Actions of Hochschild cohomology in representation theory...
Transcript of Actions of Hochschild cohomology in representation theory...
![Page 1: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/1.jpg)
Actions of Hochschild cohomologyin representation theory of finite groups
Fajar Yuliawan
Fakultas Matematika dan Ilmu Pengetahuan AlamInstitut Teknologi Bandung
8 November 2018
![Page 2: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/2.jpg)
Outline
• Group algebras and modules
• Categories in representation of finite groups
• Actions of the Hochschild cohomology
![Page 3: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/3.jpg)
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.
kG =
∑g∈G
λgg | λg ∈ k
and ∑
g∈Gλgg
(∑h∈G
µhh
)=∑g∈G
∑h∈G
λgµhgh
![Page 4: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/4.jpg)
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.
kG =
∑g∈G
λgg | λg ∈ k
and ∑g∈G
λgg
(∑h∈G
µhh
)=∑g∈G
∑h∈G
λgµhgh
![Page 5: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/5.jpg)
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.
kG =
∑g∈G
λgg | λg ∈ k
and ∑
g∈Gλgg
(∑h∈G
µhh
)=∑g∈G
∑h∈G
λgµhgh
![Page 6: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/6.jpg)
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.
Thus
kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX
![Page 7: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/7.jpg)
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
(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.
Thus
kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX
![Page 8: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/8.jpg)
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
(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.
Thus
kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX
![Page 9: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/9.jpg)
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
(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.
Thus
kG ∼= k[X ]/〈X 2 − 1〉, λ · 1 + µ · a 7→ λ+ µX
![Page 10: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/10.jpg)
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
![Page 11: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/11.jpg)
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
![Page 12: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/12.jpg)
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
![Page 13: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/13.jpg)
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
![Page 14: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/14.jpg)
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
![Page 15: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/15.jpg)
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
![Page 16: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/16.jpg)
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 ).
![Page 17: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/17.jpg)
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 ).
![Page 18: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/18.jpg)
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 ).
![Page 19: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/19.jpg)
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 ).
![Page 20: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/20.jpg)
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 ).
![Page 21: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/21.jpg)
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
![Page 22: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/22.jpg)
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
![Page 23: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/23.jpg)
Category Set
![Page 24: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/24.jpg)
Category Mod(k)
![Page 25: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/25.jpg)
Functor
![Page 26: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/26.jpg)
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.
![Page 27: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/27.jpg)
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.
![Page 28: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/28.jpg)
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.
![Page 29: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/29.jpg)
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.
![Page 30: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/30.jpg)
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 )
![Page 31: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/31.jpg)
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 )
![Page 32: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/32.jpg)
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 )
![Page 33: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/33.jpg)
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 )
![Page 34: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/34.jpg)
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}
![Page 35: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/35.jpg)
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}
![Page 36: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/36.jpg)
Morphisms in homotopy category of injectives
![Page 37: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/37.jpg)
Morphisms in homotopy category of injectives
![Page 38: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/38.jpg)
Morphisms in homotopy category of injectives
![Page 39: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/39.jpg)
Morphisms in homotopy category of injectives
![Page 40: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/40.jpg)
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 ))
![Page 41: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/41.jpg)
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 ))
![Page 42: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/42.jpg)
Homotopy category of injectives
• 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}.
![Page 43: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/43.jpg)
Homotopy category of injectives
• 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}.
![Page 44: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/44.jpg)
Homotopy category of injectives
• 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}.
![Page 45: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/45.jpg)
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
d0
−→ I 1d1
−→ I 2 → · · ·
is exact.
• In particular, iX is an object in K(InjA)
![Page 46: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/46.jpg)
Some homological algebras
• 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:
![Page 47: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/47.jpg)
Some homological algebras
• 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:
![Page 48: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/48.jpg)
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.
![Page 49: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/49.jpg)
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.
![Page 50: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/50.jpg)
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.
![Page 51: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/51.jpg)
Hochschild cohomology
• 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.
![Page 52: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/52.jpg)
Hochschild cohomology
• 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.
![Page 53: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/53.jpg)
Hochschild cohomology
• 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.
![Page 54: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/54.jpg)
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)
![Page 55: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/55.jpg)
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)
![Page 56: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/56.jpg)
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)
![Page 57: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/57.jpg)
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)
![Page 58: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/58.jpg)
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).
![Page 59: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/59.jpg)
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).
![Page 60: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/60.jpg)
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).
![Page 61: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/61.jpg)
Main Theorem
![Page 62: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/62.jpg)
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?
![Page 63: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/63.jpg)
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?
![Page 64: Actions of Hochschild cohomology in representation theory ...matematika.fmipa.unand.ac.id/images/seminar... · In ‘ordinary representation theory’: Theorem [Maschke]. If char(k)](https://reader035.fdocuments.in/reader035/viewer/2022071023/5fd8158a9bda4138976bc369/html5/thumbnails/64.jpg)
Thank You!