Applications of the Defect to Representation Theory Jeremy Russell - - PowerPoint PPT Presentation

Applications of the Defect to Representation Theory

Jeremy Russell

The College of New Jersey

Representation Theory Seminar January 23, 2015

Goals of The Talk

1 Recall Auslander’s construction of the defect

w: fp(A, Ab) − → A

Goals of The Talk

1 Recall Auslander’s construction of the defect

w: fp(A, Ab) − → A

2 Explain how the defect “detects” almost split sequences.

Goals of The Talk

1 Recall Auslander’s construction of the defect

w: fp(A, Ab) − → A

2 Explain how the defect “detects” almost split sequences. 3 Explain how the defect determines the dual of a finitely

presented functor.

Goals of The Talk

1 Recall Auslander’s construction of the defect

w: fp(A, Ab) − → A

2 Explain how the defect “detects” almost split sequences. 3 Explain how the defect determines the dual of a finitely

presented functor.

4 Show that the category fp(A, Ab) has enough injectives

whenever A has enough projectives.

Goals of The Talk

1 Recall Auslander’s construction of the defect

w: fp(A, Ab) − → A

2 Explain how the defect “detects” almost split sequences. 3 Explain how the defect determines the dual of a finitely

presented functor.

4 Show that the category fp(A, Ab) has enough injectives

whenever A has enough projectives.

5 Show how these injectives produce a functor

D: fp(Mod(Rop), Ab) − → fp(Mod(R), Ab) which restricts to Auslander’s duality for a coherent ring R

Notation

Notation: Ab - category of abelian groups.

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra.

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra. R - ring.

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra. R - ring. Mod(R) - category of right modules

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra. R - ring. Mod(R) - category of right modules Mod(Rop) - category of left modules.

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra. R - ring. Mod(R) - category of right modules Mod(Rop) - category of left modules. mod(R) - category of finitely presented right modules

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra. R - ring. Mod(R) - category of right modules Mod(Rop) - category of left modules. mod(R) - category of finitely presented right modules mod(Rop) - category of finitely presented left modules

Notation

Notation: Ab - category of abelian groups. Λ - finite dimensional k-algebra. R - ring. Mod(R) - category of right modules Mod(Rop) - category of left modules. mod(R) - category of finitely presented right modules mod(Rop) - category of finitely presented left modules A - abelian category (need not be small).

The Functor Category

The Functor Category

Definition The category (A, Ab) consists of all additive covariant functors F : A → Ab together with the natural transformations between them.

The Functor Category

Definition The category (A, Ab) consists of all additive covariant functors F : A → Ab together with the natural transformations between them. Definition A functor F : A → Ab is called representable if F ∼ = HomA(X, ) for some X ∈ A.

Notation For Representable Functors

Notation: We will abbreviate the representable functor HomA(X, ) by (X, ) and abbreviate Hom(A,Ab)(F, ) by Nat(F, )

• r

(F, ) depending on the situation.

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab.

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX The category (A, Ab) is abelian.

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX The category (A, Ab) is abelian. A natural transformation α: F → G is

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX The category (A, Ab) is abelian. A natural transformation α: F → G is

a monomorphism ⇐ ⇒ αX is a monomorphism for all X ∈ A.

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX The category (A, Ab) is abelian. A natural transformation α: F → G is

a monomorphism ⇐ ⇒ αX is a monomorphism for all X ∈ A. an epimorphism ⇐ ⇒ αX is an epimorphism for all X ∈ A.

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX The category (A, Ab) is abelian. A natural transformation α: F → G is

a monomorphism ⇐ ⇒ αX is a monomorphism for all X ∈ A. an epimorphism ⇐ ⇒ αX is an epimorphism for all X ∈ A. a kernel of β ⇐ ⇒ αX is a kernel of βX for all X ∈ A.

Properties of (A, Ab)

The functor category (A, Ab) has some interesting homological properties which are essentially inherited from the category Ab. Given α, β ∈ Nat(F, G) (α + β)X := αX + βX The category (A, Ab) is abelian. A natural transformation α: F → G is

a monomorphism ⇐ ⇒ αX is a monomorphism for all X ∈ A. an epimorphism ⇐ ⇒ αX is an epimorphism for all X ∈ A. a kernel of β ⇐ ⇒ αX is a kernel of βX for all X ∈ A. a cokernel of β ⇐ ⇒ αX is a cokernel of βX for all X ∈ A.

Exactness in (A, Ab)

A sequence in (A, Ab): F − → G − → H is exact if and only if F(X) − → G(X) − → H(X) is exact for all X ∈ A.

Exactness in (A, Ab)

A sequence in (A, Ab): F − → G − → H is exact if and only if F(X) − → G(X) − → H(X) is exact for all X ∈ A. All universal objects such as the kernel, cokernel, pullback, pushout, etc. are constructed componentwise.

Exactness in (A, Ab)

A sequence in (A, Ab): F − → G − → H is exact if and only if F(X) − → G(X) − → H(X) is exact for all X ∈ A. All universal objects such as the kernel, cokernel, pullback, pushout, etc. are constructed componentwise. For each X ∈ A, the evaluation functor evX : (A, Ab) → Ab is exact.

Yoneda’s Lemma

Lemma (Yoneda) For any X ∈ A and any F ∈ (A, Ab), Nat

• (X,

), F ∼ = F(X)

1 The isomorphism is defined by α → αX(1X). 2 The isomorphism is natural in both X and F.

Consequence of Yoneda’s Lemma

Take an exact sequence in (A, Ab): F G H

Consequence of Yoneda’s Lemma

Take an exact sequence in (A, Ab): F G H Apply Nat

• (X,

),

• :
• (X,

), F

• (X,

), G

• (X,

), H

Consequence of Yoneda’s Lemma

Take an exact sequence in (A, Ab): F G H Apply Nat

• (X,

),

• :
• (X,

), F

• (X,

), G

• (X,

), H

• F(X)

G(X) H(X) ∼ = ∼ = ∼ =

Consequence of Yoneda’s Lemma

Take an exact sequence in (A, Ab): F G H Apply Nat

• (X,

),

• :
• (X,

), F

• (X,

), G

• (X,

), H

• F(X)

G(X) H(X) ∼ = ∼ = ∼ = The bottom row is exact resulting in exactness of the top row.

Consequences of Yoneda’s Lemma

Therefore representable functors are projectives in (A, Ab).

Consequences of Yoneda’s Lemma

Therefore representable functors are projectives in (A, Ab). Proposition The Yoneda embedding Y: A → (A, Ab) defined by Y(X) = (X, ) is

1 full

Consequences of Yoneda’s Lemma

Therefore representable functors are projectives in (A, Ab). Proposition The Yoneda embedding Y: A → (A, Ab) defined by Y(X) = (X, ) is

1 full 2 faithful

Consequences of Yoneda’s Lemma

Therefore representable functors are projectives in (A, Ab). Proposition The Yoneda embedding Y: A → (A, Ab) defined by Y(X) = (X, ) is

1 full 2 faithful 3 left exact

Left Exactness of the Yoneda Embedding

Start with exact sequence 0 → A → B → C → 0

Left Exactness of the Yoneda Embedding

Start with exact sequence 0 → A → B → C → 0 Apply the left exact functor ( , X): 0 → (C, X) → (B, X) → (A, X)

Left Exactness of the Yoneda Embedding

Start with exact sequence 0 → A → B → C → 0 Apply the left exact functor ( , X): 0 → (C, X) → (B, X) → (A, X) Since the exactness of this sequence holds for all X ∈ A, the sequence 0 → (C, ) → (B, ) → (A, ) is exact in (A, Ab).

Finitely Presented Functors

Definition (Auslander) A functor F : A → Ab is called finitely presented if there exists exact sequence (Y, ) → (X, ) → F → 0 In other words, F is a cokernel of a representable transformation.

Finitely Presented Functors

Definition (Auslander) A functor F : A → Ab is called finitely presented if there exists exact sequence (Y, ) → (X, ) → F → 0 In other words, F is a cokernel of a representable transformation. Definition fp(A, Ab) = category of finitely presented functors.

Properites of Finitely Presented Functors

fp(A, Ab) (A, Ab) fp(A, Ab) is abelian

Properites of Finitely Presented Functors

fp(A, Ab) (A, Ab) fp(A, Ab) is abelian fp(A, Ab) has enough projectives and they are precisely the representable functors.

Properites of Finitely Presented Functors

fp(A, Ab) (A, Ab) fp(A, Ab) is abelian fp(A, Ab) has enough projectives and they are precisely the representable functors. All finitely presented functors have projective dimension at most 2: 0 → (Z, ) → (Y, ) → (X, ) → F → 0

Examples of Finitely Presented Functors

Proposition (Auslander)

1 Extn(X,

) ∈ fp(Mod(Rop), Ab).

Examples of Finitely Presented Functors

Proposition (Auslander)

1 Extn(X,

) ∈ fp(Mod(Rop), Ab).

2 X ⊗

∈ fp(Mod(Rop), Ab) if and only if X ∈ mod(R).

Construction of w: fp(A, Ab) → A

Auslander constructed a contravariant functor w: fp(A, Ab) → A

Construction of w: fp(A, Ab) → A

Auslander constructed a contravariant functor w: fp(A, Ab) → A Step 1: Start with presentation (Y, ) → (X, ) → F → 0

Construction of w: fp(A, Ab) → A

Auslander constructed a contravariant functor w: fp(A, Ab) → A Step 1: Start with presentation (Y, ) → (X, ) → F → 0 Step 2: By Yoneda (Y, ) → (X, ) comes from a unique morphism X → Y

Construction of w: fp(A, Ab) → A

Auslander constructed a contravariant functor w: fp(A, Ab) → A Step 1: Start with presentation (Y, ) → (X, ) → F → 0 Step 2: By Yoneda (Y, ) → (X, ) comes from a unique morphism X → Y Step 3: The exact sequence 0 → w(F) → X → Y completely determines w.

Properties

Proposition (Auslander)

1 w does not depend on any choices of presentation.

Properties

Proposition (Auslander)

1 w does not depend on any choices of presentation. 2 w is exact.

Properties

Proposition (Auslander)

1 w does not depend on any choices of presentation. 2 w is exact. 3 w(X,

) = X.

What w Measures

Take presentation of F: 0 → (Z, ) → (Y, ) → (X, ) → F → 0

What w Measures

Take presentation of F: 0 → (Z, ) → (Y, ) → (X, ) → F → 0 Apply w: 0 → w(F) → X → Y → Z → 0

What w Measures

Proposition (Auslander) For F ∈ fp(A, Ab) the following are equivalent:

What w Measures

Proposition (Auslander) For F ∈ fp(A, Ab) the following are equivalent:

1 w(F) = 0

What w Measures

Proposition (Auslander) For F ∈ fp(A, Ab) the following are equivalent:

1 w(F) = 0 2 All presentations of F arise from short exact sequences.

What w Measures

Proposition (Auslander) For F ∈ fp(A, Ab) the following are equivalent:

1 w(F) = 0 2 All presentations of F arise from short exact sequences. 3 There exists short exact sequence in A

0 → X → Y → Z → 0 such that the following is a presentation of F: 0 → (Z, ) → (Y, ) → (X, ) → F → 0

Construction of the Defect Sequence

Step 1: Choose any presentation of F: 0 − → (Z, ) − → (Y, ) − → (X, ) − → F − → 0

Construction of the Defect Sequence

Step 1: Choose any presentation of F: 0 − → (Z, ) − → (Y, ) − → (X, ) − → F − → 0 Step 2: Apply w to get a four term exact sequence in A: 0 − → w(F) − → X − → Y − → Z − → 0

Construction of the Defect Sequence

Step 3: Embed this diagram into the following commutative diagram with exact rows and columns in A:

• w(F)

w(F)

• w(F)

X

• Y

V

• Y

Construction of the Defect Sequence

Step 4:

• (Y,

) (V, )

• F0
• (Y,

) (X, )

• F
• (w(F),

)

• (w(F),

)

• F1
• F1

Zeroth Derived Functors

Proposition (Auslander) For any finitely presented functor F the defect sequence: 0 → F0 → F →

• w(F),
• → F1 → 0

is exact and the following assignments are functorial in F:

Zeroth Derived Functors

Proposition (Auslander) For any finitely presented functor F the defect sequence: 0 → F0 → F →

• w(F),
• → F1 → 0

is exact and the following assignments are functorial in F: F → F0

Zeroth Derived Functors

Proposition (Auslander) For any finitely presented functor F the defect sequence: 0 → F0 → F →

• w(F),
• → F1 → 0

is exact and the following assignments are functorial in F: F → F0 F → F1

Zeroth Derived Functors

Proposition (Auslander) For any finitely presented functor F the defect sequence: 0 → F0 → F →

• w(F),
• → F1 → 0

is exact and the following assignments are functorial in F: F → F0 F → F1 F →

• w(F),

More Properties of the Defect Sequence

Proposition (Auslander) From the defect sequence 0 → F0 → F →

• w(F),
• → F1 → 0
• ne easily verifies the following:

w(F0) = w(F1) = 0

More Properties of the Defect Sequence

Proposition (Auslander) From the defect sequence 0 → F0 → F →

• w(F),
• → F1 → 0
• ne easily verifies the following:

w(F0) = w(F1) = 0 F0, F1 vanish on injectives.

More Properties of the Defect Sequence

Proposition (Auslander) From the defect sequence 0 → F0 → F →

• w(F),
• → F1 → 0
• ne easily verifies the following:

w(F0) = w(F1) = 0 F0, F1 vanish on injectives. F and

• w(F),
• agree on injectives.

More Properties of the Defect Sequence

Proposition (Auslander) From the defect sequence 0 → F0 → F →

• w(F),
• → F1 → 0
• ne easily verifies the following:

w(F0) = w(F1) = 0 F0, F1 vanish on injectives. F and

• w(F),
• agree on injectives.

If A has enough injectives, then

• w(F),
• = R0F.

More Properties of the Defect Sequence

Proposition (Auslander) From the defect sequence 0 → F0 → F →

• w(F),
• → F1 → 0
• ne easily verifies the following:

w(F0) = w(F1) = 0 F0, F1 vanish on injectives. F and

• w(F),
• agree on injectives.

If A has enough injectives, then

• w(F),
• = R0F.

In this case F vanishes on injectives if and only if w(F) = 0.

Finite Dimensional k -Algebras

For a finite dimensional k - algebra Λ, the category mod(Λop) is abelian and has enough injectives.

Finite Dimensional k -Algebras

For a finite dimensional k - algebra Λ, the category mod(Λop) is abelian and has enough injectives. Every finitely presented left Λ-module M is a finite direct sum

• f indecomposables

M =

n

• i=1

Xi

Projective Covers

Definition Recall that an epimorphism f : P → X from a projective P to

• bject X is called a projective cover if

fh = f implies that h is an isomorphism.

Minimal Resolutions

Definition A projective resolution · · · Pk − → Pk−1 − → · · · P1 − → P0 − → X − → 0 is a minimal projective resolution if each Pn − → ΩnX is a projective cover.

fp(mod(Λop), Ab) Has Minimal Projective Resolutions

Proposition (Auslander) All finitely presented functors F : mod(Λop) → Ab have minimal projective resolutions.

fp(mod(Λop), Ab) Has Minimal Projective Resolutions

Proposition (Auslander) All finitely presented functors F : mod(Λop) → Ab have minimal projective resolutions. Given (X, ) → F → 0, one can take X to have smallest

• dimension. This will be a projective cover.

Simple Functors

Definition A functor S : A → Ab is called simple if S = 0 and any non-zero morphism F → S is an epimorphism.

Simple Functors

Definition A functor S : A → Ab is called simple if S = 0 and any non-zero morphism F → S is an epimorphism. Proposition (Auslander) For a simple functor S : mod(Λop) → Ab there exists a unique indecomposable N such that

1 S(N) = 0

Simple Functors

Definition A functor S : A → Ab is called simple if S = 0 and any non-zero morphism F → S is an epimorphism. Proposition (Auslander) For a simple functor S : mod(Λop) → Ab there exists a unique indecomposable N such that

1 S(N) = 0 2 There is a projective cover (N,

) → S → 0.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N).

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Step 2: Choose N from above to have smallest dimension.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Step 2: Choose N from above to have smallest dimension.

N will be indecomposable.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Step 2: Choose N from above to have smallest dimension.

N will be indecomposable. Otherwise N ∼ = A B.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Step 2: Choose N from above to have smallest dimension.

N will be indecomposable. Otherwise N ∼ = A B. S(N) = S(A) S(B) = 0.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Step 2: Choose N from above to have smallest dimension.

N will be indecomposable. Otherwise N ∼ = A B. S(N) = S(A) S(B) = 0. A and B have smaller dimension.

Sketch of Proof

Step 1: Find exact sequence (N, ) → S → 0.

Since S = 0, there exists N ∈ mod(Λop) such that S(N) = 0. Since S(N) = 0, there exists non-zero x ∈ S(N). x determines εx : (N, ) → S where εx(1N) = x. x = 0 implies εx = 0 Because S is simple, εx is an epimorphism.

Step 2: Choose N from above to have smallest dimension.

N will be indecomposable. Otherwise N ∼ = A B. S(N) = S(A) S(B) = 0. A and B have smaller dimension. Either S(A) = 0 or S(B) = 0.

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

This gives εx = εx(f n, ) for all n ≥ 1

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

This gives εx = εx(f n, ) for all n ≥ 1 ∴ f is not nilpotent.

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

This gives εx = εx(f n, ) for all n ≥ 1 ∴ f is not nilpotent. (N, N) is a local ring because N is indecomposable.

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

This gives εx = εx(f n, ) for all n ≥ 1 ∴ f is not nilpotent. (N, N) is a local ring because N is indecomposable. ∴ f must be an isomorphism.

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

This gives εx = εx(f n, ) for all n ≥ 1 ∴ f is not nilpotent. (N, N) is a local ring because N is indecomposable. ∴ f must be an isomorphism. ∴ (N, ) → S is a projective cover.

Sketch of Proof

Step 3: Show that εx : (N, ) → S → 0 is a projective cover.

Suppose that we have the following commutative diagram: (N, ) F (N, ) F

(f, ) 1 εx εx

This gives εx = εx(f n, ) for all n ≥ 1 ∴ f is not nilpotent. (N, N) is a local ring because N is indecomposable. ∴ f must be an isomorphism. ∴ (N, ) → S is a projective cover. Uniqueness of N follows from uniqueness of projective covers.

Simple Functors Are Finitely Presented

SN denotes simple functor S such that S(N) = 0 for indecomposable N.

Simple Functors Are Finitely Presented

SN denotes simple functor S such that S(N) = 0 for indecomposable N. Theorem (Auslander) The simple functors are finitely presented in

• mod(Λop), Ab
• .

Simple Functors Come From Exact Sequences

Recall that we are looking at the category fp(mod(Λop), Ab)

Simple Functors Come From Exact Sequences

Recall that we are looking at the category fp(mod(Λop), Ab) Proposition Suppose that N is a non-injective indecomposable. Then w(SN) = 0.

Proof

Since N is not injective, SN(I) = 0 for any indecomposable injective I.

Proof

Since N is not injective, SN(I) = 0 for any indecomposable injective I. ∴ SN(J) = 0 for any finite sum of indecomposable injectives J =

k Ik.

Proof

Since N is not injective, SN(I) = 0 for any indecomposable injective I. ∴ SN(J) = 0 for any finite sum of indecomposable injectives J =

k Ik.

All injectives in mod(Λop) are of this form.

Proof

Since N is not injective, SN(I) = 0 for any indecomposable injective I. ∴ SN(J) = 0 for any finite sum of indecomposable injectives J =

k Ik.

All injectives in mod(Λop) are of this form. ∴ SN vanishes on injectives.

Proof

Since N is not injective, SN(I) = 0 for any indecomposable injective I. ∴ SN(J) = 0 for any finite sum of indecomposable injectives J =

k Ik.

All injectives in mod(Λop) are of this form. ∴ SN vanishes on injectives. Because mod(Λop) has enough injectives, this is equivalent to w(SN) = 0

Proof

Since N is not injective, SN(I) = 0 for any indecomposable injective I. ∴ SN(J) = 0 for any finite sum of indecomposable injectives J =

k Ik.

All injectives in mod(Λop) are of this form. ∴ SN vanishes on injectives. Because mod(Λop) has enough injectives, this is equivalent to w(SN) = 0 Hence there exists a short exact sequence in mod(Λop): 0 → N → Y → Z → 0 such that SN has presentation: 0 → (Z, ) → (Y, ) → (N, ) → SN → 0

What Are Almost Split Sequences?

For N indecomposable non-injective:

What Are Almost Split Sequences?

For N indecomposable non-injective: Step 1: Take minimal projective presentation: 0 → (Z, ) → (Y, ) → (N, ) → SN → 0

What Are Almost Split Sequences?

For N indecomposable non-injective: Step 1: Take minimal projective presentation: 0 → (Z, ) → (Y, ) → (N, ) → SN → 0 Step 2: Apply w to get the short exact sequence 0 → N → Y → Z → 0

What Are Almost Split Sequences?

Start with any morphism u: N → K N Y Z K f u

What Are Almost Split Sequences?

Start with any morphism u: N → K N Y Z K E Z

push out

diagram

f u 1

Two Possibilities for u

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN G 1 (f, ) (u, )

Two Possibilities for u

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN G 1 (f, ) (u, ) α

Case 1: α = 0

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN G 1 (f, ) (u, ) α = 0

(u, ) factors through (f, )

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN G 1 (f, ) (u, ) α = 0

u factors through f

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN N Y G K 1 (f, ) (u, ) α = 0 f u

Case 2: α = epimorphism

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN G 1 (f, ) (u, ) α

(u, ) = epimorphism and hence u = section.

(Z, ) (E, ) (K, ) F (Z, ) (Y, ) (N, ) SN 1 (f, ) (u, ) α In this case u must be a section.

What Are Almost Split Sequences?

The fact that f is left minimal follows from the minimality of the presentation (Y, ) → (N, ) → SN → 0

Almost Split Sequences

Definition (Auslander, Reiten) An exact sequence N Y Z f is almost split if

1 f is left minimal, so hf = f implies h is an isomorphism. 2 If u: N → K is not a section then u = fu′.

Almost Split Sequences

Definition (Auslander, Reiten) An exact sequence N Y Z f is almost split if

1 f is left minimal, so hf = f implies h is an isomorphism. 2 If u: N → K is not a section then u = fu′.

Note: These are simply the properties of the sequences obtained by applying w to minimal projective resolutions of simple functors SN.

Left Exact Functors

The defect allows one to understand natural transformations from finitely presented functors into any left exact functor L: A → Ab

Left Exact Functors

The defect allows one to understand natural transformations from finitely presented functors into any left exact functor L: A → Ab Theorem (Auslander∗) Suppose that F ∈ fp(A, Ab) and L ∈ (A, Ab) is any left exact functor, then there are natural isomorphisms: Nat(F, L) ∼ = Nat

• (w(F),

), L ∼ = L(w(F))

The Diagrammatic Interpretation

F0 F

• w(F),
• F1

L

ϕ η

L: A → Ab is left exact

The Diagrammatic Interpretation

F0 F

• w(F),
• F1

L

ϕ η ∃!ψ

L: A → Ab is left exact

The CoYoneda Lemma

Lemma (The CoYoneda Lemma) For any F ∈ fp(A, Ab) and any X ∈ A, Nat

• F, (X,

) ∼ =

• X, w(F)
• this isomorphism being natural in F and X.

Duals

Definition (Fisher-Palmquist and Newell) For F ∈ (A, Ab), define F ∗ ∈ (Aop, Ab) F ∗(A) := Nat

• F, Hom(A,

)

Consequence of the CoYoneda Lemma

Proposition For F ∈ fp(A, Ab) F ∗ ∼ = ( , w(F))

Consequence of the CoYoneda Lemma

Proposition For F ∈ fp(A, Ab) F ∗ ∼ = ( , w(F)) This easily is seen as F ∗(X) = Nat

• F, (X,

) ∼ =

• X, w(F)

The Reinterpretation of the Defect Sequence

Proposition (Auslander) For any finitely presented module M over a Noetherian ring R, there is an exact sequence 0 − → Ext1(Tr(M), R) − → M − → M∗∗ − → Ext2(Tr(M), R) − → 0

The Reinterpretation of the Defect Sequence

Proposition For any finitely presented functor F ∈ fp(A, Ab) the defect sequence 0 − → F0 − → F − →

• w(F),
• −

→ F1 − → 0 is the following exact sequence 0 − → Ext1(Tr(F), A) − → F − → F ∗∗ − → Ext2(Tr(F), A) − → 0 Where Exti(Tr(F), A): A → Ab is the i-th right satellite of the functor which sends A ∈ A to Nat(Tr(F), ( , A)).

Free Abelian Category

Let A denote any pre-additive category. A free abelian category

• n A is an abelian category Ab(A) together with an additive

functor j : A → Ab(A) satisfying the following universal property: A Ab(A) j

Free Abelian Category

Let A denote any pre-additive category. A free abelian category

• n A is an abelian category Ab(A) together with an additive

functor j : A → Ab(A) satisfying the following universal property: A Ab(A) B j ∀a a = additive, B = abelian

Free Abelian Category

Let A denote any pre-additive category. A free abelian category

• n A is an abelian category Ab(A) together with an additive

functor j : A → Ab(A) satisfying the following universal property: A Ab(A) B j ∃!e ∀a a = additive, B = abelian, e = exact

Free Abelian Category

The following was stated first by Gruson for rings and then proved for general pre-additive A by Krause:

Free Abelian Category

The following was stated first by Gruson for rings and then proved for general pre-additive A by Krause: Theorem (Gruson, Krause) The double Yoneda embedding Y2 : A → fp

• fp(mod(A), Ab), Ab
• is the free abelian category on A.

Injective Resolutions in fp(C, Ab)

Injective Resolutions in fp(C, Ab)

Proposition (Gentle) If A has enough projectives then every finitely presented functor F ∈ fp(A, Ab) has an injective resolution: 0 → F → I0 → I1 → I2 → 0

Sketch of Proof

Consider I ⊆ fp(A, Ab) consisting of those F with presentation: (Q, ) → (P, ) → F → 0 Q, P are projectives in A

Sketch of Proof

Consider I ⊆ fp(A, Ab) consisting of those F with presentation: (Q, ) → (P, ) → F → 0 Q, P are projectives in A Properties of I:

Sketch of Proof

Consider I ⊆ fp(A, Ab) consisting of those F with presentation: (Q, ) → (P, ) → F → 0 Q, P are projectives in A Properties of I:

1 I is an additive subcategory of fp( Ab).

Sketch of Proof

Consider I ⊆ fp(A, Ab) consisting of those F with presentation: (Q, ) → (P, ) → F → 0 Q, P are projectives in A Properties of I:

1 I is an additive subcategory of fp( Ab). 2 I consists of injectives.

Sketch of Proof

Consider I ⊆ fp(A, Ab) consisting of those F with presentation: (Q, ) → (P, ) → F → 0 Q, P are projectives in A Properties of I:

1 I is an additive subcategory of fp( Ab). 2 I consists of injectives. 3 I is closed under cokernels.

Sketch of Proof

Define I to be the full subcategory of fp(A, Ab) consisting of G with copresentation: 0 → G → I0 → I1 I0, I1 ∈ I

Sketch of Proof

Define I to be the full subcategory of fp(A, Ab) consisting of G with copresentation: 0 → G → I0 → I1 I0, I1 ∈ I Since I satisfies the above properties, I is abelian.

Sketch of Proof

Define I to be the full subcategory of fp(A, Ab) consisting of G with copresentation: 0 → G → I0 → I1 I0, I1 ∈ I Since I satisfies the above properties, I is abelian. I contains all representable functors

Sketch of Proof

Define I to be the full subcategory of fp(A, Ab) consisting of G with copresentation: 0 → G → I0 → I1 I0, I1 ∈ I Since I satisfies the above properties, I is abelian. I contains all representable functors P1 → P0 → X → 0

Sketch of Proof

Define I to be the full subcategory of fp(A, Ab) consisting of G with copresentation: 0 → G → I0 → I1 I0, I1 ∈ I Since I satisfies the above properties, I is abelian. I contains all representable functors P1 → P0 → X → 0 0 → (X, ) → (P0, ) → (P1, )

Sketch of Proof

Define I to be the full subcategory of fp(A, Ab) consisting of G with copresentation: 0 → G → I0 → I1 I0, I1 ∈ I Since I satisfies the above properties, I is abelian. I contains all representable functors P1 → P0 → X → 0 0 → (X, ) → (P0, ) → (P1, ) Since I is abelian, it must be all of fp(A, Ab)

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

YR YR

YR(F) := (F(R), )

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

YR YR

YR(F) := (F(R), ) Ln(YR) = 0 for all n ≥ 1.

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

YR YR

YR(F) := (F(R), ) Ln(YR) = 0 for all n ≥ 1. Definition D := L0(YR)

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

D D

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

D D

Proposition D is exact.

Derived Functors

fp(Mod(R), Ab) fp(Mod(Rop), Ab)

D D

Proposition D is exact. Warning: D is not a duality.

Coherent Rings

Definition A ring is coherent if mod(R) is abelian.

Duality

Theorem For any coherent ring R the functor D restricts to a duality: fp(mod(R), Ab) fp(Mod(R), Ab) fp(mod(Rop), Ab) fp(Mod(Rop), Ab)

D D

Appearances of D

It was first discovered by Auslander

Appearances of D

It was first discovered by Auslander It was independently discovered by Gruson and Jensen.

Appearances of D

It was first discovered by Auslander It was independently discovered by Gruson and Jensen. For this reason it is commonly referred to as the Auslander, Gruson, Jenson duality.

Appearances of D

It was first discovered by Auslander It was independently discovered by Gruson and Jensen. For this reason it is commonly referred to as the Auslander, Gruson, Jenson duality. Hartshorne found D using an approach similar to Auslander.

Appearances of D

It was first discovered by Auslander It was independently discovered by Gruson and Jensen. For this reason it is commonly referred to as the Auslander, Gruson, Jenson duality. Hartshorne found D using an approach similar to Auslander. Krause showed how to obtain D from a universal property.

Appearances of D

It was first discovered by Auslander It was independently discovered by Gruson and Jensen. For this reason it is commonly referred to as the Auslander, Gruson, Jenson duality. Hartshorne found D using an approach similar to Auslander. Krause showed how to obtain D from a universal property. It was discovered model theoretically through work of Mike Prest, Ivo Herzog, and Kevin Burke.

Appearances of D

It was first discovered by Auslander It was independently discovered by Gruson and Jensen. For this reason it is commonly referred to as the Auslander, Gruson, Jenson duality. Hartshorne found D using an approach similar to Auslander. Krause showed how to obtain D from a universal property. It was discovered model theoretically through work of Mike Prest, Ivo Herzog, and Kevin Burke. As seen above, it can be recovered in yet another by restricting an exact functor obtained using injective resolutions to the smaller category.