Homological theory of k -idempotent ideals in dualizing varieties
Luis Gabriel Rodríguez Valdés, Valente Santiago Vargas, Martha Lizbeth Shaid Sandoval Miranda
aa r X i v : . [ m a t h . C T ] A ug HOMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS INDUALIZING VARIETIES LUS GABRIEL RODR´IGUEZ-VALD´ES, VALENTE SANTIAGO-VARGAS
Abstract.
We develop the theory of k -idempotent ideals in the setting ofdualizing varieties and we extend several results given by Auslander-Platzeck-Todorov in [4]. Given an ideal I , which is the trace of a projective module,we construct a canonical recollement, which is the analogous of a well knownrecollement in module categories over artin algebras. We study homologicalproperties of the categories involved in such recollement and we find conditionson the ideal I in order to obtain quasi-hereditary algebras in the recollement.Finally, we give an application to bounded derived categories. Introduction
The idea that additive categories are rings with several objects was developedconvincingly by Barry Mitchell (see [18]). An example of the latter is the approachthat M. Auslander and I. Reiten gave to the study of representation theory (see forexample [2], [3], [5], [6]), which gave birth to the concept of almost split sequence.This led to the notion of dualizing R -varieties, introduced and investigated in [5].Dualizing R -varieties have appeared in the context of locally bounded k -categoriesover a field k , categories of graded modules over artin algebras and also in connec-tion with covering theory. One of the advantages of the notion of dualizing R -varietydefined in [5] is that it provides a common setting for the category proj(Λ) of finitelygenerated projective Λ-modules, mod(Λ) and mod(mod(Λ)), which all play an im-portant role in the study of an artin algebra Λ.On the other hand, in [4], Auslander-Platzeck-Todorov studied homological idealsin the case of mod(Λ) where Λ is an artin algebra. They proved several fundamen-tal results related to homological ideals and they connected such a notion with thecontext of quasi-hereditary algebras. In the case that I is the trace of a projectivemodule P , they studied how the homological properties of the categories of finitelygenerated modules over Λ, Λ /I and the endomorphism ring of P are related.It is natural to extend this study to the setting of rings with several objects. Thisextension is better expressed in the language of dualizing R -varieties. In this paperwe generalize several results given in [4] to the context of dualizing varieties. In thefollowing, we describe our results in more detail.After the introduction (section 1), in section 2 we consider a preadditive category C , and we recall basic definitions and results about Mod( C ) and dualizing varieties.In section 3, we consider the notion of ideal in a preadditive category C . Westart our work by generalizing the classical adjunction for artin algebras given byHom Λ /I ( Y, Tr Λ /I ( X )) ≃ Hom Λ ( Y, X ) to the case of rings with several objects. Westudy certain derived functors, and with the help of this derived functors, we givehomological characterizations of when the functor Tr CI : Mod( C ) −→ Mod( C / I )preserves injective coresolutions of length k (see 3.13 and 3.22). Mathematics Subject Classification.
Key words and phrases.
Dualizing varieties, functor categories, ideals, recollement.The author thanks project PAPIIT-Universidad Nacional Aut´onoma de M´exico IN100520.
In section 4, we study conditions on the ideal I under which we can restrict ourprevious results to the subcategory mod( C ) of finitely presented C -modules. Weintroduce the condition A on the ideal I (see definition 4.5), and we prove that ifan ideal satisfies property A then we can restrict our attention to the case of finitelypresented modules (see 4.7). In particular, we prove that if C is a dualizing varietyand I is an ideal satisfying property A , then C / I is also dualizing (see 4.6).In section 5, we introduce the notion of k -idempotent ideal in preadditive categories(see 5.1). We describe the idempotent ideals in terms of the vanishing of certainderived functors (see 5.2 and 5.3). Moreover, if the ideal I satisfies property A and C is a dualizing R -variety, by using Auslander-Reiten duality, we give character-izations of when I is k -idempotent in terms of the functors Ext i mod( C ) ( − , − ) andTor C i ( − , − ) (see 5.9).In section 6, we prove the dual basis lemma for the category Mod( C ) (see 6.1), andgiven a projective module P , we introduce the trace ideal I := Tr P C (see 6.3) andwe prove several classical results about ideals which are trace of projective modules,for example we prove that Tr P C is an idempotent ideal (see 6.4 and 6.5). We alsoshow that if C is a dualizing R -variety, then Tr P C satisfies property A (see 6.10).We study projective resolutions of k -idempotent ideals and introduce specials sub-categories, P k and I k (see 6.14). Here we prove that I = Tr P C is k + 1-idempotentif and only if I ( C ′ , − ) ∈ P k for all C ′ ∈ C (see 6.18), which is a generalization ofthe result [4, Theorem 2.1].In section 7, we consider P = Hom C ( C, − ) ∈ mod( C ) and R P := End mod( C ) ( P ) op and we study the functor Hom mod( C ) ( P, − ). We obtain a generalization of a wellknown recollement (see for example [21, example 3.4]) to the setting of dualizing R -varieties (see 7.6). Finally in this section, we prove that mod( R P ) is equivalentto certain subcategories of mod( C ) (see 7.12).In section 8, we study some homological properties of the functor Hom mod( C ) ( P, − ) :mod( C ) → mod( R P ) and how it relates homological properties of mod( C ) andmod( R P ). We explore the relationship between injective coresolution in mod( C )and mod( R P ). We give necessary and sufficient conditions for I to be equal to I ∞ .Finally, we study the property of the ideal of being projective. This notion is im-portant because the condition of I ( C ′ , − ) being projective is part of the definitionof heredity ideal given in [19]. We show that under certain conditions we are ableto produce quasi-hereditary algebras (see 8.15). Finally, we have an application toderived categories (see 8.16), which is a generalization of a well known result forthe category Mod( R ) where R is an associative ring.In section 9, we provide some examples of k -homological ideals.2. Preliminaries
In this section we recall the basic notions of rings with several objects and thenotion of dualizing R -variety introduced by Auslander and Reiten in [5].2.1. Categorical Foundations.
We remember that a category C together withan abelian group structure on each of the sets of morphisms C ( C , C ) is called preadditive category provided all the composition maps C ( C, C ′ ) ×C ( C ′ , C ′′ ) −→C ( C, C ′′ ) in C are bilinear maps of abelian groups. A covariant functor F : C −→ C between preadditive categories C and C is said to be additive if for each pair ofobjects C and C ′ in C , the map F : C ( C, C ′ ) −→ C ( F ( C ) , F ( C ′ )) is a morphismof abelian groups. If C is a preadditive category, we always consider its oppositecategory C op as a preadditive category by letting C op ( C ′ , C ) = C ( C, C ′ ). We followthe usual convention of identifying each contravariant functor F from a category C to D with a covariant functor F from C op to D . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 3 An arbitrary category C is small if the class of objects of C is a set. An additivecategory is a preadditive category C such that every finite family of objects in C hasa coproduct. Given a small preadditive category C and D an arbitrary preadditivecategory, we denote by ( C , D ) the category of all the covariant additive functors.2.2. The category
Mod( C ) . Throughout this section C will be an arbitrary smallpreadditive category, and Mod( C ) will denote the category of additive covariantfunctors from C to the category of abelian groups Ab , called the category of C -modules. This category has as objects the functors from C to Ab , and a morphism f : M −→ M of C -modules is a natural transformation. Sometimes we will writefor short C ( − , ?) instead of Hom C ( − , ?), and when it is clear from the context we willuse just ( − , ?) . As usual, Mod( C op ) will be identified with the category of additivecontravariant functors from C to Ab .We now recall some properties of the category Mod( C ), for more details consult [2].The category Mod( C ) is an abelian category with enough injectives and projectives.For each C in C , the C -module ( C, − ) given by ( C, − )( X ) = C ( C, X ) for each X in C , has the property that for each C -module M , the map (( C, − ) , M ) −→ M ( C )given by f f C (1 C ) for each C -morphism f : ( C, − ) −→ M is an isomorphism ofabelian groups (Yoneda’s Lemma).(1) The functor Y : C −→
Mod( C ) given by Y ( C ) = ( C, − ) is fully faithful.(2) For each family { C i } i ∈ I of objects in C , the C -module ∐ i ∈ I Y ( C i ) is projective.(3) For each M ∈ Mod( C ), there exists an epimorphism ∐ i ∈ I Y ( C i ) → M for somefamily { C i } i ∈ I in C . We say that M is finitely generated if such family is finite.(4) A finitely generated projective C -module is a direct summand of ∐ i ∈ I Y ( C i )for some finite family of objects { C i } i ∈ I in C .2.3. Change of Categories.
The results that appears in this subsection are di-rectly taken from [2]. Let C be a small predditive category. There is a unique(up to isomorphism) functor ⊗ C : Mod( C op ) × Mod( C ) −→ Ab called the tensorproduct . The abelian group ⊗ C ( A, B ) is denoted by A ⊗ C B for all C op -modules A and all C -modules B . Proposition 2.1.
The tensor product has the following properties: (1) (a)
For each C -module B , the functor ⊗ C B : Mod( C op ) −→ Ab given by ( ⊗ C B )( A ) = A ⊗ C B for all C op -modules A is right exact. (b) For each C op -module A , the functor A ⊗ C : Mod( C ) −→ Ab given by ( A ⊗ C )( B ) = A ⊗ C B for all C -modules B is right exact. (2) For each C op -module A and each C -module B , the functors A ⊗ C and ⊗ C B preserve arbitrary coproducts. (3) For each object C in C we have A ⊗ C ( C, − ) = A ( C ) and ( − , C ) ⊗ C B = B ( C ) for all C op -modules A and all C -modules B . Suppose now that C ′ is a preadditive subcategory of the small category C . We usethe tensor product of C ′ -modules to describe the left adjoint C⊗ C ′ of the restrictionfunctor res C ′ : Mod( C ) −→ Mod( C ′ ) given by M M | C ′ .Define the functor C⊗ C ′ : Mod ( C ′ ) −→ Mod ( C ) by ( C ⊗ C ′ M ) ( C ) = C ( − , C ) | C ′ ⊗ C ′ M for all M ∈ Mod ( C ′ ) and C ∈ C . Using the properties of the tensor productit is not difficult to establish the following proposition.
Proposition 2.2. [2, Proposition 3.1]
Let C ′ be a subcategory of the small category C . Then the functor C⊗ C ′ : Mod ( C ′ ) −→ Mod ( C ) satisfies: (1) C⊗ C ′ is right exact and preserves coproducts; (2) The composition
Mod ( C ′ ) C⊗ C′ −→ Mod ( C ) res C′ −→ Mod ( C ′ ) is the identity; LUS GABRIEL RODR´IGUEZ-VALD´ES, VALENTE SANTIAGO-VARGAS (3)
For each object C ′ ∈ C ′ , we have C ⊗ C ′ C ′ ( C ′ , − ) = C ( C ′ , − ) ;(4) The restriction map C ( C ⊗ C ′ M, N ) −→ C ′ ( M, N | C ′ ) is an isomorphismfor each C ′ -module M and each C -module N ; (5) C⊗ C ′ is a fully faithful functor. Having described the left adjoint C⊗ C ′ of the restriction functor res C ′ : Mod ( C ) −→ Mod ( C ′ ) , we now describe its right adjoint. Define the functor C ′ ( C , − ) : Mod ( C ′ ) −→ Mod ( C ) by C ′ ( C , M ) ( X ) = C ′ ( C ( X, − ) | C ′ , M ) for all C ′ -modules M and all objects X in C . We have the following proposition.
Proposition 2.3. [2, Proposition 3.4]
Let C ′ be a subcategory of the small category C . Then the functor C ′ ( C , − ) : Mod ( C ′ ) −→ Mod ( C ) has the following properties: (1) C ′ ( C , − ) is left exact and preserves inverse limits; (2) The composition
Mod ( C ′ ) C ′ ( C , − ) −→ Mod ( C ) res C′ −→ Mod ( C ′ ) is the identity; (3) The restriction map C ( N, C ′ ( C , M )) −→ C ′ ( N | C ′ , M ) is an isomorphismfor each C ′ -module M and C -module N ; (4) C ′ ( C , − ) is a fully faithful functor. Dualizing varieties and Krull-Schmidt Categories.
Let C be an additivecategory. It is said that C is a category in which idempotents split if given e : C −→ C an idempotent endomorphism of an object C ∈ C , then e has a kernel in C . Let us denote by proj ( C ) the full subcategory of Mod( C ) consisting of all finitelygenerated projective C -modules. It is well known that proj( C ) is a small additivecategory in which idempotents split, the functor Y : C → proj( C ) given by Y ( C ) = C ( C, − ), is fully faithful and induces by restriction res : Mod(proj( C ) op ) → Mod( C )an equivalence of categories. We recall the following notion given by Auslander in[2]. A variety is a small, additive category in which idempotents split.Given a ring R , we denote by Mod( R ) the category of left R -modules and by mod( R )the full subcategory of Mod( R ) consisting of the finitely generated left R -modules.Now, we recall some notions from [5]. Definition 2.4.
Let R be a commutative artin ring. An R - category C , is anadditive category such that C ( C , C ) is an R -module, and the composition is R -bilinear. An R - variety C is a variety which is an R -category. An R -variety C is Hom - finite , if for each pair of objects C , C in C , the R -module C ( C , C ) isfinitely generated. We denote by ( C , mod( R )) , the full subcategory of ( C , Mod( R )) consisting of the C -modules such that for every C in C the R -module M ( C ) is finitelygenerated. Suppose C is a Hom-finite R -variety. If M : C −→ Ab is a C -module, for each C ∈ C the abelian group M ( C ) has a structure of End C ( C )-module, and hence asan R -module since End C ( C ) is an R -algebra. Furthermore, if f : M −→ M ′ is amorphism of C -modules, it is easy to show that f C : M ( C ) −→ M ′ ( C ) is a morphismof R -modules for each C ∈ C . Then, Mod( C ) is an R -variety, which we identify withthe category of covariant functors ( C , Mod( R )). Moreover, ( C , mod( R )) is abelianand the inclusion ( C , mod( R )) → ( C , Mod( R )) is exact. Definition 2.5.
Let C be a Hom-finite R -variety. We denote by mod( C ) the fullsubcategory of Mod( C ) whose objects are the finitely presented functors . Thatis, M ∈ mod( C ) if and only if there exists an exact sequence in Mod( C )Hom C ( C , − ) / / Hom C ( C , − ) / / M / / . Consider the functors D C op : ( C op , mod( R )) → ( C , mod( R )), and D C : ( C , mod( R )) → ( C op , mod( R )), which are defined as follows: for any object C in C , D C ( M )( C ) =Hom R ( M ( C ) , E ) where E is the injective envelope of R/ rad( R ) ∈ mod( R ). The OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 5 functor D C defines a duality between ( C , mod( R )) and ( C op , mod( R )). We knowthat since C is Hom-finite, mod( C ) is a subcategory of ( C , mod( R )). Then we havethe following definition due to Auslander and Reiten (see [5]). Definition 2.6. An Hom -finite R -variety C is dualizing , if the functor D C :( C , mod( R )) → ( C op , mod( R )) induces a duality between the categories mod( C ) and mod( C op ) . It is clear from the definition that for dualizing varieties C the category mod( C )has enough injectives and projectives. To finish, we recall the following definition: Definition 2.7.
An additive category C is Krull-Schmidt , if every object in C decomposes in a finite sum of objects whose endomorphism ring are local. Assume that R is a commutative artin ring and C is a dualizing R -variety. Sincethe endomorphism ring of each object in C is an artin algebra, it follows that C isa Krull-Schmidt category [5, p.337]. We recall the following result. Theorem 2.8.
Let C a dualizing R -variety. Then mod( C ) is a dualizing R -variety. An adjunction and some derived functors
In the article [4], Auslander-Platzeck-Todorov studied homological ideals in thecase of mod(Λ) where Λ is an artin algebra. Given a two sided ideal I of Λ theyconsider Λ /I and they studied the trace Tr Λ /I ( M ) of a Λ-module M defined asTr Λ /I ( M ) = X f ∈ Hom Λ (Λ /I,M ) Im( f ) . In order to define the analogous of Tr Λ /I ( M ) in thecategory Mod( C ) we introduced the following notions.In this section C will be a small preadditive category. Let M = { M i } i ∈ I be afamily of C -modules and set M := L i ∈ I M i . For F ∈ Mod( C ) we define Λ F :=Hom Mod( C ) ( M, F ) , and for λ ∈ Λ F we set u λ : M −→ M (Λ F ) as the λ -th inclusionof M into M (Λ F ) := L λ ∈ Λ F M . For λ ∈ Λ F we have the morphism λ : M −→ F ,then by the universal property of the coproduct, there exists a unique morphismΘ F : M (Λ F ) −→ F such that λ = Θ F ◦ u λ for every λ ∈ Λ F . Definition 3.1.
The trace of F respect to the family M = { M i } i ∈ I , denotedby Tr M ( F ) , is the image of Θ F . That is, we have the following factorization M (Λ F ) ∆ F / / Tr M ( F ) Ψ F / / F where Θ F = Ψ F ∆ F with ∆ F an epimorphism and Ψ F a monomorphism. Proposition 3.2.
For each family M = { M i } i ∈ I , we have a functor Tr M :Mod( C ) → Mod( C ) .Proof. Straightforward. (cid:3)
Now, we recall the following definitions which are essential throughout this work.
Definition 3.3.
Let C be a preadditive category. An ideal I of C is an additivesubfunctor of Hom C ( − , − ) . That is, I is a class of the morphisms in C such that: (a) I ( A, B ) = Hom C ( A, B ) ∩ I is an abelian subgroup of Hom C ( A, B ) for each A, B ∈ C ; (b) If f ∈ I ( A, B ) , g ∈ Hom C ( C, A ) and h ∈ Hom C ( B, D ) , then hf g ∈I ( C, D ) . (c) Let I and J be ideals in C . The product of ideals IJ is defined asfollows: for each A, B ∈ C we set IJ ( A, B ) := ( n X i =1 f i g i (cid:12)(cid:12)(cid:12)(cid:12) g i ∈ C ( A, C i ) , f i ∈ C ( C i , B ) for some C i ∈ C ) . LUS GABRIEL RODR´IGUEZ-VALD´ES, VALENTE SANTIAGO-VARGAS
We say that an ideal I of C is idempotent if I = I . (d) Let I be an ideal of C , we set Ann( I ) := { F ∈ Mod( C ) | F ( f ) = 0 ∀ f ∈I ( A, B ) ∀ A, B ∈ C} . Now, we recall the construction of the quotient category. Let I be an ideal in apreadditive category C . The quotient category C / I is defined as follows: it hasthe same objects as C and Hom C / I ( A, B ) :=
Hom C ( A,B ) I ( A,B ) for each A, B ∈ C / I .For f = f + I ( A, B ) ∈ Hom C / I ( A, B ) and g = g + I ( B, C ) ∈ Hom C / I ( B, C ) we set g ◦ f := gf + I ( A, C ) ∈ Hom C / I ( A, C ) . Let I be an ideal of C , we have the canonical functor π : C −→ C / I defined as: π ( A ) = A ∀ A ∈ C and π ( f ) := f = f + I ( A, B ) ∈ Hom C / I ( A, B ) ∀ f ∈ Hom C ( A, B ). Definition 3.4.
Let I be an ideal in a preadditive category C and consider thefunctor π : C −→ C / I . We have the functor π ∗ : Mod( C / I ) −→ Mod( C ) definedas follows: π ∗ ( F ) := F ◦ π for F ∈ Mod( C / I ) and π ∗ ( η ) = η for η : F −→ G in Mod( C / I ) . Now, we construct the analogous of the functor Tr Λ /I ( − ) . Definition 3.5.
Let I be an ideal in C , for C ∈ C we set M C := Hom C ( C, − ) I ( C, − ) ∈ Mod( C ) . We consider the familiy M = { M C } C ∈C and we define Tr CI := Tr M :Mod( C ) −→ Mod( C ) . Remark 3.6.
For every F ∈ Mod( C ) we have that Tr CI ( F ) ∈ Ann( I ) . Proposition 3.7.
There exists a functor
Ω : Ann( I ) −→ Mod( C / I ) . For F ∈ Ann( I ) we will use the notation F := Ω( F ) .Proof. Since F ( I ) = 0 for each A, B ∈ C , there exists F A,B : Hom C ( A,B ) I ( A,B ) −→ Hom Ab ( F ( A ) , F ( B )) morphism of abelian groups such that F A,B = F A,B ◦ π A,B where π A,B : Hom C ( A, B ) −→ Hom C ( A,B ) I ( A,B ) . We define the functor F : C / I −→ Ab asfollows: F ( A ) = F ( A ) ∀ A ∈ C ; and for f ∈ Hom C /I ( A, B ) we set F ( f ) := F ( f ) . (cid:3) The following result tells us that we can identify the category Mod( C / I ) withthe full subcategory Ann( I ) of Mod( C ). Proposition 3.8.
The functors π ∗ : Mod( C / I ) −→ Ann( I ) and Ω : Ann( I ) −→ Mod( C / I ) satisfy that π ∗ ◦ Ω = 1
Ann( I ) and Ω ◦ π ∗ = 1 Mod( C / I ) . Proof.
Let F ∈ Mod( C ) such that F ( I ) = 0, then we get F = Ω( F ) ∈ Mod( C / I ).For f : A −→ B a morphism in C we have that ( π ∗ ( F ))( f ) = ( F ◦ π )( f ) = F ( f ) = F ( f ). Then ( π ∗ ◦ Ω)( F ) = F in objects. Now, let η : F −→ G be a morphism inAnn( I ), then π ∗ Ω( η ) = π ∗ ( η ) = ηπ = η . Therefore, π ∗ ◦ Ω = 1
Ann( I ) . The otherequality is proved similarly. (cid:3) We have the following properties and the proof is left to the reader.
Proposition 3.9.
Let I be an ideal in C . (a) Let F ∈ Mod( C / I ) . Then Tr CI ( F ◦ π ) = F ◦ π . (b) Let F ∈ Mod( C ) . Then Tr CI ( F ) = F if and only if F ∈ Ann( I ) . (c) Let M C := Hom C ( C, − ) I ( C, − ) ∈ Mod( C ) , then Ω( M C ) = M C = Hom C / I ( C, − ) and M C = Hom C / I ( C, − ) ◦ π = π ∗ (Hom C / I ( C, − )) .Proof. Straightforward. (cid:3)
OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 7 Now, let Tr CI := Ω ◦ Tr CI : Mod( C ) −→ Mod( C / I ). Let us see that π ∗ is leftadjoint to Tr CI . Proposition 3.10.
The functor π ∗ : Mod( C / I ) −→ Mod( C ) is left adjoint to Tr CI := Ω ◦ Tr CI : Mod( C ) −→ Mod( C / I ) . That is, there exists a natural isomor-phism θ F,G : Hom
Mod( C ) ( π ∗ ( F ) , G ) −→ Hom
Mod( C / I ) ( F, Tr CI ( G )) for F ∈ Mod( C / I ) and G ∈ Mod( C ) .Proof. First, we construct the unit η : 1 Mod( C / I ) −→ Tr CI ◦ π ∗ of the adjunction.Indeed, we have that (Tr CI ◦ π ∗ )( F ) = (Ω ◦ Tr CI )( F ◦ π ) = Ω( F ◦ π ) = (Ω ◦ π ∗ )( F ) = F (see 3.9(a) and 3.8). Then, for each F ∈ Mod( C / I ) we define η F := 1 F : F −→ (Tr CI ◦ π ∗ )( F ) . Now, we define the counit ǫ : π ∗ ◦ Tr CI −→ Mod( C ) of the adjunction. Wenote that for G ∈ Mod( C ) we have ( π ∗ ◦ Tr CI )( G ) = ( π ∗ ◦ Ω ◦ Tr CI )( G ) = ( π ∗ ◦ Ω)(Tr CI ( G )) = Tr CI ( G ) (see 3.8). Then, for G ∈ Mod( C ) we define ǫ G := Ψ G whereΨ G : Tr CI ( G ) −→ G is the canonical inclusion given in 3.1. Now, let us see that( ǫ ◦ π ∗ ) ◦ ( π ∗ ◦ η ) = 1 π ∗ . Indeed, for F ∈ Mod( C / I ) we have that [ π ∗ ◦ η ] F = π ∗ ( η F ),but η F = 1 F (definition of the unit). Hence, we get [ π ∗ ◦ η ] F = π ∗ ( η F ) = π ∗ (1 F ) =1 π ∗ ( F ) = 1 F ◦ π . On the other hand, we have that [ ǫ ◦ π ∗ ] F = ǫ π ∗ ( F ) = ǫ F ◦ π . Inorder to compute ǫ F ◦ π , we consider the factorization of Θ F ◦ π through its im-age: M (Λ F ◦ π ) ∆ F ◦ π / / Tr CI ( F ◦ π ) Ψ F ◦ π / / F ◦ π (see 3.1). By 3.9(a), we have thatTr CI ( F ◦ π ) = F ◦ π and then we have that Ψ F ◦ π = 1 F ◦ π . Then by definitionof the counit we get that ǫ π ∗ ( F ) = ǫ F ◦ π = Ψ F ◦ π = 1 F ◦ π . Hence, we have that[ ǫ ◦ π ∗ ] F ◦ [ π ∗ ◦ η ] F = 1 F ◦ π = 1 π ∗ ( F ) . Therefore, [ ǫ ◦ π ∗ ] ◦ [ π ∗ ◦ η ] = 1 π ∗ . The othertriangular identity is proved similarly. So, π ∗ is left adjoint to Tr CI . (cid:3) For the following results of this section we are going to use the functor ⊗ C :Mod( C op ) × Mod( C ) −→ Ab which we introduced in section 2.3. Let I be an idealin a preadditive category C . We recall the following functor (for more details see[15]). Definition 3.11.
We define the functor CI ⊗ C : Mod( C ) −→ Mod( C / I ) as follows:for M ∈ Mod( C ) we set (cid:0) CI ⊗ C M (cid:1) ( C ) := C ( − ,C ) I ( − ,C ) ⊗ C M for all C ∈ C / I and (cid:0) CI ⊗ C M (cid:1) ( f ) = CI ( − , f ) ⊗ C M for all f = f + I ( C, C ′ ) ∈ Hom C / I ( C, C ′ ) . We also recall the following functor which will be fundamental in this work.
Definition 3.12.
We define the functor C ( CI , − ) : Mod ( C ) −→ Mod ( C / I ) asfollows: for M ∈ Mod( C ) we set C ( CI , M )( C ) = C (cid:16) C ( C, − ) I ( C, − ) , M (cid:17) for all C ∈ C / I and C ( CI , M )( f ) = C (cid:0) CI ( f, − ) , M (cid:1) for all f = f + I ( C, C ′ ) ∈ Hom C / I ( C, C ′ ) . It is well known that C ( CI , − ) : Mod ( C ) −→ Mod ( C / I ) is right adjoint to π ∗ and CI ⊗ C is left adjoint to π ∗ (see for example, [15, Proposition 3.9]), and hence by 3.10,we have that C ( CI , − ) ≃ Tr CI since adjoint functors are unique up to isomorphisms.Thus, we have the following result. Proposition 3.13.
Let I be an ideal in C and π : C −→ C / I the canonical functor.Then we have the following diagram Mod( C / I ) π ∗ / / Mod( C ) π ∗ o o π ! o o LUS GABRIEL RODR´IGUEZ-VALD´ES, VALENTE SANTIAGO-VARGAS where ( π ∗ , π ∗ ) and ( π ∗ , π ! ) are adjoint pairs, π ! := C ( CI , − ) ≃ Tr CI and π ∗ := CI ⊗ C . We recall that for every small preadditive category C it is well known that Mod( C )is an abelian category with enough projectives and enough injectives (see for ex-ample [17, Proposition 2.3] in p. 99 and also see p. 102 in [17]). So, we can definederived functors in Mod( C ). Definition 3.14.
Let M ∈ Mod( C ) , we denote by Ext i Mod( C ) ( M, − ) : Mod( C ) −→ Ab the i -th derived functor of Hom
Mod( C ) ( M, − ) : Mod( C ) −→ Ab . Similarly wehave Ext i Mod( C ) ( − , M ) : Mod( C ) op −→ Ab . We recall that if ( I • , ǫ N ) is an injective coresolution of N , then by definitionExt i Mod( C ) ( M, N ) = H i (Hom Mod( C ) ( M, I • )) where I • is the deleted injective cores-olution of N . In the case of Ext i Mod( C ) ( − , M ) : Mod( C ) op −→ Ab we use projectiveresolutions. Now, we can construct canonical morphisms. Proposition 3.15.
Let G ∈ Mod( C ) and F ∈ Mod( C / I ) and consider an in-jective coresolution of G in Mod( C ) : → G → I → I → · · · . Then, thereexists canonical morphisms of abelian groups ϕ iF,G : Ext i Mod( C / I ) ( F, Tr CI ( G )) −→ Ext i Mod( C ) ( π ∗ ( F ) , G ) for each i ≥ .Proof. Since Tr CI is right adjoint to π ∗ (see 3.10), we get the following complex X • : 0 / / Tr CI ( G ) / / Tr CI ( I ) / / Tr CI ( I ) / / . . . in Mod( C / I ), whereeach Tr CI ( I j ) es injective in Mod( C / I ).On the other hand, since Mod( C / I ) has enough injectives we can construct aninjective coresolution J • : 0 / / Tr CI ( G ) / / J / / J / / . . . of Tr CI ( G ).By the dual of the comparison lemma (see [22, Theorem 6.16] in p. 340), have a mor-phism of complexes h = { h i } i ≥ : J • −→ X • . Then, by applying Hom Mod( C / I ) ( F, − )to the morphism h , and because of 3.10 we have the following commutative diagramHom Mod( C ) ( π ∗ F, I ) / / Hom
Mod( C ) ( π ∗ F, I ) / / . . . Hom
Mod( C /I ) ( F, Tr CI ( I )) / / θ − F,I O O Hom
Mod( C /I ) ( F, Tr CI ( I )) / / θ − F,I O O . . . Hom
Mod( C /I ) ( F, J ) / / h ∗ O O Hom
Mod( C /I ) ( F, J ) / / h ∗ O O . . . where each θ F,I j are the adjuntion isomorphisms (see 3.10). Computing homology,we have a morphism H i ( θ − F,I ) ◦ H i ( h ∗ ) := ϕ iF,G : Ext i Mod( C / I ) ( F, Tr CI ( G )) −→ Ext i Mod( C ) ( π ∗ ( F ) , G ) , where h ∗ := { h ∗ i } i ≥ and θ − F,I = { θ − F,I i } i ≥ are the morphisms of complexes in thelast diagram. (cid:3) Now, we give the following definition which is the analogous to the multiplicationof an ideal and a module in the classical sense.
Definition 3.16.
Let G ∈ Mod( C ) and I an ideal in C . We define I G as the sub-functor of G defined as follows: for X ∈ C we set I G ( X ) := X f ∈ S C ∈C I ( C,X ) Im( G ( f )) . The following proposition is analogous to the one in modules over a ring R . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 9 Lemma 3.17. [24, Lemma 2.9]
Let G ∈ Mod( C ) and I an ideal in C . Then G/ I G ∈ Mod( C / I ) and there exists an isomorphism C / I ⊗ C G ≃ G/ I G . Definition 3.18.
Let N ∈ Mod( C op ) and consider the functor N ⊗ − : Mod( C ) −→ Ab . We define Tor C i ( N, − ) : Mod( C ) −→ Ab as the i -th left derived functor of N ⊗ − . We recall that if ( P • , γ M ) is a projective resolution of M ∈ Mod( C ), by defini-tion we have that Tor C i ( N, M ) = H i ( N ⊗ C P • ) where P • is the deleted projectiveresolution of M . Proposition 3.19.
Let F ∈ Mod(( C / I ) op ) and G ∈ Mod( C ) , and consider aprojective resolution ( P • , ǫ G ) of G . Then for each i ≥ , there exists a canonicalmorphism of abelian groups ψ iF,G : Tor C i ( F ◦ π, G ) −→ Tor C / I i ( F, G/ I G ) . Proof.
Similar to 3.15. (cid:3)
Now, we give the following definition.
Definition 3.20.
Consider the functors C ( CI , − ) , CI ⊗ C − : Mod( C ) −→ Mod( C / I ) given in the definitions 3.11 and 3.12. We denote by EXT i C ( C / I , − ) : Mod( C ) −→ Mod( C / I ) the i -th right derived functor of C ( CI , − ) and TOR C i ( C / I , − ) : Mod( C ) −→ Mod( C / I ) the i -th left derived functor of CI ⊗ C . Proposition 3.21.
Consider the functors
EXT i C ( C / I , − ) : Mod( C ) −→ Mod( C / I ) and TOR C i ( C / I , − ) : Mod( C ) −→ Mod( C / I ) . For C ∈ C / I we have the following. (a) For M ∈ Mod( C ) we get EXT i C ( C / I , M )( C ) = Ext i Mod( C ) (cid:16) Hom C ( C, − ) I ( C, − ) , M (cid:17) . (b) For M ∈ Mod( C ) we have that TOR C i ( C / I , M )( C ) = Tor C i (cid:16) Hom C ( − ,C ) I ( − ,C ) , M (cid:17) .Proof. Straightforward. (cid:3)
Now, we have the following proposition which will help us to characterize k -idempotent ideals in the forthcoming sections. Proposition 3.22.
Let I an ideal in C , π : C −→ C / I the canonical functor andconsider the diagram given in 3.13. Let G ∈ Mod( C ) , and → G → I → I →· · · → an injective coresolution of G . For ≤ k ≤ ∞ , the following conditions areequivalent. (a) 0 → Tr CI ( G ) → Tr CI ( I ) → Tr CI ( I ) → · · · → Tr CI ( I k ) is the beginning ofan injective coresolution of Tr CI ( G ) ∈ Mod( C / I ) . (b) EXT i C ( C / I , G ) = 0 for all ≤ i ≤ k . (c) For F ∈ Mod( C / I ) the morphisms given in 3.15, ϕ iF,G : Ext i Mod( C / I ) ( F, Tr CI ( G )) −→ Ext i Mod( C ) ( π ∗ ( F ) , G ) , are isomorphisms for ≤ i ≤ k .Proof. ( b ) ⇔ ( a ). By definition of the derived functor, we have that EXT i C ( C / I , G )is the i -th homology of the complex of C / I -modules C ( C /I, I ) → C ( C /I, I ) → · · · → C ( C /I, I k ) → · · · But Tr CI = C ( C /I, − ), then we have that EXT i C ( C / I , G ) = 0 for all 1 ≤ i ≤ k if andonly if the following complex is exact0 → Tr CI ( G ) → Tr CI ( I ) → Tr CI ( I ) → · · · → Tr CI ( I k )where each Tr CI ( I j ) is an injective C / I -module.( a ) ⇒ ( c ). Suppose that 0 → Tr CI ( G ) → Tr CI ( I ) → Tr CI ( I ) → · · · → Tr CI ( I k ) is the beginning of an injective coresolution of Tr CI ( G ). We can complete to aninjective coresolution: 0 → Tr CI ( G ) → Tr CI ( I ) → · · · → Tr CI ( I k ) → I ′ k +1 → I ′ k +2 → · · · . Now, by taking h i = Id for all i = 1 , . . . , k , in the proof of 3.15,we obtain that H i ( h ∗ ) is an isomorphism for all 1 ≤ i ≤ k . By 3.15 we have theisomorphism for all 1 ≤ i ≤ kH i ( θ − F,I ) ◦ H i ( h ∗ ) := ϕ iF,G : Ext i Mod( C /I ) ( F, Tr CI ( G )) −→ Ext i Mod( C ) ( π ∗ ( F ) , G ) . ( c ) ⇒ ( b ). By 3.9(c), we have that M C := Hom C ( C, − ) I ( C, − ) ∈ Mod( C ) satisfies that M C = Hom C / I ( C, − ) ◦ π = π ∗ (cid:16) Hom C / I ( C, − ) (cid:17) . Let us fix i such that 1 ≤ i ≤ k .We have that EXT i C ( C / I , G ) ∈ Mod( C / I ) is defined for C ∈ C / I as follows. By3.21, we have that EXT i C ( C / I , G )( C ) := Ext i Mod( C ) (cid:18) Hom C ( C, − ) I ( C, − ) , G (cid:19) = Ext i Mod( C ) (cid:16) π ∗ (cid:16) Hom C / I ( C, − ) (cid:17) , G (cid:17) ≃ Ext i Mod( C /I ) (cid:16) Hom C / I ( C, − ) , Tr CI ( G ) (cid:17) [hypothesis]= 0 [Hom C / I ( C, − ) es projective in Mod( C / I )] (cid:3) Now, we have the following result that is analogous to the previous result.
Proposition 3.23.
Let I be an ideal in C , π : C −→ C / I the canonical functorand consider the diagram given in 3.13. Let G ∈ Mod( C ) , and · · · → P k → · · · → P → P → G → a projective resolution of G . For ≤ k ≤ ∞ , the followingconditions are equivalent. (a) P k / I P k → · · · → P / I P → P / I P → G/ I G → is the beginning of aprojective resolution of G/ I G ∈ Mod( C / I ) . (b) TOR C i ( C / I , G ) = 0 for ≤ i ≤ k . (c) For F ∈ Mod(( C / I ) op ) the morphisms given in 3.19, ψ iF,G : Tor C i ((ˆ π ) ∗ F ) , G ) −→ Tor C / I i ( F, G/ I G ) , are isomorphisms for ≤ i ≤ k . Property A and restriction of adjunctions In this section we will use some of the notions given in 2.4. First we recall thefollowing well known result.
Proposition 4.1.
Let C be a variety and proj( C ) the category of finitely generatedprojective C -modules. Consider the Yoneda functor Y : C −→ proj( C ) defined as Y ( C ) := Hom C ( C, − ) . Then Y is a contravariant functor which is full, faithful anddense. Let C be a Hom-finite R -variety, we recall that mod( C ) denotes the full subcat-egory of Mod( C ) whose objects are the finitely presented functors (see definition2.5). The aim of this section is to restrict the functors obtained in the last section. Proposition 4.2.
Let C be a Hom-finite R -variety, I an ideal in C and π : C −→C / I the canonical functor. Consider the upper part of the diagram given in 3.13. (a) We can restrict π ∗ to a functor π ∗ : mod( C ) −→ mod( C / I ) . (b) If for every C ∈ C there exists an epimorphism Hom C ( C ′ , − ) −→ I ( C, − ) ,we can restrict the functor π ∗ to a functor π ∗ : mod( C / I ) −→ mod( C ) . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 11 (c) If for every C ∈ C there exists an epimorphism Hom C ( C ′ , − ) −→ I ( C, − ) ,we have the adjoint pair mod( C / I ) π ∗ / / mod( C ) . π ∗ o o Proof. (a) Let us see that we have π ∗ : mod( C ) −→ mod( C / I ) . Indeed, we knowthat π ∗ : Mod( C ) −→ Mod( C / I ) is right exact. Moreover, by the construc-tion of π ∗ it follows that π ∗ (Hom C ( C, − ))( C ′ ) = C ( − ,C ′ ) I ( − ,C ′ ) ⊗ Hom C ( C, − ) = C ( C,C ′ ) I ( C,C ′ ) = Hom C / I ( C, C ′ ), and thus π ∗ (Hom C ( C, − )) = Hom C / I ( C, − ) (see2.1). From this we have the restriction π ∗ : mod( C ) −→ mod( C / I ).(b) Let us see that if M ∈ mod( C / I ), then π ∗ ( M ) ∈ mod( C ). Indeed, let M ∈ mod( C / I ), then there exists an exact sequence Hom C / I ( X, − ) → Hom C / I ( Y, − ) → M → X, Y ∈ C / I . Applying π ∗ , by 3.9 we havethe following exact sequence Hom C ( X, − ) I ( X, − ) → Hom C ( Y, − ) I ( Y, − ) → π ∗ ( M ) →
0. Weassert that
Hom C ( X, − ) I ( X, − ) is finitely presented for each X ∈ C . To prove thiswe consider 0 / / I ( X, − ) / / Hom C ( X, − ) / / Hom C ( X, − ) I ( X, − ) / / I ( X, − ) is finitely generated, then by theorem[2, proposition 4.2(c)], we have that Hom C ( X, − ) I ( X, − ) is finitely presented. Thenby [2, proposition 4.2(b)], we conclude that π ∗ ( M ) is finitely presented.(c) Follows from (a) and (b). (cid:3) Since C is a Hom-finite R -variety we have the following functors (see section 2.4) D C : ( C , mod( R )) −→ ( C op , mod( R )) , D C op : ( C op , mod( R )) −→ ( C , mod( R )) . Given an ideal I in C we will consider the canonical functors π : C −→ C / I and π : C op −→ C op / I op . Since ( C , mod( R )) ⊆ Mod( C ), it is easy to show that we havefunctors ( π ) ∗ : ( C / I , mod( R )) −→ ( C , mod( R )) and ( π ) ∗ : ( C op / I op , mod( R )) −→ ( C op , mod( R )). We also have that C / I is a Hom-finite R -variety and we get afunctor D C / I : ( C / I , mod( R )) → (( C / I ) op , mod( R )). Remark 4.3.
We have the following commutative diagram ( C / I , mod( R )) ( π ) ∗ / / D C / I (cid:15) (cid:15) ( C , mod( R )) D C (cid:15) (cid:15) (( C / I ) op , mod( R )) ( π ) ∗ / / ( C op , mod( R ))Consider the functor Ω : Ann( I ) −→ Mod( C / I ) defined in 3.7, we know that Ωis an equivalence of categories with inverse π ∗ : Mod( C / I ) −→ Ann( I ). Then wehave the following proposition which tells us that we can restrict the functor π ∗ . Proposition 4.4.
Let C be a Hom-finite R -variety and I an ideal in C such thatfor each C ∈ C there exists an epimorphism Hom C ( C ′ , − ) −→ I ( C, − ) . Then, thereexists an equivalence π ∗ | mod( C / I ) : mod( C / I ) −→ mod( C ) ∩ Ann( I ) . Proof.
By 4.2(b), we have a functor π ∗ | mod( C / I ) : mod( C / I ) −→ mod( C ) ∩ Ann( I ).In order to prove that π ∗ is an equivalence it is enough to see that π ∗ is dense.Indeed, let M ∈ mod( C ) ∩ Ann( I ). Then M ( f ) = 0 for all f ∈ I . Since π ∗ :Mod( C / I ) −→ Ann( I ) is an equivalence, we have that there exists M ′ ∈ Mod( C / I )such that π ∗ ( M ′ ) = M ′ ◦ π ≃ M . By 3.13, we have that π ∗ is right adjoint to π ∗ and, moreover, we have that π ∗ is full and faithful. Then by [9, Theorem 3.4.1], we conclude that π ∗ π ∗ ≃ Mod( C / I ) . Now, by 4.2 we have that π ∗ ( M ) ∈ mod( C / I ).Then we have that M ′ ≃ π ∗ π ∗ ( M ′ ) = π ∗ ( M ) ∈ mod( C / I ), hence, proving that M ′ is finitely presented and π ∗ ( M ′ ) ≃ M . Therefore π ∗ | mod( C / I ) is dense. (cid:3) Because of the last proposition we are now interested in ideals that satisfy thehypothesis of 4.4 . So we have the following definition.
Definition 4.5.
Let C be a preadditive category. We say that an ideal I satisfiesthe property (A) if for every C ∈ C there exists epimorphisms Hom C ( X, − ) −→I ( C, − ) −→ and Hom C ( − , Y ) −→ I ( − , C ) −→ . The following result tells us that if the ideal I satisfies property A , then thecategory C / I is dualizing provided C is also dualizing. Proposition 4.6.
Let C be a dualizing R -variety and I an ideal which satisfiesproperty A . Then C / I is a dualizing R -variety and the following diagram mod( C / I ) ( π ) ∗ / / D C / I (cid:15) (cid:15) mod( C ) D C (cid:15) (cid:15) mod(( C / I ) op ) ( π ) ∗ / / mod( C op ) is commutative.Proof. Let D C : mod( C ) −→ mod( C op ) be the duality. It is enough to see that wehave a functor D C : mod( C ) ∩ Ann( I ) −→ mod( C op ) ∩ Ann( I op ) and similarly for D C op .Indeed, let M ∈ mod( C ) ∩ Ann( I ) and consider D C ( M ) ∈ mod( C op ). Let f op ∈I op ( B op , A op ) then f ∈ I ( A, B ). Therefore, D ( M )( f op )) := Hom R ( M ( f ) , I ( R/r )) =0 since M ( f ) = 0. Then C / I is dualizing and D C / I ≃ ( D C ) | mod( C ) ∩ Ann( I ) . This waywe have proved that the required diagram is commutative. (cid:3) The proof of the following result is similar to the one given in [20, Proposition2.7].
Proposition 4.7.
Let C be a dualizing R -variety and I an ideal which satisfiesproperty A . Let π : C −→ C / I be the canonical functor, then we can restrict thediagram given in 3.13 to the finitely presented modules mod( C / I ) ( π ) ∗ / / mod( C ) π ∗ o o π !1 o o Proof.
The proof given in [20, Proposition 2.7] works in this setting. (cid:3)
Now, we give some examples where the property A holds. We recall that the(Jacobson) radical of an additive category C is the two-sided ideal rad C in C definedby the formula rad C ( X, Y ) = { h ∈ C ( X, Y ) | X − gh is invertible for any g ∈C ( Y, X ) } for all objects X and Y of C . Proposition 4.8.
Let C be a dualizing R -variety and I = rad( C )( − , − ) the radicalideal. Then I satisfies the property A .Proof. See [12, Prop. 2.10 (2)] in p. 128. (cid:3)
In order to give more examples of ideals satisfying the property A we recall thefollowing definition. Definition 4.9.
Let C be small abelian R -category with the following properties. OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 13 (a) There is only a finite number of nonisomorphic simple objects in C . (b) Every object in C is of finite length.(It is well known that under this hypothesis C is a Krull-Schmidt category). Itis said that C is of finite representation type if C has only a finite number ofnon-isomorphic indecomposable objects (see [3, p. 12] ). Following the notation in [3, p. 3], an object M ∈ Mod( C ) is finite if it is bothnoetherian an artinian. That is, if M satisfies the ascending and descending chaincondition on submodules. Proposition 4.10.
Let C be of finite representation type as in definition 4.9. Thenevery ideal I ( − , − ) in C satisfies property A .Proof. By [3, 3.6 (a) and (b)], we have that Mod( C ) and Mod( C op ) are locallyfinite. By [3, 3.1], we have that each Hom C ( C, − ) and Hom C ( − , C ) are finite foreach C ∈ C . Since the subcategory of finite modules is a Serre subcategory, we havethat the submodules of Hom C ( C, − ) and Hom C ( − , C ) are finite. In particular, each I ( C, − ) and I ( − , C ) are finite. By [3, Corollary 1.7], we have that I ( C, − ) and I ( − , C ) are finitely generated. Then I satisfies property A . (cid:3) Corollary 4.11. If Λ is an artin algebra of finite representation type, then everyideal in C = mod(Λ) satisfies property A . We recall the following notions. Let A be an arbitrary category and B a fullsubcategory in A . The full subcategory B is contravariantly finite if for every A ∈ A there exists a morphism f A : B −→ A with B ∈ B such that if f ′ : B ′ −→ A is other morphism with B ′ ∈ B , then there exist a morphism g : B ′ −→ B suchthat f ′ = f A ◦ g . Dually, is defined the notion of covariantly finite . We say that B is functorially finite if B is contravariantly finite and covariantly finite. Proposition 4.12.
Let C be an additive category and X an additive full subcategoryof C . Let I = I X be the ideal of morphisms in C which factor through some objectin X . Then I satisfies property A if and only if X is functorially finite in C .Proof. ( ⇐ =). Suppose that X is contravariantly finite. Then for each C ∈ C , thereexists a right X -approximation f C : X −→ C . Thus, we have a morphismHom C ( − , f C ) : Hom C ( − , X ) −→ Hom C ( − , C ) . We assert that Im(Hom C ( − , f C )) = I ( − , C ). Indeed, let C ′ ∈ C and α ∈ I ( C ′ , C ).Since I = I X , there exists X ′ ∈ X and morphisms α ′ : C ′ −→ X ′ and α ′′ : X ′ −→ C such that α = α ′′ α ′ . Since f C is an X -approximation, there exists β : X ′ −→ X such that α ′′ = f C β . Then α = α ′′ α ′ = f C βα ′ . Then we have that α ∈ Im(Hom C ( − , f C ) C ′ ). Now, for γ : C ′ −→ X we have that (Hom C ( − , f C )) C ′ ( γ ) = f C γ ∈ I ( C ′ , C ) since f C γ factors through X ∈ X and I = I X , proving thatIm(Hom C ( − , f C )) = I ( − , C ). Thus, there exists an epimorphism Hom C ( − , f C ) :Hom C ( − , X ) −→ I ( − , C ) . Similarly, we can prove that if X is covariantly finite,then there exists and epimorphism Hom C ( Y, − ) −→ I ( C, − ) −→ C ∈ C .Therefore, we have that if X is functorially finite, then I satisfies property A .The other implication is similar and the details are left to the reader. (cid:3) Now let us consider the transfinite radical of C denoted by rad ∗C ( − , − ) (see[26] for details). Proposition 4.13.
Let C be Hom-finite R -variety and suppose that rad ∗C ( − , − ) = 0 .Let I be an idempotent ideal in C and let X = { X ∈ C | X ∈ I ( X, X ) } . If X isfunctorially finite, then I satisfies property A . Proof.
Since R is artinian and C is a Hom-finite R -variety, we have by [26], that C is Krull-Schmidt with local d.c.c on ideals (see [26, Definition 5]). By [26, Corollary10], we have that I = I X where X = { X ∈ C | X ∈ I ( X, X ) } . Now, if X isfunctorially finite, by 4.12, we have that I satisfies property A . (cid:3) Example 4.14.
Let C = mod(Λ) where Λ is a finite dimensional K -algebra overan algebraically closed field. If Λ is a standard selfinjective algebra of domesticrepresentation type or Λ is a special biserial algebra of domestic representationtype, then rad ∗C ( − , − ) = 0 (see [13] and [23] ). We recall that Λ is of domesticrepresentation type if there is a natural number N such that for each dimension d,all but finitely many indecomposable modules of dimension d belong to at most N one-parameter families. k -idempotent ideals In this section we will work in preadditive categories as well as in dualizing R -varieties. So, we will say explicitely in which context we are working on.We introduce the definition of k -idempotent ideal in C , which is the analogous tothe one given by Aulander-Platzeck-Todorov in [4] for the case of artin algebras.In order to do this we consider the morphisms given in 3.15. Let us consider F ′ ∈ Mod( C / I ) and G := π ∗ ( F ′ ). In the proof of 3.10, we have that Tr CI ( π ∗ ( F ′ )) = F ′ .Then for F, F ′ ∈ Mod( C / I ) we have canonical morphisms ϕ iF,π ∗ ( F ′ ) : Ext i Mod( C /I ) ( F, F ′ ) −→ Ext i Mod( C ) ( π ∗ ( F ) , π ∗ ( F ′ )) . Definition 5.1.
Let C be a preadditive category and I an ideal in C . (a) We say that I is k-idempotent if ϕ iF,π ∗ ( F ′ ) : Ext i Mod( C /I ) ( F, F ′ ) −→ Ext i Mod( C ) ( π ∗ ( F ) , π ∗ ( F ′ )) is an isomorphism for all F, F ′ ∈ Mod( C / I ) and for all ≤ i ≤ k . (b) We say that I is strongly idempotent if ϕ iF,π ∗ ( F ′ ) : Ext i Mod( C /I ) ( F, F ′ ) −→ Ext i Mod( C ) ( π ∗ ( F ) , π ∗ ( F ′ )) is an isomorphism for all F, F ′ ∈ Mod( C / I ) and for all ≤ i < ∞ . We note that we have defined k -idempotent ideal in Mod( C ), but this concept canalso be defined in the category of finitely presented C -modules mod( C ). Next, wehave a characterization of k -idempotent ideals in terms of the vanishing of certainderived functors. Proposition 5.2.
Let C be a preadditive category, I an ideal in C and ≤ i ≤ k .The following conditions are equivalent. (a) I es k -idempotent (b) ϕ iF,π ∗ ( F ′ ) : Ext i Mod( C /I ) ( F, F ′ ) −→ Ext i Mod( C ) ( π ∗ ( F ) , π ∗ ( F ′ )) is an isomor-phism for all F, F ′ ∈ Mod( C / I ) and for all ≤ i ≤ k . (c) EXT i C ( C / I , F ′ ◦ π ) = 0 for ≤ i ≤ k and for F ′ ∈ Mod( C / I ) . (d) EXT i C ( C / I , J ◦ π ) = 0 for ≤ i ≤ k and for each J ∈ Mod( C / I ) which isinjective.Proof. The equivalences between (a), (b) and (c) are straightforward using 3.22,and ( c ) ⇒ ( d ) is trivial.( d ) ⇒ ( c ). Let us see by induction on i that EXT i C ( C / I , F ′ ◦ π ) = 0 for all F ′ ∈ Mod( C / I ). Let us suppose that i = 1 and F ′ ∈ Mod( C / I ). Consider the exactsequence 0 / / F ′ µ / / I / / IF ′ / / OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 15 where I is an injective C -module. Since Tr CI ≃ C ( CI , − ), and by the proof of theadjunction 3.10, we have that η : 1 Mod( C / I ) −→ Tr CI ◦ π ∗ is an isomorphism. Thenwe have the following commutative and exact diagram0 / / C ( CI , F ′ ◦ π ) / / C ( CI , I ◦ π ) / / C ( CI , IF ′ ◦ π ) δ / / EXT C ( C / I , F ′ ◦ π )0 / / F ′ µ / / O O I / / O O IF ′ / / O O δ = 0. Then wehave the following exact sequence0 / / EXT C ( C / I , F ′ ◦ π ) / / EXT C ( C / I , I ◦ π ) / / EXT C ( C / I , IF ′ ◦ π ) / / By hypothesis we have that
EXT C ( C / I , I ◦ π ) = 0, then we get that EXT C ( C / I , F ′ ◦ π ) = 0, proving the case i = 1.Now, let us suppose that EXT i − C ( C / I , N ◦ π ) = 0 for all N ∈ Mod( C / I ). Let F ′ ∈ Mod( C / I ). From the long exact homology sequence we have the exact sequence EXT i − C ( C / I , IF ′ ◦ π ) / / EXT i C ( C / I , F ′ ◦ π ) / / EXT i C ( C / I , I ◦ π ) . Since IF ′ ∈ Mod( C / I ), by induction we have that EXT i − C ( C / I , IF ′ ◦ π ) = 0 and byhypothesis we have that EXT i C ( C / I , I ◦ π ) = 0, then we conclude that EXT i C ( C / I , F ′ ◦ π ) = 0, therefore, proving the proposition. (cid:3) The following proposition tells us that we can restrict the result given in 5.2 tothe category of finitely presented modules.
Proposition 5.3.
Let C be a dualizing R -variety, I an ideal which satisfies property A and ≤ i ≤ k . The following are equivalent. (a) I es k -idempotent. (b) ϕ iF, ( π ) ∗ ( F ′ ) : Ext i mod( C /I ) ( F, F ′ ) −→ Ext i mod( C ) (( π ) ∗ ( F ) , ( π ) ∗ ( F ′ )) is anisomorphism for all F, F ′ ∈ mod( C / I ) and for all ≤ i ≤ k . (c) EXT i C ( C / I , F ′ ◦ π ) = 0 for ≤ i ≤ k and for F ′ ∈ mod( C / I ) . (d) EXT i C ( C / I , J ◦ π ) = 0 for ≤ i ≤ k and for each J ∈ mod( C / I ) which isinjective.Proof. By 4.7, we can restrict the diagram given in 3.13 to the category of finitelypresented modules, then 5.2 holds for the case of finitely presented modules. (cid:3)
Now, we will work in the category Mod( C op ) and we will consider the corre-sponding canonical morphisms analogous to ϕ iF,π ∗ ( F ′ ) , which we will denote by δ iF, ( π ) ∗ ( F ′ ) : Ext i mod(( C / I ) op ) ( F, F ′ ) −→ Ext i mod( C op ) (cid:16) ( π ) ∗ ( F ) , ( π ) ∗ ( F ′ ) (cid:17) for all F, F ′ ∈ mod(( C / I ) op ) and for all 0 ≤ i ≤ k , where π : C op −→ C op / I op is theprojection. Therefore, we have that the result 5.2 holds for the category Mod( C op ). Proposition 5.4.
Let C be a dualizing R -variety and I an ideal which satisfiesproperty A . Then I is k -idempotent in C if and only if I op is k -idempotent in C op .Proof. Suppose that I is k -idempotent in C . Let us see that δ iF, ( π ) ∗ ( F ′ ) : Ext i mod(( C / I ) op ) ( F, F ′ ) −→ Ext i mod( C op ) (cid:16) ( π ) ∗ ( F ) , ( π ) ∗ ( F ′ ) (cid:17) is an isomorphism for all F, F ′ ∈ mod(( C / I ) op ) and for all 0 ≤ i ≤ k . By theproposition 5.3 it is enough to see that EXT i C op ( C op / I op , F ′ ◦ π ) = 0 for 1 ≤ i ≤ k and for F ′ ∈ mod( C op / I op ). Indeed, for C ∈ C op / I op we have that EXT i C op ( C op / I op , F ′ ◦ π )( C ) == Ext i mod( C op ) (cid:18) Hom C ( − , C ) I ( − , C ) , ( π ) ∗ ( F ′ ) (cid:19) [see 3 . i mod( C op ) (cid:16) ( π ) ∗ (cid:16) Hom C / I ( − , C ) (cid:17) , ( π ) ∗ ( F ′ ) (cid:17) [see 3 . ≃ Ext i mod( C ) (cid:16) D − C (cid:16) ( π ) ∗ ( F ′ ) (cid:17) , D − C (cid:16) ( π ) ∗ (cid:16) Hom C / I ( − , C ) (cid:17)(cid:17)(cid:17) [ D C is a duality] ≃ Ext i mod( C ) (cid:16) ( π ) ∗ ( D − C / I ( F ′ )) , ( π ) ∗ (cid:16) D − C / I (cid:16) Hom C / I ( C, − ) (cid:17)(cid:17)(cid:17) [diagram in 4 . ≃ Ext i mod( C / I ) (cid:16) D − C / I ( F ′ ) , D − C / I (cid:16) Hom C / I ( − , C ) (cid:17)(cid:17) [ I is k-f.p-idempotent]= 0 [ D − C / I (Hom C / I ( − , C )) is injective in mod( C / I ))]In the third equality we are using Ext i mod( C ) ( X, Y ) ≃ Ext i mod( C op ) (cid:16) D C ( Y ) , D C ( X ) (cid:17) for X, Y ∈ mod( C ). Hence, EXT i C op ( C op / I op , F ′ ◦ π ) = 0 for 1 ≤ i ≤ k and for F ′ ∈ mod( C op / I op ), proving by 5.2 that I op is k -idempotent. The other implicationis similar. (cid:3) Now, consider the morphism given in 3.19. For G = F ′ ◦ π with F ′ ∈ Mod( C / I )we have that G/ I G ≃ F ′ . Then for F ∈ Mod(( C / I ) op ) and F ′ ∈ Mod( C / I ) wehave the morphism ψ iF, ( π ) ∗ ( F ′ ) : Tor C i ( F ◦ π , F ′ ◦ π ) −→ Tor C / I i ( F, F ′ ) . The proofof the following two propositions are similar to 5.2 and 5.3.
Proposition 5.5.
Let C be a preadditive category, I an ideal in C and ≤ i ≤ k .The following conditions are equivalent. (a) ψ iF, ( π ) ∗ ( F ′ ) : Tor C i ( F ◦ π , F ′ ◦ π ) −→ Tor C / I i ( F, F ′ ) is an isomorphism forall ≤ i ≤ k , F ∈ Mod(( C / I ) op ) and F ′ ∈ Mod( C / I ) . (b) TOR C i ( C / I , F ′ ◦ π ) = 0 for ≤ i ≤ k and for all F ′ ∈ Mod( C / I ) . (c) TOR C i ( C / I , P ◦ π ) = 0 for ≤ i ≤ k and for all P ∈ Mod( C / I ) that isprojective. This result can be restricted to the category of finitely presented modules.
Proposition 5.6.
Let C be a dualizing R -variety, I an ideal which satisfies property A and ≤ i ≤ k . The following are equivalent. (a) ψ iF, ( π ) ∗ ( F ′ ) : Tor C i ( F ◦ π , F ′ ◦ π ) −→ Tor C / I i ( F, F ′ ) is an isomorphism forall ≤ i ≤ k , F ∈ mod(( C / I ) op ) and F ′ ∈ mod( C / I ) . (b) TOR C i ( C / I , F ′ ◦ π ) = 0 for ≤ i ≤ k and for all F ′ ∈ mod( C / I ) . (c) TOR C i ( C / I , Hom C / I ( C, − ) ◦ π ) = 0 for ≤ i ≤ k and for all Hom C / I ( C, − ) ∈ mod( C / I ) . Now, in order to relate the functors
EXT i C ( C / I , − ) and TOR C i ( C / I , − ) we needthe Auslander-Reiten duality. We have the following result due to Auslander andReiten. Proposition 5.7.
Let C be a dualizing R -variety and M ∈ mod( C ) . Then we havean isomorphism of contravariant functors from mod( C op ) to mod( R ) D mod( C op ) (Tor C i ( − , M )) ≃ Ext i mod( C op ) ( − , D C ( M )) . Proof.
See [5, Proposition 7.3] in p. 341. (cid:3)
OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 17 Now, we have the following result that characterizes k -idempotent ideals in termsof the morphisms ψ iF, ( π ) ∗ ( F ′ ) . Proposition 5.8.
Let C be a dualizing R -variety and I an ideal which satisfiesproperty A . Then I is k -idempotent if and only if ψ iF, ( π ) ∗ ( F ′ ) is an isomorphismfor every F ∈ mod(( C / I ) op ) and for every F ′ ∈ mod( C / I ) .Proof. (= ⇒ ). Let F ′ ∈ mod( C / I ). Let us see that TOR C i ( C / I , F ′ ◦ π ) = 0 for1 ≤ i ≤ k . Indeed, for C ∈ C / I we have TOR C i ( C / I , F ′ ◦ π )( C ) =:= Tor C i (cid:18) Hom C ( − , C ) I ( − , C ) , ( π ) ∗ ( F ′ ) (cid:19) = Tor C i (cid:16) ( π ) ∗ (cid:16) Hom C / I ( − , C ) (cid:17) , ( π ) ∗ ( F ′ ) (cid:17) [see 3 . R (cid:16) Ext i Mod( C ) (cid:16) ( π ) ∗ ( F ′ ) , D − C (cid:16) ( π ) ∗ (cid:16) Hom C / I ( − , C ) (cid:17)(cid:17) , E (cid:17) [by 5 . R (cid:16) Ext i Mod( C ) (cid:16) ( π ) ∗ ( F ′ ) , ( π ) ∗ (cid:16) D − C / I (cid:16) Hom C / I ( − , C ) (cid:17)(cid:17) , E (cid:17) [diagram in 4 . R (cid:16) Ext i Mod( C / I ) (cid:16) F ′ , D − C / I (cid:16) Hom C / I ( − , C ) (cid:17)(cid:17) , E (cid:17) [ I is k-idempotent] ≃ i Mod( C / I ) (cid:16) F ′ , D − C / I (cid:16) Hom C / I ( − , C ) (cid:17)(cid:17) = 0 since thefunctor D − C / I (cid:16) Hom C / I ( − , C ) (cid:17) is injective in mod( C / I ). Therefore, by 5.6 we havethat ψ iF, ( π ) ∗ ( F ′ ) is isomorphism. The other implication is similar. (cid:3) We finish this section with the following result, which is analogous to proposition1.3 in [4].
Corollary 5.9.
Let C be a dualizing R -variety, I an ideal which satisfies property A . For ≤ i ≤ k , the following are equivalent. (a) I es k -idempotent. (b) ϕ iF, ( π ) ∗ ( F ′ ) : Ext i mod( C /I ) ( F, F ′ ) −→ Ext i mod( C ) (( π ) ∗ ( F ) , ( π ) ∗ ( F ′ )) is anisomorphism for all F, F ′ ∈ mod( C / I ) and for all ≤ i ≤ k . (c) EXT i C ( C / I , F ′ ◦ π ) = 0 for ≤ i ≤ k and for F ′ ∈ mod( C / I ) . (d) EXT i C ( C / I , J ◦ π ) = 0 for ≤ i ≤ k and for each J ∈ mod( C / I ) which isinjective. (e) ψ iF, ( π ) ∗ ( F ′ ) : Tor C i ( F ◦ π , F ′ ◦ π ) −→ Tor C / I i ( F, F ′ ) is an isomorphism forall ≤ i ≤ k and F ∈ mod(( C / I ) op ) and F ′ ∈ mod( C / I ) . (f) TOR C i ( C / I , F ′ ◦ π ) = 0 for ≤ i ≤ k and for all F ′ ∈ mod( C / I ) . (g) TOR C i ( C / I , Hom C / I ( C, − ) ◦ π ) = 0 for ≤ i ≤ k and for all Hom C / I ( C, − ) ∈ mod( C / I ) .Proof. It follows from 5.3, 5.8 and 5.6. (cid:3) Projective resolutions of k -idempotent ideals In this section we will work in preadditive categories as well as in dualizing R -varieties. So, we will say explicitely in which context we are working on.In the previous section we characterized k -idempotent ideals in terms of the projec-tive resolutions of all C / I -modules. We will show here that knowing the projectiveresolutions of I ( C, − ) for all C ∈ C is enough to determine for which k the ideal I is k -idempotent. Proposition 6.1. (Dual basis Lemma) Let C be a preadditive category. An object P ∈ Mod( C ) is projective if and only if there exists a family of morphisms { β j : P −→ Hom C ( C j , − ) } j ∈ J and a family { x j } j ∈ J with x j ∈ P ( C j ) such that for all X ∈ C and for every a ∈ P ( X ) there exists a finite subset J X,a ⊆ J such that a = X j ∈ J X,a P ([ β j ] X ( a ))( x j ) . Proof. (= ⇒ ) Since { Hom C ( C, − ) } C ∈C is a generating set of projective modules,there exists an epimorphism f : L j ∈ J Hom C ( C j , − ) −→ P. We get the morphisms η j := f u j : Hom C ( C j , − ) −→ P , where u j : Hom C ( C j , − ) −→ L j ∈ J Hom C ( C j , − )is the j -th inclusion. By Yoneda’s Lemma, f u j corresponds to one element x j :=[ η j ] C j (1 C j ) ∈ P ( C j ); furthermore, η j : Hom C ( C j , − ) −→ P is such that [ η j ] X ( α ) = P ( α )( x j ) for all X ∈ C .Now, let γ = ( γ j ) j ∈ J ∈ L j ∈ J Hom C ( C j , X ). Then, there exists a finite subset J γ of J such that γ j = 0 if j / ∈ J γ . We know that f X : L j ∈ J Hom C ( C j , X ) −→ P ( X ) is defined for γ = ( γ j ) j ∈ J ∈ L j ∈ J Hom C ( C j , X ) as follows: f X (( γ j ) j ∈ J ) = P j ∈ J γ [ η j ] X ( γ j ) = P j ∈ J γ P ( γ j )( x j ) . Now, since P is projective we have that f isa split epimorphism, and then there exists g : P −→ L j ∈ J Hom C ( C j , − ) such that f g = 1 P . Let us consider the projection π j : L j ∈ J Hom C ( C j , − ) −→ Hom C ( C j , − ),then we have β j := π j g : P −→ Hom C ( C j , − ) . Then for X ∈ C we have that g X : P ( X ) −→ L j ∈ J Hom C ( C j , X ) is defined as follows: g X ( a ) = ([ β j ] X ( a )) j ∈ J ∀ a ∈ P ( X )where [ β j ] X ( a ) : C j −→ X . Since g X ( a ) ∈ L j ∈ J Hom C ( C j , X ), there exists a finitesubset J X,a ⊆ J such that [ β j ] X ( a ) = 0 if j / ∈ J X,a . Then a = f X ( g X ( a )) = f X (cid:16) ([ β j ] X ( a )) j ∈ J (cid:17) = X j ∈ J X,a P ([ β j ] X ( a ))( x j ) . The other implication is similar and is left to the reader. (cid:3)
We recall that given a family of objects F = { F i } i ∈ I and M ∈ Mod( C ), in 3.1we defined the trace in M of the family F which is denoted by Tr F ( M ). We havethe following description of the trace. Remark 6.2.
Let F = { F i } i ∈ I be a family in Mod( C ) . For each N ∈ Mod( C ) and X ∈ C we have that Tr F ( N )( X ) = X { f ∈ Hom(
F,N ) | F ∈F} Im( f X ) . In the case F = { F } is just oneobject, we will write Tr F N . We have the following definition.
Definition 6.3.
Let C be a preadditive category and F = { F i } i ∈ I a family ofobjects in Mod( C ) . For each C ∈ C consider the C -submodule Tr F (Hom C ( C, − )) of Hom C ( C, − ) . We define the subfunctor Tr F C of Hom C ( − , − ) : C op × C −→ Ab asfollows: (Tr F C )( C, C ′ ) := Tr F (Hom C ( C, − ))( C ′ ) for all C, C ′ ∈ C . This ideal will be called trace ideal . In the case that F = { P } with P a projective C -module, we will write Tr P C . It is easy to see that Tr F C ( − , − ) is a subfunctor of Hom C ( − , − ) and thus an ideal. Proposition 6.4.
Let C be a preadditive category and let P be a projective C -module. Then Tr P C defines an idempotent ideal of C . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 19 Proof.
We have to show that (Tr P C )( C, X ) ⊆ (Tr P C )( C, X ) for all
C, X ∈ C .By definition we have that (Tr P C )( C, X ) := Tr P (Hom C ( C, − ))( X ). Then, if α ∈ (Tr P C )( C, X ) we have that α = P ni =1 [ η i ] X ( a i ) for some η i : P −→ Hom C ( C, − ) and a i ∈ P ( X ) (see 6.2). Then, if we want to show that (Tr P C )( C, X ) = (Tr P C )( C, X )it is enough to show that if η : P −→ Hom C ( C, − ), then η X ( a ) ∈ (Tr P C )( C, X ) for a ∈ P ( X ). Then consider η : P −→ Hom C ( C, − ) and a ∈ P ( X ). By the dual basislemma, there exists a family of morphisms { β j : P −→ Hom C ( C j , − ) } j ∈ J and afamily { x j } j ∈ J with x j ∈ P ( C j ) such that there exists a finite subset J X,a ⊆ J suchthat a = P j ∈ J X,a P ([ β j ] X ( a ))( x j ) , with [ β j ] X ( a ) : C j −→ X . Since η is a naturaltransformation for each [ β j ] X ( a ) : C j −→ X , we get the following commutativediagram P ( C j ) η Cj / / P (cid:0) [ β j ] X ( a ) (cid:1) (cid:15) (cid:15) Hom C ( C, C j ) Hom C ( C, [ β j ] X ( a )) (cid:15) (cid:15) P ( X ) η X / / Hom C ( C, X )Then for x j ∈ P ( C j ) we have that η X (cid:16) P (cid:0) [ β j ] X ( a ) (cid:1) ( x j ) (cid:17) = Hom C (cid:16) C, [ β j ] X ( a ) (cid:17) ( η C j ( x j )) = [ β j ] X ( a ) ◦ η C j ( x j ) . Now, since η C j ( x j ) ∈ Im( η C j ) we have that η C j ( x j ) ∈ Tr P (Hom C ( C, − ))( C j ) ⊆ Hom C ( C, C j ). Similarly, [ β j ] X ( a ) ∈ Tr P (Hom C ( C j , − ))( X ) ⊆ Hom C ( C j , X ). Bydefinition of Tr P C , we have that [ β j ] X ( a ) ◦ η C j ( x j ) ∈ Tr P (Hom C ( C, − ))( X ) =(Tr P C )( C, X ). Therefore, we have that η X ( a ) = P j ∈ J X,a η X (cid:16) P (cid:0) [ β j ] X ( a ) (cid:1) ( x j ) (cid:17) = P j ∈ J X,a [ β j ] X ( a ) ◦ η C j ( x j ) ∈ (Tr P C )( C, X ) . (cid:3) Then we can define the (Tr P C · F )( X ) := P f ∈ S C ∈C (Tr P C )( C,X ) Im( F ( f )) . (see3.16). Now, we have the following result which is a generalization of the basicresult in modules over a ring R . Proposition 6.5.
Let C be a preadditive category, F ∈ Mod( C ) and Tr P C the traceideal. Then Tr P C · F = Tr P ( F ) . Proof.
Let us see that (Tr P F )( X ) ⊆ (Tr P C · F )( X ) . It is enough to see thatIm( η X ) ⊆ (Tr P C · F )( X ) for η : P −→ F . Then, let x ∈ Im( η X ) ⊆ F ( X )with η : P −→ F , then there exists a ∈ P ( X ) such that η X ( a ) = x . ByYoneda’s Lemma, there exists α : Hom C ( X, − ) −→ P such that α X (1 X ) = a and α Y ( β ) = P ( β )( a ) ∀ β ∈ Hom C ( X, Y ). By the dual basis lemma, there existsa family of morphisms { β j : P −→ Hom C ( C j , − ) } j ∈ J and a family { x j } j ∈ J with x j ∈ P ( C j ) such that ( ∗ ) : a = X j ∈ J X,a P ([ β j ] X ( a ))( x j ) , for a finite subset J X,a ⊆ J . We note that [ β j ] X ( a ) : C j −→ X and [ β j ] X ( a ) ∈ Im([ β j ] X ) and then [ β j ] X ( a ) ∈ Tr P (Hom C ( C j , − ))( X ) = (Tr P C )( C j , X ) . Applying η X to the equality ( ∗ ) we get x = η X ( a ) = P j ∈ J X,a η X (cid:16) P ([ β j ] X ( a ))( x j ) (cid:17) . Now,since η is a natural transformation we get following commutative diagram P ( C j ) P (cid:0) [ β j ] X ( a ) (cid:1) (cid:15) (cid:15) η Cj / / F ( C j ) F (cid:0) [ β j ] X ( a ) (cid:1) (cid:15) (cid:15) P ( X ) η X / / F ( X ) . Then, we get that η X (cid:16) P ([ β j ] X ( a ))( x j ) (cid:17) = F (cid:16) [ β j ] X ( a ) (cid:17) ( η C j ( x j )) and since themorphism [ β j ] X ( a ) ∈ (Tr P C )( C j , X ) we get that η X (cid:16) P ([ β j ] X ( a ))( x j ) (cid:17) ∈ (Tr P C · F )( X ) (see 6.2). Therefore, we get x = η X ( a ) = P j ∈ J X,a η X (cid:16) P ([ β j ] X ( a ))( x j ) (cid:17) ∈ (Tr P C · F )( X ) . Proving that (Tr P F )( X ) ⊆ (Tr P C · F )( X ) . The other inclusion isproved in a similar way. (cid:3)
Let M ∈ Mod( C ). We recall that add( M ) is the full subcategory of Mod( C )whose objects are direct summands of finite coproducts of the module M . That is, X ∈ add( M ) if and only if there exists a module Y such that X ⊕ Y ≃ M n for some n ∈ N . The following proposition tells us when two finitely generated projective C -modules produce the same ideal. Proposition 6.6.
Let P and Q finitely generated projective C -modules. Then Tr P C = Tr Q C if and only if add( P ) = add( Q ) .Proof. Suppose that Tr P C = Tr Q C . We have that Tr P ( P ) = P . Since Tr P C = Tr Q C by 6.5, we have that P = Tr P P = Tr P C· P = Tr Q C· P = Tr Q P . Then there exists anepimorphism η : Q ( I ) −→ P . Since P is finitely generated, we have that there existsa finite subset J ⊆ I and an epimorphism η ′ : Q J −→ P (see [2, 2.1(b)]). Since P isprojective, we have that P is a direct summand of Q J . Then P ∈ add( Q ). Similarly, Q ∈ add( P ), and thus add( P ) = add( Q ). The other implication is similar. (cid:3) Lemma 6.7.
Let C be an additive category and P = Hom C ( C, − ) ∈ Mod( C ) .Let us consider B := add( C ) ⊆ C and F := { Hom C ( C ′ , − ) } C ′ ∈B , then for each Hom C ( X, − ) ∈ Mod( C ) we have that Tr F (cid:16) Hom C ( X, − ) (cid:17) = Tr P (cid:16) Hom C ( X, − ) (cid:17) . Proof.
Straightforward. (cid:3)
Corollary 6.8.
Let C be a Hom-finite R -variety and P = Hom C ( C, − ) ∈ Mod( C ) .Then Tr P (cid:16) Hom C ( X, − ) (cid:17) = I add( C ) ( X, − ) , where I add( C ) is the ideal of the mor-phisms in C which factor through some object of add( C ) .Proof. It follows from [16, Lemma 2.3] and 6.7. (cid:3)
Now, we recall the following well known result.
Proposition 6.9.
Let C be an R -category which is Hom-finite, where R is a com-mutative ring. Then for every C ∈ C we have that add( C ) is functorially finite.Proof. The proof given in [8, Theorem 4.2] can be adapted to this setting. (cid:3)
In the following results we focus in projective modules of the form Hom C ( C, − )since in R -varieties all the finitely generated projectives are of such form (see 4.1). Proposition 6.10.
Let C be a dualizing R -variety, P = Hom C ( C, − ) a finitelygenerated projective C -module and I = Tr P C . Then I satisfies property A .Proof. By 6.8, we have that Tr P C = I add( C ) . By 6.9, we have that add( C ) isfuntorially finite. By 4.12, we have that I add( C ) satisfies the property A . (cid:3) Corollary 6.11.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and I = Tr P C . Then we can restrict the diagram given in 3.13 to the finitely presentedmodules mod( C / I ) ( π ) ∗ / / mod( C ) π ∗ o o π !1 o o OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 21 Proof.
It follows by 6.10 and 4.7. (cid:3)
For a preadditive category C , we recall the construction of the functor ( − ) ∗ :Mod( C ) −→ Mod( C op ) which is a generalization of the functor Mod( A ) −→ Mod( A op )given by M Hom A ( M, A ) for all the A -modules M , where A is a ring.Indeed, for each C -module M we define M ∗ : C −→ Ab given by M ∗ ( C ) =Hom Mod( C ) ( M, Hom C ( C, − )). Clearly M ∗ is a C op -module. In this way we ob-tain a contravariant functor ( − ) ∗ : Mod( C ) −→ Mod( C op ) given by M M ∗ .If M = Hom C ( C, − ), it can be seen that M ∗ ≃ Hom C ( − , C ); we refer the reader tosection 6 in [5] for more details. Corollary 6.12.
Let C be a Hom-finite R -variety, P = Hom C ( C, − ) ∈ Mod( C ) andconsider the ideal I = Tr P C . Then we have that I op = Tr P ∗ C op . Proof.
It follows from 6.8 and the fact that P ∗ = Hom C ( − , C ). (cid:3) For the next, we recall that if C is a dualizing R -variety by [5, Proposition 3.4]we have that mod( C ) has projective covers. Proposition 6.13.
Let C be a dualizing R -variety, P a projective C -module and F ∈ mod( C ) . Let P ( F ) be the projective cover of F , then Tr P ( F ) = F if and onlyif P ( F ) ∈ add( P ) .Proof. Now, suppose that Tr P ( F ) = F . Then there exists an epimorphism P ( I ) −→ F . Since F ∈ mod( C ), we have that F is finitely generated and, therefore, we havethat there exists an epimorphism P n −→ F −→ P ( F ) isthe projective cover of F , we have that P ( F ) is a direct summand of P n , provingthat P ( F ) ∈ add( P ). The other implication is easy to check. (cid:3) We introduce the following definition that will be used along the paper.
Definition 6.14.
Let C be a dualizing R -variety and P ∈ mod( C ) be a projectivemodule. For each ≤ k ≤ ∞ we define P k to be the full subcategory of mod( C ) consisting of the C -modules X having a projective resolution · · · P n / / P n − / / · · · / / P / / P / / X / / with P i ∈ add( P ) for ≤ i ≤ k . We have the following easy lemma.
Lemma 6.15.
Let C be a Hom-finite R -variety, P = Hom C ( C, − ) ∈ Mod( C ) and I = Tr P C . Consider π : C −→ C / I . Then we have that Hom mod( C ) ( Q, ( π ) ∗ ( Y )) =0 for all Q ∈ add( P ) and for all Y ∈ mod( C / I ) .Proof. It is enough to see that Hom mod( C ) ( P, ( π ) ∗ ( Y )) = 0 for Y ∈ mod( C / I ).Indeed, by Yoneda we have that Hom mod( C ) ( P, ( π ) ∗ ( Y )) ≃ ( π ) ∗ ( Y )( C ) = Y ( C ) =0, because 1 Y ( C ) = Y (1 C ) = 0 since C ∈ add( C ), I = I add( C ) and Y ∈ mod( C / I ). (cid:3) Corollary 6.16.
Let C be a Hom-finite R -variety, P = Hom C ( C, − ) ∈ Mod( C ) , I = Tr P C and consider π : C −→ C / I . Let X ∈ mod( C ) . Then X ∈ P if and onlyif Hom mod( C ) ( X, ( π ) ∗ ( Y )) = 0 for all Y ∈ mod( C / I ) .Proof. (= ⇒ ). Suppose that X ∈ P , then there exists an epimorphism γ : P −→ X with P ∈ add( P ). Let α : X −→ ( π ) ∗ ( Y ) be, then we have that αγ ∈ Hom mod( C ) ( Q, ( π ) ∗ ( Y )). By 6.15, we have that αγ = 0. Then α = 0, since γ is anepimorphism; proving that Hom mod( C ) ( X, ( π ) ∗ ( Y )) = 0.( ⇐ =). Suppose that Hom mod( C ) ( X, ( π ) ∗ ( Y )) = 0 for all Y ∈ mod( C / I ). Let P ( X ) be the projective cover of X . We assert that P ( X ) ∈ add( P ). Let us consider the exact sequence 0 / / I X u / / X q / / X I X / / . We have that X I X ∈ Ann( I ). By 3.8, we have that X I X = ( π ) ∗ (Ω( X I X )) with Ω( X I X ) ∈ mod( C / I ).Then, by hypothesis we have that Hom mod( C ) ( X, X I X ) = 0. We conclude that q = 0 and, therefore, u is an isomorphism; that is, we have that I X = X . But I X = Tr P C · X = Tr P ( X ) (see 6.5); proving that Tr P ( X ) = X and, by 6.13, weconclude that P ( X ) ∈ add( P ). Therefore, we have that X ∈ P . (cid:3) In the following proposition we give a characterization of the modules in P k thatwill be used in the rest of the paper. Proposition 6.17.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and I = Tr P C . For ≤ k ≤ ∞ and X ∈ mod( C ) , the following are equivalent. (a) X ∈ P k . (b) Ext i mod( C ) ( X, ( π ) ∗ ( Y )) = 0 for all Y ∈ mod( C / I ) and i = 0 , . . . , k . (c) Ext i mod( C ) ( X, ( π ) ∗ ( J )) = 0 for all J ∈ mod( C / I ) injective and i = 0 , . . . , k .Proof. ( a ) ⇒ ( b ). When applying Hom( − , ( π ) ∗ Y ) to a projective resolution of X ,by 6.15 and 6.16, we have a complex with the first k terms zero.( b ) ⇒ ( a ). We will proceed by induction on k . Suppose that k = 1. That is,suppose that Ext i mod( C ) ( X, ( π ) ∗ ( Y )) = 0 for all Y ∈ mod( C / I ) and i = 0 ,
1. Bythe proof of 6.16, we have that the projective cover P ( X ) of X belongs to add( P ).Now, let us consider the exact sequence 0 / / L / / P ( X ) / / X / / . Applying Hom mod( C ) ( − , ( π ) ∗ ( Y )) to the last sequence we have the exact sequence Hom mod( C ) ( P ( X ) , ( π ) ∗ ( Y )) Hom mod( C ) ( L , ( π ) ∗ ( Y )) Ext C ) ( X, ( π ) ∗ ( Y )) . Since P ( X ) ∈ add( P ), we have that Hom mod( C ) ( P ( X ) , ( π ) ∗ ( Y )) = 0 (see 6.15)and, by hypothesis, we have that Ex C ) ( X, ( π ) ∗ ( Y )) = 0. Thus, we concludethat Hom mod( C ) ( L , ( π ) ∗ ( Y ) = 0 for all Y ∈ mod( C / I ). Then, in the same wayas we did for X , we get that P ( L ) ∈ add( P ). Hence, we have an exact sequence P ( L ) → P ( X ) → X → , with P ( L ) , P ( X ) ∈ add( P ), proving that X ∈ P .Suppose that is true for k −
1. Let X ∈ mod( C ) such that Ext i mod( C ) ( X, ( π ) ∗ ( Y )) =0 for all Y ∈ mod( C / I ) and i = 0 , . . . , k . In particular, Ext i mod( C ) ( X, ( π ) ∗ ( Y )) = 0for all Y ∈ mod( C / I ) and i = 0 , . . . , k −
1. Then, by induction, there exists aresolution · · · / / P k − d k − / / · · · / / P d / / P d / / X / / P i ∈ add( P ) for all i = 0 , . . . , k −
1. Consider the exact sequence0 / / L k − = Ker( d k − ) / / P k − / / Ker( d k − ) = L k − / / mod( C ) ( − , ( π ) ∗ ( Y )) we have the exact sequence Hom mod( C ) ( P k − , ( π ) ∗ ( Y )) Hom mod( C ) ( L k − , ( π ) ∗ ( Y )) Ext C ) ( L k − , ( π ) ∗ ( Y )) By shifting lemma, we get Ext C ) ( L k − , ( π ) ∗ ( Y )) ≃ Ext k mod( C ) ( X, ( π ) ∗ ( Y ))= 0. Since P k − ∈ add( P ), we conclude that Hom mod( C ) ( P k − , ( π ) ∗ ( Y )) = 0 (see6.15); and hence Hom mod( C ) ( L k − , ( π ) ∗ ( Y )) = 0 for all Y ∈ mod( C / I ). In thesame way as we did for the case k = 1, we get that P ( L k − ) ∈ add( P ). Then wecan construct · · · → P k = P ( L k − ) → P k − → · · · → P → P → X → , which isexact with P i ∈ add( P ) for all i = 0 , . . . , k , proving that X ∈ P k .( b ) ⇒ ( c ). Trivial.( c ) ⇒ ( b ). Follows by induction on i . (cid:3) OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 23 Proposition 6.18.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and I = Tr P C . Let ≤ k ≤ ∞ , then I is ( k +1) -idempotent if and only if I ( C ′ , − ) ∈ P k for all C ′ ∈ C .Proof. First, we note that by 6.10, we have that I satisfies property A .Let π : C −→ C / I be canonical functor. By 6.11, we have that π ∗ (Hom C / I ( C ′ , − )) = Hom C ( C ′ , − ) I ( C ′ , − ) ∈ mod( C ). Consider the following exact sequence in Mod( C )0 / / I ( C ′ , − ) / / Hom C ( C ′ , − ) / / Hom C ( C ′ , − ) I ( C ′ , − ) / / . Since C is a dualizing R -variety, we have that mod( C ) is abelian subcategory ofMod( C ) and thus I ( C ′ , − ) ∈ mod( C ) (see [5, Theorem 2.4]).Since I = Tr P C , there exists an epimorphism γ : P n −→ I ( C ′ , − ) for some n ∈ N ,this is because I ( C ′ , − ) is finitely generated (see [2, 2.1(b)]). Let Y ∈ mod( C / I ).If there exists a non zero morphism α : I ( C ′ , − ) −→ π ∗ ( Y ) we have that αγ = 0,which contradicts 6.15. Therefore, we have that Hom mod( C ) (cid:16) I ( C ′ , − ) , π ∗ ( Y ) (cid:17) = 0.On the other hand, applying Hom mod( C ) ( − , π ∗ ( Y )) to the last exact sequence weget an isomorphism for i ≥ ∗ ) : Ext i mod( C ) (cid:16) I ( C ′ , − ) , π ∗ ( Y ) (cid:17) −→ Ext i +1mod( C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , π ∗ ( Y ) (cid:17) . We know that I is ( k + 1)-idempotent if and only if EXT i C (cid:16) C / I , π ∗ ( Y ) (cid:17) = 0 for1 ≤ i ≤ k + 1 and for all Y ∈ mod( C / I ) (see 5.3). Now, the result follows from 5.3,6.17. (cid:3) Corollary 6.19.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and I = Tr P C . Then I is strongly idempotent if and only if I ( C ′ , − ) ∈ P ∞ ∀ C ′ ∈ C . Let C be a dualizing R -variety. Given M ∈ mod( C ), we recall that rad( M )denotes the radical of M , that is, rad( M ) is the intersection of the maximal sub-modules of M . Definition 6.20.
Let C be a dualizing R -variety, P ∈ mod( C ) a projective moduleand J := I ( P rad( P ) ) ∈ mod( C ) the injective envelope of P rad( P ) . For each ≤ k ≤ ∞ we define I k to be the full subcategory of mod( C ) consisting of the C -modules Y having an injective coresolution / / Y / / J / / J / / · · · with J i ∈ add( J ) for ≤ i ≤ k . Let C be a dualizing R -variety. Since the endomorphism ring of each object in C is an artin algebra, it follows that C is a Krull-Schmidt category [5, p.337]. By 4.1,we conclude that P ∈ proj( C ) is indecomposable if and only if P ≃ Hom C ( C, − )where C ∈ C is indecomposable (see also [16, Lemma 2.2 (b)]). Lemma 6.21.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) anindecomposable projective module, J := I ( P rad( P ) ) ∈ mod( C ) and I = Tr P C . Con-sider the projection π : C −→ C / I . Then Hom mod( C ) (( π ) ∗ ( X ) , J ) = 0 for all X ∈ mod( C / I ) .Proof. It can be seen that Tr CI ( J ) = 0, and hence Tr CI ( J ) = Ω(Tr CI ( J )) = 0. Now,by adjunction (see 3.10), Hom mod( C ) (( π ) ∗ ( X ) , J ) ≃ Hom mod( C / I ) ( X, Tr CI ( J )) =Hom mod( C / I ) ( X,
0) = 0 . (cid:3) Proposition 6.22.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) an indecomposable projective module, J := I ( P rad( P ) ) ∈ mod( C ) and I = Tr P C .Consider the projection π : C −→ C / I . Then J ≃ D − C (Hom C ( − , C )) . Proof.
Since J is the injective envelope of the simple P rad( P ) , we have that J is inde-composable. Then we have that D C ( J ) ∈ mod( C op ) is an indecomposable projective C -module. By 6.21 and since J is injective, we have that Ext i mod( C ) (( π ) ∗ ( X ) , J ) = 0for all X ∈ mod( C / I ) for i = 0 ,
1. Then we have thatExt i mod( C ) (( π ) ∗ ( X ) , J ) = Ext i mod( C op ) ( D C ( J ) , D C (( π ) ∗ ( X )))= Ext i mod( C op ) ( D C ( J ) , ( π ) ∗ D C / I ( X )))where the last equality is by the diagram in 4.6. Since D C / I is a duality, we havethat Ext i mod( C op ) ( D C ( J ) , ( π ) ∗ ( X ′ )) = 0 for all X ′ ∈ mod( C op / I op ). By 6.12, wehave that I op = Tr P ∗ C op . Then, by 6.17 (for the dualizing R -variety C op ), wehave that there exists an exact sequence Q / / Q / / D C ( J ) / / Q i ∈ add( P ∗ ) = add(Hom C ( − , C )). Since D C ( J ), is projective we conclude that D C ( J ) is a direct summand of Q and therefore D C ( J ) ∈ add(Hom C ( − , C )). SinceHom C ( − , C ) is indecomposable and mod( C ) is Krull-Schmidt (because mod( C ) is adualizing R -variety), we conclude that D C ( J ) ≃ Hom C ( − , C ). (cid:3) The following shows that the last proposition holds for every finitely generatedprojective C -module. Proposition 6.23.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) aprojective module, J := I ( P rad( P ) ) ∈ mod( C ) the injective envelope of P rad( P ) and I = Tr P C . Then J ≃ D − C (Hom C ( − , C )) . Proof.
Since mod( C ) is Krull-Schmidt, we have that P = L ni =1 P i where each P i = Hom C ( C i , − ) is indecomposable and C = ⊕ ni =1 C i . The result follows from6.22. (cid:3) Similarly, the result given in 6.21 holds for every finitely generated projective C -module. That is, we have the following result. Remark 6.24.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) a pro-jective module, J := I ( P rad( P ) ) ∈ mod( C ) and I = Tr P C . Consider the projection π : C −→ C / I . Then Hom mod( C ) (( π ) ∗ ( X ) , J ) = 0 for all X ∈ mod( C / I ) . Corollary 6.25.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) a projective module, J := I ( P rad( P ) ) ∈ mod( C ) and I = Tr P C . Consider theprojection π : C −→ C / I . Then Y ∈ I if and only if Hom mod( C ) (( π ) ∗ ( X ) , Y ) = 0 for all X ∈ mod( C / I ) .Proof. If follows by duality using 6.12, 6.16 and 6.23. (cid:3)
Proposition 6.26.
Let P = Hom C ( C, − ) ∈ mod( C ) be a projective module, I =Tr P C and ≤ k ≤ ∞ . Let π : C −→ C / I the canonical projection. The followingconditions are equivalent for Y ∈ mod( C ) . (a) Y ∈ I k . (b) Ext i mod( C ) (( π ) ∗ ( X ) , Y ) = 0 for all X ∈ mod( C / I ) and i = 0 , . . . , k . (c) Ext i mod( C ) (( π ) ∗ ( Q ) , Y ) = 0 for all Q ∈ mod( C / I ) projective and i =0 , . . . , k .Proof. It follows from 6.17 using the duality. (cid:3)
OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 25 Given any class of objects C in an abelian category A , we define C ⊥ := { A ∈A | Hom A ( C, A ) = 0 ∀ C ∈ C} and ⊥ C := { A ∈ A | Hom A ( A, C ) = 0 ∀ C ∈ C} . Werecall that a torsion theory for A is a pair ( T , F ) of classes of objects of A suchthat the following conditions hold: (i) Hom C ( T, F ) = 0 for all T ∈ T and for all F ∈ T ; (ii) T ⊥ = F and ⊥ F = T . A class of objects T is a TTF class if thereexists torsion theories of the form ( D , T ) and ( T , F ). For more details related totorsion theories we refer the reader to chapter 6 in [25]. Proposition 6.27.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) aprojective module and I = Tr P C . Then ( P , mod( C / I ) , I ) is a TTF triple .Proof. By 6.16 and 6.25, we have that ⊥ (mod( C / I )) = P and (mod( C / I )) ⊥ = I .We have that mod( C / I ) ≃ Ann( I ) is a hereditary torsion theory which is closedunder arbitrary products. By [25, Proposition 8.1], we have that mod( C / I ) is aTTF-class, and hence ( P , mod( C / I )) and (mod( C / I ) , I ) are torsion pairs. (cid:3) A recollement
Let C be a preadditive category. Throught this section P will be a finitelygenerated projective module in Mod( C ) and R P := End Mod( C ) ( P ) op . In this sectionwe will study the functor Hom Mod( C ) ( P, − ) : Mod( C ) −→ Mod( R P ). The followingremark is straightforward and we left the details to the reader. Remark 7.1.
Let C be a preadditive category and P ∈ proj( C ) . (a) Consider the Yoneda embedding Y : C −→ proj( C ) defined as Y ( C ) :=Hom C ( C, − ) . Then Y ∗ : Mod((proj( C )) op ) −→ Mod( C ) is an equivalence. (b) Consider the inclusion { P } op ⊆ proj( C ) op and the restriction functor res :Mod((proj( C )) op ) −→ Mod( { P } op ) given as: M M | { P op } . Then thefollowing diagram is commutative ( ∗ ) : Mod((proj( C )) op ) res / / Y ∗ (cid:15) (cid:15) Mod( { P } op ) e P (cid:15) (cid:15) Mod( C ) Hom
Mod( C ) ( P, − ) / / Mod( R P ) where R P := End Mod( C ) ( P ) op and e P is the evaluation functor defined asfollows: e P ( M ) = M ( P ) for M ∈ Mod( { P } op ) . (c) We define a functor P ⊗ R P − : Mod( R P ) −→ Mod( C ) as follows: ( P ⊗ R P M )( C ) = P ( C ) ⊗ R P M for all M ∈ Mod( R P ) and C ∈ C . Then thefollowing diagram commutes Mod( { P } op ) (proj( C ) op ) ⊗ { P } op − / / e ′ P (cid:15) (cid:15) Mod(proj( C )) op ) Y ∗ (cid:15) (cid:15) Mod( R P ) P ⊗ RP − / / Mod( C ) where e ′ P is the evaluation and (proj( C ) op ) ⊗ { P } op − is the functor definedin 2.2. (d) Recall that P ∗ ∈ Mod( C op ) is given by P ∗ ( C ) := Hom Mod( C ) ( P, Hom C ( C, − )) .Now we can construct a functor Hom R P ( P ∗ , − ) : Mod( R P ) −→ Mod( C ) where for M ∈ Mod( R P ) we define Hom R P ( P ∗ , M ) : C −→ Ab as follows: (cid:16) Hom R P ( P ∗ , M ) (cid:17) ( C ) := Hom R P ( P ∗ ( C ) , M ) . Then the following diagram commutes Mod( { P } op ) { P } op (proj( C ) op , − ) / / e ′ P (cid:15) (cid:15) Mod(proj( C )) op ) Y ∗ (cid:15) (cid:15) Mod( R P ) Hom RP ( P ∗ , − ) / / Mod( C ) where e ′ P is the evaluation and { P } op (proj( C ) op , − ) is the functor definedin 2.3. We also have the following result.
Proposition 7.2.
Let C be an R -category and P = Hom C ( C, − ) ∈ proj( C ) . Con-sider add( P ) ⊆ proj( C ) and I = I add( P ) the ideal of morphisms in proj( C ) thatfactor through objects in add( P ) . The Yoneda embedding Y induces a functor Y : C / I add( P ) −→ proj( C ) / I add( P ) , such that the following diagram commutes Mod((proj( C ) / I add( P ) ) op ) / / Y ∗ (cid:15) (cid:15) Mod((proj( C )) op ) Y ∗ (cid:15) (cid:15) Mod( C / I add( C ) ) / / Mod( C ) where the vertical functors are equivalences.Proof. Straightforward. (cid:3)
Remark 7.3.
We have that
Mod( { P } op ) ≃ Mod(add( P ) op ) . Indeed, this followsfrom [2, Proposition 2.5] . Now, we give the following definition which encodes the information of severaladjunctions.
Definition 7.4.
Let A , B and C be abelian categories. Then the diagram B i ∗ = i ! / / A j ! = j ∗ / / i ∗ o o i ! o o C j ! o o j ∗ o o is called a recollement , if the additive functors i ∗ , i ∗ = i ! , i ! , j ! , j ! = j ∗ and j ∗ satisfy the following conditions: (R1) ( i ∗ , i ∗ = i ! , i ! ) and ( j ! , j ! = j ∗ , j ∗ ) are adjoint triples, i.e. ( i ∗ , i ∗ ) , ( i ! , i ! )( j ! , j ! ) and ( j ∗ , j ∗ ) are adjoint pairs; (R2) j ∗ i ∗ = 0 ; (R3) i ∗ , j ! , j ∗ are full embedding functors. Next, we will see that we can construct a recollement.
Proposition 7.5.
Let C be an R -category, P = Hom C ( C, − ) ∈ Mod( C ) a finitelygenerated projective module, B = add( C ) ⊆ C and R P = End Mod( C ) ( P ) op . Then,there exists a recollement of the form Mod( C / I B ) π ∗ / / Mod( C ) Hom
Mod( C ) ( P, − ) / / C / I B ⊗ C − o o C ( C / I B , − ) o o Mod( R P ) P ⊗ RP − o o Hom RP ( P ∗ , − ) o o where I B is the ideal of morphisms in C which factor through objects in B . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 27 Proof.
Let us take C ′ = proj( C ) op and B ′ = add( P ) op . By [15, Theorem 3.10], wehave the recollementMod( C ′ / I B ′ ) res C′ / / Mod( C ′ ) res B′ / / C ′ / I B′ ⊗ C′ − o o C ′ ( C ′ / I B′ , − ) o o Mod( B ′ ) C ′ ⊗ B′ − o o B ′ ( C ′ , − ) o o where I B ′ = I op add( P ) is the ideal of morphisms in C ′ which factor through objects inadd( P ). Considering the projection π : C −→ C / I B we can construct the diagramgiven in 3.13. By 7.2 we can identify π ∗ with res C ′ . Then, we can identify C / I B ⊗ C − with C ′ / I B ′ ⊗ C ′ − , and C ( C / I B , − ) with C ′ ( C ′ / I B ′ , − ), since adjoint functors areunique up to isomorphisms. The result follows from 7.1, and 7.3. (cid:3) We can restrict the last recollement to the finitely presented modules. So, wehave the following result that is an analogous to the one given in artin algebras.
Proposition 7.6.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) , B = add( C ) and R P = End Mod( C ) ( P ) op . Then, there exist a recollement mod( C / I B ) π ∗ / / mod( C ) Hom mod( C ) ( P, − ) / / C / I B ⊗ C − o o C ( C / I B , − ) o o mod( R P ) P ⊗ RP − o o Hom RP ( P ∗ , − ) o o where I B is the ideal of morphisms in C which factor through objects in B . Moreover,we have that I B = Tr P C .Proof. Let us take C ′ = proj( C ) op and B ′ = add( P ) op . In the proof of 7.5 wehave a recollement where I B ′ = I add( P ) is the ideal of morphisms in C ′ whichfactor through objects in add( P ). The Yoneda’s embedding gives us an equivalence C ≃ proj( C ) op (see 4.1), and hence C ′ = proj( C ) op is a dualizing R -variety. Now, B ′ = add( P ) op is a functorially finite subcategory of C ′ (see 6.9). Therefore, by [20,Theorem 2.5] and 7.5, we can restrict the last recollement to the finitely presentedmodules where I B is the ideal of morphisms in C which factor through objects in B , since mod( R P ) coincides with the finitely presented R -modules (because R P isan artin R -algebra). Finally, by 6.8, we have that I B = Tr P C . (cid:3) Remark 7.7.
Consider the adjoints in 7.6. Consider the counit ǫ ′ and unit η ′ ofthe adjoint pair (cid:16) ( P ⊗ R P − ) , Hom
Mod( C ) ( P, − ) (cid:17) ; and also the counit ǫ and the unit η of the adjoint pair (cid:16) Hom
Mod( C ) ( P, − ) , Hom R P ( P ∗ , − ) (cid:17) . By [2, Propositions 3.1and 3.4] , we have that ǫ and η ′ are isomorphisms. We have the following definition due to Auslander [2].
Definition 7.8.
Let C a dualizing R -category and P = Hom C ( C, − ) ∈ mod( C ) . Let M ∈ mod( C ) . (a) It is said that M is proyectively presented over P if ǫ ′ M is an isomor-phism. Let us denote by F . P . P ( P ) the full subcategory of mod( C ) consistingof the projectively presented modules (b) It is said that M is inyectively copresented over P if η M is an isomor-phism. Let us denote by F . I . C ( P ) the full subcategory of mod( C ) consistingof the injectively copresented modules. We recall that in the case of a dualizing R -variety every finitely generated pro-jective C -module is of the form Hom C ( C, − ) (see 4.1). Proposition 7.9.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .For M ∈ mod( C ) , the following are equivalent. (a) M ∈ F . P . P ( P ) . (b) There exists a module X ∈ mod( R P ) such that M ≃ P ⊗ R P X . (c) There exists an exact sequence P → P → M → with P , P ∈ add( P ) . (d) Hom Mod( C ) ( M, N ) → Hom R P (cid:16) Hom
Mod( C ) ( P, M ) , Hom
Mod( C ) ( P, N ) (cid:17) is anisomorphism for each module N ∈ mod( C ) .Proof. First, we note that since C is a dualizing R -variety we have the adjunctionsin the diagram of 7.6. The result follows from [2, Proposition 3.2], considering thesubcategories { P } op ⊆ proj( C ) op , using that mod(add( P ) op ) ≃ mod( { P } op ) andusing the identifications given in 7.1. (cid:3) The next result give us a characterization of the the categories F . P . P ( P ) and F . I . C ( P ) which will help us in the forthcoming section. Proposition 7.10.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .Then the following conditions hold. (a) M ∈ F . P . P ( P ) if and only if Hom mod( C ) ( M, N ) = 0 and
Ext C ) ( M, N ) =0 for all N ∈ mod( C ) with N ∈ Ker(Hom mod( C ) ( P, − )) . (b) N ∈ F . I . C ( P ) if and only if Hom
Mod( C ) ( M, N ) = 0 and
Ext C ) ( M, N ) =0 for all M ∈ mod( C ) with M ∈ Ker(Hom mod( C ) ( P, − )) .Proof. The proof given in [2, Proposition 3.7] works for the finitely presented mod-ules, since mod( C ) is an abelian subcategory of Mod( C ) with enough injectives andprojectives and we have the adjunctions in 7.6. (cid:3) Remark 7.11.
Since we have the recollement of 7.6, we have that
Im( π ∗ ) =Ker(Hom mod( C ) ( P, − )) . Proposition 7.12.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .Consider the functor Hom mod( C ) ( P, − ) : mod( C ) −→ mod( R P ) . The following hold. (a) Hom mod( C ) ( P, − ) | P : P −→ mod( R P ) and Hom mod( C ) ( P, − ) | I : I −→ mod( R P ) are equivalences, where P and I are the categories defined in6.14 and 6.20. (b) Let
Hom mod( C ) ( X, Y ) ρ X,Y / / Hom R P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, Y ) (cid:17) . Then: (i) ρ X,Y is a monomorphism if either X ∈ P or Y ∈ I , (ii) ρ X,Y is an isomorphism if X ∈ P and Y ∈ I , (iii) ρ X,Y is an isomorphism if either X ∈ P or Y ∈ I . (c) The functor
Hom mod( C ) ( P, − ) induces an equivalence of categories between add( P ) an the category of projective R P -modules, and between add( J ) andthe category of injective R P -modules.Proof. ( a ). The fact that Hom mod( C ) ( P, − ) | P : P −→ mod( R P ) is an equivalencefollows from [2, Proposition 3.3], since F . P . P ( P ) = P .By 7.11, 7.10 and 6.26, we have that F . I . C ( P ) = I . Then, by [2, Proposition 3.6],we have that Hom mod( C ) ( P, − ) | I : I −→ mod( R P ) is an equivalence.( bi ). It follows from 7.9(d) and the proof is left to the reader.( bii ). Since X ∈ P , we have by ( bi ) that ρ X,Y : Hom mod( C ) ( X, Y ) −→ Hom R P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, Y ) (cid:17) is a monomorphism. Since X ∈ P and Y ∈ I , there exists an epimorphism Q π / / X / / / / Y µ / / I with Q ∈ add( P )and I ∈ add( J ). By definition of P and I , we have that Q ∈ P and I ∈ I . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 29 Let us see that ρ X,Y is surjective.Let ϕ : Hom mod( C ) ( P, X ) −→ Hom mod( C ) ( P, Y ) be a morphism of R P -modules.Consider the morphism Hom mod( C ) ( P, µ ) : Hom mod( C ) ( P, Y ) −→ Hom mod( C ) ( P, I )and then we get Hom mod( C ) ( P, µ ) ◦ ϕ : Hom mod( C ) ( P, X ) −→ Hom mod( C ) ( P, I ).Since I ∈ I = F . I . C ( P ), we have that ρ X,I : Hom mod( C ) ( X, I ) −→ Hom R P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, I ) (cid:17) is an isomorphism (see dual of 7.9). Then there exists a morphism λ : X −→ I suchthat Hom mod( C ) ( P, λ ) = Hom mod( C ) ( P, µ ) ◦ ϕ . We also consider Hom mod( C ) ( P, π ) :Hom mod( C ) ( P, Q ) −→ Hom mod( C ) ( P, X ) and then we have ϕ ◦ Hom mod( C ) ( P, π ) :Hom mod( C ) ( P, Q ) −→ Hom mod( C ) ( P, Y ). Since Q ∈ P , we have that ρ Q,Y : Hom mod( C ) ( Q, Y ) −→ Hom R P (cid:16) Hom mod( C ) ( P, Q ) , Hom mod( C ) ( P, Y ) (cid:17) is an isomorphism (see 7.9(d)). Then there exists a morphism β : Q −→ Y suchthat ϕ ◦ Hom mod( C ) ( P, π ) = Hom mod( C ) ( P, β ).Then we have two morphisms λπ, µβ : Q −→ I and it is straigthforward to checkthat λπ = µβ . Let us consider the factorization X p / / K δ / / I of λ throughits image. Then we have that µβ = λπ = δpπ = δ ( pπ ) with pπ an epimorphismand δ a monomorphism. Then we have that δ is the image of λπ . Since µ is amonomorphism, by the universal property of the image, there exists ψ : K −→ Y such that δ = µψ . Now, it is easy to see that ϕ = Hom mod( C ) ( P, ψ ◦ p ). Therefore,we conclude that ρ X,Y is surjective, and then an isomorphism.( biii ) Follows from 7.9 and its dual since F . P . P ( P ) = P and F . C . I ( P ) = I .( c ) Since Hom mod( C ) ( P, P ) op = R P , we have that Hom mod( C ) ( P, − ) : add( P ) −→ add( R P ) = proj( R P ) is an equivalence.We have that J = D − C (Hom C ( − , C )) is an injective C -module. Then J is injectivein the subcategory I of mod( C ). Since Hom mod( C ) ( P, − ) : I −→ mod( R P ) is anequivalence, we have that Hom mod( C ) ( P, J ) is an injective R P -module. Using theadjoint pair (cid:16) Hom mod( C ) ( P, − ) , Hom R P ( P ∗ , − ) (cid:17) , it is easy to see that if I is aninjective R P -module, then Hom R P ( P ∗ , I ) ∈ add( J ). Then we have the equivalenceHom mod( C ) ( P, − ) : add( J ) −→ inj( R P ) with inverse Hom R P ( P ∗ , − ) : inj( R P ) −→ add( J ) . (cid:3) Remark 7.13.
Since
Hom mod( C ) ( P, − ) : mod( C ) −→ mod( R P ) has a full andfaithfull right adjoint (this functor is part of a recollement), we conclude from theGabriel localization theory that mod( C ) / Ker(Hom mod( C ) ( P, − )) ≃ mod( R P ) . Extension over the endomorphism ring of a projective module
In this section we will study some homological properties of the additive functorHom mod( C ) ( P, − ) : mod( C ) → mod( R P ) and how it relates to k -idempotent ideals,in particular to I := Tr P C . We will explore the relationship between injectivecoresolution in mod( C ) and mod( R P ). For each X, Y ∈ mod( C ) consider the map ρ X,Y : Hom mod( C ) ( X, Y ) −→ Hom R P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, Y ) (cid:17) defined as ρ X,Y ( f ) = Hom mod( C ) ( P, f ) for all f ∈ Hom mod( C ) ( X, Y ). It is easy tosee that ρ X,Y is functorial in X and Y, then we have the following construction. Proposition 8.1.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .For each X, Y ∈ mod( C ) and for all i ≥ we have canonical morphisms Φ iX,Y : Ext i mod( C ) ( X, Y ) −→ Ext iR P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, Y ) (cid:17) where Φ X,Y = ρ X,Y .Proof.
It is straightforward, using injective coresolutions of Y and Hom mod( C ) ( P, Y ),and the comparison lemma (see dual of [22, Theorem 6.16]). (cid:3)
We give conditions in order to know when the morphisms Φ iX,Y are isomorphisms.
Proposition 8.2.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .The map Φ nX,Y : Ext n mod( C ) ( X, Y ) −→ Ext nR P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, Y ) (cid:17) above defined is an isomorphism for all n ≥ , provided one of the three followingconditions holds: (a) X ∈ P i , Y ∈ I j and n ≤ i + j , (b) X ∈ mod( C ) and Y ∈ I n +1 , (c) X ∈ P n +1 and Y ∈ mod( C ) .Proof. The prove given in [4, Theorem 3.2] works for this setting. (cid:3)
We recall that the projective dimension of an object M in an abelian category A with enough projectives is the length of the shortest projective resolution of M ,and it is denoted by pd( M ). Corollary 8.3.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .The following conditions hold. (a) If X ∈ P ∞ then pd( X ) = pd R P (cid:16) ( P, X ) (cid:17) . (b) If X ∈ I ∞ then id( X ) = id R P (cid:16) ( P, X ) (cid:17) .Proof. It follows from 8.2. (cid:3)
The global dimension of A is the supremum of the projective dimensionsp . d( M ) with M ∈ A ; and it is denoted by gl . dim( A ). Proposition 8.4.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .If P = P ∞ or I = I ∞ then gl . dim(mod( R P )) ≤ gl . dim(mod( C )) .Proof. We know that Hom mod( C ) ( P, − ) : P −→ mod( R P ) and Hom mod( C ) ( P, − ) : I −→ mod( R P ) are equivalences (see 7.12(a)). By 8.3, for each M ∈ mod( R P )there exists X ∈ mod( C ) such that p . d mod( C ) ( X ) = p . d mod( R P ) ( M ). This impliesthat gl . dim(mod( R P )) ≤ gl . dim(mod( C )). (cid:3) Proposition 8.5.
Let C be a dualizing R -variety with cokernels and consider P =Hom C ( C, − ) ∈ mod( C ) . If P = P ∞ or I = I ∞ then gl . dim( R P ) ≤ . In particular, R P is a quasi-hereditary algebra.Proof. If C has cokernels we know that gl . dim(mod( C )) ≤ . dim( R P ) ≤
2. By a well known result of Dlab-Ringel (see[10, Theorem 2]), we now that every artin algebra with global dimension less orequal to 2 is quasi-hereditary. (cid:3)
The following proposition gives us a relation of the canonical morphisms Φ iX,Y :Ext i mod( C ) ( X, Y ) −→ Ext iR P (cid:16) Hom mod( C ) ( P, X ) , Hom mod( C ) ( P, Y ) (cid:17) in the categorymod( C ) and mod( C op ) . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 31 Proposition 8.6.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .There exists a commutative diagram Ext i mod( C ) ( X, Y ) D C / / Φ iX,Y (cid:15) (cid:15) Ext i mod( C op ) ( D C ( Y ) , D C ( X )) Φ i D C ( Y ) , D C ( X ) (cid:15) (cid:15) Ext iR P (( P, X ) , ( P, Y )) D RP / / Ext iR opP (( P ∗ , D C ( Y )) , ( P ∗ , D C ( X )) where the horizontal maps are isomorphisms, P ∗ = Hom C ( − , C ) and Φ i D C ( Y ) , D C ( X ) is the analogous to the morphism Φ iX,Y but constructed in the category mod( C op ) .Proof. Straightforward. (cid:3)
If we work in the category mod( C op ), with the projective P ∗ = Hom C ( − , C ) and J ∗ := I (cid:16) P ∗ rad( P ∗ ) (cid:17) , the injective envelope of P ∗ rad( P ∗ ) in mod( C op ); we can define P ∗ k and I ∗ k in a similar way to definitions 6.14 and 6.20. Proposition 8.7.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) . (a) Then we have that X ∈ P k if and only if D C ( X ) ∈ I ∗ k . (b) Then we have that X ∈ I k if and only if D C ( X ) ∈ P ∗ k .Proof. It follows from duality using 6.12, 6.26 and 6.17. (cid:3)
Proposition 8.8.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .Let ≤ k ≤ ∞ . Then (a) Y ∈ I k if and only if Φ iX,Y : Ext i mod( C ) ( X, Y ) −→ Ext iR P (cid:16) ( P, X ) , ( P, Y ) (cid:17) is an isomorphism for all ≤ i ≤ k − and for all X ∈ mod( C ) . (b) X ∈ P k if and only if Φ iX,Y : Ext i mod( C ) ( X, Y ) −→ Ext iR P (cid:16) ( P, X ) , ( P, Y ) (cid:17) is an isomorphism for all ≤ i ≤ k − and for all Y ∈ mod( C ) .Proof. ( a ) ( ⇒ ). Suppose that Y ∈ I k . Then we have that Y ∈ I i for all 1 ≤ i ≤ k .By 8.2(b), we have that Φ iX,Y is an isomorphism for all 0 ≤ i ≤ k − X ∈ mod( C ).( ⇐ ). Consider the ideal I := Tr P C and the exact sequence in mod( C )( ∗ ) : 0 / / I ( C ′ , − ) u / / Hom C ( C ′ , − ) / / Hom C ( C ′ , − ) I ( C ′ , − ) / / P, u ) : ( P, I ( C ′ , − )) −→ ( P, Hom C ( C ′ , − )) is an isomorphism. ThenExt iR P (cid:16) ( P, Hom C ( C ′ , − )) , ( P, Y ) (cid:17) / / Ext iR P (cid:16) ( P, I ( C ′ , − )) , ( P, Y ) (cid:17) is an isomorphism for all i ≥
0. On the other hand, we have the commutativediagram Ext i mod( C ) (Hom C ( C ′ , − ) , Y ) / / Φ i ( C ′ , − ) ,Y (cid:15) (cid:15) Ext i mod( C ) ( I ( C ′ , − ) , Y ) Φ i I ( C ′ , − ) ,Y (cid:15) (cid:15) Ext iR P (cid:16) ( P, Hom C ( C ′ , − )) , ( P, Y ) (cid:17) / / Ext iR P (cid:16) ( P, I ( C ′ , − )) , ( P, Y ) (cid:17) By hypothesis, we have that the vertical maps are isomorphisms for all 0 ≤ i ≤ k − ≤ i ≤ k − ∗∗ ) : Ext i mod( C ) (Hom C ( C ′ , − ) , Y ) / / Ext i mod( C ) ( I ( C ′ , − ) , Y ) . For i = 0, we get the isomorphism (cid:16) Hom C ( C ′ , − ) , Y (cid:17) → (cid:16) I ( C ′ , − ) , Y (cid:17) . Ap-plying Hom mod( C ) ( − , Y ) to the sequence ( ∗ ) we get that (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , Y (cid:17) = 0 =Ext C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , Y (cid:17) . For 1 ≤ i ≤ k −
1, using the isomorphism ( ∗∗ ), we havethat Ext i mod( C ) ( I ( C ′ , − ) , Y ) = 0 since Hom C ( C ′ , − ) is a projective C -module. Fromthe sequence ( ∗ ) we get that Ext i mod( C ) (cid:16) I ( C ′ , − ) , Y (cid:17) → Ext i +1mod( C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , Y (cid:17) is an isomorphism for i ≥
1. This implies that Ext i mod( C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , Y (cid:17) = 0 for2 ≤ i ≤ k . Then, Ext i mod( C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , Y (cid:17) = Ext i mod( C ) (cid:16) ( π ) ∗ (Hom C / I ( C ′ , − )) , Y (cid:17) = 0 for 0 ≤ i ≤ k . Therefore, we have that Ext i mod( C ) (( π ) ∗ ( Q ) , Y ) = 0 for all Q ∈ mod( C / I ) projective and i = 0 , . . . , k (in a dualizing variety the finitely gen-erated projectives are of the form Hom C ( C ′ , − )). By 6.26, we have that Y ∈ I k .( b ) It follows from (a) using duality, 8.6 and 8.7. (cid:3) Proposition 8.9.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .Let X ∈ I and k ≥ . Then X ∈ I k if and only if Ext iR P ( P ∗ ( C ′ ) , ( P, X )) = 0 forall ≤ i ≤ k − and for all C ′ ∈ C .Proof. First note that P ∗ ( C ′ ) = Hom mod( C ) (cid:16) P, Hom C ( C ′ , − ) (cid:17) . Then we have that P ∗ ( C ′ ) ∈ mod( R P ).( ⇒ ). Suppose that X ∈ I k . Then, we have the following Ext iR P ( P ∗ ( C ′ ) , ( P, X )) =Ext iR P (cid:16)(cid:16) P, ( C ′ , − ) (cid:17) , (cid:16) P, X (cid:17)(cid:17) ≃ Ext i mod( C ) (cid:16) ( C ′ , − ) , X (cid:17) = 0 for all 1 ≤ i ≤ k − C ′ ∈ C .( ⇐ ). Suppose that Ext iR P ( P ∗ ( C ′ ) , ( P, X )) = 0 for all 1 ≤ i ≤ k − C ′ ∈ C . Let us see by induction on k that X ∈ I k . For k = 1, by hypothesis,we have that X ∈ I . So let us check the first non trivial case, then supposethat k = 2. Consider the ideal I = Tr P C and π : C −→ C / I the projection.Since I ( C ′ , − ) = Tr P (Hom C ( C ′ , − )), we have that I ( C ′ , − ) ∈ P . By 8.2(a),we have an isomorphism Ext C ) ( I ( C ′ , − ) , X ) −→ Ext R P (cid:16) ( P, I ( C ′ , − )) , ( P, X ) (cid:17) .Since I ( C ′ , − ) = Tr P (Hom C ( C ′ , − )), there exists an isomorphism ( P, I ( C ′ , − )) ≃ ( P, Hom C ( C ′ , − )) . Then we have thatExt R P (cid:16) P ∗ ( C ′ ) , ( P, X ) (cid:17) = Ext R P (cid:16) ( P, Hom C ( C ′ , − )) , ( P, X ) (cid:17) ≃ Ext R P (cid:16) ( P, I ( C ′ , − )) , ( P, X ) (cid:17) ≃ Ext C ) ( I ( C ′ , − ) , X ) ≃ Ext C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , X (cid:17) By hypothesis we have that Ext R P (cid:16) P ∗ ( C ′ ) , ( P, X ) (cid:17) = 0, then we conclude thatExt C ) (cid:16) Hom C ( C ′ , − ) I ( C ′ , − ) , X (cid:17) = 0. This implies that Ext C ) (cid:16) ( π ) ∗ ( Q ) , X (cid:17) = 0 forall projective module Q ∈ mod( C / I ). Since X ∈ I (hypothesis), we have thatExt i mod( C ) (cid:16) ( π ) ∗ ( Q ) , X (cid:17) = 0 for all projective module Q ∈ mod( C / I ) and i = 0 , X ∈ I .Suppose that the theorem is true for k − k − ≥ X ∈ mod( C ) such that Ext iR P ( P ∗ ( C ′ ) , ( P, X )) = 0 for all 1 ≤ i ≤ k − C ′ ∈ C . In particular, we have that Ext R P ( P ∗ ( C ′ ) , ( P, X )) = 0 for all C ′ ∈ C .Then, by the case k = 2 just proved above, we have that X ∈ I . Then, we havean exact sequence: ( ⋆ ) : 0 → X → I → L → I ∈ add( J ) and L ∈ I . OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 33 Applying Hom mod( C ) ( P, − ) we get an exact sequence 0 → ( P, X ) → ( P, I ) → ( P, L ) → . Since I ∈ add( J ), we have that ( P, I ) is an injective R P -module (see7.12(c)). Then, applying Hom R P ( P ∗ ( C ′ ) , − ) to the last exact sequence, we havean isomorphism Ext iR P ( P ∗ ( C ′ ) , ( P, L )) ≃ Ext i +1 R P ( P ∗ ( C ′ ) , ( P, X ))for all i ≥
1. By hypothesis, we can conclude that Ext iR P ( P ∗ ( C ′ ) , ( P, L )) = 0 for all i = 1 , . . . , k −
2. Since L ∈ I , we can apply the induction to L . Then we concludethat L ∈ I k − . From the exact sequence ( ⋆ ) we conclude that X ∈ I k . (cid:3) Proposition 8.10.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .Let X ∈ P and k ≥ . Then X ∈ P k if and only if Tor R P i ( P ( C ′ ) , ( P, X )) = 0 forall ≤ i ≤ k − and for all C ′ ∈ C .Proof. Follows from 8.9 using duality, 8.7 and the fact that ( P ∗ ) ∗ ≃ P . (cid:3) Proposition 8.11.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .The following holds. (a) I = I ∞ if and only if P ∗ ( C ′ ) = Hom mod( C ) ( P, Hom C ( C ′ , − )) is a projective R P -module for all C ′ ∈ C . (b) P = P ∞ if and only if P ( C ′ ) ≃ Hom mod( C ) (Hom C ( C ′ , − ) , P ) is a projective R opP -module for all C ′ ∈ C .Proof. ( ⇐ ). By definition we have that I ∞ ⊆ I . Now, let X ∈ I and sup-pose that P ∗ ( C ′ ) is a projective R P -module for all C ′ ∈ C . Then we have thatExt iR P ( P ∗ ( C ′ ) , ( P, X )) = 0 for all i ≥
1. By 8.9, we have that X ∈ I ∞ , provingthat I = I ∞ .( ⇒ ). Suppose that I = I ∞ . Consider Z ∈ mod( R P ). Since Hom mod( C ) ( P, − ) | I : I −→ mod( R P ) is an equivalence (see 7.12), there exists a X ∈ I such that Z ≃ ( P, X ). Then, Ext iR P ( P ∗ ( C ′ ) , Z ) ≃ Ext iR P ( P ∗ ( C ′ ) , ( P, X )) = 0 for all i ≥ X ∈ I = I ∞ and 8.9). This proves that P ∗ ( C ′ ) is a projective R P -module.The proof of b is similar. (cid:3) Proposition 8.12.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and M ∈ mod( C ) . (a) There exists an exact sequence / / K u / / P ⊗ R P Hom
Mod( C ) ( P, M ) ǫ ′ M / / M p / / K / / with M := P ⊗ R P Hom
Mod( C ) ( P, M ) ∈ P and K , K ∈ Ker(Hom( P, − )) . (b) Hom mod( C ) ( P, ǫ ′ M ) : Hom mod( C ) ( P, M ) −→ Hom mod( C ) ( P, M ) is an iso-morphism.Proof. ( a ) and ( b ) . We consider the following exact sequence0 / / K u / / P ⊗ R P Hom mod( C ) ( P, M ) ǫ ′ M / / M p / / K / / Hom mod( C ) ( P,M ) = Hom mod( C ) ( P, ǫ ′ M ) ◦ η ′ Hom mod( C ) ( P,M ) . By 7.7, we have that η ′ Hom mod( C ) ( P,M ) is an isomorphism, then weconclude that Hom mod( C ) ( P, ǫ ′ M ) is an isomorphism. Applying Hom mod( C ) ( P, − )to the last exact sequence and using that Hom mod( C ) ( P, − ) is exact, we concludethat Hom mod( C ) ( P, K ) = 0 = Hom mod( C ) ( P, K ) . Now, by 7.9 we conclude that P ⊗ R P Hom mod( C ) ( P, M ) ∈ P . (cid:3) Now, we give other necessary and sufficient conditions for I to be equal to I ∞ . Proposition 8.13.
Let C be a dualizing R -variety and P = Hom C ( C, − ) ∈ mod( C ) .The following are equivalent (a) I = I ∞ (b) P ⊗ R P Hom
Mod( C ) ( P, Hom C ( C ′ , − )) is a projective C -module for all C ′ ∈ C .Proof. ( b ) ⇒ ( a ). Let M = Hom C ( C ′ , − ) and we consider the module M := P ⊗ R P Hom
Mod( C ) ( P, Hom C ( C ′ , − )) ∈ mod( C ). By 8.12, there exists a morphism ǫ ′ M : M −→ M such that Hom mod( C ) ( P, ǫ ′ M ) : Hom mod( C ) ( P, M ) −→ Hom mod( C ) ( P, M )is an isomorphism. Suppose that M is a projective C -module. Since M ∈ P ,we have that there exists an epimorphism P n −→ M . Then we have that M ∈ add( P ). By 7.12, we have that Hom mod( C ) ( P, M ) is a projective R P -module. ButHom mod( C ) ( P, M ) = P ∗ ( C ′ ). Then we have that P ∗ ( C ′ ) is a projective R P -modulefor all C ′ ∈ C . By 8.11, we have that I = I ∞ . The other implication is similar. (cid:3) Proposition 8.14.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and I = Tr P C . The following are equivalent. (a) I is -idempotent and I = I ∞ ; (b) I ( C ′ , − ) is a projective C -module for all C ′ ∈ C .Proof. ( a ) ⇒ ( b ). Let C ′ ∈ C , by 6.18, we have that I ( C ′ , − ) ∈ P for all C ′ ∈ C .Then, by 7.12 we have that I ( C ′ , − ) = P ⊗ R P Hom mod( C ) ( P, I ( C ′ , − )).But, since I ( C ′ , − ) = Tr P (Hom C ( C ′ , − )), we have the following isomorphism of R P -modules Hom mod( C ) ( P, I ( C ′ , − )) = Hom mod( C ) ( P, Hom C ( C ′ , − )). Then P ⊗ R P Hom mod( C ) ( P, Hom C ( C ′ , − )) = I ( C ′ , − ) . Since I = I ∞ by 8.13, we conclude that I ( C ′ , − ) is projective. The other implication is similar. (cid:3) Proposition 8.15.
Let C be a dualizing R -variety with cokernels and consider P = Hom C ( C, − ) ∈ mod( C ) . If I ( C ′ , − ) is projective for all C ′ ∈ C . Then we havethat R P is quasi-hereditary.Proof. It follows from 8.5 and 8.14. (cid:3)
Proposition 8.16.
Let C be a dualizing R -variety, P = Hom C ( C, − ) ∈ mod( C ) and I = Tr P C . Consider the functor π ∗ : mod( C / I ) −→ mod( C ) . If I ( C ′ , − ) isprojective for all C ′ ∈ C , we have a full embedding D b ( π ∗ ) : D b (mod( C / I )) −→ D b (mod( C )) between its bounded derived categories.Proof. Since I = Tr P C , we get an epimorphism P n −→ I ( C ′ , − ) for each C ′ ∈ C .Since I ( C ′ , − ) is projective for all C ′ ∈ C , we have that I ( C ′ , − ) ∈ add( P ) ⊆ P ∞ .Then, by 6.19, we have that I is strongly idempotent. That is, ϕ iF,π ∗ ( F ′ ) : Ext i mod( C /I ) ( F, F ′ ) −→ Ext i mod( C ) ( π ∗ ( F ) , π ∗ ( F ′ ))is an isomorphism for all F, F ′ ∈ mod( C / I ) and for all 0 ≤ i < ∞ (see definition5.1). By [11, Theorem 4.3], we have the required full embedding. (cid:3) Some examples
Consider an algebraically closed field K and the infinite quiver Q : 1 α / / α / / · · · / / k α k / / k + 1 / / · · · / / · · · Consider C := KQ/ h ρ i , the path category associated to Q where ρ is given by therelations α i +1 α i = 0 for all i ≥
1. By construction, we have that C is a Hom-finite K -category (for more details see for example [16, Proposition 6.6]).It is well known that the category of representations Rep( Q, ρ ) is equivalent to
OMOLOGICAL THEORY OF K -IDEMPOTENT IDEALS IN DUALIZING VARIETIES 35 Mod( C ). In this case, the projective and simple representations associated to thevertex k are of the form P k : k (cid:15) (cid:15) k + 1 , S k : k (1a) Consider P = ⊕ kj =1 P j and I = Tr P C . In this case, we have that Hom C ( i, − ) I ( i, − ) =0 for 1 ≤ i ≤ k and Hom C ( i, − ) I ( i, − ) ≃ P i Tr P ( P i ) = P i for i ≥ k + 1. Then for all j ≥ j Mod( C ) ( Hom C ( i, − ) I ( i, − ) , F ′ ◦ π ) = 0 for all i ∈ C = F Q/ h ρ i and for all F ′ ∈ Mod( C / I ). By 5.2, we have that I is strongly idempotent.(1b) Consider the projective P := ⊕ kj =2 P j and let I := Tr P C . We assert thatTr P C is ( k − Hom C (1 , − ) I (1 , − ) ≃ P Tr P ( P ) ≃ S , where S is thesimple representation associated to the vertex 1. Moreover, we have thatTr P ( P i ) = P i for 2 ≤ i ≤ k . Then we have that Hom C ( i, − ) I ( i, − ) ≃ P i Tr P ( P i ) = 0for 2 ≤ i ≤ k . We also have that Tr P ( P i ) = 0 for all i ≥ k + 1; and hence Hom C ( i, − ) I ( i, − ) ≃ P i Tr P ( P i ) = P i for i ≥ k + 1.By the above discussion and by 5.2, it is enough that Ext j Mod( C ) ( S , F ′ ◦ π ) =0 for all 0 ≤ j ≤ k − ∀ F ′ ∈ Mod( C / I ). We have a projective resolutionof S · · · / / P k +1 / / P k / / · · · / / P / / S / / P i is the projective associated to the vertex i . Using this res-olution we can see that Ext j Mod( C ) ( S , F ′ ◦ π ) = 0 for all 1 ≤ j ≤ k − I is ( k − b ) is not k -idempotent. Indeed, we have that S k = Ω k − ( S ) (the k − S ). By the shifting lemma, we have thatExt k Mod( C ) ( S , S k +1 ) ≃ Ext C ) ( S k , S k +1 ). We have the exact sequence0 → S k +1 → P k → S k → k Mod( C ) ( S , S k +1 ) = 0. Then, I is not k -idempotent.(2) Let I be a heredity ideal in C , according to definition 3.2 in [19]. Thenwe have that I ( C, − ) is a projective C -module for all C ∈ C and I isidempotent. Then by 5.2, we have that Ext C ) ( Hom C ( C, − ) I ( C, − ) , F ′ ◦ π ) = 0for all F ′ ∈ Mod( C / I ) and for all C ∈ C .Now, since the projective dimension of each Hom C ( C, − ) I ( C, − ) is less or equal to 1,we have that Ext j Mod( C ) ( Hom C ( C, − ) I ( C, − ) , F ′ ◦ π ) = 0 for all F ′ ∈ Mod( C / I ) andfor all C ∈ C . Then, by 5.2, we have that I is strongly idempotent. References [1] M. Auslander.
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Trans.Amer. Math. Soc. 362, No. 3, 1475-1489 (2010).Luis Gabriel Rodr´ıguez Vald´es:Departamento de Matem´aticas, Facultad de Ciencias, Universidad Nacional Aut´onoma de M´exicoCircuito Exterior, Ciudad Universitaria, C.P. 04510, Ciudad de M´exico, MEXICO. [email protected]