Actions of Hochschild cohomology in representation theory...

Actions of Hochschild cohomologyin representation theory of finite groups

Fajar Yuliawan

Fakultas Matematika dan Ilmu Pengetahuan AlamInstitut Teknologi Bandung

8 November 2018

Outline

• Group algebras and modules

• Categories in representation of finite groups

• Actions of the Hochschild cohomology

Group algebras

• k field, e.g Q,R,Fp = Z/pZ,Fpn , etc

• G finite group, e.g. {1}, Cp∼= Z/pZ, Cp × Cp, Sn, etc

• Group algebra kG : vector space with basis G , andmultiplication ‘induced’ by multiplication in G , i.e.

∑g∈G

λgg | λg ∈ k

and ∑

g∈Gλgg

(∑h∈G

)=∑g∈G

∑h∈G

λgµhgh

Group algebras

∑g∈G

λgg | λg ∈ k

and ∑g∈G

(∑h∈G

)=∑g∈G

∑h∈G

λgµhgh

Group algebras

∑g∈G

λgg | λg ∈ k

and ∑

g∈Gλgg

(∑h∈G

)=∑g∈G

∑h∈G

λgµhgh

Group algebras

Examples

(1) G = {1}, kG = {λ · 1 | λ ∈ k} and (λ · 1)(µ · 1) = (λµ) · 1.

ThuskG ∼= k, λ · 1 7→ λ

(2) G = C2 = {1, a} with a2 = 1, kG = {λ · 1 + µ · a | λ, µ ∈ k},and

(λ1 · 1 + µ1 · a)(λ2 · 1 + µ2 · a)

=λ1λ2 · 1 + λ1µ2 · a + µ1λ2 · a + µ1µ2 · 1=(λ1λ2 + µ1µ2) · 1 + (λ1µ2 + λ2µ1) · a.

kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX

Group algebras

Examples

(1) G = {1}, kG = {λ · 1 | λ ∈ k} and (λ · 1)(µ · 1) = (λµ) · 1.Thus

kG ∼= k, λ · 1 7→ λ

(λ1 · 1 + µ1 · a)(λ2 · 1 + µ2 · a)

kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX

Group algebras

Examples

kG ∼= k, λ · 1 7→ λ

(λ1 · 1 + µ1 · a)(λ2 · 1 + µ2 · a)

kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX

Group algebras

Examples

kG ∼= k, λ · 1 7→ λ

(λ1 · 1 + µ1 · a)(λ2 · 1 + µ2 · a)

kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX

Group algebras

(3) G = C2 and char(k) = 2 then kG ∼= k[X ]/〈X 2〉.

In general, G = (Cp)n = Cp × Cp × · · ·Cp and char(k) = p,then

kG ∼= k[X1,X2, . . . ,Xn]/〈X p1 ,X

p2 , . . . ,X

pn 〉

Observation: kG is commutative if and only if G is Abelian, e.g.kS3 is not commutative

Group algebras

kG ∼= k[X1,X2, . . . ,Xn]/〈X p1 ,X

p2 , . . . ,X

pn 〉

Group algebras

kG ∼= k[X1,X2, . . . ,Xn]/〈X p1 ,X

p2 , . . . ,X

pn 〉

Modules over group algebras

k field, G finite group, kG group algebra

• (left) kG -module: vector space M together with‘multiplication’

kG ×M → M, (a,m) 7→ a ·m,

satisfying the ‘usual’ modules axioms, e.g. 1 ·m = m,(a + b)m = am + bm, (ab)m = a(bm),a(λm + µn) = λam + µan, etc

Example

1. The zero vector space (of dimension 0)

2. k is a kG -module with(∑

g∈G λg · g)· µ =

∑g∈G λgµ

3. kG is a kG -module with the usual multiplication

kG ×M → M, (a,m) 7→ a ·m,

Example

g∈G λg · g)· µ =

∑g∈G λgµ

kG ×M → M, (a,m) 7→ a ·m,

Example

g∈G λg · g)· µ =

∑g∈G λgµ

Homomorphism of kG -modules

k field, G finite group, kG group algebraM,N be kG -modules

• kG -homomorphism: k-linear map f : M → N s.t.f (a ·m) = a · f (m) for all a ∈ kG , m ∈ M,

or equivalently f (g ·m) = g · f (m) for all g ∈ G and m ∈ M.

• Composition of two kG -homomorphisms is again akG -homomorphism, and

the class of all kG -modules together with kG -homomorphismbetween them form a category, denoted by Mod(kG ).

Definition of categories

A category C consists of

• a ‘class’ of objects Ob(C)

• a set of morphism HomC(X ,Y ) for each pair X ,Y ∈ Ob(C)

• a function, called composition

◦ : HomC(X ,Y )× HomC(Y ,Z )→ HomC(X ,Z )

(f , g) 7→ g ◦ f

for each triple X ,Y ,Z ∈ Ob(C)

satisfying certain axioms.

A ‘morphism’ between categories is called a functor

Definition of categories

A category C consists of

• a ‘class’ of objects Ob(C)

• a set of morphism HomC(X ,Y ) for each pair X ,Y ∈ Ob(C)

• a function, called composition

◦ : HomC(X ,Y )× HomC(Y ,Z )→ HomC(X ,Z )

(f , g) 7→ g ◦ f

for each triple X ,Y ,Z ∈ Ob(C)

satisfying certain axioms.

A ‘morphism’ between categories is called a functor

Category Set

Category Mod(k)

Functor

Category of kG -modules

• Usual objective: Classify the indecomposable in mod(kG ).

Theorem [Krull-Schmidt]. Every f.d. module M in mod(kG )can be decomposed uniquely into direct sum ofindecomposable modules,

M = M1 ⊕M2 ⊕ · · · ⊕Mn.

• In ‘ordinary representation theory’:Theorem [Maschke]. If char(k) - |G |, every indecomposablekG -module is irreducible/simple, i.e. kG is semisimple

• In ’modular representation theory’:If char(k) | |G |, the group algebra kG is usually of ‘wildrepresentation type’, hence it is impossible to classify theindecomposables modules.

• Usual objective: Classify the indecomposable in mod(kG ).Theorem [Krull-Schmidt]. Every f.d. module M in mod(kG )can be decomposed uniquely into direct sum ofindecomposable modules,

M = M1 ⊕M2 ⊕ · · · ⊕Mn.

Categories in modular representation theory

k field, G finite group and char(k) | |G |Fact: kG is f.d. self-injective algebra, hence projective = injective

1. Stable module category StMod(kG ) or Mod(kG )• objects: kG -modules• morphisms: HomkG (M,N) = Hom(M,N)/PHom(M,N),

PHom(M,N) = {f | f factors through some proj. ob.}

2. Derived category D(Mod kG ) of the module category• objects: ‘complexes’ of kG -modules• morphisms: equivalence classes of ‘roof’, i.e. M → X ← N

with N → X a ‘quasi-isomorphism’

3. Homotopy category of injective kG -modules K(Inj kG )

Homotopy category of injectives

• Objects: complexes of injective kG -modules

where I n are injective kG -modules and dn ◦ dn−1 = 0.

• Morphism: chain maps (of degree 0) modulo null-homotopymaps

HomK(Inj kG)(I , J) ={chain maps I → J}

{null-homotopic maps I → J}

• Objects: complexes of injective kG -modules

where I n are injective kG -modules and dn ◦ dn−1 = 0.

• Morphism: chain maps (of degree 0) modulo null-homotopymaps

HomK(Inj kG)(I , J) ={chain maps I → J}

{null-homotopic maps I → J}

Morphisms in homotopy category of injectives

Theorem (Krause 2005)

There exists a diagram of six functors

satisfying some ‘nice’ properties.

RemarkThe category K (Inj kG ) contains a copy of Mod(kG ) and twocopies of D(Mod(kG ))

Theorem (Krause 2005)

There exists a diagram of six functors

satisfying some ‘nice’ properties.

RemarkThe category K (Inj kG ) contains a copy of Mod(kG ) and twocopies of D(Mod(kG ))

• Aim: Study K(Inj kG )

• Plan: Give some algebraic structure on the set of morphismbetween two objects

Benson-Iyengar-Krause theory (2008):If R is a graded-commutative notherian ring ‘acting’ onK(Inj kG ) then there is a ‘local cohomology’ functor

Γp : K(Inj kG )→ K(Inj kG )

for each homogeneous prime ideals p ⊆ R.Define the support of an object X in K(Inj kG ) to be

suppR(X ) = {p ∈ Spech(R) | Γp(X ) = 0}.

for each homogeneous prime ideals p ⊆ R.

Define the support of an object X in K(Inj kG ) to be

for each homogeneous prime ideals p ⊆ R.Define the support of an object X in K(Inj kG ) to be

Some homological algebras

A an f.d. algebra over a field k and X ,Y A-modules

• an injective resolution of X is a complex of injective modules

iX = · · · → 0→ I 0d0

−→ I 1d1

−→ I 2 → · · ·

together with a map η : X → I 0 such that the sequence

0→ Xη−→ I 0

−→ I 1d1

−→ I 2 → · · ·

is exact.

• In particular, iX is an object in K(InjA)

• The n-th extension group Extn(X ,Y ) is

ExtnA(X ,Y ) = HomK(InjA)(iX , iY [n]),

where iX , iY are injective resolution of X ,Y , resp.

Thus, an element of ExtnA(X ,Y ) looks like:

• The n-th extension group Extn(X ,Y ) is

ExtnA(X ,Y ) = HomK(InjA)(iX , iY [n]),

where iX , iY are injective resolution of X ,Y , resp.Thus, an element of ExtnA(X ,Y ) looks like:

Hochschild cohomology

A an f.d. algebra over a field k , Ae the enveloping algebra of A(i.e. (A,A)-bimodules = Ae-modules)

• M an (A,A)-bimodule. The n-th Hochschild cohomology of Awith coefficient in M is

HH(A,M) = ExtnAe (A,M) = HomK(InjAe)(iA, iM[n])

• The Hochschild cohomology ring of A is the graded ring

HH∗(A,A) =⊕n∈Z

HHn(A,A)

with multiplication defined using composition and shift.

HHn(A,A)

• If f ∈ HHm(A,A), i.e. f : iA→ iA[m]and g ∈ HHn(A,A), i.e. g : iA→ iA[n],

then f · g is defined as the composition

iAf−→ iA[m]

g [m]−−−→ iA[m + n]

in HHm+n(A,A)

Theorem [Gerstenhaber 1963]. The Hochschild cohomology ring ofA is a graded-commutative ring.

Theorem [Evens 1961, Venkov 1959, Ginzburg-Kumar 1993,Friedlander-Suslin 1997]. The Hochschild cohomology ring of A isnoetherian.

iAf−→ iA[m]

g [m]−−−→ iA[m + n]

in HHm+n(A,A)

iAf−→ iA[m]

g [m]−−−→ iA[m + n]

in HHm+n(A,A)

The Action

k field and G finite group, A = kG

• Idea: Use tensor product over AIf X is an (A,A)-bimodule and M an A-module, then X ⊗A Mis an A-module.

• Extend to tensor product on the homotopy category:

−⊗A − : K(ModAe)× K(ModA)→ K(ModA)

• Restrict to the homotopy category of injectives

−⊗A − : K(InjAe)× K(InjA)→ K(InjA)

The Action

The ActionA = kG , K = K(InjA),

Hom∗K(X ,Y ) =⊕n∈Z

Hom∗K(X ,Y [n])

• Apply the bifunctor to the morphisms, get:

HH∗(A,A)× Hom∗K(X ,Y )→ Hom∗K(X ,Y ),

where a ‘homogeneous’ pair f : iA→ iA[m] and g : X → Y [n]is mapped to

X ∼= iA⊗A Xf⊗g−−→ iA[m]⊗A Y [n] ∼= Y [m + n]

• Get (graded) HH∗(A,A)-module structure on Hom∗K(X ,Y ) foreach pair X ,Y ∈ K = K(InjA).

Hom∗K(X ,Y [n])

Main Theorem

Conclusion and Outlook

• The Hochschild cohomology ring acts on the homotopycategory of injective kG -modules

• Since the stable category and the derived category is‘contained’ in the homotopy category of injectives, we get alsoactions on both categories (for free)

• The above action can be generalized to arbitrary f.d.self-injective algebra over a field k

• Using B-I-K’s machinery, get local cohomology functor andsupport theory for all categories above

An open problem:

• Does the above action generalize to arbitrary f.d. algebras?

Conclusion and Outlook

• The Hochschild cohomology ring acts on the homotopycategory of injective kG -modules

• Since the stable category and the derived category is‘contained’ in the homotopy category of injectives, we get alsoactions on both categories (for free)

• The above action can be generalized to arbitrary f.d.self-injective algebra over a field k

• Using B-I-K’s machinery, get local cohomology functor andsupport theory for all categories above

An open problem:

• Does the above action generalize to arbitrary f.d. algebras?

Thank You!

Actions of Hochschild cohomology in representation theory...

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