# Notes on Relative Extension Functors

Authors: Xu QingBing; Yang Liu; Mohammad Munir; Ayoub Youssefi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The objective of this paper is to study the relative homological properties of contravariantly and covariantly finite subcategories. Some sufficient conditions forExt¯YnA,B≃Ext¯XnA,B are obtained. We also give the conditions under which the stable categories C/W/X/W are one-side triangulated categories.


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## Body

## 1. Introduction

The notions of contravariantly and covariantly finite subcategories were introduced in [1] by Auslander and Smalø in connection with studying the problem of which subcategories of an Artin algebra have almost split sequences. Since then, contravariantly and covariantly finite subcategories are widely used in representation theory and relative homological algebra. In [2], Beligiannis studied the relative homological algebra induced by a pair C,X consisting of an additive category C and a contravariantly finite subcategory X of C. Suppose that X is contravariantly finite and any X-epic has a kernel in C. Then, ∀B∈C; the contravariantly X− extension functors Ext¯Xn−,B:Cop⟶Ab,∀n≥0 are defined as the right X-derived functor of C−,C. The covariantly Y-extension functors Ext¯YnA,− are defined dually. Under some conditions, if Ext¯YnY,B=Ext¯XnA,X=0, then Ext¯YnA,B≃Ext¯XnA,B. In the present paper, some results in [2] will be generalized. In [3], Beligiannis and Marmaridis constructed the left and right triangulated structures on the stable categories of additive categories induced from some homological finite subcategories. Recently, Li extended their results to more general settings [4]. Let C be an additive category and X be a full additive subcategory of C. If X is contravariantly finite in C and any special X-epic has a kernel in C, then the stable category C/X has a left triangulated structure induced by X. If Y is covariantly finite in C and any special Y-monic has a cokernel in C, then the stable category C/Y has a right triangulated structure induced by Y. In Section 3, let C be an abelian category, W be a contravariantly finite subcategory of C and W⊆X⊆C. We prove that the stable category C/W/X/W also has a left triangulated structure.In this paper, unless otherwise stated, we assume that all considered categories are skeletally small and additive, and their subcategories are full, additive, closed under direct summands and isomorphisms. Functors between categories are supposed to be additive. The following undefined symbols can be referred in [2, 5]. The latest related profound research conclusions on this subject can be found in [6–10].


## 2. Relative Homology

Homology provides an algebraic picture of topological spaces, and complexes provide a mean of calculating homology. LetC be an abelian category, X is a full subcategory of C, which is closed under direct summands and isomorphisms. Consider a complex
(1)A·:⋯⟶Ai+1⟶Ai⟶⋯,in C. The complex A· is called covariantlyX-exact, if the induced complex
(2)CX,A·:⋯⟶CX,Ai+1⟶CX,Ai⟶⋯is exact in an abelian category. For example, when X is a contravariantly finite subcategory of C, for any right X-approximation of an object A in C, 0⟶ΩA⟶X⟶A is covariantly X-exact, where ΩA is called the first sysygy of A. If A∈GenX=B∈C there exists an epimorphism X⟶B, for a X∈X, then the right X-approximation of A, 0⟶ΩA⟶X⟶A⟶0 is an exact sequence. Dually, the complex A· is contravariantlyX-exact, if the induced complex
(3)CA·X:⋯⟶CAi−1,X⟶CAi,X⟶⋯is exact in an abelian category. If CA·,X and CA·,X are both exact, the complex A· is called functorially X-exact.LetX be a contravariantly finite subcategory of C and A∈C. The X-resolution of A is the following complex:
(4)XA·:⋯⟶XAi+1⟶XAi⟶⋯⟶XA1⟶XA0⟶A⟶0,where XAi+1⟶XAi is the composition XAi+1⟶Ωi+1A⟶XAi. The morphism XAi+1⟶Ωi+1A is a right X-approximation of the i+1th sysygy Ωi+1A of A. Dually, if Y is a covariantly finite subcategory of C and B∈C, the Y-coresolution of B is the complex
(5)YB.:0⟶B⟶Y0B⟶Y1B⟶⋯⟶YiB⟶Yi+1B⟶⋯,where YiB⟶Yi+1B is the composition YiB⟶Ω−1B⟶Yi+1B. The morphism Ω−1B⟶Yi+1B is a left Y-approximation of the ith cosysygy Ω−1B of B. So X-resolution of A is covariant X-exact complex, Y-coresolution of B is contrvariant Y-exact complex. If the category C has enough projective objects, and these projective objects are contained in X, then any X-resolution is an exact sequence. Similarly, if the category C has enough injective objects, which are contained in Y, then any Y-coresolution is an exact sequence.For any objectA∈C, we denote XA·^ by the deleted complex of the X-resolutions XA·(6)X⌢A⋅:⋯⟶XAi+1⟶fi+1XAi⟶⋯⟶XA0⟶0.ForB∈C, the right derived X-functors Ext¯Xn−,B:Cop⟶Ab,∀n≥0 of C−,B are defined to be
(7)Ext¯XnA,B=KerCfn+1,BImCfn,B,n=0,1,2,⋯,where the Cop denoted the opposite category of C. Dually, Y·B^ denotes the deleted complex of Y-coresolutions Y·B(8)Y⌢⋅B:0⟶Y0B⟶Y1B⟶⋯⟶YiB⟶giYi+1B⟶⋯.ForA∈C, the right derived Y-functors Ext¯YnA,−:C⟶Ab,∀n≥0 of CA,− are defined to be
(9)Ext¯YnA,B=KerCA,gnImCA,gn−1,n=0,1,2,⋯.It is similar to the extension functor in homological algebra, by the comparison theorem [11], Ext¯XnA,B does not depend on the choice of X-resolutions of A. Ext¯YnA,B has nothing to do with the choice of Y-coresolutions of B. If A∈GenX or X contains all projective objects, then Ext¯X0A,B≃CA,B. If B∈CogenY=B∈C there exists Y∈Y, such that B⟶Y is a monomorphism or Y contains all injective objects, then Ext¯Y0A,B≃CA,B.Example LetR be an algebra over a field K determined the following quiver 1⟶α2⋅⟶β3¯ with relation βα=0. Then, its AR-quiver is Scheme 1Scheme 1LetSi and Pi be the indecomposable simple module and projective module at vertex i, respectively, i=1,2,3. Put T=S1, X=addT are direct sums of direct summand-s of T, then X=addT is a contravariantly finite subcategory of R–mod categories and Ext¯X1T,−=0, but Ext¯X1T,−≠0, since 0⟶S2⟶P1⟶S1⟶0 is a nonsplit exact sequence. If X=addS3;P2, then by Prop.1.2 in [12], X is a covariantly finite subcategory, Ext¯X1P1,P3≃EndRP3≠0,Ext¯X1P1,P3=0.Proposition 1.
LetX,Y be full subcategories of C. Suppose to be closed under direct sums and direct summands. If X is contravariantly finite subcategory and Y is a covariantly finite subcategory, then
(1)
for any covariantlyX-exact complex 0⟶A⟶B⟶C⟶0, there is a corresponding X- resolution exact sequence, 0⟶X^A.⟶X^B.⟶X^C.⟶0. In particular, for any n, XBn≃XAn⨁XCn(2)
for any contravariantlyY-exact complex, 0⟶A⟶B⟶C⟶0, there is a corresponding Y-coresolution exact sequence, 0⟶Y^.A⟶Y^.B⟶Y^.C⟶0, and for any n, YnB≃YnA⊕YnCProof.
It is only necessary to prove (1) because (2) is its dual. LetfA0:XA0⟶A,fC0:XC0⟶C be the right X-approximations. The morphism iA:A⟶B and pc:B⟶C are corresponding morphisms in X-exact complex 0⟶A⟶B⟶C⟶0. By X-exact property, there exists g:XC0⟶B such that fC0=pcg. Since iAfA0:XA0⟶B by definition of direct sum, there exists a unique morphism d:XA0⊕XC0⟶B, such that iAfA0=dixA0,g=dixC0. So, we have the following commutative diagram Scheme 2.
Next, it is proved thatd:XA0⊕XC0⟶B is a right X-approximation.
Puts:X⟶B and X∈X, there is l:X⟶XC0, such that pcs=fC0l. Since pcdixC0l=fC0l=pcs,u:X⟶A, so dixC0l−s=iAu, thus, there is t:X⟶XA0 such that u=fA0t, hence dixC0l−ixA0t=s.Scheme 2Finally, it is proved that the complex0⟶KerfA0⟶Kerd⟶KerfC0⟶0 is covariantly X-exact. Obviously, we just need to prove that for all X∈X,CX,Kerd⟶CX,KerfC0 is an epimorphism. Then, XB0≃XA0⊕XC0 is obtained by the fact that 0⟶CX,XA0⟶CX,XA0⊕XC0⟶CX,XC0⟶0 is a split exact sequence. Repeating the above procedure Scheme 3.Scheme 3We haveXBn≃XAn⊕XCn.By Proposition1, if 0⟶A⟶B⟶C⟶0 is a covariantly X-exact complex, then for any D∈C, there is a long exact sequence
(10)0⟶Ext¯X0C,D⟶Ext¯X0B,D⟶Ext¯X0A,D⟶Ext¯X1C,D⟶⋯If0⟶A⟶B⟶C⟶0 is a contravariantly Y-exact complex, then for any D∈C, there is a long exact sequence
(11)0⟶Ext¯Y0D,A⟶Ext¯Y0D,B⟶Ext¯Y0D,C⟶Ext¯Y1D,A⟶⋯.Example 1.
has shown that, in general,Ext¯YnD,C≄Ext¯XnD,C, even though X=Y is a functorially finite subcategory. But, we have the following conclusions.Lemma 2.
LetX be a contravariantly finite subcategory, Y be a covariantly finite subcategory. If A,ΩA∈GenX and B,Ω−1B∈CogenY, then
(1)
IfExt¯Y1X,B=0 and Ext¯X1A,Y=0, then there are isomorphisms(12)Ext¯Y1A,B≃Ext¯X1A,B,Ext¯X1A,Ω−1B≃Ext¯Y1ΩA,B(2)
IfExt¯X1A,Y=0, then there are epimorphism(13)Ext¯Y1ΩA,B⟶Ext¯X1A,Ω−1B,and morphism
(14)Ext¯X1A,B⟶Ext¯Y1A,B(3)
IfExt¯Y1X,B=0, then there are epimorphism(15)Ext¯X1A,Ω−1B⟶Ext¯Y1ΩA,B,and morphism
(16)Ext¯Y1A,B⟶Ext¯X1A,BProof.
We only prove (1). Similarly, (2) and (3) can be proved. LetK0=ΩA,L1=Ω−1B, 0⟶K0⟶X0⟶A⟶0 be a right X-approximation of A, and 0⟶B⟶Y0⟶L1⟶0 be a left Y-approximation of B. Therefore, there is the following commutative diagram Scheme 4.
By the snake lemma, we have Scheme5.
SinceCA,B⟶τCK0,B⟶Ext¯X1A,B⟶0 is exact, so
(17)Cokerα≃Ext¯Y1A,B≃Cokerτ≃Ext¯X1A,B.
By the above commutative diagram,γσ=ηβ. Since Ext¯Y1X,B=0 and Ext¯X1A,Y=0, so σ,β are epimorphism. Then,
(18)Cokerγ≃Ext¯Y1K0,B≃Cokerη≃Ext¯X1A,L1.Theorem 3.
LetX be a contravariantly finite subcategory, Y be a covariantly finite subcategory. If A,ΩnA∈GenX; B,Ω−nB∈CogenY, n=1,2,3,⋯; Ext¯Y1X,CogenY=0 and Ext¯X1GenX,Y=0, then for all natural number n, there is an isomorphism
(19)Ext¯YnA,B≃Ext¯XnA,B.Proof.
Let0⟶ΩjA⟶Xj⟶Ωj−1A⟶0 be a covariantly X-exact sequence, where Ω−1A=A. 0⟶Ω−iB⟶Yi⟶Ω−i−1B⟶0 be a contravariantly Y-exact sequence, where Ω0B=B. Repeating the procedure of Lemma 2, we have
(20)Ext¯X1ΩjA,Ω−iB≃Ext¯Y1ΩjA,Ω−iB≃Ext¯X1Ωj−1A,Ω−i−1B.
Hence,(21)Ext¯Yn+1A,B≃Ext¯YnA,Ω−1B≃⋯Ext¯Y1A,Ω−nB≃Ext¯X1A,Ω−nB≃Ext¯Xn+1A,B.Scheme 4Scheme 5The subcategoryA of C is called pre-X-resolving if X⊆A, and A is closed under kernels of X-epic. The subcategory B of C is called pre-Y-coresolving if Y⊆B, and B is closed under cokernels of Y-monic.Corollary 4.
LetX be a contravariantly finite subcategory, Y be a covariantly finite subcategory. If GenX is pre-X-resolving, CogenY is pre-Y-coresolving, A∈GenX, B∈CogenY, Ext¯Y1X,CogenY=0, Ext¯X1GenX,Y=0, then for any natural number n, there is an isomorphism
(22)Ext¯YnA,B≃Ext¯XnA,B.The concepts of dimension and codimension are given in [2]. Put A∈C, if A has X-resolution
(23)0⟶Xn⟶Xn−1⟶⋯⟶X1⟶X0⟶A⟶0,in this case, the least such integer n is called X-dimension of A, denote by dimXA=n. If this nonnegative integer n does not exist, we call dimXA=∞. Obviously, if A∈X, then dimXA=0. But the verse is not true. The global X-dimension of the category C is defined by gl.dimXC=supdimXAA∈C. Dually, using the definition of Y-coresolution, we can define the Y-codimension of object B∈C, denoted by dimYB. The global Y-codimension of the category C, denoted by gl.dimYC=supdimYBB∈C. The following proposition can be found in [2].Proposition 5.
1gl.dimXC=0 if and only if the inclusion functor X°C admits right adjoint functor R:C⟶X
2gl.dimYC=0 if and only if the inclusion functor Y°C admits left adjoint functor L:C⟶YCorollary 6.
IfX,Y is a torsion pair in C, then gl.dimXC=gl.dimYC=0.Proposition 7.
LetX be a contravariantly finite subcategory, if Ωn+1A∈GenX, then dimXA≤n if and only if Ωn+1A=0.Proof.
Since0⟶Ωn+1A⟶Xn⟶ΩnA is a covariantly X-exact sequence, where Xn⟶ΩnA is a right X-approximation, thus CX,Ωn+1A=0 if and only if Ωn+1A=0.Similarly, we have the following proposition.Proposition 8.
LetY be a covariantly finite subcategory. If Ω−n−1A∈CogenY, then dimYA≤n if and only if Ω−n−1A=0.There should be an equivalence relation on the set of short exact sequences. SoYext should be the set of equivalence classes; If Yext¯X1A,B represents all the covariantly X-short exact sequence of the form of 0⟶B⟶Z⟶A⟶0, Yext¯Y1A,B represents all the contravariantly Y-short exact sequence of the form of 0⟶B⟶Z⟶A⟶0, then there is the corresponding Yoneda lemma.Lemma 9 (N. Yoneda).
IfA,ΩA,Ω2A∈GenX, there is a one-to-one correspondence
(24)Yext¯X1A,B⇔Ext¯X1A,B.
IfB,Ω−1B,Ω−2B∈CogenY, there is a one-to-one correspondence
(25)Yext¯Y1A,B⇔Ext¯Y1A,B.Note: ifC is a left R-module category, X is a subcategory of projective module category, Y is a subcategory of injective module category, then C=GenX=CogenY.


## 3. Stable Categories

Let us recall the stable categoryC/X. The objects of C/X are the objects of C. If A,B are objects of C/X, C/XA,B=CA,B/XA,B.Lemma 10.
LetX be a contravariantly finite subcategory, Y be a covariantly finite subcategory. A∈GenX, B∈CogenY, for all C∈C,there is an epimorphism ϕ:Ext¯X1A,C⟶C/XΩA,C, and there is an epimorphism ψ:Ext¯Y1C,B⟶C/YC,Ω−1B.Proof.
1 Let 0⟶ΩA⟶X0⟶A⟶0 be a covariantly X-exact sequence, whereX0∈X, so there is a long exact sequence:
(26)0⟶CA,C⟶CX0,C⟶CΩA,C⟶Ext¯X1A,C⟶0.
If letg:CX0,C⟶CΩA,C be a corresponding morphism, then Img⊆XΩA,C, thus
(27)ϕ:Ext¯X1A,C=CΩA,CImg⟶CXA,C=CΩA,CXΩA,C.
(2) is obtained dually
LetW⊆X⊆C be the contravariantly finite subcategories. α:C⟶C/W,β:C⟶C/X, γ:X/W⟶C/W/X/W are the canonical functors, for any A,B∈C, let αA=Aα and βB=Bβ, f:A⟶B is denoted by αf=fα:Aα⟶Bα, we have the following proposition.Proposition 11.
There is an isomorphismF:X/WAα,Bβ⟶XA,B/WA,B.Proof.
Putf′∈X/WAα,Bβ, there is f∈CA,B such that fα=f′. Then X∈X, such that f′ factor through Xα, i.e. h∈CA,X and t∈CX,B such that fα=tαhα=thα. Thus, f-th∈WA,B. Put
(28)F:XWAα,Bβ⟶XA,BWA,Bf′↦f+WA,B.,Conversely, putf+W∈XA,B/WA,B, where f∈XA,B. Thus, there exists X∈X such that f=th, where h∈CA,X,t∈CX,B. Therefore, fα=tαhα∈X/WAα,Bβ. Put G:XA,B/WA,B⟶X/WAα,Bβ such that Gf+WA,B=fα. Hence, GF=1 and FG=1.Theorem 12.
F:C/X⟶C/W/X/W is an equivalence of an additional category.Proof.
PutF:Aβ↦Aγα, for all A,B∈C,
(29)C/WX/WAγα,Bγα≃C/WAα,BβX/WAα,Bα≃CA,B/WA,BXA,B/WA,B≃CA,BXA,B≃C/XAβ,Bβ.LetC be an additive category and Ω an additive covariant endofunctor on C. Let Δ be a class of left triangles of the form Ωw⟶fu⟶gv⟶hw. The pair Ω,Δ is called a left triangulated structure on C if Δ is closed under isomorphisms and satisfies the following four axioms:Lt1 For any morphism f:v⟶w there is a left triangle in Δ of the form Ωw⟶u⟶v⟶fw. For any object u∈C, the left triangle 0⟶u⟶1uu⟶0 is in ΔLt2 Rotation axiom: for any left triangle Ωw⟶fu⟶gv⟶hw in Δ, the left triangle Ωv⟶−ΩhΩw⟶fu⟶gv is also in ΔLt3If the figure below is the commutative diagram of left triangle inΔScheme 6Scheme 6Then, there is morphismα:a1⟶b1, which makes the figure above continuous to be a commutative diagram.Lt4 Octahedral axiom: for any two left triangles Ωw⟶fu⟶gv⟶hw and Ωz⟶ix⟶lw⟶kz in Δ, there is a left triangle Ωz⟶jp⟶mv⟶khz and two morphisms α:u⟶p,β:p⟶x, such that the graph below is a commutative diagram Scheme 7, where the second column from the left is a left triangle in ΔScheme 7Dually, we can define the right triangulated structureΣ,Δ′ on C, where Σ is a covariant additive endofunctor of C and Δ′ is a class of right triangles of the form u⟶v⟶w⟶Σu. which satisfies the dual right triangulated axioms.An additive categoryC is called a left triangulated category if there is a left triangulated structure on C. Dually, an additive category C is called a right triangulated category [13] if there is a right triangulated structure on it. The data of a right triangulated category was first introduced by Bernhard Keller in [14]. If the endofunctor Ω (respectively, Σ) is an autoequivalence, the left (respectively, right) triangulated category C,Ω,Δ (respectively, C,Σ,Δ′) is a triangulated category. Left and right triangulated categories are natural generalization of triangulated categories.LetX be a contravariantly finite subcategory of an additive category C. A morphism f:B⟶A in C is called an X-epic if for any X∈X the induced map HomCX,g:HomCX,B⟶HomCX,A is surjective. A morphism f:B⟶A in C is called a special X-epic if it is of the following form:
(30)B⊕XA⟶g,pAA,where g is a morphism of C and pA is a right X-approximation of A. By definition, a right X-apprixomation is a special X-epic and a special X-epic is an X-epic. Dually, if Y is covariantly finite in C, we have the notions of an Y-monic and a special Y-monic. Let C be an additive category and X an additive subcategory of C. Assume that X is contravariantly finite in C and any special X-epic has a kernel. Then C/X is a left triangulated category. If Y is covariantly finite in C and any special Y-monic has a cokernel. Then, C/Y is a right triangulated category.LetX be a contravariantly finite subcategory of additional category C, by [4], the stable category C/X has a natural left triangulated structure. ΩX:C/X⟶C/X is the loop functor, which is defined as follows: ΩXAβ=ΩAβ, for any morphism f:A⟶B in C, ΩXfβ:ΩXAβ⟶ΩXBβ such that ΩXfβ=Ωfβ. For any covariantly X− exact complex in C/X,
(31)0⟶C⟶hA⟶gB,if
(32)0⟶ΩB⟶iXB0⟶d0B,is a right X-approximation of B, then we have the following commutative diagram Scheme 8.Scheme 8So,(33)ΩXBβ⟶γβCβ⟶iβAβ⟶dβ0Bβ,is a standard triangle in C/X.Theorem 13.
LetC be an abelian category, X and Y are subcategories of C.
(1)
IfW⊆X is a contravariantly finite subcategory of C and any special X-epic has a kernel in C, then F:C/X⟶C/W/X/W is the equivalence of a left triangulated category(2)
IfV⊆Y is a covariantly finite subcategory of C and any special Y-monic has a cokernel in C, then F¯:C/Y⟶C/V/Y/V is the equivalence of a right triangulated categoryProof.
We only prove (1), and (2) can be obtained similarly. For anyA∈C,
(34)FΩXAβ=FΩAβ=ΩAγα=ΩX/WAγα=ΩX/WFAβ,
iff:A⟶B, then
(35)FΩXfβ=FΩfβ=Ωfγα=ΩX/Wfγα=ΩX/WFfβ,if
(36)ΩXBβ⟶chgCβ⟶hβAβ⟶gβBβ,
is a triangle induced by specialX-epimorphism g:A⟶B in C/X, then
(37)ΩX/wBγα⟶Cγα⟶hγαAγα⟶gγαBγα,is the left triangle in C/W/X/W, and that is
(38)FΩXBβ⟶FCβ⟶FhβFAβ⟶FgβFBβ.


## 4. Conclusions

As a further generalization of the Proposition 2.8 in [2], we introduced the notion of GenX and CogenY, some sufficient conditions for Ext¯YnA,B≃Ext¯XnA,B are given. The left and right triangulated structures on the stable categories induced from some homological finite subcategories are discussed. Let C be an abelian category, WV be a contravariantly (covariantly) finite subcategory of C and W⊆X⊆C(V⊆Y⊆C), we have that the stable category C/W/X/W (C/V/X/V) also has a left triangulated structure (right triangulated structure).



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