coherent analytic sheaves over formal neighborhoods · shilin yu coherent analytic sheaves over...

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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA

Cohesive Modules

Coherent Analytic Sheaves over Formal Neighborhoods

Shilin Yu

Department of Mathematics, Penn State University

Algebraic Geometry Seminar

U. Wisconsin, Madison

Mar 9, 2012

Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods

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Cohesive Modules

Outline

1 Introduction

2 Dolbeault DGA of Formal Neighborhood

Review of Formal Neighborhood

Dolbeault DGA of Formal Neighborhood

3 Geometric Description of Dolbeault DGA

Goal

Kapranov’s Result

The Case of General Embeddings

4 Cohesive Modules

Review of DG-categories

Cohesive Modules over a DGA

Application to Complex Analytical Geometry


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Motivations

The infinitesimal geometry of closed embeddings i : X ↪→ Y of complex

manifolds is encoded by the formal neighborhood X(∞)

Y .

Eventual Goals: understand

• The Yoneda algebra

Ext•OY(i∗OX , i∗OX) ' Ext•O

X(∞)Y

(i∗OX , i∗OX)

and its relation with Lie theory. (Kontsevich, Caldararu,

Calaque-Caldararu-Tu, etc.)

• The derived self-intersection X×RY X. (Arinkin-Caldararu)

• Infinitesimal deformations of X inside Y. (Deligne, Drinfeld, Feigin,

Kontsevich-Soibelman, Manetti, etc.)


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Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood

Formal Neighborhood

Let Y be a complex manifold and let X be a closed submanifold. Denote by

i : X ↪→ Y the closed embedding.

The rth formal neighborhood X(r)Y of X in Y is X equipped with the structure

sheaf

OX(r)

Y:= OY/I

r+1 ,

where I is the ideal sheaf of holomorphic functions vanishing on X.

Similarly, the (complete) formal neighborhood X(∞)

Y is defined as X equipped

with the structure sheaf

OX(∞)

Y:= lim←−

rOY/I

r+1 .


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Examples

• Let Y = Cn ×Cm = {(z, w)}, X = 0×Cm = {(0, w)}. Then

OX(∞)

Y' OXJz1 , · · · , znK.

• Suppose E→ X is a holomorphic vector bundle and let Y be its total

space. X is embedded in Y as the zero section. Then

OX(∞)

Y' S(E∨) = ∏

i≥0SiE∨ ,

which is the complete symmetric algebra generated by E∨

• Consider the diagonal embedding ∆ : X ↪→ X× X. Denote by

πi : X× X → X, i = 1, 2, the projections. Then OX(∞)

X×X, or rather

π1∗OX(∞)X×X

, is the holomorphic jet bundle J∞X of X.


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Dolbeault DGA

Take k = C.

• A differential graded algebra (= dga) (A• , d) is a Z≥0-graded associativealgebra A• over k with a derivation d : A• → A•+1, s.t.

• Leibniz’s rule: d(a · b) = d(a) · b + (−1)|a|a · d(b), a, b ∈ A•;• Integrability: d2 = 0.

• The Dolbeault dga (A•(Z), ∂) of a complex manifold Z:

• A•(Z) is the graded algebra of all C∞ (0, q)-forms on Z with the wedge

product.• ∂ is the (0, 1)-part of the de Rham differential.

• It is well known that the sheafy version A •Z of the Dolbeault dga gives a

soft, flat (Malgrange) resolution of OZ.


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• Question: Is there a notion of Dolbeault dga for formal neighborhoods?

• First attempt: Take the Dolbeault resolution of OX(r)

Yas an OY-module:

A •X(r)

Y= A •Y ⊗OY O

X(r)Y

= A •Y /(I r+1 ·A •Y )

and

A •X(∞)

Y= lim←−

rA •

X(r)Y

• In the case of Y = Cm+n = {(z, w)} and X = Cm = {(0, w)} ⊂ Y, such

defined A•X(∞)

Y

looks like

A•X(∞)

Y= A(X)Jzi , ziK⊗C ∧•(Cdz j ⊕Cdz j)

Too big for our applications...


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We want to build an ’economical’ Dolbeault dga A•(X(∞)

Y ) or its sheafy

version A •X(∞)

Y

for an arbitrary embedding X ↪→ Y, s.t.

• Locally it should look like A•(X ∩U)Jz1 , · · · , znK, no zi ’s or dzi ’s.

• If Y is the total space of a holomorphic vector bundle E→ X and X ↪→ Y

the zero section, then we should have

A•(X(∞)

Y ) ' A0,•X (S(E∨)),

where the RHS is the Dolbeault resolution of the (infinite-dimensional)

vector bundle S(E∨).

• In the case of diagonal embedding, A•(X(∞)

X×X) should be naturally

identified with A0,•X (J∞

X ), the Dolbeault resolution of the jet bundle J∞X .


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Review of Jet Bundles

• At each point p ∈ X, the fiber of the rth order jet bundle J rX is the space

of r-jets [ f ]rp of holomorphic functions defined about p:

J rp := OX,p/m

r+1p ,

with mp being the maximal ideal of OX,p.

• Holomorphic (resp. C∞-)sections of J rX are locally of the form (for

simplicity, assume dim X = 1):

s(p) = [a0(p) + a1(p)(z− p) + · · ·+ ar(p)(z− p)r]p ∈ J rp ,

for any p ∈ X, where ai are holomorphic (resp. C∞-)functions over U.


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Review of Jet Bundles (cont’d)

• Given a local C∞-section

s(p) = [a0(p) + a1(p)(z− p) + · · ·+ ar(p)(z− p)r]p ∈ J rp ,

the ∂-connection on J rX can be written locally as:

∂s = ∂a0 ⊗ [1] + ∂a1 ⊗ [z− p] + · · ·+ ∂ar ⊗ [(z− p)r] ∈ A0,1X (J r

X).

• C∞-sections of J rX can be thought of as (equiv. classes of) functions

defined near the diagonal X∆ ⊂ X× X, that are smooth in the first factor

of X× X and holomorphic in the second.

• However, such a perspective uses the special features of X× X and is

not available in the general case of an embedding i : X ↪→ Y.


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• To overcome this, we realize A0X(J

rX) as a quotient of the algebra

A0(X× X) by some equiv. relation.

Define a0r to be the ideal of A0(X× X) consisting of all functions f on

X× X s.t.

V1V2 · · ·Vk f |X∆= 0, 0 ≤ k ≤ r,

where Vj are smooth (1, 0)-vector fields in the direction to the second on

X× X. (The condition when k = 0 meas f |X∆= 0.)

Then there is an isomorphism σ : A0(X× X)/a0r'−→ AX(J r

X) defined by

[ f ] 7→ s, s(p) = [ f |{p}×X ]rp , ∀ p ∈ X.

• Observation: the Vj’s in the condition above can be chosen as being of

any direction.


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Dolbeault DGA of X(r)Y

For a closed embedding i : X ↪→ Y, define aqr (q ≥ 0), to be the subset of

A0,q(Y) consisting of (0,q)-formsω satisfying:

i∗(LV1LV2 · · · LVkω) = 0, 0 ≤ k ≤ r,

where Vj are any smooth (1, 0)-vector fields on Y and LVj denote the Lie

derivatives.

Lemma

a•r is a dg-ideal of (A0,•(Y), ∂).

Definition (Yu)

The dga A•(X(r)Y ) := A0,•(Y)/a•r with the inherited differential ∂ is called the

Dolbeault dga of the rth formal neighborhood X(r)Y .


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Dolbeault DGA of X(r)Y

We have inclusions of ideals a•r+1 ⊂ a•r and hence a projective system of

dgas A•(X(r+1)

Y )→ A•(X(r)Y ).

Definition (Yu)

The Dolbeault dga of the complete formal neighborhood X(∞)

Y is defined to be

A•(X(∞)

Y ) := lim←−rA•(X(r)

Y ).

Remark: In fact, we have a natural isomorphism

A0,•(Y)/ ⋂

r≥0a•r'−→ A•(X(∞)

Y ),

based on the fact that any formal power series is the Taylor series of some

C∞-function (E. Borel).


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Making A•(X(∞)

Y ) into a Sheaf

To make A•(X(∞)

Y ) into a sheaf of dgas A •X(∞)

Y

(over either Y or X), we define

over each open subset U ⊆ Y,

A •X(∞)

Y(U) := A•((U ∩Y)(∞)

U ).

In local charts AX(∞)

Ylooks like

A •X(∞)

Y(U) ' A•(U ∩Y)Jz1 , · · · , znK

We then obtain an exact sequence of sheaves

0→ OX(∞)

Y→ A 0

X(∞)Y

∂−→ A 1X(∞)

Y

∂−→ · · · ∂−→ A mX(∞)

Y→ 0,

where m = dim X. In other words, A •X(∞)

Y

is a soft resolution of OX(∞)

Y.


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Also we have the following:

Proposition

Let ∆ : X ↪→ X× X be the diagonal embedding. There are natural

isomorphisms of dgas

(A•(X(r)X×X), ∂)

'−→ (A0,•X (J r

X), ∂)

and

(A•(X(∞)

X×X), ∂)'−→ (A0,•

X (J∞X ), ∂).


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GoalKapranov’s ResultThe Case of General Embeddings

Geometric Description of Dolbeault DGA

• Question: How to describe A•(X(∞)

Y ) in terms of the differential

geometry of the embedding?

• Answer: We want to construct an isomorphism of dgas

(A•(X(∞)

Y ), ∂) ' (A0,•X (S(N∨)),D)

where N∨ = N∨X/Y is the conormal bundle of X, and write down the

differential D = ∂+ ? using connections, curvatures, etc.

• The answer in the case of diagonal embedding is due to M. Kapranov.


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Kapranov’s Result

Assume X is Kähler. Then the (1, 0)-part ∇ of the Levi-Civita connection on

TX is

• torsion-free: ∇XY−∇YX = [X, Y], ∀ X, Y ∈ Γ(TX).

• flat: [∇,∇] = 0

The curvature of the full Levi-Civita connection ∇ = ∇+ ∂ is

R = [∂,∇] ∈ A1,1(End(TX)) = A0,1(Hom(TX⊗ TX, TX)),

such that

• In fact, R ∈ A0,1(Hom(S2TX, TX) by torsion-freeness of ∇.

• ∂R = 0 and the corresponding class [R] ∈ Ext1OX

(S2TX, TX) is the

Atiyah class of TX.


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Kapranov’s Result

Define

R2 := R, Rn := ∇i−2R ∈ A0,1(Hom(TX⊗n , TX)), ∀ n > 2.

In fact, by flatness of ∇ we have

Rn ∈ A0,1(Hom(SnTX, TX)), ∀ n ≥ 2.

Now take the transpose of Rn:

R∗n ∈ A0,1(Hom(T∗X, SnT∗X))

and extend R∗n to derivations R∗n of degree +1 on A0,•(S(T∗X)) and set

D = ∂ + ∑n≥2

R∗n .


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Kapranov’s Result

Theorem (Kapranov)

Let ∆ : X ↪→ X× X. There is an isomorphism of dgas

(A•(X(∞)

X×X), ∂) ' (A0,•X (S(T∗X)),D)

where D = ∂ + ∑n≥2 R∗n. In particular, D2 = 0.

Originally in his paper, Kapranov constructed a ‘formal exponential map’

exp : X(∞)

TX → X(∞)

X×X

using the Kähler structure. The isomorphism above can be regarded as the

pullback of functions along exp∗.


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Sketch of Proof

We reformulate the proof by constructing the pullback map exp∗ directly.

Consider ∇ as the connection on T∗X:

∇ : A(T∗X)→ A(T∗X⊗ T∗X)

and lift it to a constant family of (1, 0)-connections on X× X in the direction

of the second component.

By the torsion-freeness and flatness of ∇, set

exp∗([ω]) = (∆∗ω, ∆∗∇ω, ∆∗∇2ω, · · · , ∆∗∇nω, · · · ) ∈ A0,•(S(T∗X))

for any [ω] ∈ A•(X(∞)

X×X), where ∇ω = ∂ω and ∇nω = ∇n−1∂ω.

Then check that

exp∗([∂ω])− ∂ exp∗([ω]) = ∑n≥2

R∗n(exp∗([ω]))

Q.E.D.


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The Case of General Embeddings

Theorem (Yu)

Let Y be a Kähler manifold and let X be a submanifold. Then there exists an

isomorphism of dgas

(A•(X(∞)

Y ), ∂) ' (A0,•X (S(N∨)),D)

in which D can be explicitly expressed in geometric terms.


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Some Differential Geometry...

• Choose a C∞-splitting τ : i∗TY → TX and ρ : TX → i∗TY of the ex. seq.

0→ TX → i∗TY → N → 0

and define

β := ∂τ ∈ A0,1X (Hom(N, TX)) = A0,1

X (Hom(T∗X, N∨)).

• The shape operator A : TX⊗ N → TX or T∗X → T∗X⊗ N∨ is defined

by

Aµ(V) = −τ(∇Vρ(µ)), ∀ µ ∈ C∞(N), V ∈ C∞(TX).

• Decompose the curvature R ∈ A0,1Y (Hom(T∗Y, S2T∗Y)) of Y over X to

get

R⊥ ∈ A0,1X (Hom(N∨ , S2 N∨)) and R> ∈ A0,1

X (Hom(T∗X, S2 N∨)).


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Low-Order Terms of D

• D acts on f ∈ A0,0X (S0 N∨) = A0,0(X) by

D f = ∂ f +β ◦ ∂ f + (β · A + R>) ◦ ∂ f + · · · .

• D acts on µ ∈ A0,0X (S1 N∨) = A0,0

X (N∨) by

Dµ = ∂µ + (β · ∇Nµ + R⊥ ◦µ) + · · · ,

where ∇N is the induced connection on N∨.

• If you try to show D2 = 0 for these terms by hand, e.g.,

∂((β · ∇N + R⊥) ◦µ) + (β · ∇N + R⊥) ◦ ∂µ = 0,

then you will see the Ricci equation!

• If you try harder, you will get all the Gauss-Codazzi-Ricci equations.


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Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry

DG-Categories

Definition

• A differential graded (dg-)category C is a category in which the set of

morphisms C(X, Y) between two objects has the structure of a complex,

in such a way that the composition maps

C(Y, Z)⊗k C(X, Y)→ C(X, Z), X, Y, Z ∈ Obj(C)

are chain maps of complexes

• A dg functor F : C1 → C2 is given by chain maps of complexes

F : C1(X, Y)→ C2(F(X), F(Y)), X, Y ∈ Obj(C)• The homotopy category Ho C of the dg category C has

• objects: same as Obj C• morphisms: Ho C(X, Y) := H0(C(X, Y))


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Example: DG Category of Complexes

Let R be a k-algebra. Define Cdg(R) to be the dg-category of complexes of

(left) R-modules, with

• Objects: complexes (M• , dM) of R-modules

• Morphisms: Homk(M• , N•) := ∏i∈Z HomR(Mi , Ni+k)

· · · dM> Mi dM

> Mi+1 dM> · · ·

· · ·dN> Ni+n

dN>

f>

Ni+n+1dN>

f>

· · ·

with the differential

dHom( f ) = dN ◦ f − (−1)n f ◦ dM

Then Ho Cdg(R) ' K(R) = the category of complexes of R-modules with

homotopy classes of chain maps as morphisms.


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Definition (J. Block)

Let A = (A• , d) be a dga. A cohesive module E = (E• ,E) over A is:

1 a bounded, finitely generated projective Z-graded (left) module E• over

A = A0, with

2 a superconnection E : A• ⊗A E• → A• ⊗A E• of total degree 1,satisfying

• Leibniz rule: E(ω · e) = dω · e + (−1)|ω|ω ·E(1⊗ e)• Integrability: E ◦E = 0.

By the Leibniz rule, E is determined by its values on E• ⊆ A• ⊗ E•. Thus

E = E0 + E1 + E2 + · · · ,

where the kth component is

Ek : E• → Ak ⊗A E•−k+1 .


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All Ek are A-linear except for E1, which satisfies the Leibniz rule.

Under this decomposition, the integrability condition E ◦ E = 0 can be written

as:

E0 ◦ E0 = 0 (E• ,E0) is a chain complex

E0E1 + E1E0 = 0 E0 (anti-)commutes with E1

E0E2 + E1E1 + E2E0 = 0 E1 is flat up to a homotopy E2

· · · ‘higher homotopies’


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Cohesive Modules over the Dolbeault dga

Let A = (A0,•(X), ∂) be the Dolbeault dga of a (compact) complex manifold.

Given a cohesive module (E• ,E) over (A0,•(X), ∂), then:

• Each Ek can be regarded as a C∞-vector bundle over X.

• (E• ,E0) is a complex of C∞-vector bundles with (non-flat)

(0, 1)-connections.

• If Ek = 0, ∀ k ≥ 2, which forces E1 ◦ E1 = 0, then we have a complex of

holomorphic vector bundles.

• In general, (E• ,E) is sort of ‘twisted’ complex of holomorphic vector

bundles.


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Perfect Category of Cohesive Modules

Definition (J. Block)

The perfect category of cohesive modules PA over a dga A = (A• , d) is the

dg-category that consists of

• Objects: cohesive modules over (A• , d)

• Morphisms:

P kA(E1 , E2) =

φ : A• ⊗A E•1 → A• ⊗A E•2

∣∣∣∣∣∣∣∣deg(φ) = k,

φ(a · e) = a ·φ(e),∀ a ∈ A• .

equipped with a differential d of degree 1 s.t.

d(φ)(e) = E2(φ(e))− (−1)|φ|φ(E1(e))


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Advantages

Why cohesive modules?

• On a smooth projective variety, every coherent sheaf can be resolved by

vector bundles. This is no longer true in complex analytical geometry...

But we have

Theorem (J. Block)

Let A = (A0,•(X), ∂) be the Dolbeault dga of a compact complex manifold X.

Then HoPA is equivalent to Dbcoh(X), the bounded derived category of

OX-modules with coherent cohomology.

• It also works in noncommutative geometry, where local constructions are

unavailable. (e.g., noncommutative T-duality, J. Block & C. Daenzer)

• ∞-version of Riemann-Hilbert correspondence (J. Block & A. Smith)


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Sketch of Proof

Part 1: Making a sheaf: For E = (E• ,E) ∈ PA, define the sheaves E p,q by

E p,q(U) = A0,q(U)⊗A(X) Ep

Define a fully faithful functor

α : HoPA −→ Dbcoh(X)

(E• ,E) 7−→ (E • ,E) = (∑p+q=• E p,q ,E)

Then it can be shown that (E • ,E) has coherent cohomology and

ExtkOX

(E •1 , E •2 ) ' Hk(PA(E1 , E2)).


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Sketch of Proof (cont’d)

Part 2: Given (E • , d) ∈ Dbcoh(X), we need to find a cohesive module

E = (E• ,E) s.t. there is quasi-isomorphism α(E) '−→ (E• , d).

Try: set E •∞ = E • ⊗OX AX, then we have a quasi-isomorphism

(E • , d) '−→ (A •X ⊗OX E • , ∂⊗ 1 + 1⊗ d).

Want to set (E• ,E) = (Γ(X, E •∞), ∂⊗ 1 + 1⊗ d).

But... it is only a quasi-cohesive module over A (i.e., without finiteness and

projectivity conditions).


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Sketch of Proof (cont’d)

However, there is a (strictly) perfect complex (E• ,E0) of A(X)-modules and a

quasi-isomorphism

e0 : (E• ,E0)→ (Γ(X, E •∞), 1⊗ d)

since (E •∞ , 1⊗ d) is a perfect complex of AX-modules (Illusie).

Block showed that E0 can be extended to a Z-connection E and e0 can be

extended at the same time to a closed morphism

e : (E• ,E)→ (Γ(X, E •∞), d⊗ 1 + 1⊗ ∂)

which induces a quasi-isomorphism on the level of sheaves. Thus the

cohesive module (E• ,E) serves our purpose. Q.E.D.


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Dolbeault Resolutions over Formal Neighborhood

It is not clear if AX(∞)

Yis flat over O

X(∞)Y

. However, we can still show the

following:

Theorem (Yu)

The functor F 7→ AX(∞)

Y⊗O

X(∞)Y

F from Coh(X(∞)

Y ) to the category of

AX(∞)

Y-modules is exact.

Theorem (Yu)

Suppose F ∈ Coh(X(∞)

Y ), then we have an exact sequence of

OX(∞)

Y-modules:

0→ F → A 0X(∞)

Y⊗O

X(∞)Y

F∂−→ A 1

X(∞)Y⊗O

X(∞)Y

F∂−→ · · · ∂−→ A m

X(∞)Y⊗O

X(∞)Y

F → 0,

where m = dim X.


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Cohesive Modules over Formal Neighborhood

Theorem (Yu)

A holomorphic vector bundle E over X(∞)

Y (i.e., a locally free OX(∞)

Y-module of

finite type) is equivalent to a locally free AX(∞)

Y-module of finite type E∞

equipped with a ∂-connection ∂E : E∞ → A 1X(∞)

Y

⊗AX(∞)

Y

E∞ s.t. ∂2E = 0.

Combining these results and follow the same argument of Block, we get

Theorem (hopefully...)

Suppose X is a closed compact submanifold of a complex manifold Y. Let

A = (A•(X(∞)

Y ), ∂) be the Dolbeault dga of the formal neighborhood X(∞)

Y and

PA be the perfect category of cohesive modules over A. Then

HoPA ' Dbcoh(X(∞)

Y ).


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The End

Thank You!


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