support varieties for linear algebraic groups · support varieties for linear algebraic groups eric...

SupportVarieties for

LinearAlgebraic

groups

Eric M.Friedlander Support Varieties for Linear Algebraic groups

Eric M. Friedlander

April, 2016

HAPPY BIRTHDAY, SASHA!!


LinearAlgebraic

groups

Eric M.Friedlander

Support Varieties

Strategy: Given a finite group G , associate to each kG -moduleM a geometric object M 7→ V (G )M which reflects some of theproperties of extensions in the abelian category of G -modules.Here, k is an algebraically closed field of characteristic pdividing the order of G .

Two quite different constructions for finite groups which leadto same theory:

Cohomological varieties:V coh(G ) = Spec H•(G , k), V coh(G )M ⊂ V coh(G )is the subvariety of the annihilator of Ext∗G (M,M).

π-points spaces:Π(G ) = {[α] : α : k[t]/tp → kG π-point},Π(G )M = {[α] : α∗(M) is not free }.


LinearAlgebraic

groups

Eric M.Friedlander

Earlier “applications”

Study extensions and cohomology of finite group schemes,NOT irreducibility.

Examples of elementary abelian p-groups. Category ofkE -modules is wild. Carlson rank varieties. Vector bundles ofprojective spaces

Examples of restricted Lie algebras. Kac-Weisfeiler conjectureproved by Premet.

Arbitrary finite group schemes. Classification of thick,tensor-closed subcategories. Modules of constant Jordan type.


LinearAlgebraic

groups

Eric M.Friedlander

Linear Algebraic Groups

We consider a linear algebraic group G , a reduced, irreducibleaffine group scheme of finite type over an algebraically closedfield k of characteristic p > 0.

A rational G -module is a comodule for the coalgebra k[G ].

1-parameter subgroup of G ≡ homomorphism Ga → G .

k[G ] ≡ coordinate algebra of G .

g ≡ Lie(G ), a (restricted) Lie algebra.

kG ≡ “group algebra” (i.e., algebra of distributions at id).

G(r), the r -th Frobenius kernel ker{F r : G → G (r)}.

kG(1) is the restricted enveloping algebra of g.


LinearAlgebraic

groups

Eric M.Friedlander

Infinitesimal 1-parameter subgroups

A third construction of support varieties due to [SFB] usinginfinitesimal 1-parameter subgroups:V (G(r)) = {µ : Ga(r) → G(r)}, affine scheme.

V (G(r))M = {µ : (µ∗ ◦ εr )∗M is not free as a k[t]/tp-module}.

Consider Cr (Np(g)) ⊂ Np(g)×r , the variety of r -tuples ofp-nilpotent, pairwise commuting elements of g.

For G classical, have ψ : k[Cr (Np(g))]→ H•(G(r), k) which is ap-isogeny.


LinearAlgebraic

groups

Eric M.Friedlander

Challenge: Extend theory to linear algebraic groups

We use a mixture of the approaches of cohomology,1-parameter subgroups, and π-points. There are issues toovercome.

If G is a simple algebraic group, then its rationalcohomology is trivial.

There is no appropriate connection between V (G(r))M andV (G(r+1))M for a G(r+1)-module M.

The map εr : k[t]/tp → kGa(r) depends upon r .

The category of rational G -modules has no projectives; allinjectives are infinite dimensional.


LinearAlgebraic

groups

Eric M.Friedlander

Applications for representation theory of G

(•) Extend theory of support varieties to linear algebraic groups.

(•) Introduce interesting classes of rational G -modules.

(•) Inform computations of the rational cohomology ofunipotent algebraic groups.

(•) Construct “Geometric slices” of the category of rationalrepresentations.


LinearAlgebraic

groups

Eric M.Friedlander

Linear algebraic groups of exponential type

Definition

A structure of exponential type on a linear algebraic group G isa morphism of schemes

E : Np(g)×Ga → G , (B, s) 7→ EB(s)

satisfying

1 Each EB : Ga → G is a 1-parameter subgroup.

2 [EB(s), EB′(s ′)] = 1 if (B,B ′) = 0.

3 Eα·B(s) = EB(α · s).

4 Every 1-parameter subgroup can be written as a finiteproduct EB =

∏s=0(EBs ◦ F s).

5 Cr (Np(g))∼→ V (G(r)).


LinearAlgebraic

groups

Eric M.Friedlander

Consequences

If a structure of exponential type for G exists, then it isunique up to automorphism of Np(g).

Vr (G ) = {∏r−1

s=0(EBs ◦ F s)} ⊂ V (G ) is an exhaustivefiltration.

Vr (G )∼→ V (G(r)), EB 7→ µB ≡ EB ◦ ir .

Spec H•(G(r), k) ' Vr (G ).

[SFB] Any classical simple group or a “standard” parabolicsubgroup of such a simple group or the unipotent radical of astandard parabolic is a group of exponential type

[Sobaje] If G is reductive and if p > h(G ) (perhaps “separablygood” suffices), then G , its standard parabolic subgroups, andtheir unipotent radicals admit a structure of exponential type.


LinearAlgebraic

groups

Eric M.Friedlander

Definition of M 7→ V (G )M

Assume that G is equipped with a structure of exponential typeE : Ga ×Np(g)→ G and let M be a rational G -module.

Definition

For B = (B0, . . .) ∈ V (G ), define

αB : k[t]/tp → kG , t 7→∑s≥0

(EBs )∗(us)

(where us : Ga = k[T ]→ k , us(T n) = δps ,n). We define

V (G )M ≡ {B : α∗BM not free as a k[t]/tp-module} ⊂ V (G ).

If M finite dimensional, same information as V (G(r))M forr >> 0 (through a subtle twist).


LinearAlgebraic

groups

Eric M.Friedlander

Properties

Let G be a linear algebraic group of exponential type andM,N,Mi be rational G -modules.

V (G )M ⊂ V (G ) is G (k)-stable.

If M is finite dimensional, the V (G )M ⊂ V (G ) is closed.

V (G )M⊗N = V (G )M ∩ V (G )N .

if 0→ M1 → M2 → M3 → 0 be exact. Then for anypermutation σ of {1, 2, 3},V (G )Mσ(2) ⊂ V (G )Mσ(1) ∪ V (G )Mσ(3).

If M is rationally injective, then V (G )M = 0.

If M = k , then V (G )M = V (G ).

NOTE: this theory can be “refined” by taking into account t heJordan types of the k[t]/tp-modules α∗B(M).


LinearAlgebraic

groups

Eric M.Friedlander

Relationship to V (G(r))M

Key Observation: For B = (B0, . . . ,Br−1), the π-points

αB : k[t]/tp → kG(r), µΛr (B) ◦ εr : k[t]/tp → kG(r)

are equivalent, where Λr (B0, . . .) = (Br−1,Br−2 . . . ,B0).

Theorem

If M is a rational G -module and r > 0 such that∆M : M → M ⊗ k[G ]→ M ⊗ k[Np(g)]⊗ k[Ga] projects tok[Ga]≤pr , then

V (G )M = Λ−1r V (G(r))M).

For any finite dimensional M, there exists such an r .


LinearAlgebraic

groups

Eric M.Friedlander

Examples

G = GLn, and M a polynomial representation of G ofdegree d . If r is sufficiently large that pr > (p − 1)d , thenV (GLn)M = Λ−1

r V (GLn(r))M).

If M is rationally injective, then V (G )M = {0}.

if H ⊂ G is a normal algebraic subgroup and theprojection q : G → G/H is a map of linear algebraicgroups of exponential type, then for any rationalG -module M, V (G )q∗M = q−1(V (G/H)M).

Question

For H ⊂ G a map of linear algebraic groups of exponential typeand N a rational H-module, what can we say about V (G )M forM = indG

H (N)?


LinearAlgebraic

groups

Eric M.Friedlander

Filtrations

Study rational G -modules by considering their restrictions tosub-coalgebras of k[G ] (contrast with restriction to Frobeniuskernels which are given by subalgebras of kG .)

Definition

For any d ≥ 0, the sub-coalgebra (k[G ])[d ] ⊂ k[G ] consists offunctions f ∈ k[G ] whose restrictions along all 1-parametersubgroups EB = E(B,0,0,...) : Ga → G , B [p] = 0 have degree≤ d ; i.e., (k[G ])[d ] is the pre-image of k[G ]⊗ k[T ]≤d for

(1⊗ EB) ◦∆ : k[G ]→ k[G ]⊗ k[G ]→ k[G ]⊗ k[T ]).

For any rational G -module M, define M[d ] ⊂ M to bemaximal k[G ][d ] sub-comodule of M.


LinearAlgebraic

groups

Eric M.Friedlander

Mock Injectives

Injectivity Criterion: M is injective if and only if M[d ] is aninjective (k[G ])[d ]-comodule for all d ≥ 0.

Definition

A rational G -module M is said to be mock injective ifV (G )M = 0; this is equivalent to the condition that M isG(r)-injective for all r > 0.

Surprise! Even for G = Ga, there are mock injective which arenot injective. These are difficult to write down explicitly.

Question

Construct a theory of “Picard groups”: mock injectives whichare sub-modules of k[G ].


LinearAlgebraic

groups

Eric M.Friedlander

Possible refinement

Let B̂ = (B0, . . . ,Bn, . . .) be an infinite sequence of p-nilpotentelements Bn ∈ g which are pair-wise commuting. Then for anyrational G -module M, the sum (

∑∞s=0(EBs )∗(us))(m) is finite

for any m ∈ M, determining a p-nilpotent operator on M.

Define V̂ (G ) = lim←−sVs(G ), and define V̂ (G )M to be the

subset of V̂ (G ) of those B̂ such that the action of(∑∞

s=0(EBs )∗(us)) on M is a sum of blocks of size p.

Question

Does V̂ (G )M = 0 imply that M is injective?


LinearAlgebraic

groups

Eric M.Friedlander

New Classes of Rational G -modules

Definition

M is said to have exponential degree ≤ d if M = M[d ].

If M is finite dimensional, then M = M[d ], d >> 0.

If M has exponential degree ≤ pr − 1, then V (G )M isdetermined by V (G(r))M .

Definition

M is said to be mock trivial if E∗BM is a trivial Ga-module forall EB ∈ V (G ). In other words, if M = M[0].

Observation If G 6= Up(G ), then k[G ][0] ⊂ k[G ] is a sub-Hopfalgebra which is a non-trivial indecomposable mock trivialmodule.


LinearAlgebraic

groups

Eric M.Friedlander

The functor (−)[d ]

The functor

(−)[d ] : (G −Mod)→ (k[G ][d ])− coMod)

is left exact, left adjoint to the exact, fully faithful the inclusionfunctor

ι[d ] : (k[G ][d ])− coMod)→ (G −Mod).

Proposition

The isomorphism of functors

H0(G ,−) ' Hom(k[G ][d ]−comod)(k ,−) ◦ (−)[d ]

leads to Grothendieck spectral sequences converging toH∗(G ,M).


LinearAlgebraic

groups

Eric M.Friedlander

Projective spectrum for rational cohomology

Definition

We define Proj V coh(G ) as the image in Proj H•(G , k) oflim−→r

Proj H•(G(r), k).

Theorem

For G a linear algebraic group of exponential type,

ΘG : Proj′ V (G )→ Proj V coh(G ), EB 7→ ker{α∗B}

is well defined, surjective, and factors through the co-invariantsof the action of G on Proj′ V (G ).


LinearAlgebraic

groups

Eric M.Friedlander

Cohomological Support Varieties

Definition

We define Proj V coh(G )M ⊂ Proj V coh(G ) to consist of thoseprime ideals p ⊂ H•(G , k) which are of the form p = ρ−1

r (qr )for homogeneous prime ideal qr ⊂ H•(G(r), k) which containsthe annihilator of Ext∗G(r)

(M,M).

Proposition

If G is a linear algebraic group of exponential type, then

ΘG : Proj′ V (G ) → Proj V coh(G )

restricts to the surjective map

ΘG ,M : Proj′ V (G )M → Proj V coh(G )M .


LinearAlgebraic

groups

Eric M.Friedlander

Example of G = Ga

Proj′ V (Ga) = A∞.

Observation: The restriction map k[Ga]→ k[Ga(r)] is split as amap of coalgebras by k[Ga]<pr ⊂ k[Ga].

Fact: A rational Ga-module is a k-vector space V equippedwith the structure of a k[u0, . . . , un, . . .]-module such that∀v ∈ V , ∃rv with us acting trivially on v for s ≥ rv .

In particular, (Ga(r) −Mod) ' (kZ/pr −Mod) embedsnaturally in (Ga −Mod) (adjoint pairs, etc). This enablesrealization of many subspaces of V (Ga) as support varieties.

Proposition

For M a finite dimensional Ga-module,

ΘGa : Proj′ V (Ga)M∼→ Proj V coh(Ga)M .


LinearAlgebraic

groups

Eric M.Friedlander

Coinvariants

Question

For which linear algebraic groups G of exponential type is it thecase that ΘG : Proj V (G )→ Projcoh V (G ) induces

? (Proj′ V (G ))/G∼→ Projcoh V (G ) ?

Remark

ΘUN: Proj′ V (UN)→ Projcoh V (UN), EB 7→ ker{α∗B}

appears to induce an isomorphism

(Proj′ V (UN))/UN∼→ Projcoh V (UN).


LinearAlgebraic

groups

Eric M.Friedlander

H•(U3(r), k)

1st IngredientIn the Lyndon-Hochschild-Serre spectral sequence for thecentral extension 1→ Z ' Ga → U3 to U3 ' G2

a → 1, the

Steenrod operation β ◦ Ppj applied to (x(i)1,3)p

jequals

−(x(i)1,2)p

j+1 · x (i+1+j)2,3 + (x

(i)2,3)p

j+1 · x (i+1+j)1,2 .

2nd Ingredient [SFB] provide a purely inseparable isogenyψ : H•(U3(r), k)→ k[Vr (U3)] for any r > 0.

Conclusion Explicit computation of H•(U3(r), k)red compatible

with r 7→ r + 1. Moreover, (x(i)1,3)p

jrepresents a permanent

cycle if and only if i + j + 1 ≥ r .


LinearAlgebraic

groups

Eric M.Friedlander

H•(U3, k)

Theorem

The commutative algebra H•(U3, k)red is generated by classes

{x (i)1,2, x

(i ′)2,3 , i , i ′ ≥ 0} modulo the relations

−(x(i)1,2)p

j+1 · x (i+1+j)2,3 + (x

(i)2,3)p

j+1 · x (i+1+j)1,2 , i , j ≥ 0.

Thus, there is a natural closed embeddingProj V coh(U3) → Proj V coh(G×2

a ) = A∞ × A∞.

Corollary

ΘU3 : Proj′ V (U3) → Proj V coh(U3) is the coinvariant map.


LinearAlgebraic

groups

Eric M.Friedlander

Cohomological supports for rational U3-modules

Example

Let N be a finite dimensional rational G×2a module.

Denote by M the “inflation” of N via U3 � U3 ' G×2a .

Then V (U3)M equals the pre-image of V (G×2a )N under the

projection V (U3)→ V (G×2a ).

Moreover, Proj V coh(U3)M equals equals the intersection of

(Proj′ V (G×2a )N ∩ Proj V coh(U3)) ⊂ Proj′ V (G×2

a ).


LinearAlgebraic

groups

Eric M.Friedlander

Extension to UN

Employ the TN -equivariant Hochschild-Serre spectral sequence

E ∗,∗2 = H∗(UN/Γv−1, k)⊗ H∗(Γv−1/Γv , k) ⇒ H∗(UN/Γv , k)

for terms of the descending central series for UN .

Compute differentials such as

βPpj (∑v−1

t=1 (x(i)s,s+t)

pj ⊗ y(i+1+j)s+t,s+v − (x

(i)s+t,s+v )p

j ⊗ y(i+1+j)s,s+t )

=v−1∑t=1

(x(i)s,s+t)

pj+1 ⊗ x(i+1+j)s+t,s+v − (x

(i)s+t,s+v )p

j+1 ⊗ x(i+1+j)s,s+t

CONCLUSION: Computation for V coh(UN) extending caseN = 3.

support varieties for linear algebraic groups · support varieties for linear algebraic groups eric...

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