SLIDE 1
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Theorem Let A be a commutative Q-algebra
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1 / 24 Theorem Let A be a commutative Q -algebra 1 / 24 Theorem - - PowerPoint PPT Presentation
1 / 24 Theorem Let A be a commutative Q -algebra 1 / 24 Theorem - - PowerPoint PPT Presentation
1 / 24 Theorem Let A be a commutative Q -algebra 1 / 24 Theorem Let A be a commutative Q -algebra, a , b A such that a 3 b + a = b 3 a + b = a 2 b 2 + 1 = 0 . 1 / 24 Theorem Let A be a commutative Q -algebra, a , b A such that a 3 b + a
SLIDE 2
SLIDE 3
Theorem Let A be a commutative Q-algebra, a, b ∈ A such that a3b + a = b3a + b = a2b2 + 1 = 0.
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SLIDE 4
Theorem Let A be a commutative Q-algebra, a, b ∈ A such that a3b + a = b3a + b = a2b2 + 1 = 0. Then 1 = 0 in A.
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Theorem Let A be a commutative Q-algebra, a, b ∈ A such that a3b + a = b3a + b = a2b2 + 1 = 0. Then 1 = 0 in A. Proof
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Theorem Let A be a commutative Q-algebra, a, b ∈ A such that a3b + a = b3a + b = a2b2 + 1 = 0. Then 1 = 0 in A. Proof By a Gröbner basis computation, the theorem holds for U := Q[x, y] x3y + x, y3x + y, x2y2 + 1
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SLIDE 7
Theorem Let A be a commutative Q-algebra, a, b ∈ A such that a3b + a = b3a + b = a2b2 + 1 = 0. Then 1 = 0 in A. Proof By a Gröbner basis computation, the theorem holds for U := Q[x, y] x3y + x, y3x + y, x2y2 + 1 Claim follows from the universal property of U
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SLIDE 8
Theorem Let A be a commutative Q-algebra, a, b ∈ A such that a3b + a = b3a + b = a2b2 + 1 = 0. Then 1 = 0 in A. Proof By a Gröbner basis computation, the theorem holds for U := Q[x, y] x3y + x, y3x + y, x2y2 + 1 Claim follows from the universal property of U: {x, y} U A ∃! algebra homomorphism
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SLIDE 9
Q: Can we apply a similiar proof strategy for the Snake lemma?
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Applications of computable free abelian categories
Sebastian Posur July 18, 2019
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SLIDE 11
1
The Adelman category
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SLIDE 12
1
The Adelman category
2
Software demo
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SLIDE 13
1
The Adelman category
2
Software demo
3
Towards computable Serre quotients of the Adelman category
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SLIDE 14
1
The Adelman category
2
Software demo
3
Towards computable Serre quotients of the Adelman category
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SLIDE 15
Free abelian categories
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SLIDE 16
Free abelian categories
Let A be an additive category.
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SLIDE 17
Free abelian categories
Let A be an additive category. Data of the free abelian category
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Free abelian categories
Let A be an additive category. Data of the free abelian category an abelian category U
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Free abelian categories
Let A be an additive category. Data of the free abelian category an abelian category U an additive functor A → U
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Free abelian categories
Let A be an additive category. Data of the free abelian category an abelian category U an additive functor A → U For every additive functor A → B into an abelian category:
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Free abelian categories
Let A be an additive category. Data of the free abelian category an abelian category U an additive functor A → U For every additive functor A → B into an abelian category: A U B
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Free abelian categories
Let A be an additive category. Data of the free abelian category an abelian category U an additive functor A → U For every additive functor A → B into an abelian category: A U B ∃! up to nat. iso., exact
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SLIDE 23
How to construct free abelian categories
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SLIDE 24
How to construct free abelian categories
Existence: Peter Freyd
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SLIDE 25
How to construct free abelian categories
Existence: Peter Freyd Explicit construction: Murray Adelman
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SLIDE 26
How to construct free abelian categories
Existence: Peter Freyd Explicit construction: Murray Adelman
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SLIDE 27
Homology of composable arrows
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SLIDE 28
Homology of composable arrows
How to compute homology in an abelian category:
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SLIDE 29
Homology of composable arrows
How to compute homology in an abelian category: a b c α β
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) α β
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) cok(α) α β
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) cok(α) α β
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) cok(α) H(a
α
− → b
β
− → c) α β
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) cok(α) H(a
α
− → b
β
− → c) α β
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) cok(α) H(a
α
− → b
β
− → c) α β This construction makes sense even if we don’t have α · β = 0.
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Homology of composable arrows
How to compute homology in an abelian category: a b c ker(β) cok(α) H(a
α
− → b
β
− → c) α β This construction makes sense even if we don’t have α · β = 0. Adelman’s observation Equipping an additive category with "homologies" makes it abelian.
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SLIDE 37
Adelman category
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Adelman category
Let A be additive.
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Adelman category
Let A be additive. Adelman category: data structures
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data:
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A.
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SLIDE 42
Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms:
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ β
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ β
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ β
?
=
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ γ β
?
=
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ γ ρ β
?
=
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SLIDE 50
Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ γ ρ β σ1 σ2
?
=
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Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ γ ρ β σ1 σ2
?
= σ1 · ρ + γ · σ2 = β
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SLIDE 52
Adelman category
Let A be additive. Adelman category: data structures The Adelman category Adel(A) is given by the following data: An object in Adel(A) is a composable pair (a → b → c) in A. Morphisms: a b c a′ b′ c′ γ ρ β σ1 σ2 = σ1 · ρ + γ · σ2 = β
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SLIDE 53
Features of Adel(A)
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Features of Adel(A)
Canonical embedding A − → Adel(A)
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Features of Adel(A)
Canonical embedding A − → Adel(A) a → (0 → a → 0)
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Features of Adel(A)
Canonical embedding A − → Adel(A) a → (0 → a → 0) Duality
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Features of Adel(A)
Canonical embedding A − → Adel(A) a → (0 → a → 0) Duality Adel(A)
∼
− → Adel(Aop)op
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Features of Adel(A)
Canonical embedding A − → Adel(A) a → (0 → a → 0) Duality Adel(A)
∼
− → Adel(Aop)op (a
ρ
− → b
γ
− → c) → (c
γop
− − → b
ρop
− − → a)
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Features of Adel(A)
Canonical embedding A − → Adel(A) a → (0 → a → 0) Duality Adel(A)
∼
− → Adel(Aop)op (a
ρ
− → b
γ
− → c) → (c
γop
− − → b
ρop
− − → a) if Adel(A) has cokernels, then it also has kernels
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SLIDE 60
How to compute cokernels
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How to compute cokernels
Construction of the cokernel object
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How to compute cokernels
Construction of the cokernel object Input: a b c a′ b′ c′ γ ρ′ γ′ β
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How to compute cokernels
Construction of the cokernel object Input: a b c a′ b′ c′ γ ρ′ γ′ β Output: a′ ⊕ b b′ ⊕ c c′ ⊕ c ρ′ β γ
- γ′
idc
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How to compute cokernels
Construction of the cokernel object Input: a b c a′ b′ c′ γ ρ′ γ′ β Output: a′ ⊕ b b′ ⊕ c c′ ⊕ c ρ′ β γ
- γ′
idc
- Formal bookkeeping of data relevant for the cokernel.
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How to compute cokernels
Construction of the cokernel object Input: a b c a′ b′ c′ γ ρ′ γ′ β Output: a′ ⊕ b b′ ⊕ c c′ ⊕ c ρ′ β γ
- γ′
idc
- Formal bookkeeping of data relevant for the cokernel.
Adel(A) has cokernels and kernels.
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How to compute cokernels
Construction of the cokernel object Input: a b c a′ b′ c′ γ ρ′ γ′ β Output: a′ ⊕ b b′ ⊕ c c′ ⊕ c ρ′ β γ
- γ′
idc
- Formal bookkeeping of data relevant for the cokernel.
Adel(A) has cokernels and kernels. Even better: it is abelian.
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Adelman’s theorem
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Adelman’s theorem
Let A be an additive category.
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Adelman’s theorem
Let A be an additive category. Theorem
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A.
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows:
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows: A Adel(A)
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows: A Adel(A) B F
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows: A Adel(A) B F (a α − → b
β
− → c)
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows: A Adel(A) B F (a α − → b
β
− → c) − →
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows: A Adel(A) B F (a α − → b
β
− → c) − → H(Fa Fα − − → Fb
Fβ
− − → Fc)
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Adelman’s theorem
Let A be an additive category. Theorem A ֒ → Adel(A) is the free abelian category of A. Its universal property is given as follows: A Adel(A) B F (a α − → b
β
− → c) − → H(Fa Fα − − → Fb
Fβ
− − → Fc)
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Computability of Adelman categories
Adel(A) provides a computable model of the free abelian category whenever we can solve 2-sided linear systems in A.
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1
The Adelman category
2
Software demo
3
Towards computable Serre quotients of the Adelman category
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SLIDE 80
Software demo
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SLIDE 81
1
The Adelman category
2
Software demo
3
Towards computable Serre quotients of the Adelman category
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SLIDE 82
The membership problem
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The membership problem
How do we model the free abelian category of sequences F1 F2 F3 F4 α β γ that are exact at F2 and F3?
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The membership problem
How do we model the free abelian category of sequences F1 F2 F3 F4 C : α · β = 0 β · γ = 0 α β γ that are exact at F2 and F3?
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The membership problem
How do we model the free abelian category of sequences F1 F2 F3 F4 C : α · β = 0 β · γ = 0 α β γ that are exact at F2 and F3? By a Serre quotient: Adel(AdditiveClosure(C)) H(α, β) ⊕ H(β, γ)Serre
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The membership problem
How do we model the free abelian category of sequences F1 F2 F3 F4 C : α · β = 0 β · γ = 0 α β γ that are exact at F2 and F3? By a Serre quotient: Adel(AdditiveClosure(C)) H(α, β) ⊕ H(β, γ)Serre Theorem (Barakat, Lange-Hegermann) Let C ⊆ A be a Serre subcategory of a computable abelian category with an algorithm for deciding A ∈ C. Then A
C is computable abelian.
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SLIDE 87
The membership problem
How do we model the free abelian category of sequences F1 F2 F3 F4 C : α · β = 0 β · γ = 0 α β γ that are exact at F2 and F3? By a Serre quotient: Adel(AdditiveClosure(C)) H(α, β) ⊕ H(β, γ)Serre Theorem (Barakat, Lange-Hegermann) Let C ⊆ A be a Serre subcategory of a computable abelian category with an algorithm for deciding A ∈ C. Then A
C is computable abelian.
study the membership problem in Serre subcategories of free abelian categories generated by a single object.
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SLIDE 88
The membership problem
How do we model the free abelian category of sequences F1 F2 F3 F4 C : α · β = 0 β · γ = 0 α β γ that are exact at F2 and F3? By a Serre quotient: Adel(AdditiveClosure(C)) H(α, β) ⊕ H(β, γ)Serre Theorem (Barakat, Lange-Hegermann) Let C ⊆ A be a Serre subcategory of a computable abelian category with an algorithm for deciding A ∈ C. Then A
C is computable abelian.
study the membership problem in Serre subcategories of free abelian categories generated by a single object. Note: A ∈ BSerre ⇐ ⇒ A ⊕ BSerre = BSerre
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SLIDE 89
Free abelian categories as functor categories
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SLIDE 90
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV.
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SLIDE 91
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation
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SLIDE 92
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups
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SLIDE 93
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab
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Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab A-mod: subcat. of A-Mod gen. by finitely presented functors
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Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab A-mod: subcat. of A-Mod gen. by finitely presented functors A Adel(A) (A-mod)-mod
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SLIDE 96
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab A-mod: subcat. of A-Mod gen. by finitely presented functors A Adel(A) (A-mod)-mod a − → eva
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SLIDE 97
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab A-mod: subcat. of A-Mod gen. by finitely presented functors A Adel(A) (A-mod)-mod a − → eva where eva : A-mod → Ab : m → m(a)
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SLIDE 98
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab A-mod: subcat. of A-Mod gen. by finitely presented functors A Adel(A) (A-mod)-mod a − → eva where eva : A-mod → Ab : m → m(a)
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SLIDE 99
Free abelian categories as functor categories
To understand Serre subcategories of Adel(A), we change our POV. Notation Ab: category of abelian groups A-Mod: category of additive functors A → Ab A-mod: subcat. of A-Mod gen. by finitely presented functors A Adel(A) (A-mod)-mod a − → eva ∼ where eva : A-mod → Ab : m → m(a)
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SLIDE 100
Definable subcategory
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SLIDE 101
Definable subcategory
Advantage of new POV: functors can be evaluated.
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SLIDE 102
Definable subcategory
Advantage of new POV: functors can be evaluated. Definition F ∈ (A-mod)-mod, we define its associated definable subcategory V(F) := {m ∈ A-Mod | − → F (m) ≃ 0} ⊆ A-Mod where − → F : A-Mod → Ab denotes the extension by filtered colimits of F.
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SLIDE 103
Definable subcategory
Advantage of new POV: functors can be evaluated. Definition F ∈ (A-mod)-mod, we define its associated definable subcategory V(F) := {m ∈ A-Mod | − → F (m) ≃ 0} ⊆ A-Mod where − → F : A-Mod → Ab denotes the extension by filtered colimits of F. Theorem For F1, F2 ∈ (A-mod)-mod, we have F1Serre = F2Serre ⇐ ⇒ V(F1) = V(F2)
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SLIDE 104
The case of Dedekind domains
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SLIDE 105
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain.
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SLIDE 106
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2)
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SLIDE 107
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
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SLIDE 108
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
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SLIDE 109
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
2
lim ← −i R/pi,
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SLIDE 110
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
2
lim ← −i R/pi,
3
E(R/p) (injective hull),
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SLIDE 111
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
2
lim ← −i R/pi,
3
E(R/p) (injective hull), for p ∈ mSpec(R)
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SLIDE 112
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
2
lim ← −i R/pi,
3
E(R/p) (injective hull), for p ∈ mSpec(R), and
4
Quot(R).
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SLIDE 113
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
2
lim ← −i R/pi,
3
E(R/p) (injective hull), for p ∈ mSpec(R), and
4
Quot(R). Remark (Z-mod)-mod is the free abelian category generated by a single object.
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SLIDE 114
The case of Dedekind domains
The case of Dedekind domains Let R be a Dedekind domain. For F1, F2 ∈ (R-mod)-mod, we have V(F1) = V(F2) if and only if − → F1, − → F2 vanish on the same modules of the following list:
1
R/pn , n ≥ 1,
2
lim ← −i R/pi,
3
E(R/p) (injective hull), for p ∈ mSpec(R), and
4
Quot(R). Remark (Z-mod)-mod is the free abelian category generated by a single object. Already in this case calculating V(F1) = V(F2) seems non-trivial.
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SLIDE 115
Algorithmic test for Quot(R) and E(R/p)
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SLIDE 116
Algorithmic test for Quot(R) and E(R/p)
FB : (A-mod)-mod → (A-mod)op : cok(Hom(α, −)) → ker(α)
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SLIDE 117
Algorithmic test for Quot(R) and E(R/p)
FB : (A-mod)-mod → (A-mod)op : cok(Hom(α, −)) → ker(α) Lemma Suppose given F1, F2 ∈ (R-mod)-mod for R a Dedekind domain.
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SLIDE 118
Algorithmic test for Quot(R) and E(R/p)
FB : (A-mod)-mod → (A-mod)op : cok(Hom(α, −)) → ker(α) Lemma Suppose given F1, F2 ∈ (R-mod)-mod for R a Dedekind domain.Then FB(F1)Serre = FB(F2)Serre as Serre subcategories of (R-mod)op
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SLIDE 119
Algorithmic test for Quot(R) and E(R/p)
FB : (A-mod)-mod → (A-mod)op : cok(Hom(α, −)) → ker(α) Lemma Suppose given F1, F2 ∈ (R-mod)-mod for R a Dedekind domain.Then FB(F1)Serre = FB(F2)Serre as Serre subcategories of (R-mod)op if and only if − → F1, − → F2 vanish on the same modules of the set {Quot(R), E(R/p) | p ∈ mSpec(R)}
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SLIDE 120
Algorithmic test for Quot(R) and E(R/p)
FB : (A-mod)-mod → (A-mod)op : cok(Hom(α, −)) → ker(α) Lemma Suppose given F1, F2 ∈ (R-mod)-mod for R a Dedekind domain.Then FB(F1)Serre = FB(F2)Serre as Serre subcategories of (R-mod)op if and only if − → F1, − → F2 vanish on the same modules of the set {Quot(R), E(R/p) | p ∈ mSpec(R)} Deciding equality of such Serre subcategories in (R-mod)op boils down to computing supports in Spec(R) algorithm
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SLIDE 121
Algorithmic test for Quot(R) and E(R/p)
FB : (A-mod)-mod → (A-mod)op : cok(Hom(α, −)) → ker(α) Lemma Suppose given F1, F2 ∈ (R-mod)-mod for R a Dedekind domain.Then FB(F1)Serre = FB(F2)Serre as Serre subcategories of (R-mod)op if and only if − → F1, − → F2 vanish on the same modules of the set {Quot(R), E(R/p) | p ∈ mSpec(R)} Deciding equality of such Serre subcategories in (R-mod)op boils down to computing supports in Spec(R) algorithm Remark: the case {Quot(R), lim ← −i R/pi | p ∈ mSpec(R)} is similar.
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SLIDE 122
Algorithmic test for R/pn
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SLIDE 123
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length.
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SLIDE 124
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn))
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SLIDE 125
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant.
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SLIDE 126
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod
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SLIDE 127
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n)
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SLIDE 128
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn)
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SLIDE 129
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n
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SLIDE 130
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n)
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SLIDE 131
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n) = lengthR Hom(R/ql, R/pn)
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SLIDE 132
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n) = lengthR Hom(R/ql, R/pn) = δq,p · min(l, n)
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SLIDE 133
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n) = lengthR Hom(R/ql, R/pn) = δq,p · min(l, n) We can compute a projective resolution (length ≤ 2) of F
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SLIDE 134
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n) = lengthR Hom(R/ql, R/pn) = δq,p · min(l, n) We can compute a projective resolution (length ≤ 2) of F, thus, we can compute HF
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SLIDE 135
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n) = lengthR Hom(R/ql, R/pn) = δq,p · min(l, n) We can compute a projective resolution (length ≤ 2) of F, thus, we can compute HF, from which we read off the zeros
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SLIDE 136
Algorithmic test for R/pn
If F ∈ (R-mod)-mod, then F(R/pn) is an R-module of finite length. Definition (Hilbert function) HF : (p, n) → lengthR (F(R/pn)) Remark: HF is an additive invariant. HF of projectives in (R-mod)-mod HHom(R,−)(p, n) = lengthR Hom(R, R/pn) = n HHom(R/ql,−)(p, n) = lengthR Hom(R/ql, R/pn) = δq,p · min(l, n) We can compute a projective resolution (length ≤ 2) of F, thus, we can compute HF, from which we read off the zeros algorithm
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SLIDE 137
How do we computationally model the free abelian category of sequences F1 F2 F3 F4 C : α · β = 0 β · γ = 0 α β γ that are exact at F2 and F3?
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SLIDE 138
References I
Murray Adelman, Abelian categories over additive ones, J. Pure
- Appl. Algebra 3 (1973), 103–117. MR 0318265
Mohamed Barakat and Markus Lange-Hegermann, Gabriel morphisms and the computability of Serre quotients with applications to coherent sheaves, (arXiv:1409.2028), 2014. Peter Freyd, Representations in abelian categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 95–120. MR 0209333 Mike Prest, Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009. MR 2530988
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