Gorenstein objects in the n-Trivial extensions of abelian categories
aa r X i v : . [ m a t h . K T ] J un GORENSTEIN OBJECTS IN THE n -TRIVIALEXTENSIONS OF ABELIAN CATEGORIES. DIRAR BENKHADRA
CeReMAR Center, Department of Mathematics, Faculty of Sciences, B.P.1014, Mohammed V University in Rabat, [email protected]
Abstract.
Given a left n − trivial extension A ⋉ n F of an abelian cat-egory A by a family of quasi-perfect endofunctors F := ( F i ) ni =1 . Wepropose a categorical aspect of the strongly Gorenstein objects in A andgive the transfer properties to the category A ⋉ n F . Then we fully char-acterize the Gorenstein projective and injective objects of A ⋉ n F viathe strongly Gorenstein ones. Therefore establishing a categorical frame-work in which we characterize Gorenstein flat module over the categoryof modules over R ⋉ n M . Moreover we give a recollement situation ofthe stable category of Gorenstein projective objects over A ⋉ n F . n -Trivial extensions of abelian categories, Gorenstein objects, recollement dia-gram, triangular matrix rings Introduction
Enochs et al., in [8], introduced the so-called Gorenstein projective and in-jective modules over a ring R and developed the first properties of Gorensteinhomological algebra. Since then, many others studied several properties of Goren-stein modules and objects for more details see [5], [6], [8], [9], [10], [11], [20], [22],[15].Namely, in [6] Mahdou and Bennis introduced strongly Gorenstein projective, in-jective and flat modules as an analogue structure to free modules. Therefore, theygave a characterization of Gorenstein modules via strongly Gorenstein modules.The trivial extension R ⋉ M of a ring R by a bimodule M is considered the mainfield of application of the notion. In [18] for instance, the authors studied thetransfer of gorestein projective ( resp. injective) properties and dimensions from R to R ⋉ M .Furthermore, in [1] Anderson et al., gave a generalisation of R ⋉ M , raisinga new ring called n − trivial ring extension by a family of modules M = ( M i ) ni =1 denoted by R ⋉ n M . Also they extensively studied its various algebraic proper-ties . Recently in [4] the author et al., established a categorical generalisation of R ⋉ n M by constructing a new category, called the left n − trivial extension ofabelian category A by a family of endofunctors F := ( F i ) i, ∈ I denoted A ⋉ n F .Using a set of functors T , C , U and Z among other tools. They provided a widerange of properties such as: generators, projective, injective and flat objects. Alsoensuring the existence of limits and colimits in such categories. Lately in [20] Y. Peng et al, established a categorical framework to study Goren-stein projective and injective objects in Comma category which is an exampleof trivial extension. In addition they were able the give an equivalent character-ization of projective and injective objects. Moreover in [20] they constructed arecollement situation of the stable category of Gorenstein projectives objects overthe Comma category.My own work is to propose a categorical approach to some (Strongly) Goren-stein properties. The main result is the description of GP ( A ⋉ n F ) (resp. GI ( A ⋉ n F ) )the category of Gorenstein projective (resp. injective) objects in A ⋉ n F via thecharacterization of SGP ( A ⋉ n F ) (resp. SGI ( A ⋉ n F ) ) Theorem 1.1.
We assume that the F i ’s are quasi-perfect. Then the fol-lowing statement hold. a) if X is in SGP ( A ) then T ( X ) ∈ SGP ( A ⋉ n F ) . b) Let ( X, f ) in SGP ( A ⋉ n F ) then C ( X, f ) ∈ SGP ( A ) . c) Then an object ( X, f ) in A ⋉ n F is Gorenstein projective if, and only if, C ( X, f ) is also Gorenstein projective and ( X, f ) ∼ = T ( C ( X, f )) .By inverting the arrows we get the injective case in the category G ⋊ n A . Specifically, in section 3, we define at first the meaning of strongly Gorensteinobjects in an abelian category relatively to a class of objects called SP (resp. SI ).Motivated by the utility of perfect functors in [20], we define quasi-(co)perfectfunctors that to give an equivalent characterization of strongly Gorenstein pro-jective and injective in A ⋉ n F via objects in A . Using the new categorical tools,we give in section 4 some results concerning Gorenstein homological dimensions.This approach generalize perfectly the well known results in the setting of mod-ules over rings see section 5. Further we prove when R ⋉ n M is n − Gorenstein,and under this condition we characterize Gorenstein flat modules over R ⋉ n M .We end the section by an application of our results on triangular matrix rings ofthe form R M R M M R , where R i are associative rings for i ∈ { , , } , M is an R − R module, R − R module and M is an R − R module.A recollement of triangulated category introduced in [2] plays an important rolein algebraic geometry and representation theory. In [22] P. Zhang proved whenthe stable category GP (cid:18) R M S (cid:19) with R , S associative rings and M is a com-patible R − S module. Recently in [20] the authors gave a similar results in thecase of left Comma category which lead us the second main result of this paper.In the last section, we prove the existence of a left recollement of GP ( A ⋉ n F ) thestable category of GP ( A ⋉ n F ) that we filled to a recollement. Theorem 1.2.
Let ( F i ) i ∈ I be quasi-perfect and preserve projective object and A is Gorenstein category then there exist a recollement as follows GP ( A ) Z / / GP ( A ⋉ n F ) C (cid:127) (cid:127) N ` ` U / / GP ( A ) L ` ` T (cid:127) (cid:127) ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.3 Where N (( X ⊕ X , g )) = X and L ( X ) = ( X ⊕ ni =1 P i , f ) where f is of matrixform defined by a morphism F i X / / P i and zero otherwise, for i ∈ I , with ( P i ) ni =1 a familly of projective objects in A . Preliminary
Through this section we will fix the notations and give the background mate-rials that will be necessary in the sequel. For more details, we refer the reader to[18, 17, 11, 14, 4].Along this paper R will be an associative ring with identity and all moduleswill be, unless otherwise specified, unital left R -modules. The category of all left(right if needed) R -modules will be denoted by R M od ( M od R ).Let M := ( M i ) ni =1 a family of R bimodules and ϕ = ( ϕ i,j ) i,j n − a familyof bilinear maps such that each ϕ i,j is written multiplicatively ϕ i,j : M i ⊗ M j ( m i ,m j ) / / M i + jϕ ( m i ,m j ):= m i .m j whenever i + j ≤ n with, ϕ i + j,k ( ϕ i,j ( m i , m j ) , m k ) = ϕ i,j + k (( m i , ϕ i,k ( m j , m k )) . The n -trivial extension ring of R by M = ( M i ) ni =1 denoted by R ⋉ n M is the additive group R ⊕ M ⊕ · · · ⊕ M n with the multiplication defined by: ( m , ..., m n )( m ′ , ..., m ′ n ) = P i + j = n m i m ′ j . Notice that when n = 1 , R ⋉ n M is nothing but the classical trivial extension of R by M which is denoted by R ⋉ M . We maintain the notation R ⋉ n M for the n -trivial extension of the R by M = ( M i ) ni =1 .Let A be a Grothendieck category with enough projective objects. The categoryof Gorenstein projective (resp. injective) object over A will be denoted GP ( A ) (resp. GI ( A ) ).Let F := (F i ) i ∈ I , with I = { , . . . , n } , a family of additive covariant end-ofunctors of A equipped with a family of natural transformations Φ = (Φ i,j :F i F j −→ F i+j ) . For every i, j, k ∈ I such that i + j + k ∈ I , we have the followingcommutative diagram: F i F j F k X F i (Φ j,k ) X / / (Φ i,j ) Fk X (cid:15) (cid:15) F i F j+k X (Φ i,j ) X (cid:15) (cid:15) F i+j F k X (Φ i + j,k ) X / / F i + j + k X In [4] the author et al., introcuded the concept of the n -trivial extension of anabelian category by a family of functors. Definition 2.1. ( [4, p:4-6] ) The right n − trivial extension of A by F is the cate-gory whose objects are of the form ( X, f ) with X is an object of A and f = ( f i ) i ∈ I DIRAR BENKHADRA is a family of morphisms f i : F i X −→ X such that ,for every i, j ∈ I , the diagram F i F j X F i f j (cid:15) (cid:15) (Φ i,jX ) X / / F i+j X f i + j (cid:15) (cid:15) F i X f i / / X commutes when i + j ∈ I , if not, the composition F i F j X F i f j / / F i X f i / / X is zero.A morphism γ : ( X, f ) −→ ( Y, g ) in A ⋉ n F is a morphism X −→ Y in A suchthat, for every i ∈ I , the diagram F i X f i (cid:15) (cid:15) F i γ / / F i Y g i (cid:15) (cid:15) X γ / / Y commutes.By a dual statement, the left n − trivial extension of A by a family G = (G i ) i ∈ I ofadditive covariant endofunctors was also defined and denoted by G ⋊ n A . In [4], the transfer of properties between A and A ⋉ n F was extensively studiedusing the following pairs of functors(2.1) A Z A ⋉ n F C y y U A T w w A Z & & G ⋊ n A K e e U % % A H f f The functor T : A −→ A ⋉ n F given, for every object X , by T ( X ) = ( X ⊕ ( M i ∈ I F i X ) , κ ) , where κ = ( κ i ) such that, for every i ∈ I , κ i = ( a iα,β ) : F i X M j ∈ I F i F j X −→ X ⊕ ( M j ∈ I F j X ) defined by a ii +1 , = id F i X ,a ii + j +1 ,j +1 = Φ ijX for j with i + j ∈ I, otherwise . ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.5 and for morphisms by: T ( α ) = α . . . ... F α ... . . . . . . . . . F n α . The cokernel functor C : A ⋉ n F −→ A is given by C (( X, f )) = Coker( f : ⊕ i ∈ I F i X −→ X ) , for every object ( X, f ) . For a morphism α : ( X, f ) −→ ( Y, g ) in A ⋉ n F , C ( α ) isthe induced map.The functor H : A −→ G ⋊ n A given, for every X ∈ A , by H ( X ) = ( G n X ⊕ G n − X ⊕ · · · ⊕ G X ) ⊕ X, λ = ( λ i )) , where, for every i ∈ I , λ i = ( a iα,β ) : G i G n X ⊕ G i G n − X ⊕· · ·⊕ G i G X ⊕ G i X −→ G n X ⊕ G n − X ⊕· · ·⊕ G X ⊕ X defined by a in +1 ,n +1 − i = id G i X ,a in − j +1 ,n +1 − ( i + j ) = Ψ ijX for j where i + j ∈ I, otherwise . and on morphisms by H ( α ) = G n α . . . ... . . . ... . . . G α . . . α . The kernel functor K : G ⋊ n A −→ A is given, for every object in G ⋊ n A by K ( X, g ) = Ker( g : X −→ ⊕ i ∈ I G i X ) . For a morphism β in G ⋊ n A , K ( β ) is the induced morphism. U is the underlying functor and Z is the zero functor. It is shown in [4] that ( T, U ) , ( C, Z ) , ( U, H ) and ( K, Z ) are adjoint pairs verifying the following equali-ties CT ∼ = id A and KH ∼ = id A . Gorenstein object
In this section, we begin by introducing a special class of objects necessary topropose a categorical definition of Strongly Gorenstein projective (resp. injective)objects. Then we establish an equivalent characterization of strongly projective(resp. injective ) objects over A . Definition 3.1.
An object in A is called SP (resp. SI )-object if it admits a shortexact sequence of the form −→ X −→ P −→ X −→ (resp. −→ X −→ I −→ X −→ ), with P (resp. I ) a projective (resp injective) object of A .We say that an endofunctor F of A is SP -left( resp. SI -right) exact if −→ F X −→ F P −→ F X (resp.
F X −→ F I −→ F X −→ ) is still exact. We propose the following definition of Strongly Gorenstein objects over A . DIRAR BENKHADRA
Definition 3.2.
We say that an object X in A is strongly Gorenstein projectiveor simply X ∈ SGP ( A ) , if X is SP and the functor Hom A ( − , Q ) is SP − leftexact for any projective object Q in A . One can remark that this definition is a categorical analogue of the one men-tioned [5] in the category of modules.In [20] perfect functors were introduced to study Gorenstein objects over theComma category. In this paper we show that the strongly Gorenstein case re-quires a lesser condition and we give the following definition.
Definition 3.3.
An endofunctor F : A −→ A is said to be quasi-perfect endo-functor if (GP1): F is left SP-exact. (GP2): Hom A ( X, F Q ) is exact with X is SP -object and Q projective object of A .The Strongly Gorenstein injective case is dual to this statement and called quasi-coperfect. For the category of R modules, if pd R ( M i ) is finite, for i ∈ I , then all M i ⊗ R − are quasi-perfect. Lemma 3.4. [20, Lemma 3.4]
The following statements are equivalent: • F i , for i ∈ I , satisfies ( GP . • Ext A ( X, F i Q ) = 0 , for any X Strongly Gorenstein projective and Q aprojective object of A . • Ext n ≥ A ( X, F i Q ) = 0 , for any X Strongly Gorenstein projective and Q aprojective object of A . We start by characterizing objects in A ⋉ n F of the form T ( X ) with X in A . Proposition 3.5.
We assume that the F i ’s are quasi-perfect, if X is in SGP ( A ) then T ( X ) ∈ SGP ( A ⋉ n F ) . Proof. If X ∈ SGP ( A ) then there is an exact sequence / / X δ / / P δ / / X / / with P is a projective object in A . Then, we get T ( X ) T ( δ ) / / T ( P ) T ( δ ) / / T ( X ) / / , where T ( δ ) = δ . . . ... F δ ... . . . . . . . . . F n δ . Since we assume (GP1), we have in-deed the following exact sequence. / / T ( X ) T ( δ ) / / T ( P ) T ( δ ) / / T ( X ) / / . Now, for every projective object T ( Q ) in A ⋉ n F , we have / / Hom ( T ( X ) , T ( Q )) T ∗ ( δ ) / / Hom ( T ( P ) , T ( Q )) T ∗ ( δ ) / / Hom ( T ( X ) , T ( Q )) . By (GP2) and Lemma 3.4 we have
Ext ( T ( X ) , T ( Q )) ∼ = Ext A (X , UT(Q)) = Ext n A (X , UT(Q)) = 0 . ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.7 Hence the following short exact sequence is exact / / Hom ( T ( X ) , T ( Q )) T ∗ ( δ ) / / Hom ( T ( P ) , T ( Q )) T ∗ ( δ ) / / Hom ( T ( X ) , T ( Q )) / / . Corollary 3.6.
Suppose that the functor T is fully faithful. For every object X in A , If T ( X ) is strongly Gorenstein projective then X is also strongly Gorensteinprojective. Theorem 3.7.
We assume that ( F i ) i ∈ I are quasi-perfect functors. Let ( X, f ) in SGP ( A ⋉ n F ) then C ( X, f ) ∈ SGP ( A ) . Proof.
Let ( X, f ) ∈ SGP ( A ⋉ n F ) then there exist a projective object P in A such that we have the following commutative diagram(3.1) / / ( X, f ) δ / / T ( P ) δ / / ( X, f ) / / . From the commutative diagram / / ⊕ i ∈ I F i X / / f (cid:15) (cid:15) ⊕ i ∈ I F i P / / (cid:15) (cid:15) ⊕ i ∈ I F i X / / f (cid:15) (cid:15) / / X / / P ⊕ i ∈ I F i P / / X / / we deduce that the following commutative diagram / / Imf / / α (cid:15) (cid:15) X π / / (cid:15) (cid:15) C ( X, f ) C ( δ ) (cid:15) (cid:15) / / / / ⊕ i ∈ I F i P / / (cid:15) (cid:15) P ⊕ i ∈ I F i P / / (cid:15) (cid:15) P / / C ( δ ) (cid:15) (cid:15) / / Imf / / (cid:15) (cid:15) X / / (cid:15) (cid:15) C ( X, f ) / / (cid:15) (cid:15)
00 0 0 by snake lemma we get the exact sequence, / / Ker ( δ ) / / Imf
Ker ( π ) / / X / / C ( X, f ) / / . Since
Ker ( π ) is a monomorphism, we get that Ker ( δ ) = 0 . Hence we have theshort exact sequence. / / C ( X, f ) / / P / / C ( X, f ) / / . For every projective object Q of A , we have Ext A (C(X , f) , Q) ∼ = Ext A ((X , f) , (Q , .Since ( X, f ) is in SGP ( A ⋉ n F ) , then the second term is zero. Therefore the short DIRAR BENKHADRA sequence / / Hom( C ( X, f ) , Q ) γ / / Hom(
P, Q ) γ / / Hom( C ( X, f ) , Q ) / / is exact, where γ = C ∗ ( δ ) and γ = C ∗ ( δ ) is exact.By the duality pointed in [4, Theorem 2.5] between A ⋉ n F and G ⋊ n A , with-out stating the proofs we give the strongly Gorenstein injective context. Similarlyto 3.2, We assume that ( G i ) ( i ∈ I ) are quasi-coperfect:(GI1) ( G i ) ( i ∈ I ) are right SI − exact;(GI2) Hom A ( G i Q, X ) is exact, for any strongly Gorenstein injective object X in A and Q injective object of A . Definition 3.8.
We say that an object X in A is strongly Gorenstein injectiveor simply X ∈ SGI ( A ) , if X is SI and for any injective object Q , the functor Hom ( Q, − ) is SI − exact for any injective object Q in A . Proposition 3.9. • If X ∈ SGI ( A ) , then H ( X ) is also in SGI ( A ⋉ n F ) .The converse holds if H is fully faithful. • We assume that G i ’s are quasi-coperfect functors. Let ( X, f ) be a in SGI (G ⋊ n A ) then K ( X, f ) ∈ SGI ( A ) . Inspired by [5] we adopt the following definition of Gorenstein projective (resp.injective ) objects.
Definition 3.10.
An object is Gorenstein projective (resp. injective) if, and onlyif, it is a direct summand of a Strongly Gorenstein projective (resp. injective)objects.
From [4], we have the characterization of the class
Add ( T ( P )) of objects thatare isomorphic to direct sums of copies of T ( P ) , where P is an object of A . Duallywe gave a similar results concerning the class P rod ( H ( P )) , of product of copiesof H ( P ) . Lemma 3.11. [4, Theorem 3.1]a)
Let P be an object of A . An object ( X, f ) of A ⋉ n F is in
Add ( T ( P )) if,and only if, C ( X, f ) ∈ Add ( P ) and ( X, f ) ∼ = T ( C ( X, f )) . ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.9 b) Let I be an object of A . An object ( X, f ) of G ⋊ n A is in P rod ( H ( I )) if,and only if, K ( X, f ) ∈ P rod ( I ) and ( X, f ) ∼ = H ( K ( X, f )) . Remark 1.
One could remark that if we take P (resp. I ) a projective(resp. in-jective) object in A the above theorem then give a characterization of projective (resp. injective) objects in A ⋉ n F( resp. G ⋊ n A ). By combining theorem 3.10 and lemma 3.11 we get the following
Theorem 3.12. a) We assume that the F i ’s are quasi-perfect. Then an ob-ject ( X, f ) in A ⋉ n F is Gorenstein projective if, and only if, C ( X, f ) isalso Gorenstein projective and ( X, f ) ∼ = T ( C ( X, f )) .Dually, we assume that (GI1), (GI2) are satisfied in G ⋊ n A . b) An object ( X, f ) in G ⋊ n A is Gorenstein injective if, and only if, K ( X, f ) is also Gorenstein injective and ( X, f ) ∼ = H ( K ( X, f )) . Gorestein dimension
In the following, we focus on the Goresntein homological dimensions. First, werecall the definition of Gorenstein projective (resp. injective) dimension.
Definition 4.1. [15, Definition 2.8]
The Gorenstein projective dimension,
Gpd R M ,of an R − module M is defined by declaring that Gpd R M ≤ n ( n ∈ N ) if, and onlyif, M has a Gorenstein projective resolution of length n . Similarly, one defines theGorenstein injective dimension. Proposition 4.2.
We assume that ( F i ) ( i ∈ I ) ,for i ∈ I , are quasi-perfect.a) Let ( X, f ) be an object in A ⋉ n F such that X = X ⊕ X with Im( f i ) ∈ X ,for all i ∈ I , then Gpd A (X ) ≤ Gpd A ⋉ n F (X , f) . We assume G i ’s are quasi-coperfect.b) Let ( X, f ) be an object in G ⋊ n A such that X = X ⊕ X with X ∈ Ker ( f i ) ,for all i ∈ I , then Gid A (X ) ≤ Gid G ⋉ n A (X , f) . Proof.
Since a ) and b ) are proved dually, we assume that Gpd A ⋉ n F (X , f) = m < ∞ . For m = 0 , we have ( X, f ) is a Gorenstein projective object of A ⋉ n F . Hence C ( X, f ) is also Gorenstein projective in A by theorem 3.7. Since Im( f i ) ∈ X and X = X ⊕ X , we get that X is a direct summand of C ( X, f ) . Therefore X , isGorenstein projective in A .Now assume that m > . Consider two exact sequences in A , / / K / / P ǫ / / X / / P ǫ / / X / / with P is Gorenstein projective and P is projective in A . Since Im f i ∈ X , wecan set f i = ( f i, , f i, ) with f i, : F i X −→ X and f i, = F i X −→ X . Now, let L to be the kernel of the morphism: λ = (( f i, ◦ F i ǫ ) i ∈ I , ǫ , ( f i, ◦ F i ǫ ) i ∈ I ) : M i ∈ I F i P ⊕ P M i ∈ I F i P −→ X and consider µ = (cid:18) ǫ λ (cid:19) : P M i ∈ I F i P ⊕ P M i ∈ I F i P −→ X ⊕ X . Then we obtain the following exact sequence: ( K ⊕ L, α ) Ker µ / / T ( P ) ⊕ T ( P ) µ / / ( X, f i ) / / with α = ( α i, : F i K −→ L, α i, : F i L −→ L ) . The middle term represents aGorenstein projective object in A ⋉ n F , hence Gpd A ⋉ n F (X , f i ) = 1+Gpd A ⋉ n F (K ⊕ L , α i ) . By the induction on m we have Gpd A ⋉ n F (K ⊕ L , α i ) ≥ Gpd A K . Therefore Gpd A ⋉ n F (X , f) ≥ A K = Gpd A X . Proposition 4.3.
Supposing that L j F i = 0 (resp. R j G i ), for i ∈ I , j > , then Gpd A (X) = Gpd A ⋉ n F (T(X)) (resp. Gid A (X) = Gid G ⋉ n A (H(X)) ). Corollary 4.4.
For every object X ∈ A , we have : • Gpd A X ≤ min(Gpd A ⋉ n F Z(X) , Gpd A ⋉ n F T(X)) . • Gid A X ≤ min(Gid G ⋉ n A Z(X) , Gid G ⋉ n A H(X)) . • G − gldim A ( A ) ≤ G − gldim A ⋉ n F ( A ⋉ n F ) Considering the ring R ⋉ n M and observing that R ⋉ n M = R ( ⊗ R R ⋉ n M ) .These results have the following concretisation. Corollary 4.5. G − gldim A (R) ≤ G − gldim A ⋉ n F (R ⋉ n M) .Supposing that M i are flat, for all i ∈ I , then G − gldim A (R) = G − gldim A ⋉ n F (R ⋉ n M) . Application on the category of modules over n-trivialextension
Now, we look upon the realisation of the Gorenstein categorical properties onthe category of modules over the ring of the n − trivial extension. Throughout thispart, we adopt the following notations: Let R be an associative ring and ( M i ) i ∈ I family of R bimodules such that S := R ⋉ n M is the n − trivial extension definedin [1].In [4],they identified the category of modules over S to the category A ⋉ n F , bychoosing A = R Mod and F i = − ⊗ M i , for i ∈ I , with Φ ( i,j ) = id A ⊗ ˜Φ ( i,j ) . Theyalso proved the following theorem. Theorem 5.1. ( [4, Theorem 2.2] ) The category R M od ⋉ n ( M i ⊗ − ) i ∈ I is isomor-phic to the category of left modules over R ⋉ n M . Remark 2.
Denoting G i = Hom( M i , − ) , for i ∈ I , to be the adjoint of F i . In [4] , it was proved that A ⋉ n F ∼ = G ⋊ n A . Also, the functors T , U , C , H and K between R M od and S M od have a concrete realisation as follows C = − ⊗ S R , T = − ⊗ R S , K = Hom S ( R, − ) , H = Hom R ( S, − ) , see [3] .Under these identification, we investigate some aspect of strongly Gorenstein ho-mological properties of the n − trivial extension and extending some of the resultestablished in [18] . We recall the results in [18] that are the subject of our applica-tions. Theorem 5.2. [18, theorem 2.1]
Let N be an R − module. Then: ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.11
1) a)
Suppose that pd R (M) < ∞ . If N is a strongly Gorenstein projec-tive R − modules, then M ⊗ R S is a strongly Gorenstein projective S − module. b) Conversely, suppose that M is flat R − module. If N ⊗ R S is a stronglyGorenstein projective S − module, then N is a strongly Gorensteinprojective R − module Suppose that
Ext pR ( S, M ) = 0 for all p ≥ and f d R ( S ) < ∞ . If N is astrongly Gorenstein injective R − module, then Hom R ( S, N ) is a stronglyGorenstein projective R − moduleFor sake of simplicity, we will restrain to the case of R ⋉ M ⋉ M , since theresults are the same for R ⋉ n M with n ∈ N . Remark 3.
One should remark that if an R − module M is of finite projectivedimension then M ⊗ R − is quasi-perfect. Also if M is flat then Hom( M, − ) isquasi-coperfect.We establish the transfer of strongly Gorenstein between R and R ⋉ M ⋉ M generalizing theorem (5.2). The following corollaries comes directly from proposi-tion (3.5), theorem (3.7) and remark (3) . Corollary 5.3.
Let N be an R − module and Then: a) Assume that − ⊗ R M i is quasi-perfect, for i ∈ { , } , if N is stronglyGorenstein projective as an R − module then N ⊗ R S is also strongly Goren-stein projective as an S − module. b) Conversely, suppose that ( M i ) ( i ∈ I ) are flat R − modules, if N ⊗ R S isstrongly Gorenstein projective then so that N as an R − module. Corollary 5.4.
Let N be an R − module. For i ∈ I , we assume that f d R ( M i ) < ∞ ,then: a) If N is strongly Gorenstein injective as an R − module then Hom R ( S, N ) is also strongly Gorenstein injective S − module. b) Conversely if N is strongly Gorenstein injective S − module then Hom S ( R, N ) is Strongly Gorenstein injective. Corollary 5.5.
We suppose that ,for every i ∈ I , M i ⊗ R − is quasi-perfect .An S − module N is Gorenstein projective if and only if N (cid:30) M i N are Gorensteinprojective and M i ⊗ R M i ⊗ R N M i ⊗ f i / / M i ⊗ R N f i / / N / / Coker ( f i ) / / is exact, for i ∈ I .The dual statement is the injective case. Gorenstein flat modules.
Always under the realisation established in re-mark 2 and for simplicity sake, we will prove the following result in the case of n = 2 , since iteratively we can have the general case. Also we keep using thecategorical frame work. Proposition 5.6.
Let ( X, f ) represent an object of G ⋊ n A of finite injectivedimension. If R i K ( X, f ) = 0 for i ≥ , then id G ⋉ n A (X , f) = id A (Ker(X , f)) Proof. If id G ⋉ n A (X , f) = 0 , it is straightforward from the construction of injectiveobjects. We shall continue by way of induction on id G ⋉ n A (X , f) = n ≥ . We takea one step injective resolution of ( X, f ) . (5.1) / / ( X, f ) / / H ( I ) / / ( B, β ) / / . This is equivalent to the following commutative diagram, / / X / / I ⊕ i ∈ I G i I / / B / / / / ⊕ i ∈ I G i X / / f O O ⊕ i ∈ I G i ( I ⊕ i ∈ I G i I ) / / O O ⊕ i ∈ I G i B β O O / / Since R i K ( X, f ) = 0 for i ≥ , and KH ( I ) ∼ = I we get the exact sequence / / K ( X, f ) / / I / / K ( B, β ) / / , and R i K ( B, β ) = 0 . By induction we have id G ⋉ n A (X , f) = 1 + id G ⋉ n A (B , β ) , = 1 + id A (K(B , β )) , = id A (K(X , f)) . Proposition 5.7. If Ext iS ( R, S ) = 0 for i ≥ , then id S ( S ) = id R ( B ⊕ B ⊕ M ) . where • B = Ann R ( M ) ∩ Ann R ( M ) , • B = { (0 , m , ∈ S/ (0 , m , . (0 , m ′ ,
0) = 0 for m ′ ∈ M } . Proof.
We shall present the proof within the framework of the category G ⋊ n A ,One also can remark that S as an S − module is represented by the object T ( R ) .Under the category isomorphism in theorem 5.1, T ( R ) is represented as object of G ⋊ n A by T op ( R ) = ( R ⊕ M ⊕ M , κ = ( κ , κ )) , with κ : R ⊕ M ⊕ M / / G R ⊕ G M ⊕ G M ,κ : R ⊕ M ⊕ M / / G R ⊕ G M ⊕ G M . The morphisms κ and κ are of matrix form of order 3, they are defined asfollows: • κ , : R / / G M and κ , : R / / G M , • κ , : M / / G M and zero otherwise.An observation on the following maps R ⊕ M ⊕ M κ / / G R ⊕ G M ⊕ G M G κ / / G G R ⊕ G G M ⊕ G G M ,R ⊕ M ⊕ M κ / / G R ⊕ G M ⊕ G M G κ / / G G R ⊕ G G M ⊕ G G M , yields the following formulas, Ker ( κ ⊕ κ ) = B ⊕ B ⊕ M . ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.13 Corollary 5.8. If B ⊕ B ⊕ M have a finite injective dimension then S := R ⋉ n M is n − Gorenstein ring.
Lemma 5.9. [4, (Theorem 3.8)]
Let ( X, f ) represent a left S − module. The fol-lowing assertions are equivalent: • ( X, f ) ∼ = lim −→ T ( P i ) where P i ∈ A , for i ∈ I . • C ( X, f ) ∼ = lim −→ P i and the following sequence is exact: F i F i X F i f i / / F i X f i / / X / / Coker ( f i ) / / Proposition 5.10.
We assume that S is n-Gorenstein S − module and M i ⊗ R − are quasi-perfect, for all i ∈ I . Let ( X, f ) represent a left S − module. ( X, f ) isGorenstein flat if and only if Coker ( f ) is Gorenstein flat and F i F i X F i f i / / F i X f i / / X / / Coker ( f i ) / / is exact, for every i ∈ I . Proof.
From [9, Theorem 10.3.8] we have that ( X, f ) ∼ = lim −→ T ( P i ) where P i arefinitely generated Gorenstein projective. The proposition 3.2 in [4] ensure that T ( P i ) still finitely generated in R ⋉ n M . By lemma 5.9, we get the wanted result. Triangular matrix ring.
Let R := ( R i ) i =1 be a family of rings, ( M , M ) a couple of modules such that M is an ( R j , R i ) ≤ i We suppose that M i ⊗ − are quasi-perfect for i ∈ { , } . Let V = XYZ be left T − module. The following assertions are equivalent: i) V is Gorenstein projective left T − module, ii) X , Coker ( f ) and Coker ( f ) are Gorenstein projective left module re-spectively over R , R and R . For i ∈ { , } , f i are injective and f issurjective. Recollement We prove in the following the existence of a left recollement of GP ( A ⋉ n F ) the stable category of GP ( A ⋉ n F ) . Then we show that it can be filled into arecollement of triangulated categories. Definition 6.1. [21] A recollement denoted by ( A, B, C ) of abelian categories is a diagram A i ∗ / / B i ! { { i ∗ a a j ∗ / / C j ! a a j ∗ { { of abelian categories and additive functors such that (1) ( i ∗ , i ∗ ) , ( i ∗ , i ! ) , ( j ! , j ∗ ) and ( j ∗ , j ∗ ) are adjoints pairs, (2) i ∗ , j ! are fully faithful, (3) Imi ∗ = Kerj ∗ . Remark 4. The stable category GP ( A ) of GP ( A ) is the triangulated categorywhere GP ( A ) has the same objects as GP ( A ) and Hom GP ( A ) ( X, Y ) := Hom A ( X, Y ) /Hom ( X, P ( A ) , Y) ,with Hom ( X, P ( A ) , Y) := { f ∈ Hom ( X, Y ) such that f factors through a projective object } ( see [22, p 74] ) . Definition 6.2 ([2]) . Let A , B and C be triangulated categories. The diagram ofexact functors (6.1) A i ∗ / / B i ! { { i ∗ a a j ∗ / / C j ! a a j ∗ { { is a recollement of B relative to A and C , if the following four conditions aresatisfied. R1) ( i ∗ , i ∗ ) , ( i ∗ , i ! ) , ( j ! , j ∗ ) and ( j ∗ , j ∗ ) are adjoints pairs, R2) i ∗ , j ! are fully faithful, R3) j ∗ i ∗ = 0 , R4) For each object X ∈ C , the counits and units give rise to the followingdistinguished triangles j ! j ∗ ( X ) ǫ X / / X η X / / i ∗ i ∗ ( X ) / / j ! j ∗ ( X )[1] ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.15 i ∗ i ! ( X ) ω X / / X ζ X / / j ∗ j ∗ ( X ) / / i ∗ i ! ( X )[1] where [1] is the shift functor.The following lemma makes the construction of recollements easier, and uni-fies the definitions of a recollement of an abelian category defined in [13] and oftriangulated categories. For more details we refer to [22, Theorem 3.2] . Lemma 6.3. Let (6.1) be a diagram of triangulated categories. Then the followingstatements are equivalent. (1) The diagram (6.1) is a recollement. (2) The condition ( R , ( R and Imi ∗ = Kerj ∗ are satisfied. (3) The condition ( R , ( R and Imj ! = Keri ∗ are satisfied. (4) The condition ( R , ( R and Imj ∗ = Keri ! are satisfied.Let D ( A ⋉ n F ) be the subcategory of A ⋉ n F such that objects are of the form ( X ⊕ X , f ) with Imf ⊆ X and a morphism φ : ( X ⊕ X , f ) / / ( Y ⊕ Y , g ) is given by (cid:18) a a b (cid:19) . From the diagram (2.1) mentioned in section 2 we candeduce the following diagram of adjoint pairs A Z D ( A ⋉ n F ) C x x U A T v v with Z ( X ) = (0 ⊕ X, , U ( X ⊕ X , f ) = X .We remark from theroem (3.12) that objects in GP ( A ⋉ n F ) have the above form,also for X in A , Z ( X ) is in D ( A ⋉ n F ) . Therefore if we consider the subcategory GP ( A ) of A then the diagram associated to D ( A ⋉ n F ) induces a diagram of GP ( A ⋉ n F ) and we have the following result of a weaker version of a recollementcalled left recollement. Definition 6.4. A left recollement of A relative to B and C is a diagram of exactfunctors consisting of the upper two rows in the diagram 6.1 satisfying all theconditions which involve only the function i ∗ , i ∗ , j ! , j ∗ . Proposition 6.5. We suppose that ( F i ) i ∈ I are all quasi-perfect, then we have aleft recollement (6.2) GP ( A ) Z > > GP ( A ⋉ n F ) C (cid:127) (cid:127) U > > GP ( A ) T (cid:127) (cid:127) Proof. In this proof we fallow similar steps as in [20, Theorem 4.6] and [22,Theorem 3.3] . By theorem (3.12) the following T , Z , U are fully faithful. Since U and Z are exact functor and ( T, U ) , ( C, Z ) and adjoint pairs it fallows from [16, Lemma 8.3] that T and C are exact. We have KerU = { ( X ⊕ X , f ) ∈GP ( A ⋉ n F ) /X is projective } . Let ( X ⊕ X , f ) ∈ KerU then f is the zeromorphism according to the equivalence relation, hence ( X ⊕ X , f ) = Z ( X ⊕ X ) ∈ ImZ . By construction ImZ ⊂ KerU . By lemma (6.3) we have that thediagram (6.2) is a left recollement. Now we prove that this left rocollement can be completed into a recollment oftriangulated categories. First we give a lemma which the proof is straitforwardfrom the definition(3.2) and it is identical to the one in [20, Lemma 4.7] . Lemma 6.6. Let ( F i ) i ∈ I be quasi-perfect and preserve projective object, if A isGorenstein category then F i ’s preserve (Strongly)Gorenstein projective objects Theorem 6.7. Let ( F i ) i ∈ I be quasi-perfect and preserve projective object and A is Gorenstein category then there exist a recollement as follows (6.3) GP ( A ) Z / / GP ( A ⋉ n F ) C (cid:127) (cid:127) N ` ` U / / GP ( A ) L ` ` T (cid:127) (cid:127) Where N (( X ⊕ X , g )) = X and L ( X ) = ( X ⊕ ni =1 P i , f ) where f is of matrixform defined by a morphism F i X / / P i and zero otherwise ,for i ∈ I , with ( P i ) ni =1 a familly of projective objects in A . Proof. By theorem (6.5) we have the upper part of diagram (6.3) so it is left toprove that N and L are well defined functor verifying the condition of a recolement.We start by the functor N , if a morphism ( X ⊕ X , f ) / / ( Y ⊕ Y , g ) factorsthrough T ( Q ) with Q is projective in A , by construction we have X / / Y factors throught ⊕ i ∈ I F i Q which is also projective since F i ’s preserve projectiveobjects. Also from theorem 3.12 we have that X = ⊕ i ∈ I F i ( Coker ( f )) by lemma6.6it is in GP ( A ) . Hence N is a well defined functor. The adjunction between Z and N is clear from the fact that Hom GP ( A ⋉ n F ) (0 ⊕ X, , ( Y ⊕ Y , g ) ∼ = Hom GP ( A ) ( X, Y ) . Now we prove the claim that L is a fully faithful functor.Let f : X / / X ′ be in GP ( A ) , by lemma (6.6) F i X , F i X ′ are in GP ( A ) , foreach i ∈ I , also there exist a projective objects a family of ( P i ) i ∈ I and ( P ′ i ) i ∈ I suchthat the following sequences are exact / / F i X ϕ i / / P i π i / / Cokerϕ i / / , / / F i X ′ ϕ ′ i / / P ′ i π ′ i / / Cokerϕ ′ i / / , where Cokerϕ i and Cokerϕ ′ i are in GP ( A ) , since Ext ( Cokerϕ, P ′ i ) = 0 we havethe following commutative diagram, / / F i X F i f (cid:15) (cid:15) ϕ i / / P ih i (cid:15) (cid:15) / / Cokerϕ i (cid:15) (cid:15) / / / / F i X ′ ϕ ′ i / / P ′ i / / Cokerϕ ′ i / / (6.4) If there exist another f ′ : X / / X ′ such that the diagram (6.4) is commutative,thus f − f ′ factors through Cokerϕ i . Moreover since Cokerϕ i ∈ GP ( A ) thereexist a projective object ¯ P and a monomorphism ε : Cokerϕ i / / ¯ P such that ORENSTEIN OBJECTS IN THE n -TRIVIAL EXTENSIONS OF ABELIAN CATEGORIES.17 the following diagram is commutative in the inner triangle P i h i − h ′ i / / (cid:15) (cid:15) P ′ i Cokerϕ i / / ; ; ✇✇✇✇✇✇✇✇✇✇ ¯ P since Ext A ( Cokerϕ, P ′ i ) = 0 , there exist a morphism ¯ P / / P ′ i completing theabove diagram to a commutative square, therefore h = h ′ in GP ( A ) . We point outthat taking f = id X proves that considering the object X ⊕ ni =1 P i is independentto the choice of P i ’s. Now we assume that f factor trough a projective object Q in A i.e f = f ◦ f . We have that L ( f ) is given by X ⊕ ni =1 P i f 00 ( h i ) i ! / / X ′ ⊕ ni =1 P ′ i . Proving that L ( f ) factors trough a certain projective object is equivalent to com-pleting to following diagram / / F i X ϕ i / / " " (cid:15) (cid:15) P i } } (cid:15) (cid:15) π i / / Cokerϕ i { { / / (cid:15) (cid:15) F i Q | | ¯ P i (cid:127) (cid:127) _ _ / / F i X ′ ϕ ′ i / / P ′ i π ′ / / Cokerϕ i / / Such that all the diagram are commutative where ¯ P i projective, for i ∈ I . We startwith the first square. We have f factors trough Q then F i Q is projective and alsoinjective ( A is Gorentein category), thus there exist a morphism α : P i / / F i Q such that F i X (cid:15) (cid:15) F i f / / F i QP i α < < ①①①①①①①①① Now lets factorize the second square, g ◦ ϕ i = ϕ ′ i ◦ F i f = ϕ ′ i ◦ F i f ◦ F i f = ϕ ′ i ◦ F i f ◦ α ◦ ϕ i . We have then ( g − ϕ i ◦ F i f ◦ α ) ◦ ϕ i = 0 , by universal property of Cokernels thereexist a morphism ¯ g i : Cokerϕ i / / P ′ i such that g − ϕ i ◦ F i f ◦ α = ¯ g i ◦ π . 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