pt}). A lot of the conditions we imposeon an ABCD-approximation become quite simple in this case; e. g.the stratifications Sn are trivial, so every Bn is irreducible and simplyconnected.

Proposition 100. There is an ABCD-approximation for GLn.

Proof. Recall that an n-frame in a vector space is an n-tuple oflinearly independent vectors. Let Ei be the Stiefel variety of n-framesin Cn+i. We identify such n-frames with (n+i)×n-matrices of maximalrank. The group GLn acts on Ei as follows: Given a matrix A ∈ Eiand a group element g ∈ GLn, we define g.A := Ag−1. Let Bi be theGrassmannian variety of n-dimensional subspaces of Cn+i. The map qi :Ei → Bi, A 7→ spanA, that maps an n-frame to the subspace spannedby its components is easily seen to be a GLn-Zpfb. The Stiefel varietyE := Ei is 2i-connected (e. g. [Hat02]). The long exact sequence ofhomotopy groups of a fiber bundle and the connectedness of GLn showthat Bi is simply connected. We prove in Lemma 101 below that Ei is2i-acyclic (cf. [BL94, 3.1]).

The varieties Ei and Bi are nonsingular and irreducible. The stan-dard closed embeddings Cn+i ↪→ Cn+i+1, x 7→ (x, 0) induce morphismsof Zpfbs fi : Ei → Ei+1. The maps fi : Bi → Bi+1 are closed em-beddings of varieties and normally nonsingular inclusions ([GM88,

I.1.11]). So (Bq←− E, f) is an AB-approximation for GLn.

Let (e1, . . . , eN) be the standard basis of CN , where N = n + i.In the obvious way, the group GLN acts transitively on Bi. Thestabilizer P of the space span(e1, . . . , en) is a parabolic subgroup ofGLN and contains the standard Borel subgroup B ⊂ GLN , consistingof upper triangular matrices. The orbit map yields an isomorphismGLN /P

∼−→ Bi, and the B-orbits of GLN /P form a cell-stratification.We define Ti to be the corresponding stratification of Bi. Obviously,each fi : (Bi, Ti)→ (Bi+1, Ti+1) is a closed embedding of cell-stratifiedvarieties. It is easy to see that qi can be trivialized around each T ∈ Ti.Condition C3 (cf. Remark 38) is satisfied, since IC(Bi) = [dBi

]Bi. (Asan overkill argument, we could also use Corollary 42.) It follows that

(Bq←− E, f, T ) is an ABCD-approximation for GLn. �

Lemma 101. The Stiefel variety Ei of n-frames in Cn+i is 2i-acyc-lic.

Proof. The Stiefel variety E := Ei is 2i-connected and we obtainH0(E; R) = R and Hj(E; R) = 0 for 1 ≤ j ≤ 2i (e. g. [Hat02]). In


order to apply Corollary 89 we have to prove that

(130) Hj(E;EV ) =

{R for j = 0,

0 for 1 ≤ j ≤ 2i,

for every vector space V , where EV is the constant sheaf on E withstalk V . For finite dimensional V this is obvious.

Let F be the set of orthonormal n-frames in Cn+i. This is a compactsubspace of E. The Gram-Schmidt process shows that the inclusionF ⊂ E is a homotopy equivalence. By homotopy invariance of sheafcohomology it is sufficient to show (130) for Hj(F ;F V ) = Hj

c(F ;F V ).But cohomology with compact support commutes with arbitrary directsums. �

Lemma 102. Let Gi be affine algebraic groups and let (X i,S i) beGi-stratified varieties that have ABCD-approximations (i = 1,2). Thenthe G1×G2-stratified variety (X1×X2,S1×S2) has an ABCD-approxi-mation.

Proof. The product of ABCD-approximations is an ABCD-ap-proximation. This follows from the following observations. Finite prod-ucts of Zpfbs ((Whitney) stratifications, (Zariski) resolutions, closedembeddings, normally nonsingular inclusions, ...) are again Zpfbs (stra-tifications, resolutions, ...). By Lemma 87, the product of two n-acyclicresolutions is n-acyclic. If S and T are strata of stratified varieties X

and Y respectively, then IC(S×T ) = IC(S)� IC(T ) in MHM(X×Y ).Hence C3 is satisfied for the product approximation. �

Proposition 103. Let P be an affine algebraic group. Assume thatthere are closed subgroups R and V of P such that

(a) multiplication V ×R→ P is an isomorphism,(b) R has an ABCD-approximation and(c) V is ∞-acyclic (in the classical topology).

Then there is an ABCD-approximation for P . In particular, ABCD-approximations exist for

(a) B a connected solvable affine algebraic group.(b) P ⊂ GLn a parabolic subgroup of some general linear group.

Proof. Let (BqR

←− ER, T , fR) be an ABCD-approximation for R.Define EP

i := indPR ERi = P ×R ER

i , and let fPi := indPR fRi . The

morphisms P×ERi → Bi, (p, e) 7→ qRi (e) induce P -Zpfbs qPi : EP

i → Bi.

Using the isomorphism V × R∼−→ P , we get an isomorphism of

P -varieties EPi = P ×R ER

i∼← V × ER

i . Since V is ∞-acyclic, EPi is i-

acyclic (Lemma 87). So (BqP

← EP , T , fP ) is an ABCD-approximationfor P .


Let B be a connected solvable affine algebraic group and T ⊂ B amaximal torus. The set U ⊂ B of unipotent elements is a closed sub-group, and multiplication U × T → B is an isomorphism. By Propo-sition 100 and Lemma 102, T has an ABCD-approximation. Since wework over the complex numbers, the exponential map LieU → U is anisomorphism (even of varieties). So U is contractible in the classicaltopology, hence ∞-acyclic.

Let now P ⊂ GLn be a parabolic subgroup of a general lineargroup and B ⊂ GLn the standard Borel subgroup consisting of uppertriangular matrices. We may assume that P is a standard parabolicsubgroup, i. e. B ⊂ P . Let V be the unipotent radical of P , andR a Levi factor, so V o R

∼−→ P by multiplication. All Levi factorsare conjugate by an element of V and there is an obvious Levi factorthat is isomorphic to a product of general linear groups. By the abovearguments, R has an ABCD-approximation and V is ∞-acyclic in theclassical topology. �

Proposition 104. Let G be an affine algebraic group and (X,S)a G-stratified variety. Assume that

(R1) S is a G-stratification into cells,

(R2) IC(S) is S-pure, for every S ∈ S,(R3) G has an ABCD-approximation,

(R4) there is a G-stratified variety (Y, S) together with(a) a G-equivariant locally closed embedding v : X → Y sat-

isfying v(S) := {v(S) | S ∈ S} ⊂ S, and(b) a G-equivariant closed embedding of Y in a smooth G-

manifold M .

Then (X,S) has an ABCD-approximation.

Remark 105. Condition (R4) will be used for the proof of B3.Possibly it is redundant. It is satisfied for Schubert varieties and moregenerally for unions of Borel-orbits that are locally closed in the flagvariety (take as Y and M the flag variety).

For a normal projective G-variety Y and connected G, (R4) (b) issatisfied by [Sum74] or [Mum65].

Proof. If (Br←− E, f, T ) is an ABCD-approximation for G, its i-

th subdatum (Biri←− Ei, Ti) satisfies an obvious subset of the conditions

A1-D.So let (B

r←− E, T ) be the i-th subdatum of an ABCD-approxima-tion for G. Consider the commutative diagram

(131) E ×G Xπ

��

E ×Xqoo p //

π

��

X

��B E

roo c // pt.


Here q is the quotient map for the diagonal action of G on E × X, pis the second projection, π the first projection and π the induced mapon quotient spaces.

We claim that the upper row of diagram (131) together with

r−1(T )×G S := {r−1(T )×G S | T ∈ T , S ∈ S}

satisfies the conditions imposed on the i-th subdatum of an ABCD-ap-proximation of (X,S).

The square on the left in (131) is cartesian, q is a Zpfb and π isa (Zariski locally trivial) fiber bundle with fiber X. These statementsare also true in the classsical topology. They can be deduced from localtrivializations of r. If τ : G× U ∼−→ r−1(U) is such a trivialization (cf.(115)) over an open subvariety U of B, diagram (131) restricts to

(132) U ×XprU

��

G× U ×X(u,g−1x)←[(g,u,x)

oo p //

prG×U

��

X

��U G× U

prUoo c // pt.

Here we consider U×X as an open subvariety of E×GX, the inclusiongiven by (u, x) 7→ [τ(1, u), x].

Since c is i-acyclic and smooth, the same holds for p (A1, B1).If S ∈ S is a stratum, the intersection of E ×G S = q(p−1(S)) withU ×X ⊂ E ×G X is U × S. Hence

E ×G S := {E ×G S | S ∈ S}

is a stratification of E×GX (A2) (the irreducibility of the strata will beestablished below). It results from the long exact sequence of homotopygroups for the fiber bundle π : E ×G S → B with fiber S that E ×G Sis simply connected (A3), since S is a cell and B is simply connectedby assumption.

Let S ∈ S and T ∈ T be strata. The intersection of r−1(T ) ×G Swith U × X is (T ∩ U) × S, so r−1(T ) ×G S is a stratification ofE ×G X. By D, we find a local trivialization τ of r as above withT ⊂ U . Then τ × idX is a local trivialization of q over U × X, andr−1(T )×G S = T × S ⊂ U ×X is a cell (D, C1).

By A2, B is irreducible. Let T ∈ T be dense in B. Then r−1(T ) isdense in E and the cell r−1(T )×G S is dense in E ×G S, for all S ∈ S,showing the irreducibility of each stratum E ×G S (A2).

Let S ∈ S. We prove C3. By Remark 38 is is sufficient to show that

IC(E ×G S) is (r−1(T ) ×G S)-∗-pure of weight dB + dS. Let R ∈ S,T ∈ T . We choose U ⊂ B open as above containing T . Then the

inclusion r−1(T )×GRl↪→ E×GX looks like T×R t×r

↪→ U×Xj↪→ E×GX.

Since j is an open embedding, j∗ preserves weights and we obtain

(133) j∗(IC(E ×G S)) ∼= IC(U × S) ∼= IC(U) � IC(S)


Let lT : T ↪→ B be the inclusion and IC(B) the Hodge intersection

cohomology sheaf on B. Then l∗T (IC(B)) ∼= t∗(IC(U)), and restrictionof (133) yields

l∗(IC(E ×G S)) ∼= t∗(IC(U)) � r∗(IC(S)) ∼= l∗T (IC(B)) � r∗(IC(S)),

and this is pure of weight dB + dS by assumptions C3 and (R2).

Let (Br←− E, f, T ) be an ABCD-approximation for G. Let

(Ei ×G Xqi←− Ei ×X

pi−→, r−1i (Ti)×G S)

be the i-th subdatum constructed by the above method. We claim thatthe sequence of these subdata together with the sequence of morphisms

fi × idX : Ei ×X → Ei+1 ×X

defines an ABCD-approximation of (X,S). Conditions B2 and C2 areobviously satisfied.

A slight modification of the above arguments (replace cell by irre-

ducible variety) shows that (Ei ×G Y,Ei ×G S) is a stratified variety.Consider the diagram

Ei+1 ×G Xid×Gv // Ei+1 ×G Y // Ei+1 ×GM

Ei ×G Xid×Gv //

fi×GidX

OO

Ei ×G Y //

fi×GidY

OO

Ei ×GM,

fi×GidM

OO

where both squares are cartesian. In the smooth manifold Ei+1×GM ,the smooth submanifold Ei ×GM is transverse to each stratum of theclosed stratified variety (Ei+1×GY,Ei+1×G S). It follows from [GM88,I.1.11] that Condition B3 is satisfied. (For each S ∈ S, fi×idS obviouslyis a normally nonsingular inclusion of the same codimension as fi. Ifthis implies the same statement on quotient spaces, we can do without(R4).) �

Remarks 106. We give some motivation for the terminology “ap-proximation”:

(a) Let (Bq←− E

p−→ X, T , f) be an ABCD-approximation for(X,S). At least in the examples we have constructed above,the direct limit morphism p∞ : E∞ → X (in the classicaltopology) of G-spaces is an ∞-acyclic resolution.

(b) Let (Bq←− E, T , f) be an ABCD-approximation for G. If the

direct limit q∞ : E∞ → B∞ (in the classical topology) is anumerable G-pfb and if E∞ is contractible, then q∞ is theuniversal numerable G-pfb ([Dol63, 7.5]). This is the case inall our constructions. (One may use: If all Bi are compact,then B∞ is paracompact ([Dug78, Morita’s Theorem 6.5]).Any G-pfb over a paracompact base is numerable.)


5.7. Formality of Equivariant Flag Varieties

This section is the equivariant analog of Section 2.16.Let G ⊃ P ⊃ B be respectively a complex connected reductive

affine algebraic group, a parabolic subgroup and a Borel subgroup.

Theorem 107. If S is the stratification of G/P into B-orbits, thereis a t-exact equivalence of triangulated categories

DbB(G/P,S) ∼= dgPrae(Ext(ICB(S))),

and we can replace dgPrae by dgPer. Restriction to the hearts inducesan equivalence

PervB(G/P,S) ∼= dgFlag(Ext(ICB(S))).

Proof. The B-stratified variety (G/P,S) has an ABCD-approxi-mation by Corollary 42 and Propositions 103 and 104. Hence we canapply Theorem 98. �

Remark 108. Proposition 94 shows thatDbB(G/P,S) = Db

B,c(G/P ).Similarly, PervB(G/P,S) = PervB(G/P ).

By the quotient equivalence ([BL94, 2.6.2, 4.1, 5.1], using thatG→G/P is Zariski locally trivial [Jan87, II.1.10(5)]), we have equivalences

DbB,c(G/P )

∼−→ DbB×P,c(G)

and

PervB(G/P )∼−→ [−dP ] PervB×P (G).

Remark 109. We claim that the diagram

(134) DbB(G/P,S)

∼ //

For��

dgPrae(Ext(ICB(S)))

?L⊗

Ext(ICB(S))Ext(IC(S))

��Db(G/P,S)

∼ // dgPrae(Ext(IC(S)))

is commutative (up to natural isomorphism). The horizontal equiva-lences are those from Theorems 107 and 43, the vertical functors arethe forgetful functor and the induced extension of scalars functor.

The upper horizontal equivalence was established as the inverselimit of a sequence of equivalences (Formn)n∈N of categories (cf. proofof Theorem 98). The lower horizontal equivalence can be choosen equalto Form0. (If we use our method for constructing the ABCD-approxi-mation for (G/P,S) we have E0 = B ×T T × G/P = B × G/P forT ⊂ B a maximal torus, and B\E0 = G/P has the stratificationS0 = S.) Hence diagram (134) is commutative.

Since all functors in diagram (134) are triangulated and t-exact, weobtain by restriction a commutative diagram relating the hearts.

Bibliography

[BBD82] A. A. Beılinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Anal-ysis and topology on singular spaces, I (Luminy, 1981), Asterisque,vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171. MR MR751966(86g:32015)

[Beı87] A. A. Beılinson, On the derived category of perverse sheaves, K-theory,arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math.,vol. 1289, Springer, Berlin, 1987, pp. 27–41. MR MR923133 (89b:14027)

[BGS96] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszulduality patterns in representation theory, J. Amer. Math. Soc. 9 (1996),no. 2, 473–527. MR MR1322847 (96k:17010)

[BL94] Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors,Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994.MR MR1299527 (95k:55012)

[BL06] Tom Braden and Valery A. Lunts, Equivariant-constructible Koszul du-ality for dual toric varieties, Adv. Math. 201 (2006), no. 2, 408–453.MR MR2211534 (2006m:14069)

[BM01] Tom Braden and Robert MacPherson, From moment graphs to in-tersection cohomology, Math. Ann. 321 (2001), no. 3, 533–551.MR MR1871967 (2003g:14030)

[Bor91] Armand Borel, Linear algebraic groups, second ed., GraduateTexts in Mathematics, vol. 126, Springer-Verlag, New York, 1991.MR MR1102012 (92d:20001)

[CMSP03] James Carlson, Stefan Muller-Stach, and Chris Peters, Period mappingsand period domains, Cambridge Studies in Advanced Mathematics,vol. 85, Cambridge University Press, Cambridge, 2003. MR MR2012297(2005a:32014)

[Del71] Pierre Deligne, Theorie de Hodge. II, Inst. Hautes Etudes Sci. Publ.Math. (1971), no. 40, 5–57. MR MR0498551 (58 #16653a)

[Del77] P. Deligne, Cohomologie etale, Springer-Verlag, Berlin, 1977, Seminairede Geometrie Algebrique du Bois-Marie SGA 4 1

2 , Avec la collaborationde J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, LectureNotes in Mathematics, Vol. 569. MR MR0463174 (57 #3132)

[Del94] Pierre Deligne, Structures de Hodge mixtes reelles, Motives (Seattle,WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc.,Providence, RI, 1994, pp. 509–514. MR MR1265541 (95a:14011)

[DGMS75] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan,Real homotopy theory of Kahler manifolds, Invent. Math. 29 (1975),no. 3, 245–274. MR MR0382702 (52 #3584)

[DMOS82] Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih,Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math-ematics, vol. 900, Springer-Verlag, Berlin, 1982, Philosophical StudiesSeries in Philosophy, 20. MR MR654325 (84m:14046)

103

104 BIBLIOGRAPHY

[Dol63] Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. ofMath. (2) 78 (1963), 223–255. MR MR0155330 (27 #5264)

[Dug78] James Dugundji, Topology, Allyn and Bacon Inc., Boston, Mass., 1978,Reprinting of the 1966 original, Allyn and Bacon Series in AdvancedMathematics. MR MR0478089 (57 #17581)

[GM83] Mark Goresky and Robert MacPherson, Intersection homology. II, In-vent. Math. 72 (1983), no. 1, 77–129. MR MR696691 (84i:57012)

[GM88] Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergeb-nisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathe-matics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988.MR MR932724 (90d:57039)

[Gui05] Stephane Guillermou, Equivariant derived category of a complete sym-metric variety, Represent. Theory 9 (2005), 526–577 (electronic).MR MR2176937 (2006f:16018)

[Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, Cam-bridge, 2002. MR MR1867354 (2002k:55001)

[Huy05] Daniel Huybrechts, Complex geometry, Universitext, Springer-Verlag,Berlin, 2005, An introduction. MR MR2093043 (2005h:32052)

[Jan87] Jens Carsten Jantzen, Representations of algebraic groups, Pure andApplied Mathematics, vol. 131, Academic Press Inc., Boston, MA, 1987.MR 89c:20001

[Kal05] V. Yu. Kaloshin, A geometric proof of the existence of Whitney stra-tifications, Mosc. Math. J. 5 (2005), no. 1, 125–133. MR MR2153470(2006d:14069)

[Kel94] Bernhard Keller, Deriving DG categories, Ann. Sci. Ecole Norm. Sup.(4) 27 (1994), no. 1, 63–102. MR MR1258406 (95e:18010)

[Kel98] Bernhard Keller, On the construction of triangle equivalences, De-rived equivalences for group rings, Lecture Notes in Math., vol. 1685,Springer, Berlin, 1998, pp. 155–176. MR MR1649844

[KL80] David Kazhdan and George Lusztig, Schubert varieties and Poincareduality, Geometry of the Laplace operator (Proc. Sympos. Pure Math.,Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math.,XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203.MR MR573434 (84g:14054)

[KS94] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds,Grundlehren der Mathematischen Wissenschaften [Fundamental Princi-ples of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1994,With a chapter in French by Christian Houzel, Corrected reprint of the1990 original. MR MR1299726 (95g:58222)

[Lan02] Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol.211, Springer-Verlag, New York, 2002. MR MR1878556 (2003e:00003)

[LC07] Jue Le and Xiao-Wu Chen, Karoubianness of a triangulated category,J. Algebra 310 (2007), no. 1, 452–457. MR MR2307804 (2007k:18016)

[Lun95] Valery Lunts, Equivariant sheaves on toric varieties, Compositio Math.96 (1995), no. 1, 63–83. MR MR1323725 (96e:14060)

[Mum65] David Mumford, Geometric invariant theory, Ergebnisse der Mathe-matik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag,Berlin, 1965. MR MR0214602 (35 #5451)

[Nee92] Amnon Neeman, The connection between the K-theory localization the-orem of Thomason, Trobaugh and Yao and the smashing subcategoriesof Bousfield and Ravenel, Ann. Sci. Ecole Norm. Sup. (4) 25 (1992),no. 5, 547–566. MR MR1191736 (93k:18015)

BIBLIOGRAPHY 105

[Rav84] Douglas C. Ravenel, Localization with respect to certain periodic homol-ogy theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR MR737778(85k:55009)

[Sai89] Morihiko Saito, Introduction to mixed Hodge modules, Asterisque(1989), no. 179-180, 10, 145–162, Actes du Colloque de Theorie deHodge (Luminy, 1987). MR MR1042805 (91j:32041)

[Sai90] Morihiko Saito, Extension of mixed Hodge modules, Compositio Math.74 (1990), no. 2, 209–234. MR MR1047741 (91f:32046)

[Sai94] Morihiko Saito, On the theory of mixed Hodge modules [ MR1150484(93e:14014)], Selected papers on number theory, algebraic geometry,and differential geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 160,Amer. Math. Soc., Providence, RI, 1994, pp. 47–61. MR MR1308540

[Sch73] Wilfried Schmid, Variation of Hodge structure: the singularities of theperiod mapping, Invent. Math. 22 (1973), 211–319. MR MR0382272(52 #3157)

[Soe89] Wolfgang Soergel, n-cohomology of simple highest weight moduleson walls and purity, Invent. Math. 98 (1989), no. 3, 565–580.MR MR1022307 (90m:22037)

[Soe90] Wolfgang Soergel, Kategorie O, perverse Garben und Moduln uber denKoinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2,421–445. MR MR1029692 (91e:17007)

[Soe92] Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules, J.Reine Angew. Math. 429 (1992), 49–74. MR MR1173115 (94b:17011)

[Soe01] Wolfgang Soergel, Langlands’ philosophy and Koszul duality, Algebra—representation theory (Constanta, 2000), NATO Sci. Ser. II Math. Phys.Chem., vol. 28, Kluwer Acad. Publ., Dordrecht, 2001, pp. 379–414.MR MR1858045 (2002j:22019)

[Soe07] Wolfgang Soergel, Werkbank, Lecture Notes, Freiburg, Version of 11December 2007 (2007).

[Sum74] Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14(1974), 1–28. MR MR0337963 (49 #2732)

[Ver96] Jean-Louis Verdier, Des categories derivees des categories abeliennes,Asterisque (1996), no. 239, xii+253 pp. (1997), With a preface by LucIllusie, Edited and with a note by Georges Maltsiniotis. MR MR1453167(98c:18007)

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