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- 1/1
Algebraic topology computationsand
Representation theory of GLn
Antoine Touze
Universite Paris 13
Arolla 2012
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 2/1
In this talk...
k is a PID (ex : algebraically closed field, Fp or Z).
We will deal with two kinds of functors :
I Ordinary functors, that is :
F : (Free) k-mod→ k-mod .
Huge number of examples.
• ⊗n : V 7→ V ⊗k · · · ⊗k V︸ ︷︷ ︸n times
.
• Sn : V 7→ Sn(V ) = (V⊗n)Sn .
I Strict polynomial functors, that is :
F : Free k-mod→ k-mod .
+ additional algebraic structure.
They come from representation theory of GLn.
• ⊗n and Sn have a canonical structure of strict polynomialfunctors.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen] [Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen] [Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen] [Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen]
[Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen]
[Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen] [Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 3/1
In this talk...
Algebraictopology
Representationsof GLn
Ordinary functors Strict polynomial functors
derived functor of FLF
Ringel dual of FΘF
[Dold-Puppe, Quillen] [Ringel, Donkin, Cha lupnik]
Plan :
I. Functors in algebraic topology.
II. Functors in representation theory of GLn
III. Thm : LF and ΘF coincide.
IV. Some applications.
V. The homology of EML spaces.
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :
1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),
2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)
3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .
4. Use functorial constructions :Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n)
, πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n)
, HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n)
,πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .
What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .
What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 4/1
I. Functors in algebraic topology (1)
Let A be an abelian group.
I Spaces X (A) constructed from A :1. EML spaces K (A, n),2. Moore spaces M(A, n)3. Embedding spaces Emb(M,A⊗Z R). . .4. Use functorial constructions :
Σ, Ω, SPk , ∨, ∧, ×, DnF ,. . .
I Functors h : Top→ Ab : πi , πsi ,Hi , . . .
I Plenty of ordinary functors : Ab → AbA 7→ h(X (A))
πkM(A, n), HkK (A, n), πkΣK (A, n), HkSP iK (A, n),πkK (A, n) ∨ K (A, n + 1), . . .What a mess !
1. How are they related ?
2. Which ones are isomorphic ?
3. Can we describe them from simpler functors ?Ex : H10K (A, 3) = A⊗ A/3A (A free abelian)
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
HiNFKPM [n]Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
HiNFKPM [n]Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
HiNFKPM [n]Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
HiNFK
PM [n]Free (or projective) resolution of M, shifted
Free simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
HiNF
KPM [n]
Free (or projective) resolution of M, shifted
Free simplicial k-mod, with homology M[n]
Simplicial k-modChain complex of k-modk-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
HiN
FKPM [n]
Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]
Simplicial k-mod
Chain complex of k-modk-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) =
Hi
NFKPM [n]
Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-mod
Chain complex of k-mod
k-modRk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) = HiNFKPM [n]
Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-mod
k-mod
Rk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F :
Free
k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) = HiNFKPM [n]
Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-mod
Rk : Result does not depend on the choice of PM
- 5/1
I. Functors in algebraic topology (2)
Derived functors of ordinary functors [Dold-Puppe]
I They help to explain and organize the whole picture
1. Dold Kan correspondence :Normalized Chain functor :N : s(k-mod)→ Ch≥0(k-mod)
is an equivalence of categories, with explicit inverseK : Ch≥0(k-mod)→ s(k-mod) .
N ,K preserve homotopy equivalences.
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd), M ∈ k-mod.
Def : LiF (M; n) = HiNFKPM [n]
Free (or projective) resolution of M, shiftedFree simplicial k-mod, with homology M[n]Simplicial k-modChain complex of k-modk-mod
Rk : Result does not depend on the choice of PM
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n])
= πiLF (K (M, n)) [Q]Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n])
= πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n])
= πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n])
= πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n])
= πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961] , later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961] , later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 6/1
I. Functors in algebraic topology (3)
2. Derived functors :F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n)) [Q]
Comments :
I n = ‘height of derivation’.The higher n is, the more complex LiF (M, n) is.
I If F additive, LiF (M, n) = Li−nF (M).
I Definition : [Dold Puppe 1961]
, later generalized [Quillen]
LiF (M; n) = πiLF (K (M, n)) [Q] (K (M, n) = simpl. EML.)
3. Examples (k = Z, A abelian group) :
1. GR(A) := Z[A]. Then L∗GR(A; n) = H∗K (A, n).
2. Thm [Dold] L∗S(A; n) ' H∗K (A, n) (A free).
3. Thm [Dold] H∗(SPdX ) computed from H∗(X ) and LiSd .
4. Curtis spectral sequence (1965) :
E 1i ,j(A) = LiLj(A, n) =⇒ πi+1M(A, n + 1) .
free Lie functor
- 7/1
I. Functors in algebraic topology (4)
To sum up :
I F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))
I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations
I We want to know more about them !
I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]
But there remains work : for ex, still not clear what LiSd(A, n)
are !
- 7/1
I. Functors in algebraic topology (4)
To sum up :
I F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))
I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations
I We want to know more about them !
I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]
But there remains work : for ex, still not clear what LiSd(A, n)
are !
- 7/1
I. Functors in algebraic topology (4)
To sum up :
I F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))
I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations
I We want to know more about them !
I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]
But there remains work : for ex, still not clear what LiSd(A, n)
are !
- 7/1
I. Functors in algebraic topology (4)
To sum up :
I F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))
I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations
I We want to know more about them !
I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]
But there remains work : for ex, still not clear what LiSd(A, n)
are !
- 7/1
I. Functors in algebraic topology (4)
To sum up :
I F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))
I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations
I We want to know more about them !
I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]
But there remains work : for ex, still not clear what LiSd(A, n)
are !
- 7/1
I. Functors in algebraic topology (4)
To sum up :
I F : Free k-mod→ k-mod (Ex : Sd).
LiF (−, n) : k-mod → k-modM 7→ HiNFK(PM [n]) = πiLF (K (M, n))
I Derived functors (of Sd , GR, L,. . . ) have topologicalinterpretations
I We want to know more about them !
I 60es-70es : lot of work has been done to compute them[Bott, Bousfield, Curtis, Quillen, Schlessinger . . . ]
But there remains work : for ex, still not clear what LiSd(A, n)
are !
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-mod
Evaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ :
GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is
M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups
(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ :
GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is
M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups
(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ : GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is
M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups
(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ : GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.
A representation of GLn,k isM ∈ k-mod + ρ : GLn,k → GLk(M)
morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups
(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ : GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is
M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups
(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ : GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is
M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 8/1
II. Functors in representation theory of GLn (1)
1. Functors → representations of GLn(k)Fk := category of ordinary functors :
F : Free k-mod→ k-modEvaluation functor :ev : Fk → GLn(k)-mod
F 7→ F (kn) +ρ : GLn(k) → GL(F (kn))g 7→ F (g)
2. Representations of GLn,kNow we think of GLn,k as an algebraic group scheme.A representation of GLn,k is
M ∈ k-mod + ρ : GLn,k → GLk(M)morphism of algebraic group schemes
Forgetful functor : Alg. Group Schemes→ Groups
(G ,OG ) 7→ G
Forget that ρ is a morphism of algebraic group schemes :
U : GLn,k-mod → GLn(k)-mod(M, ρ) 7→ (M, ρ)
- 9/1
II. Functors in representation theory of GLn (2)
Examples of representations of GLn,k :
k = Fp.
(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.
(kn, Id) is a representation of GLn,k
(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j
p]defines an algebraic group scheme morphism.
(kn, ρFrob) is a representation of GLn,k
(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).
Hence
I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod
- 9/1
II. Functors in representation theory of GLn (2)
Examples of representations of GLn,k :
k = Fp.
(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.
(kn, Id) is a representation of GLn,k
(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j
p]defines an algebraic group scheme morphism.
(kn, ρFrob) is a representation of GLn,k
(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).
Hence
I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod
- 9/1
II. Functors in representation theory of GLn (2)
Examples of representations of GLn,k :
k = Fp.
(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.
(kn, Id) is a representation of GLn,k
(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j
p]defines an algebraic group scheme morphism.
(kn, ρFrob) is a representation of GLn,k
(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).
Hence
I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod
- 9/1
II. Functors in representation theory of GLn (2)
Examples of representations of GLn,k :
k = Fp.
(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.
(kn, Id) is a representation of GLn,k
(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j
p]defines an algebraic group scheme morphism.
(kn, ρFrob) is a representation of GLn,k
(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).
Hence
I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod
- 9/1
II. Functors in representation theory of GLn (2)
Examples of representations of GLn,k :
k = Fp.
(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.
(kn, Id) is a representation of GLn,k
(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j
p]defines an algebraic group scheme morphism.
(kn, ρFrob) is a representation of GLn,k
(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).
Hence
I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod
- 9/1
II. Functors in representation theory of GLn (2)
Examples of representations of GLn,k :
k = Fp.
(a) Id : GLn,k → GLn,k is algebraic group scheme morphism.
(kn, Id) is a representation of GLn,k
(b) The polynomial formula ρFrob GLn,k → GLn,k[ai ,j ] 7→ [ai ,j
p]defines an algebraic group scheme morphism.
(kn, ρFrob) is a representation of GLn,k
(c) Observe that ρFrob and Id :I are different as morphisms of algebraic groups,I coincide as maps between the sets GLn(k)→ GLn(k).
Hence
I (kn, ρFrob) 6= (kn, Id) in GLn,k-modI U(kn, ρFrob) = U(kn, Id) in GLn(k)-mod
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors =
GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ Fk
I For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f ) (+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 10/1
II. Functors in representation theory of GLn (3)
3. Strict polynomial functors
strict polynomial functors = GLn,k-mod
U
Fkev // GLn(k)-mod
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) with
I F ∈ FkI For all free k-mod M,N, polynomial
FM,N : Homk(M,N)→ Homk(F (M),F (N))
such that FM,N(f ) = F (f )
(+ technical axiom)
Def : A strict polynomial functor is homogeneous of degree d if allthe FM,N are homogeneous of degree d .
Ex : Sd , ⊗d , Ld are canonically strict polynomial functors,homogeneous of degree d .
- 11/1
II. Functors in representation theory of GLn (4)
3. Strict polynomial functors
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).
Notation : Pd ,k = full subcategory of Pkwith objects homogeneous functors of degree d .
I Strict polynomial functors are interesting because :
Thm : [Friedlander-Suslin] If n ≥ d , iso :
Ext∗Pd,k(F ,G ) ' Ext∗GLn,k(F (kn),G (kn))
Allows explicit ExtGLn,k-computations !
Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].
- 11/1
II. Functors in representation theory of GLn (4)
3. Strict polynomial functors
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).Notation : Pd ,k = full subcategory of Pk
with objects homogeneous functors of degree d .
I Strict polynomial functors are interesting because :
Thm : [Friedlander-Suslin] If n ≥ d , iso :
Ext∗Pd,k(F ,G ) ' Ext∗GLn,k(F (kn),G (kn))
Allows explicit ExtGLn,k-computations !
Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].
- 11/1
II. Functors in representation theory of GLn (4)
3. Strict polynomial functors
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).Notation : Pd ,k = full subcategory of Pk
with objects homogeneous functors of degree d .
I Strict polynomial functors are interesting because :
Thm : [Friedlander-Suslin] If n ≥ d , iso :
Ext∗Pd,k(F ,G ) ' Ext∗GLn,k(F (kn),G (kn))
Allows explicit ExtGLn,k-computations ! Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].
- 11/1
II. Functors in representation theory of GLn (4)
3. Strict polynomial functors
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).Notation : Pd ,k = full subcategory of Pk
with objects homogeneous functors of degree d .
I Strict polynomial functors are interesting because :
Thm : [Friedlander-Suslin] If n ≥ d , iso :
Ext∗Pd,k(F ,G ) ' Ext∗GLn,k(F (kn),G (kn))
Allows explicit ExtGLn,k-computations !
Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].
- 11/1
II. Functors in representation theory of GLn (4)
3. Strict polynomial functors
strict polynomial functors = Pkev //
U
GLn,k-mod
U
Fkev // GLn(k)-mod
Def : Strict polynomial functor = pair (F , FM,N) (with F ∈ Fk)Ex : Sd , ⊗d , Ld (homogeneous of degree d).Notation : Pd ,k = full subcategory of Pk
with objects homogeneous functors of degree d .
I Strict polynomial functors are interesting because :
Thm : [Friedlander-Suslin] If n ≥ d , iso :
Ext∗Pd,k(F ,G ) ' Ext∗GLn,k(F (kn),G (kn))
Allows explicit ExtGLn,k-computations ! Ingredient for finitegeneration theorems [FS, 97] [T, Van der Kallen, 2010].
- 12/1
II. Functors in representation theory of GLn (5)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functors :F ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
related to alg. top.computations
4. Ringel Duality Θ
I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k
Ringel duality linked with theory of tilting modules
- 12/1
II. Functors in representation theory of GLn (5)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functors :F ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
related to alg. top.computations
4. Ringel Duality Θ
I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k
Ringel duality linked with theory of tilting modules
- 12/1
II. Functors in representation theory of GLn (5)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functors :F ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
related to alg. top.computations
4. Ringel Duality Θ
I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k
Ringel duality linked with theory of tilting modules
- 12/1
II. Functors in representation theory of GLn (5)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functors :F ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
related to alg. top.computations
4. Ringel Duality Θ
I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k
Ringel duality linked with theory of tilting modules
- 12/1
II. Functors in representation theory of GLn (5)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functors :F ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
related to alg. top.computations
4. Ringel Duality Θ
I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k
Ringel duality linked with theory of tilting modules
- 12/1
II. Functors in representation theory of GLn (5)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functors :F ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
related to alg. top.computations
4. Ringel Duality Θ
I [Ringel, 91] repres. finite dim. algebrasI [Donkin, 93] the GLn,k-mod caseI [Cha lupnik, 2008] translated to Pd ,k
Ringel duality linked with theory of tilting modules
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =R
Hom(Λd ,−) :
Db
Pd ,k →
Db
Pd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =R
Hom(Λd ,−) :
Db
Pd ,k →
Db
Pd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =R
Hom(Λd ,−) :
Db
Pd ,k →
Db
Pd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =R
Hom(Λd ,−) :
Db
Pd ,k →
Db
Pd ,k (left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =R
Hom(Λd ,−) :
Db
Pd ,k →
Db
Pd ,k (left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =
RHom(Λd ,−) : DbPd ,k → DbPd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =RHom(Λd ,−) : DbPd ,k → DbPd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =RHom(Λd ,−) : DbPd ,k → DbPd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.
Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =RHom(Λd ,−) : DbPd ,k → DbPd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =RHom(Λd ,−) : DbPd ,k → DbPd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)
Θ(Γd) = big complex !
- 13/1
II. Functors in representation theory of GLn (6)
4. Ringel Duality Θ
A) Functional Homs
Notation : F ∈ Pd ,k, N free k-mod :FN = M 7→ F (N ⊗k M) ∈ Pd ,k
Hom(G ,F ) : N 7→ HomPd,k(G ,FN) ∈ Pd ,k
If G fixed, functional Hom yields :
Hom(G ,−) : Pd ,k → Pd ,kF 7→ Hom(G ,F )
B) Case G = Λd (exterior power).
Def : Ringel duality operatorΘ =RHom(Λd ,−) : DbPd ,k → DbPd ,k
(left exact)
Recall the derived category DbPd ,k :I Objects : bounded complexes of objects of Pd ,k.I Morphisms : chain maps with quasi-isos inversed.
Thm : [RDC] Θ is an equivalence of categories.Ex : Θ(Sd) = Λd ,
Θ(Λd) = Γd , Γd(M) = (M⊗d)Sd (M free k-mod)Θ(Γd) = big complex !
- 14/1
III. ΘF and LF coincide (1)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functorsF ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
Ringel duals :Θ(F ) ∈ DbPd ,k
(alg. top. computations) (tilting modules)
Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Rq : Stronger statement on the level of derived categories.Cor : Ld−iF (M, 1) = ExtiPd,k
(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
- 14/1
III. ΘF and LF coincide (1)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functorsF ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
Ringel duals :Θ(F ) ∈ DbPd ,k
(alg. top. computations) (tilting modules)
Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Rq : Stronger statement on the level of derived categories.Cor : Ld−iF (M, 1) = ExtiPd,k
(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
- 14/1
III. ΘF and LF coincide (1)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functorsF ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
Ringel duals :Θ(F ) ∈ DbPd ,k
(alg. top. computations) (tilting modules)
Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Rq : Stronger statement on the level of derived categories.Cor : Ld−iF (M, 1) = ExtiPd,k
(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
- 14/1
III. ΘF and LF coincide (1)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functorsF ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
Ringel duals :Θ(F ) ∈ DbPd ,k
(alg. top. computations) (tilting modules)
Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Rq : Stronger statement on the level of derived categories.
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
- 14/1
III. ΘF and LF coincide (1)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functorsF ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
Ringel duals :Θ(F ) ∈ DbPd ,k
(alg. top. computations) (tilting modules)
Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Rq : Stronger statement on the level of derived categories.Cor : Ld−iF (M, 1) = ExtiPd,k
(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
- 14/1
III. ΘF and LF coincide (1)
Algebraictopology
Representationsof group scheme GLn,k
Fk = ordinary functors :F : Free k-mod→ k-mod
Sd , ⊗d , Ld ,. . .
Pd ,k = strict polynomial functorsF ∈ Fk + additional structure
Sd , ⊗d , Ld ,. . .
derived functors [DP,Q] :LiF (M, n)
Ringel duals :Θ(F ) ∈ DbPd ,k
(alg. top. computations) (tilting modules)
Prop : If M is free k-mod, F ∈ Pd ,k then M 7→ LiF (M, n) ∈ Pd ,k.
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Rq : Stronger statement on the level of derived categories.Cor : Ld−iF (M, 1) = ExtiPd,k
(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
- 15/1
III. ΘF and LF coincide (2)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
Why is this theorem interesting ?
Left hand side Right hand side
Simplicial methodsIntuition from alg. top.
Representation theory methods :• Highest weight categories• Tilting modules• Block theory. . .
Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).
- 15/1
III. ΘF and LF coincide (2)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
Why is this theorem interesting ?
Left hand side Right hand side
Simplicial methodsIntuition from alg. top.
Representation theory methods :• Highest weight categories• Tilting modules• Block theory. . .
Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).
- 15/1
III. ΘF and LF coincide (2)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
Why is this theorem interesting ?
Left hand side Right hand side
Simplicial methodsIntuition from alg. top.
Representation theory methods :• Highest weight categories
• Tilting modules• Block theory. . .
Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).
- 15/1
III. ΘF and LF coincide (2)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
Why is this theorem interesting ?
Left hand side Right hand side
Simplicial methodsIntuition from alg. top.
Representation theory methods :• Highest weight categories• Tilting modules
• Block theory. . .
Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).
- 15/1
III. ΘF and LF coincide (2)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
Why is this theorem interesting ?
Left hand side Right hand side
Simplicial methodsIntuition from alg. top.
Representation theory methods :• Highest weight categories• Tilting modules• Block theory. . .
Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).
- 15/1
III. ΘF and LF coincide (2)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
Why is this theorem interesting ?
Left hand side Right hand side
Simplicial methodsIntuition from alg. top.
Representation theory methods :• Highest weight categories• Tilting modules• Block theory. . .
Different insights on computations.Well-known computations on the right hand side are unknown onthe left hand side (and vice versa).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),
I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy.
Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),
I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),
I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),
I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),I Ext∗S(M)(k,k) = Λ(M[1]),
I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)
3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 16/1
III. ΘF and LF coincide (3)
Thm : [T] If M is free k-mod, F ∈ Pd ,k :
Lnd−iF (M; n) ' H i (ΘnF )(M)
Cor : Ld−iF (M, 1) = ExtiPd,k(Λd ,FM),
L2d−iF (M, 2) = ExtiPd,k(Sd ,FM)
The proof
Easy. Ingredients :
1. Classical computation : L∗Sd(M, 1) ' Λd(M)[d ]
equivalent forms :I Koszul dual of S(M) is Λ(M),I Ext∗S(M)(k,k) = Λ(M[1]),I ΩCP∞ ' S1,
2. Standard simplicial algebra (EZ-thm. . .)3. Elementary homological algebra in Pd ,k (Yoneda lemma).
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :
Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .
Question : Can we compute Ext∗Pnd,k(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H,Sn ⊗d) is not easy.
- 17/1
IV. Applications (1)
1. Application to GLn,k-modules.
A) Plethysm problem.
Let M ∈ GLn,k-mod.Plethysm = GLn,k-mod of the form F (M), F functor.
Difficult Problem : understand F (M).• Composition factors ?• Extensions Ext∗GLn,k
(N,F (M)) ?
B) Translation to Pk :Plethysm = functor of the form F GProblem : understand F G
Example : Sn ⊗d .Question : Can we compute Ext∗Pnd,k
(H,Sn ⊗d) ?
Rk : no injective resolution of Sn ⊗d is known,already HomPnd,k(H, Sn ⊗d) is not easy.
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(H
Λnd
,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).
I We get : L∗(Sn ⊗d)(k, 1) = L∗S
n(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k,
HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k, HiΘ(G ) = 0 if i 6= 0
(Sd , Λd , ⊗d , Schur functors. . .)ExtiPnd,k
(Λnd ,F GM) ' (HiΘdF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k, HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k, HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 18/1
IV. Applications (2)
Question : Can we compute Ext∗Pnd,k(
H
Λnd ,Sn ⊗d) ?
C) Translation in derived functors
Question : Can we compute L∗(Sn ⊗d)(k, 1) ?
I Recall L∗(Sn ⊗d)(k, 1) = homology of
N (Sn ⊗d)K(k[1])
NSn(K(k[1])⊗d)
I EZ-thm + DK correspondence imply : (K(k[1])⊗d) ≈ K(k[d ]).I We get : L∗(S
n ⊗d)(k, 1) = L∗Sn(k, d)
D) In general :
Thm : F ∈ Pn,k, G ∈ Pd ,k, HiΘ(G ) = 0 if i 6= 0(Sd , Λd , ⊗d , Schur functors. . .)
ExtiPnd,k(Λnd ,F GM) ' (HiΘ
dF ) (H0ΘG ) (M)
Ex : Ext∗Pnd,k(Λnd ,F Sd) = 0
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.
• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.
• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.
I By univ. coeff. thm, it suffices to prove L∗YdFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.
• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.
• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.
• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.
• A abelian category (+finiteness hyp),blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.• A abelian category (+finiteness hyp),
blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.
• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.• A abelian category (+finiteness hyp),
blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]
• Λd → (1d), Y d → (2, 1d−2).
- 19/1
IV. Applications (3)
2. Application to derived functors.k = Z.Let Y d = kernel of Λd−1 ⊗ Λ1 mult−−−→ Λd Y d ∈ Pd ,Z.
The Y ds appear in the study of the Curtis spectral sequence.
Thm : If p 6 |d , then the p-torsion part of L∗Yd(Z, 1) is zero.
Proof :I We have to prove L∗Y
d(Z, 1)⊗ Fp = 0.I By univ. coeff. thm, it suffices to prove L∗Y
dFp
(Fp, 1) = 0.
I So it suffices to prove Ext∗Pd,Fp(Λd ,Y d
Fp) = 0.
I This follows without any computation from block theory.• A abelian category (+finiteness hyp),
blocks=simples/ ≡, S ≡ S ′ if Ext∗(S ,S ′) 6= 0A = ⊕Ab.• case A = Pd ,Fp : simples indexed by partitions of dSλ ≡ Sµ iff λ and µ have same p-core. [Donkin]• Λd → (1d), Y d → (2, 1d−2).
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)
(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)
(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)
(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)
(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2),
in general ? ?
- 20/1
V. The homology of EML spaces (1)
Work in progress with. L. Breen and R. Mikhailov.
k = Z, A is an abelian group.
1. LiSd(A, n) important for representation theory of GLn,Z
(A free abelian), ex :I ExtiPd,Z
(Λd , ΓdA) = L3d−iS
d(A, 3)
I ExtiPnd,Z(Λnd ,Sn ⊗d) = Lnd−iS
n(A⊗d , d)
2. Sd
I is one of the less complicated functor,I appears everywhere,
3. Thm : [Dold-Puppe] H∗K (A, n) ' L∗S(A, n)(A free abelian group, otherwise iso up to filtration)
We want to compute L∗Sd(A, n) !
Easy cases : L∗Sd(A, 1), L∗S
d(A, 2), in general ? ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗
Xp(A, n)
φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)
φp−→ H∗K (A, n)
Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.
I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations
, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !
I Hi (A, n) '⊕
d≥0 LiSd(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]”
”. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !I Hi (A, n) '
⊕d≥0 LiS
d(A, n), how do we get the summands ?
- 21/1
V. The homology of EML spaces (2)
Problem : Compute L∗Sd(A, n).
L∗S(A, n) ' H∗K (A, n), and we know homology of EML spaces.
”everything is computed in the Cartan Seminar [54]””. . .but not under an easy-to-use form !”
[Cartan seminar, tome 7, 1954, expose. 11] :
Enormous DGA
H∗Xp(A, n)φp−→ H∗K (A, n)Enormous algebra
I φp surjects on p-primary part.I kerφp is generated by 4 pages of relations.
With a lot of sweat one can (theoretically) extract a description ofHiK (A, n) by (infinite number of) generators and relations, but :
I Hi (A, n) is finitely generated abelian group !I Hi (A, n) '
⊕d≥0 LiS
d(A, n), how do we get the summands ?
- 22/1
V. The homology of EML spaces (3)
In 1954, Cartan could not use :
1. theory of derived functors,2. the fact that H∗K (A, n) ' LiS
d(A, n) is strict polynomial,3. representation theory of GLn,Z.
Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS
d(A, 3))
Thank You !
- 22/1
V. The homology of EML spaces (3)
In 1954, Cartan could not use :
1. theory of derived functors,
2. the fact that H∗K (A, n) ' LiSd(A, n) is strict polynomial,
3. representation theory of GLn,Z.
Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS
d(A, 3))
Thank You !
- 22/1
V. The homology of EML spaces (3)
In 1954, Cartan could not use :
1. theory of derived functors,2. the fact that H∗K (A, n) ' LiS
d(A, n) is strict polynomial,
3. representation theory of GLn,Z.
Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS
d(A, 3))
Thank You !
- 22/1
V. The homology of EML spaces (3)
In 1954, Cartan could not use :
1. theory of derived functors,2. the fact that H∗K (A, n) ' LiS
d(A, n) is strict polynomial,3. representation theory of GLn,Z.
Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS
d(A, 3))
Thank You !
- 22/1
V. The homology of EML spaces (3)
In 1954, Cartan could not use :
1. theory of derived functors,2. the fact that H∗K (A, n) ' LiS
d(A, n) is strict polynomial,3. representation theory of GLn,Z.
Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS
d(A, 3))
Thank You !
- 22/1
V. The homology of EML spaces (3)
In 1954, Cartan could not use :
1. theory of derived functors,2. the fact that H∗K (A, n) ' LiS
d(A, n) is strict polynomial,3. representation theory of GLn,Z.
Hope :Using these additional tools, we hope to get usable results.(ex : complete results for LiS
d(A, 3))
Thank You !