1 / 24 Theorem Let A be a commutative Q -algebra 1 / 24 Theorem - PowerPoint PPT Presentation

Features of Adel( A ) Canonical embedding A − → Adel( A ) a �→ ( 0 → a → 0 ) Duality ∼ → Adel( A op ) op Adel( A ) − γ op ρ op ρ γ ( a → b − − → c ) �→ ( c − − → b − − → a ) � if Adel( A ) has cokernels, then it also has kernels 10 / 24

How to compute cokernels 11 / 24

How to compute cokernels Construction of the cokernel object 11 / 24

How to compute cokernels Construction of the cokernel object γ a c b Input : β ρ ′ γ ′ a ′ b ′ c ′ 11 / 24

How to compute cokernels Construction of the cokernel object γ a c b Input : β ρ ′ γ ′ a ′ b ′ c ′ Output : � ρ ′ � � γ ′ � 0 0 β γ 0 id c a ′ ⊕ b b ′ ⊕ c c ′ ⊕ c 11 / 24

How to compute cokernels Construction of the cokernel object γ a c b Input : β ρ ′ γ ′ a ′ b ′ c ′ Output : � ρ ′ � � γ ′ � 0 0 β γ 0 id c a ′ ⊕ b b ′ ⊕ c c ′ ⊕ c Formal bookkeeping of data relevant for the cokernel. 11 / 24

How to compute cokernels Construction of the cokernel object γ a c b Input : β ρ ′ γ ′ a ′ b ′ c ′ Output : � ρ ′ � � γ ′ � 0 0 β γ 0 id c a ′ ⊕ b b ′ ⊕ c c ′ ⊕ c Formal bookkeeping of data relevant for the cokernel. Adel( A ) has cokernels and kernels. 11 / 24

How to compute cokernels Construction of the cokernel object γ a c b Input : β ρ ′ γ ′ a ′ b ′ c ′ Output : � ρ ′ � � γ ′ � 0 0 β γ 0 id c a ′ ⊕ b b ′ ⊕ c c ′ ⊕ c Formal bookkeeping of data relevant for the cokernel. Adel( A ) has cokernels and kernels. Even better: it is abelian. 11 / 24

Adelman’s theorem 12 / 24

Adelman’s theorem Let A be an additive category. 12 / 24

Adelman’s theorem Let A be an additive category. Theorem 12 / 24

Adelman’s theorem Let A be an additive category. Theorem A ֒ → Adel( A ) is the free abelian category of A . 12 / 24

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: 12 / 24

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: Adel( A ) A 12 / 24

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: Adel( A ) A F B 12 / 24

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: β ( a α Adel( A ) − → b − → c ) A F B 12 / 24

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: β ( a α Adel( A ) − → b − → c ) �− A → F B 12 / 24

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: β ( a α Adel( A ) − → b − → c ) �− A → F H( Fa F α F β B − − → Fb − − → Fc ) 12 / 24

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: β ( a α Adel( A ) − → b − → c ) �− A → F H( Fa F α F β B − − → Fb − − → Fc ) 12 / 24

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 . 13 / 24

The Adelman category 1 Software demo 2 Towards computable Serre quotients of the Adelman category 3 14 / 24

Software demo 15 / 24

The Adelman category 1 Software demo 2 Towards computable Serre quotients of the Adelman category 3 16 / 24

The membership problem 17 / 24

The membership problem How do we model the free abelian category of sequences β γ α F 3 F 1 F 2 F 4 that are exact at F 2 and F 3 ? 17 / 24

The membership problem How do we model the free abelian category of sequences β γ α · β = 0 α F 3 C : F 1 F 2 F 4 β · γ = 0 that are exact at F 2 and F 3 ? 17 / 24

The membership problem How do we model the free abelian category of sequences β γ α · β = 0 α F 3 C : F 1 F 2 F 4 β · γ = 0 that are exact at F 2 and F 3 ? By a Serre quotient: Adel( AdditiveClosure ( C )) � H( α, β ) ⊕ H( β, γ ) � Serre 17 / 24

The membership problem How do we model the free abelian category of sequences β γ α · β = 0 α F 3 C : F 1 F 2 F 4 β · γ = 0 that are exact at F 2 and F 3 ? 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. 17 / 24

The membership problem How do we model the free abelian category of sequences β γ α · β = 0 α F 3 C : F 1 F 2 F 4 β · γ = 0 that are exact at F 2 and F 3 ? 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. 17 / 24

The membership problem How do we model the free abelian category of sequences β γ α · β = 0 α F 3 C : F 1 F 2 F 4 β · γ = 0 that are exact at F 2 and F 3 ? 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 ∈ � B � Serre ⇐ ⇒ � A ⊕ B � Serre = � B � Serre 17 / 24

Free abelian categories as functor categories 18 / 24

Free abelian categories as functor categories To understand Serre subcategories of Adel( A ) , we change our POV. 18 / 24

Free abelian categories as functor categories To understand Serre subcategories of Adel( A ) , we change our POV. Notation 18 / 24

Free abelian categories as functor categories To understand Serre subcategories of Adel( A ) , we change our POV. Notation Ab : category of abelian groups 18 / 24

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 18 / 24

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 18 / 24

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 Adel( A ) A ( A - mod ) - mod 18 / 24

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 Adel( A ) A a �− → ev a ( A - mod ) - mod 18 / 24

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 Adel( A ) A a �− → ev a ( A - mod ) - mod where ev a : A - mod → Ab : m �→ m ( a ) 18 / 24

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 Adel( A ) A a �− → ev a ( A - mod ) - mod where ev a : A - mod → Ab : m �→ m ( a ) 18 / 24

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 Adel( A ) A ∼ a �− → ev a ( A - mod ) - mod where ev a : A - mod → Ab : m �→ m ( a ) 18 / 24

Definable subcategory 19 / 24