Commutative algebra andrepresentations of finite groups

Srikanth Iyengar

University of Nebraska, Lincoln

Genoa, 7th June 2012

The goal

Describe a bridge between two, seemingly different, worlds:

Representation theory of finite groups in char p > 0

and

Modules over polynomial rings

This has had, and continues to have, an impact on both fields:Representation theory and commutative algebra.

Another connection: Algebraic topology (via classifying spaces)

I will not touch upon this today.

Outline

Part I - Groups to polynomial rings

Modular representations of finite groups

Elementary abelian groups in char 2 - Koszul duality

Elementary abelian groups in char p ≥ 3

Part II - Applications

Testing projectivity

Classifying modular representations

Based primarily on the following articles:

Homology of perfect complexesL. L. Avramov, R-O. Buchweitz, -, C. Miller, Adv. Math., 2010

Stratifying modular representations of finite groupsD. Benson, - , H. Krause, Ann. of Math., 2011

Representations of finite groups

Throughout, G will be a finite group and k will be a field

A representation of G is a k-vectorspace V with a G -action: amap G × V 7→ V such that for g , h ∈ G , u, v ∈ V , and c ∈ K

g(u + v) = gu + gv and g(cv) = cg(v)

g(hv) = (gh)v and 1(v) = v

Equivalently, there is a homomorphism G → Glk(V ) of groups.

The sum of representations V ,W is the vectorspace V ⊕W with

g(v + w) = gv + gw for g ∈ G , v ∈ V and w ∈W .

A representation is indecomposable if it cannot be expressed as anon-trivial sum.

Krull-Remak-Schmidt theorem

I will focus on finite dimensional representations: rankk V <∞.

Evidently, given a representation V of G , one can decompose it as

V =⊕

i=1,...,s

W eii with ei ≥ 1

where the Wi are indecomposable and Wi 6∼= Wj for i 6= j .

This decomposition of V is essentially unique:

Theorem (Krull-Remak-Schmidt; Schur in char 0)

The Wi that appear and their multiplicities, ei , depend only on V .

Up shot: One can focus on indecomposable representations.

Classical theory

Consider the k-vectorspace⊕

h∈G kh with G action defined by

g(∑h∈G

chh) =∑h∈G

chgh

This is the regular representation of G .

Theorem (Maschke)

If char k = 0, then any indecomposable representation of G is(isomorphic to) a direct summand of the regular representation.

This result holds also when char k does not divide the order of G .

Corollary

G has only finitely many indecomposable representations, in char 0.

Character theory gives an efficient way to describe them all.In char 0, representation theory thus has a combinatorial flavor.

Modular case

All this fails when char k divides |G |; this is the modular case.

The trivial representation of G is k with g(c) = c for all g ∈ G .

Example

G := Z/2 = 〈g | g 2 = 1〉 and char k = 2. Then the trivialrepresentation is not a direct summand of the regular one.

Reason: The G -action on the regular representation, k ⊕ kg , isgiven by

g 7→[

0 11 0

]If k were a direct summand, this matrix would be diagonalizable,which it is not.

In the example above, the only indecomposable representations arek and the regular one.

This is not the case with the next one.

Kronecker group

G = 〈g1, g2 | g 21 = 1 = g 2

2 , g1g2 = g2g1〉; thus G ∼= Z/2× Z/2.

Let k algebraically closed of char 2.

Kronecker classified all the indecomposable representations of G :

The trivial representation.

For each odd integer n ≥ 3, there are two indecomposablerepresentation of dimension n.

For each even integer n ≥ 2, there is a family ofindecomposable representations parameterized by k.

The two dimensional representations are given by:

g1 7→[

1 01 1

]and g2 7→

[1 0λ 1

]λ ∈ k .

Note: For G = (Z/2)3, the indecomposables cannot be classified.

Group algebras

In char p > 0, it not possible to classify indecomposablerepresentations; nor is it a meaningful endeavor, in general.

In this context, representation theory has a different flavor (moduletheoretic aspects dominate), and requires new tools: cohomology.

The group algebra of G , over k , is the k-vectorspace

kG :=⊕g∈G

kg

with multiplication induced from G .Observe: This multiplication is commutative ⇔ G is abelian.

Example

For G = 〈g | gd = 1〉, the cyclic group of order d , one has

kG = ⊕d−1i=0 kg i ∼= k[x ]/(xd − 1)

Modules over kG

Let V be a representation of G . The G action on V extends to kG :

(∑g∈G

cgg)v =∑g∈G

cggv

In this way, V becomes a module over kG .

Representation of G are the same as modules over kG .

Regular representation becomes kG viewed as a module over itself.

Definition

A kG -module P is projective if it is a direct summand of a freekG -module, i.e. some (kG )n.

Maschke’s theorem can be phrased as: When char k does notdivide |G |, the order of G , every kG -module is projective.

Projective modules

In general, Krull-Remak-Schmidt implies:

1. An indecomposable is projective ⇐⇒ it is a summand of kG .

2. There are only finitely many indecomposable projectives.

Lemma (char k = p > 0)

If |G | = pn, then kG is the only indecomposable projective.

In summary: The projectives are the “well-understood” part ofmodular representation theory.

There are more indecomposables than projectives; recall Kronecker!

There is a useful construction that allows one to focus on the rest.This is the stable module category, and it is the main actor inmodular representation theory.

Stable module category

Let G be a finite group and k a field; henceforth char k divides |G |.

Give kG -modules M and N, we write:

HomkG (M,N) for the kG -linear maps f : M → N, and

PHomkG (M,N) for the f that factor through a projective module:

P

��@@

Mf //

>>}}

N

with P a projective module. This is a k-subspace of HomkG (M,N).

Note: PHomkG (M,N) = HomkG (M,N) if M or N is projective.

The stable module category of kG , denoted stmodkG has forObjects: all kG -modules (finite dimensional over k)Morphisms: Given kG -modules M,N, set

HomkG (M,N) = HomkG (M,N)/PHomkG (M,N)

These are the stable maps from M to N.

Lemma

A kG -module M is 0 in stmodkG ⇔ M is projective.

In any category an object X is 0⇔ idX , the identity map on X is 0.

Proof of Lemma.

idM = 0 in stmodkG ⇔ as a map of kG -modules it factors as

P

AA

MidM //

>>}}

M

with P projective, ⇔ M is a direct summand of a projective.

Passage to elementary abelian groups

Definition

Groups of the form (Z/p)c are called elementary abelian p-groups;the integer c is its rank.

Let H ≤ G be a subgroup and M be a kG -module.

M ↓H := the kH-module obtained by restricting the G action to H.

Quillen (1971), Alperin and Evens (1981): The properties of akG -module M are controlled by those of M ↓E , where E spans overthe elementary abelian p-subgroups of G ; here p = char k .

There are precise results; here is a easily stated special case.

Theorem (Chouinard, 1972)

A kG -module M is projective if and only if M ↓E is projective forevery elementary abelian p-subgroup E ≤ G .

Group algebra of an elementary abelian

Set k[x1, · · · , xc ] := the ring of polynomials in variables x1, . . . , xc .

Example

E := Z/2 and char k = 2. Then kE = k ⊕ kg with g 2 = 1, so

kE ∼= k[x ]/(x2 − 1) ∼= k[z ]/(z2) where z = x − 1

Fact: If char k = p > 0, and R is a commutative k-algebra, then

(a + b)p = ap + bp for all a, b ∈ R

Example

For E := (Z/p)c and char k = p > 0, one has

kE ∼= k[x1, . . . , xc ]/(xp1 −1, . . . , xp

c −1) ∼= k[z1, . . . , zc ]/(zp1 , . . . , z

pc )

where zi = xi − 1.

Elementary abelian groups in char 2

E = (Z/2)c and k a field of characteristic 2; thus

kE ∼= k[z1, . . . , zc ]/(z21 , . . . , z

2c )

This is a Koszul algebra. Note that |zi | = 0 for i = 1, . . . , c .

The Koszul dual of kE is the algebra

Ext∗kE (k , k) = k[x1, . . . , xc ]

with |xi | = 1 for i = 1, . . . , c .

The collection of kE -modules and of modules over this polynomialring are different ways of looking at the “same thing”.

To make this precise, one has to bring in one further ingredient.

Observe that Proj(k[x1, . . . , xc ]) = Pc−1.

A differential sheaf on Pc−1 is a coherent sheaf F with a OPc−1

linear map d : F → F(1) such that d2 = 0.

Diff Pc−1 is an abelian category, with obvious notion of morphisms.

Theorem (J. Moore (?), Priddy 1970, Carlsson 1983)

There is an equivalence of categories

stmodkE≡−−→ Db(Diff Pc−1)

Under this equivalence, k maps to OPc−1 .

The equivalence can be made explicit.

stmodkE involves all kE -modules; not only the graded-ones.

This is why we get Db(Diff Pc−1) and not Db(CohPc−1).

Elementary abelian groups: general case

E = (Z/p)c and k a field of characteristic p, so

kE ∼= k[z1, . . . , zc ]/(zp1 , . . . , z

pc )

This is not Koszul algebra if p ≥ 3.

Set X = Proj k[y1, . . . , yc ], with |yi | = 2 for i = 1, . . . , c .

X ∼= Pc−1 as spaces, but the grading is important.

Theorem (Avramov, Buchweitz, - ,Miller, 2010)

There is a functor F : stmodkE −→ Db(Diff X ) such that

F (k) 'c⊕

i=0

OX (−i)(ci) .

The functor F is not an equivalence, but comes close.

For example, for any kE -module M, the following are equivalent:

M = 0 in stmodkE ; i.e. M is projective.

F (M) = 0 in Db(Diff X ); that is, H(F (M)) = 0.

This says that the functor F is faithful.

Basic idea: Study modules over kE by passing to X , using F .

Premise: Sheaves over X are easier to understand.

Remark: One is really dealing with modules over k[y1, . . . , yc ]!

I will illustrate this technique on a fundamental question:

How to test whether a kG -module is projective.

Recall Chouinard: A kG module M is projective ⇐⇒ M ↓E isprojective for each elementary abelian p-subgroup E ≤ G

So we can focus on modules over elementary abelian groups.

Henceforth: k will be algebraically closed, of char p > 0.

Testing projectivity

kE = k[z1, . . . , zc ]/(zp1 , . . . , z

pc ). For each a ∈ Pc−1, set

za =∑c

i=1 aizi where a = [a1, . . . , ac ].

This is well-defined, up to a scalar in k .

Lemma (Dade, 1978)

A kE -module M is projective ⇔ it is projective as a module overthe sub-algebra k[za] for each a ∈ Pc−1.

Proof.

M is projective ⇔ it is 0 in stmodkE .Recall X = Proj(k[y1, . . . , yc ]) and that X ∼= Pc−1. Using F , thedesired statement translates to a standard fact:

A sheaf F on Pc−1 is zero if and only if its restriction, Fa, to eachpoint a ∈ Pc−1 is zero.

The proof is something of a gross over-simplification!

It is not Dade’s original proof.

kE = k[z1, . . . , zc ]/(zp1 , . . . , z

pc ). Fix a = [a1, . . . , ac ] ∈ Pc−1

Since (za)p = (∑

i aizi )p =

∑i api zp

i = 0, as k-algebras

k[za] ∼= k[t]/(tp)

Note that za defines a k-linear map on M. It is not hard to show:

M free over k[za] ⇐⇒ za has maximal rank, (p−1)p rankk M

Example (Kronecker algebra)

kE = k[z1, z2]/(z21 , z

22 ). For each λ ∈ k , module Mλ with action

z1 =

[0 01 0

]z2 =

[0 0λ 0

]Matrix representing za = a1z1 + a2z2 is[

0 0a1 + λa2 0

]This has maximal rank (i.e. 1) exactly when [a1, a2] 6= [λ,−1].

Varieties for modules over kE

Dade’s theorem is the starting point of Carlson’s theory of ‘rankvarieties’ for modules over kE : For any kE -module M, set

VkE (M) = {a ∈ Pc−1 | M not projective over k[za] }

By Dade, this is a closed of Pc−1 under the Zariski topology.

Example (Kronecker algebra)

kE = k[z1, z2]/(z21 , z

22 ). Then, by the previous computation

VkE (Mλ) = {[λ, 1] ∈ P1} for any λ ∈ k

In particular, Mλ 6∼= Mλ′ , for λ 6= λ′.

Dade’s Lemma can be strengthened, with the same proof, to

VkE (M) = Supp H(F (M))

where SuppF denotes the support of a (coherent) sheaf F .

Classifying modular representations

Fix a kG -module M, where G is a finite group.Question: What kG -modules can one build out of M, in stmodkG ?

One is allowed the following operations:

1. Finite direct sums: if N1, . . . ,Nt are built out of M, so is ⊕iNi .2. Direct summands: if L⊕ N is built out of M, so are L and N.3. If L and N are built out of M and f : L→ N is a kE -linear map,pick a projective cover π : P � N and consider the exact sequence

0 −→ Ker(f , π) −→ L⊕ P(f ,π)−−−−→ N −→ 0

Then Ker(f , π) is built out of M; this is the mapping fiber of f .

1 and 3 define make stmodkG a triangulated category.

Definition

Thick(M) :=subcategory of stmodkG of modules built out of M.

Properties of M are inherited by modules in Thick(M).

E be an elementary abelian p-group, and k a field with char k = p

Theorem (Benson, Carlson, Rickard, 1997)

If VkE (M) ⊆ VkE (N), then M is in Thick(N).

Proof: Use F to pass to Db(Diff X ) and apply the result below.

Here X = Proj S and F ,G ∈ Db(Diff X ), differential sheaves on X .

Theorem (Hopkins, 1984; Carlson, Iyengar, 2012)

If Supp H(F) ⊆ Supp H(G), then F ∈ Thick(G).

Again, I am over-simplifying but the proof is new and simpler.

By Quillen theory, one can extend the BCR theorem to any G .

This is remarkable, for it means that properties of kG -modulesare (to a large extent) controlled by their support.

A more refined picture

kE ∼= k[z1, . . . , zc ]/(zp1 , . . . , z

pc )

S = k[y1, . . . , yc ], with |yi | = 2 for i = 1, . . . , c

X = Proj S

The functor F is induced by a functor F̃ :

stmod(kE ) oooo

F��

Db(kE )

F̃��

Db(Diff X ) oooo Db(S)

Db(kE ) :=derived category of finite dimensional kE -modules

Db(S) :=derived category of finitely generated dg S-modules

The horizontal arrows are quotient functors (a la Verdier):

the top one is a quotient by projective kE -modules;

the lower one is a quotient by dg modules with finite homology.

A closer look at F̃

K := Koszul complex on z1, . . . , zc over kE , as a dg algebra.

Λ := exterior algebra (over k) on ζ1, . . . , ζc with |ζi | = −1, viewedas a dg algebra with dΛ = 0.

kEi // K oo

q

' Λ ooKoszul

pair//____ S

These give rise to exact functors of triangulated categories:

Db(kE )K⊗kE− //

F̃

44Db(K )≡ // Db(Λ)

≡ // Db(S)

S is also viewed as a dg algebra with dS = 0.

Db(K ) := derived category of dg K -modules with finitelygenerated cohomology.

Db(Λ) and Db(S), the corresponding derived categories.

A more elaborate picture

The functors F and F̃ are shadows of one on a larger category:

K(Free kE ) :=homotopy category of complexes of free kE -modules

Db(S) :=full derived category of dg S-modules

Then there is a commutative diagram:

stmod(kE ) oooo

F��

Db(kE )

F̃��

� � // K(Free kE )

��Db(Diff X ) oooo Db(S) � � // D(S)

For deeper applications, one needs the rightmost functor.

Final remark: kE is a (very special) complete intersection ring.

The diagram above remains valid when kE is replaced by anarbitrary local complete intersection ring.

A partial list of references

Benson, Carlson, and Rickard, Thick subcategories of thestable module category, Fundamenta Mathematicae 153(1997), 59–80.

Chouinard, Projectivity and relative projectivity over grouprings, J. Pure & Applied Algebra 7 (1976), 278–302.

Dade, Endo-permutation modules over p-groups, II, Ann. ofMath. 108 (1978), 317–346.

Hopkins, Global methods in homotopy theory, HomotopyTheory, Durham 1985, Lecture Notes in Mathematics, vol.117, Cambridge University Press, 1987.

Neeman, The chromatic tower for D(R), Topology 31 (1992),519–532.

Quillen, The spectrum of an equivariant cohomology ring, I &II, Ann. of Math. 94 (1971), 549–572, 573–602.