Auslander's defect formula and a commutative triangle in an exact category
aa r X i v : . [ m a t h . R T ] J u l AUSLANDER’S DEFECT FORMULA AND A COMMUTATIVETRIANGLE IN AN EXACT CATEGORY
PENGJIE JIAO
Abstract.
We prove the Auslander’s defect formula in an exact category,and obtain a commutative triangle involving the Auslander bijections and thegeneralized Auslander-Reiten duality. Introduction
Throughout k denotes a commutative artinian ring.We consider a k -linear Hom-finite Krull-Schmidt exact category C , which is skeletally small.Recall that Auslander’s defect formula appeared as [1, Theorem III.4.1] for thefirst time. Krause [7] gave a short proof for the formula on modules categories.Ringel [9] introduced the notion of Auslander bijection on module categories. Chen[3] established a commutative triangle involving Auslander bijections, universalextensions and the Auslander-Reiten duality.More recently, the notion of generalized Auslander-Reiten duality was introducedin [5, Section 3]. Using this notion, we prove the Auslander’s defect formula in anexact category C based on some results in [3]; see Theorem 3.3. Compared with theproof of [4, Theorem 3.7], it seems that the treatment here can be applied to thestudy of higher Auslander-Reiten theory.We generalize the commutative triangle established by Chen [3, Theorem 4.6];see Theorem 4.2. The Auslander bijection in the commutative triangle suggests theusage of universal extensions in the study of morphisms determined by objects inan exact category.In Section 2, we recall the convariant defect and the contravariant defect. Sec-tions 3 and 4 are dedicated to the proofs of Theorems 3.3 and 4.2, respectively.2. Convariant defect and contravariant defect
Let k be a commutative artinian ring and ˇ k be the minimal injective cogener-ator. We denote by mod k the category of finitely generated k -modules and by D = Hom k ( − , ˇ k ) the Matlis duality. We recall the convariant defect and the con-travariant defect on an exact category.Let C be a k -linear Hom-finite Krull-Schmidt exact category, which is skeletallysmall. Recall that an exact category is an additive category C together with acollection E of exact pairs ( i, d ), which satisfies the axioms in [6, Appendix A].Here, an exact pair ( i, d ) means a sequence of morphisms X i −→ Y d −→ Z such that Date : October 26, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Auslander’s defect formula, morphisms determined by objects, Aus-lander bijections. i is the kernel of d and d is the cokernel of i . An exact pair ( i, d ) in E is called a conflation , while i is called an inflation and d is called a deflation . For a pair ofobjects X and Y in C , we denote by Ext C ( X, Y ) the set of equivalence classes ofconflations Y → E → X .Let ξ : X → E → Y be a conflation. For each morphism f : Z → Y we let ξ.f be the conflation obtained by pullback of ξ along f ; for each morphism g : X → Z we let g.ξ be the conflation obtained by pushout of ξ along g .Recall from [8, Section 2] that a morphism f : X → Y is called projectively trivial if for each object Z , the induced map Ext C ( f, Z ) : Ext C ( Y, Z ) → Ext C ( X, Z ) is zero.We observe that f is projectively trivial if and only if f factors through any deflationending at Y . Dually, we call f injectively trivial if for each object Z , the inducedmap Ext C ( Z, f ) : Ext C ( Z, X ) → Ext C ( Z, Y ) is zero.Given a pair of objects X and Y , we denote by P ( X, Y ) the set of projectivelytrivial morphisms X → Y . Then P forms an ideal of C . We set C = C / P . Given amorphism f : X → Y , we denote by f its image in C . We denote by Hom C ( X, Y ) =Hom C ( X, Y ) / P ( X, Y ) the set of morphisms X → Y in C .Dually, we denote by I ( X, Y ) the set of injectively trivial morphisms X → Y .Set C = C / I . Given a morphism f : X → Y , we denote by f its image in C . Wedenote by Hom C ( X, Y ) = Hom C ( X, Y ) / I ( X, Y ) the set of morphisms X → Y in C .Let ξ : X i −→ E d −→ Y be a conflation and K be an object in C . We have the connecting map c ( ξ, K ) : Hom C ( X, K ) −→ Ext C ( Y, K ) , f f.ξ. We mention that c ( ξ, K ) is natural in both ξ and K . Moreover, we have the exactsequence in functor categoryHom C ( E, − ) Hom C ( i, − ) −−−−−−−→ Hom C ( X, − ) c ( ξ, − ) −−−−→ Ext C ( Y, − ) . Recall from [2, Section IV.4] that the convariant defect of the conflation ξ is aconvariant functor ξ ∗ : C → mod k satisfying the following exact sequence0 → Hom C ( Y, − ) Hom C ( d, − ) −−−−−−−→ Hom C ( E, − ) Hom C ( i, − ) −−−−−−−→ Hom C ( X, − ) u −→ ξ ∗ → ξ ∗ ≃ Im c ( ξ, − ) . For any injectively trivial morphism f : K → K ′ , we have the following commu-tative diagramHom C ( E, K ) Hom C ( X, K ) ξ ∗ ( K ) 0Hom C ( E, K ′ ) Hom C ( X, K ′ ) ξ ∗ ( K ′ ) 0 . Hom C ( E,f ) Hom C ( i,K )Hom C ( X,f ) u K ξ ∗ ( f )Hom C ( i,K ′ ) u K ′ We observe that for any morphism g : X → K , the morphism f ◦ g is injectivelytrivial, and hence factors through i . Therefore f ◦ g lies in Im Hom C ( i, K ′ ). Weobtain u K ′ ( f ◦ g ) = 0. It follows that( ξ ∗ ( f ) ◦ u K )( g ) = ( u K ′ ◦ Hom C ( X, f ))( g )= u K ′ ( f ◦ g )= 0 . EFECT FORMULA AND A COMMUTATIVE TRIANGLE 3
We obtain ξ ∗ ( f ) ◦ u K = 0. Therefore ξ ∗ ( f ) = 0 since u K is surjective. We obtainthe induced functor C → mod k , which we still denote by ξ ∗ .We observe that c ( ξ, K )( f ) = 0 for any object K and any injectively trivialmorphism f : X → K . We obtain the natural transformation c ( ξ, − ) : Hom C ( X, − ) −→ Ext C ( Y, − ) . Dually, We have the connecting map c ( K, ξ ) : Hom C ( K, Y ) −→ Ext C ( K, X ) , f ξ.f. We mention that c ( K, ξ ) is natural in both ξ and K .The contravariant defect of the conflation ξ is a contravariant functor ξ ∗ : C → mod k satisfying the following exact sequence0 → Hom C ( − , X ) → Hom C ( − , E ) → Hom C ( − , Y ) → ξ ∗ → ξ ∗ ≃ Im c ( − , ξ ) . We mention that ξ ∗ vanishes on projectively trivial morphisms and c ( K, ξ )( f ) =0 for any object K and any projectively trivial morphism f : K → Y . We thenhave the induced functor ξ ∗ : C → mod k and the natural transformation c ( − , ξ ) : Hom C ( − , Y ) −→ Ext C ( − , X ) . Auslander’s defect formula
Recall from [5, Section 2] two full subcategories C r = (cid:8) X ∈ C (cid:12)(cid:12) the functor D Ext C ( X, − ) : C → mod k is representable (cid:9) and C l = (cid:8) X ∈ C (cid:12)(cid:12) the functor D Ext C ( − , X ) : C → mod k is representable (cid:9) . We have a pair of mutually quasi-inverse equivalences τ : C r ∼ −→ C l and τ − : C l ∼ −→ C r . For each object Y ∈ C r , we have a natural isomorphism φ Y : Hom C ( − , τ Y ) ∼ −→ D Ext C ( Y, − ) . We let α Y = φ Y,τY (Id τY ) and µ Y = ψ − τY,Y ( α Y ) . For each object X ∈ C l , we have a natural isomorphism ψ X : Hom C ( τ − X, − ) ∼ −→ D Ext C ( − , X ) . We let β X = ψ X,τ − X (Id τ − X ) and ν X = φ − τ − X,X ( β X ) . The following lemma is contained in the proof of [5, Proposition 3.4].
Lemma 3.1.
For each object Y ∈ C r , we have α Y = β τY ◦ Ext C ( µ Y , τ Y ); for each object X ∈ C l , we have β X = α τ − X ◦ Ext C ( τ − X, ν X ) . PENGJIE JIAO
Proof.
We only prove the first equality. We observe that α Y = ψ τY,Y ( µ Y ). Considerthe following commutative diagramHom C ( τ − τ Y, τ − τ Y ) D Ext C ( τ − τ Y, τ Y )Hom C ( τ − τ Y, Y ) D Ext C ( Y, τ Y ) . ψ τY,τ − τY Hom C ( τ − τY,µ Y ) D Ext C ( µ Y ,τY ) ψ τY,Y By a diagram chasing, we obtain α Y = ψ τY,Y ( µ Y )= ( ψ τY,Y ◦ Hom C ( τ − τ Y, µ Y ))(Id τ − τY )= ( D Ext C ( µ Y , τ Y ) ◦ ψ τY,τ − τY )(Id τ − τY )= D Ext C ( µ Y , τ Y )( β τY )= β τY ◦ Ext C ( µ Y , τ Y ) . Here, the third equality holds by the commutative diagram, and the fourth equalityholds by the definition of β τY . (cid:3) We mention the following fact. Here, we give a proof.
Lemma 3.2 ([3, Lemma 4.3]) . Let ξ : X → E → Y be a conflation in C . (1) For each object K ∈ C r , there exists a commutative diagram Hom C ( X, τ K ) Ext C ( Y, τ K ) D Ext C ( K, X ) D Hom C ( K, Y ) , c ( ξ,τK ) φ K,X Dc ( K,ξ ) D ( ψ τK,Y ◦ Hom C ( µ K ,Y )) which is natural in both ξ and K . (2) For each object K ∈ C l , there exists a commutative diagram Hom C ( τ − K, Y ) Ext C ( τ − K, X ) D Ext C ( Y, K ) D Hom C ( X, K ) , c ( τ − K,ξ ) ψ K,Y Dc ( ξ,K ) D ( φ τ − K,X ◦ Hom C ( X,ν K )) which is natural in both ξ and K .Proof. We only prove (1). We set ψ ′ = ψ τK,Y ◦ Hom C ( µ K , Y ) : Hom C ( K, Y ) −→ D Ext C ( Y, τ K ) . We observe that each conflation ζ in Ext C ( Y, τ K ) induces a k -linear map D Ext C ( Y, τ K ) −→ ˇ k, f f ( ζ ) . EFECT FORMULA AND A COMMUTATIVE TRIANGLE 5
For any morphisms g : X → τ K and h : K → Y , we have( Dψ ′ ◦ c ( ξ, τ K ))( g )( h ) = ( c ( ξ, τ K )( g ) ◦ ψ ′ )( h )= ( g.ξ )( ψ ′ ( h ))= ( ψ ′ ( h ))( g.ξ )= ψ τK,Y ( h ◦ µ K )( g.ξ ) . Here, the second equality holds by the definition of c ( ξ, τ K ), and the third equalityholds by the canonical isomorphism Ext C ( Y, τ K ) ≃ DD Ext C ( Y, τ K ). It follows bythe naturality of ψ τY that ψ τK,Y ( h ◦ µ K )( g.ξ ) = β τY ( g.ξ.h.µ K ) . On the other hand, we have( Dc ( K, ξ ) ◦ φ K,X )( g )( h ) = ( φ K,X ( g ) ◦ c ( K, ξ ))( h )= φ K,X ( g )( ξ.h ) . It follows by the naturality of φ K that φ K,X ( g )( ξ.h ) = α K ( g.ξ.h ) . By Lemma 3.1, we have α K = β τK ◦ Ext C ( µ K , τ K ) and then α K ( g.ξ.h ) = ( β τK ◦ Ext C ( µ K , τ K ))( g.ξ.h )= β τK ( g.ξ.h.µ K ) . It follows that ( Dψ ′ ◦ c ( ξ, τ K ))( g )( h ) = ( Dc ( K, ξ ) ◦ φ K,X )( g )( h ) . By the arbitrariness of h and g , we obtain Dψ ′ ◦ c ( ξ, τ K ) = Dc ( K, ξ ) ◦ φ K,X . The naturality is a direct verification. (cid:3)
Now, we prove Auslander’s defect formula; compare [1, Theorem III.4.1] and [7,Theorem].
Theorem 3.3.
Let ξ : X i −→ E d −→ Y be a conflation in C . (1) For each object K ∈ C r , there exists an isomorphism ξ ∗ ( τ K ) ≃ Dξ ∗ ( K ) ,which is natural in both ξ and K . (2) For each object K ∈ C l , there exists an isomorphism ξ ∗ ( τ − K ) ≃ Dξ ∗ ( K ) ,which is natural in both ξ and K .Proof. We only prove (1). By Lemma 3.2(1), we obtain the following commutativediagram Hom C ( X, τ K ) Ext C ( Y, τ K ) D Ext C ( K, X ) D Hom C ( K, Y ) . c ( ξ,τK ) ≃ Dc ( K,ξ ) ≃ Then the result follows by the above commutative diagram, since we have thenatural isomorphisms (2.1) and (2.2). (cid:3)
PENGJIE JIAO A commutative triangle
Recall that two morphism f : X → Y and f ′ : X ′ → Y are called right equivalent if f factors through f ′ and f ′ factors through f . We denote by [ f i the rightequivalence class containing f , and by [ → Y i the class of right equivalence classesof morphisms ending to Y . We mention that [ → Y i is a set and has a naturallattice structure; see [9, Section I.2]. We denote by [ → Y i epi the subclass of [ → Y i formed by deflations.For an object K in C , we set Γ( K ) = End C ( K ), and denote by add K the categoryof direct summands of finite direct sums of K . For a Γ( K )-module M , we denote bysub Γ( K ) M the lattice of Γ( K )-submodules of M . We obtain a morphism of posets η C,Y : [ → Y i epi −→ sub Γ( C ) op Hom C ( C, Y ) , [ d i 7→ Im Hom C ( C, d ) . Recall that a morphism f : X → Y is called right C -determined if the followingcondition is satisfied: each morphism g : T → Y factors through f , provided thatfor each morphism h : C → T the morphism g ◦ h factors through f .For an object K in C , we denote by K [ → Y i epi the subclass of [ → Y i epi formed byright K -determined deflations, and by K [ → Y i epi the subclass of [ → Y i epi formedby deflations whose kernel lies in add K . We mention that [ f i ∈ K [ → Y i epi meansthat there exists some deflation f ′ : X ′ → Y with f ′ ∈ [ f i and Ker f ′ ∈ add K .For a deflation d : X → Y , we denote by ξ d the conflation Ker d → X d −→ Y .We set δ K,Y ([ d i ) = Im c ( ξ d , K ), which is a Γ( K )-submodule of Ext C ( Y, K ). Wemention the following fact; see [3, Propositions 2.4 and 4.5].
Lemma 4.1.
Let K be an object in C . (1) There exists an anti-isomorphism of posets δ K,Y : K [ → Y i epi −→ sub Γ( K ) Ext C ( Y, K ) . (2) If K ∈ C l , then K [ → Y i epi = τ − K [ → Y i epi . (cid:3) We mention that the natural isomorphism ψ K,Y : Hom C ( τ − K, Y ) −→ D Ext C ( Y, K )induces an anti-isomorphism of posets γ K,Y : sub Γ( K ) Ext C ( Y, K ) −→ sub Γ( τ − K ) op Hom C ( τ − K, Y ) L ψ − K,Y ( D (Ext C ( Y, K ) /L )) . The following result establishes Auslander bijection. This is a slight generaliza-tion of [3, Theorem 4.6]. The proof is similar.
Theorem 4.2.
Let Y ∈ C and K ∈ C l be objects. Then we have the followingcommutative bijection triangle sub Γ( K ) Ext C ( Y, K ) τ − K [ → Y i epi sub Γ( τ − K ) op Hom C ( τ − K, Y ) . δ K,Y η τ − K,Y γ K,Y
EFECT FORMULA AND A COMMUTATIVE TRIANGLE 7
Proof.
It is sufficient to show that the triangle is commutative. For each deflation d : X → Y , we have an exact sequenceHom C ( τ − K, X ) Hom C ( τ − K,d ) −−−−−−−−−→ Hom C ( τ − K, Y ) c ( τ − K,ξ d ) −−−−−−−→ Ext C ( τ − K, Ker d )in mod k . We obtain(4.1) η τ − K,Y ([ d i ) = Im Hom C ( τ − K, d ) = Ker c ( τ − K, ξ d ) . Applying D to the exact sequenceHom C ( X, K ) c ( ξ d ,K ) −−−−−→ Ext C ( Y, K ) → Ext C ( Y, K ) / Im c ( ξ d , K ) → k , we obtain D (Ext C ( Y, K ) / Im c ( ξ d , K )) = Ker Dc ( ξ d , K ) . By the commutative diagram in Lemma 3.2(2), we obtain(4.2) ψ K,Y (Ker c ( τ − K, ξ d )) = Ker Dc ( ξ d , K ) = D (Ext C ( Y, K ) / Im c ( ξ d , K )) . It follows that( γ K,Y ◦ δ K,Y )([ d i ) = ψ − K,Y ( D (Ext C ( Y, K ) / Im c ( ξ d , K )))= Ker c ( τ − K, ξ d )= η τ − K,Y ([ d i ) . Here, the first equality holds by the definitions of δ K,Y and γ K,Y . The second andthird equalities are just (4.2) and (4.1), respectively. (cid:3)
Acknowledgements
The author thanks his supervisor Professor Xiao-Wu Chen for his guidance andencouragement.
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