2-recollements of singualrity categories and Gorenstein defect categories over triangular matrix algebras
aa r X i v : . [ m a t h . R T ] A ug ∗† Huanhuan Li a , Dandan Yang a , Yuefei Zheng b and Jiangsheng Hu c ‡ a School of Mathematics and Statistics, Xidian University, Xi’an 710071, Shaanxi Province, China b College of Science, Northwest A & F University, Yangling 712100, Shaanxi Province, China c School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, Jiangsu Province, ChinaEmail addresses: [email protected], [email protected], [email protected] and [email protected]
Abstract
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with its corner algebras A and B Ar-tinian and A M B an A - B -bimodule. The 2-recollement structures for singularity categories andGorenstein defect categories over T are studied. Under mild assumptions, we provide necessaryand sufficient conditions for the existences of 2-recollements of singularity categories and Goren-stein defect categories over T relative to those of A and B . Parts of our results strengthen andunify the corresponding work in [27, 28, 34]. The singularity category was introduced by Buchweitz, known back then as the stable derived cat-egory, in his famous unpublished paper [9]. As an initial purpose, Buchweitz used this categoryto study the stable homological algebra and Tate cohomology for certain rings. In the setting ofalgebraic geometry, this category was reconsidered by Orlov [30] and turned out to have a closedrelation with the “Homological Mirror Symmetry Conjecture” due to Kontsevich. Recall that,for a given algebra R , the singularity category D sg ( R ) of R is defined to be the Verdier quotient D sg ( R ) := D b (mod R ) / K b (proj R ), where D b (mod R ) is the bounded derived category of finitely gen-erated R -modules and K b (proj R ) is the bounded homotopy category of finitely generated projective R -modules (i.e., the subcategory of perfect complexes). It measures the “regularity” of R in sensethat D sg ( R ) = 0 if and only if R is of finite global dimension. By the fundamental result in [9],the singularity category contains Gproj R (the stable category of finitely generated Gorenstein pro-jective R -modules) as a triangulated subcategory. This means there exists a fully faithful trianglefunctor F : Gproj R → D sg ( R ); besides, F is a triangle-equivalence provided that R is Goren-stein [9,19]. Motivated by this, Bergh, Jørgensen and Oppermann [8] introduced the Verdier quotient D def ( R ) := D sg ( R ) / Im F , and they called it the Gorenstein defect category of R . This categorymeasures how far the algebra R is from being Gorenstein. More precisely, R is Gorenstein if and onlyif D def ( R ) is trivial. Recently, singularity categories and related topics have been studied by manyauthors, see for example [13, 24–28, 32, 34, 36].Recollements of triangulated categories and abelian categories arise constantly in algebraic ge-ometry and representation theory [1, 5, 10–12, 15, 18, 29, 31, 32]. Roughly speaking, a recollement is a ∗ : 18E30, 18E35, 18G20. † Keywords : recollement; 2-recollement; singularity category; Gorenstein defect category; triangular matrix algebra. ‡ Corresponding author n steps downwards, then the diagram involving is called an n -recollement [33].While the diagram involving is called a ladder [1, 6], if the recollement could be extended upwardsand downwards. Generous evidences indicate that a recollement behaves better when it admits someextra adjoint functors, see [1, 6, 33, 35] and references therein for instance.Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with its corner algebras A and B Artinian and A M B an A - B -bimodule. The study of singularity theory over T by recollements has been consideredby many people. For instance, Zhang characterized in [34] the class of Gorenstein projective T -modules. As an application, he showed that if T is a Gorenstein algebra and A M is projective,then there exists a recollement of Gproj T relative to Gproj A and Gproj B . Later on, Liu-Lu [27]and Lu [28] generalized this to consider the singularity categories and Gorenstein defect categories,respectively. More precisely, they provided sufficient conditions for the existence of a recollement D sg ( T ) (resp. D def ( T )) relative to D sg ( A ) (resp. D def ( A )) and D sg ( B ) (resp. D def ( B )). However, theresults mentioned about provided only sufficient conditions for the existences of certain recollements.So we wonder whether or not we can get necessary and sufficient conditions for the existences of suchrecollements. Besides, the recollement structures over the triangular matrix algebra might be enrichedin some suitable settings. For example, it was shown in [35] that if A , B and T are finite-dimensionalGorenstein algebras, then there exists a unbounded ladder of period 1 for the stable categories ofGorenstein projective modules (and hence for the singularity categories). Meanwhile, we note thatthe recollements under consideration in [27, 28, 34] are initially from the following 2-recollement ofmodule categories: mod A Hom A ( e A T, − ) A A i eA * * mod T Hom T ( B, − ) @ @ S eA j j A ⊗ T − (cid:0) (cid:0) S eB + + mod B, i eB j j T e B ⊗ B − (cid:0) (cid:0) (3 . T are expected. In this present paper, we aim to solve these questions. More precisely,we get the following main results. Theorem 1.1.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B an A - B -bimodule.Assume that pd A M < ∞ , pd M B < ∞ and M ∈ ⊥ A . Then we have the following 2-recollement of ingularity categories: D sg ( A ) R Hom A ( e A T, − ) @ @ D b ( i eA ) + + D sg ( T ) R Hom T ( B, − ) @ @ D b ( S eA ) k k A ⊗ L T − (cid:0) (cid:0) D b ( S eB ) + + D sg ( B ) D b ( i eB ) k k T e B ⊗ L B − (cid:0) (cid:0) (1 . if and only if pd B Hom A ( M, A ) < ∞ , where all these functors are initially from (3.1) (see Propositions3.3 and 3.5 for the detailed descriptions). Recall from [34] that A M B is compatible if M ⊗ B − carries every acyclic complex of projective B -modules to acyclic A -complex and M ∈ (Gproj A ) ⊥ . We call an A - B -bimodule A M B left Gorensteinsingular if Gpd B Hom A ( M, F ) < ∞ for any F ∈ Gproj A ; while we call A M B right Gorensteinsingular if Gpd A M ⊗ B G < ∞ for any G ∈ Gproj B . A M B is said to be Gorenstein singular if it isboth left and right Gorenstein singular. We have the following equivalent characterizations for theexistence of a 2-recollement of Gorenstein defect categories.
Theorem 1.2.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B compatible. Assumethat pd A M < ∞ , M ∈ ⊥ Gproj A and pd M B < ∞ . Then we have the following 2-recollement ofGorenstein defect categories: D def ( A ) ^R Hom A ( e A T, − ) ? ? ^D b ( i eA ) + + D def ( T ) ^R Hom T ( B, − ) ? ? ^D b ( S eA ) k k ^ A ⊗ L T − (cid:127) (cid:127) ^D b ( S eB ) + + D def ( B ) ^D b ( i eB ) k k ^ T e B ⊗ L B − (cid:127) (cid:127) (1 . if and only if M is Gorenstein singular, where all these functors are initially from (3.1) (see Propo-sitions 4.3 and 4.7 for the detailed descriptions). In the procedure of proving Theorems 1.1 and 1.2, we obtain equivalent characterizations for theexistences of recollements of singularity categories and Gorenstein defect categories over T , whichgeneralize the corresponding results in [27, 28] (see Propositions 3.3 and 4.3). As a consequence, therecollements (see Corollaries 4.4 and 4.8) and 2-recollements (see Corollary 4.9) of stable categoriesof Gorenstein projective modules over T are obtained accordingly, where Corollary 4.4 generalizesZhang’s result to a more general case (compare [34, Theorem 3.5]).The contents of this paper are outlined as follows. In Section 2, we fix some notations and recallsome basic definitions and facts that are needed in the later proofs. In Section 3, we consider the2-recollements of singularity categories over the triangular matrix algebra and prove Theorem 1.1.In Section 4, the 2-recollements for Gorenstein defect categories and stable categories of Gorensteinprojective modules over the triangular matrix algebra are studied, including the proof of Theorem1.2. 3 Preliminaries
In this section, we briefly recall some basic definitions, facts and notations needed in the sequel.
Throughout, all algebras are Artin algebras over a fixed commutative Artinian ring and all modulesare finitely generated. For a given algebra R , denote by mod R the category of left R -modules; right R -modules are viewed as left R op -modules, where R op is the opposite algebra of R . We use proj R todenote the subcategory of mod R consisting of projective modules. The ∗ -bounded derived category ofmod R and homotopy category of proj R will be denoted by D ∗ (mod R ) and K ∗ (proj R ) respectively,where ∗ ∈ { blank, + , − , b } .Usually, we use R M (resp. M R ) to denote a left (resp. right) R -module M , and the projectivedimension of R M (resp. M R ) will be denoted by pd R M (resp. pd M R ). For a subclass X ofmod R . Denote by X ⊥ (resp. ⊥ X ) the subcategory consisting of modules M ∈ mod R such thatExt nR ( X, M ) = 0 (resp. Ext nR ( M, X ) = 0) for any X ∈ X and n ≥ X • = · · · → X − d − −−→ X d −→ X → · · · be a complex in mod R . For any integer n , we set Z n ( X • ) = Ker d n , B n ( X • ) = Im d n − and H n ( X • ) = Z n ( X • ) /B n ( X • ). X • is called acyclic (or exact) if H n ( X • ) = 0 for any n ∈ Z . Recall from [2, 4, 20] that an acyclic complex X • is called totally acyclic if each X i ∈ proj R andHom R ( X • , R ) is acyclic. A module M ∈ mod R is Gorenstein projective if there exists some totallyacyclic complex X • such that M ∼ = Z ( X • ). Denote by Gproj R the subcategory of mod R consistingof Gorenstein projective modules. Given a module M ∈ mod R , the Gorenstein projective dimension
Gpd R M of M is defined to be Gpd R M = inf { n : there exists an exact sequence 0 → G n → · · · → G → G → M →
0, where each G i ∈ Gproj R } . Definition 2.1. (compare [25])
A complex X • ∈ D b (mod R ) is said to be Gorenstein perfect if X • is isomorphic to some bounded complex consisting of Gorenstein projective modules in D b (mod R ).Denote by Gperf( R ) the subcategory of D b (mod R ) consisting of Gorenstein perfect complexes. Remark 2.2.
A Gorenstein perfect complex is called a complex with finite Gorenstein projectivedimension in [25]. Here we use the name of “Gorenstein perfect” because we find this kind ofcomplexes reflects as the Gorenstein version of perfect complexes. For instance, it is not hard tosee an R -module M is Gorenstein perfect if and only if Gpd R M < ∞ . Besides, Gperf( R ) is thesmallest thick subcategory of D b (mod R ) containing Gproj R . For more details, we refer the readerto appendix in [25]. Lemma 2.3. (see [25, Proposition A.4]) Let X • ∈ D b (mod R ) . If each X i is of finite Gorensteinprojective dimension as an R -module, then X • ∈ Gperf( R ) . Recall that the singularity category D sg ( R ) of R is defined to be the verdier quotient D sg ( R ) := D b (mod R ) / K b (proj R ) , K b (proj R ) (up to isomorphisms) are the so-called perfect complexes . This cat-egory was first introduced by Buchweitz [9], and later reconsidered by a lot of authors [7, 19, 30].It is well-known that Gproj R is a Frobenius category, and hence its stable category Gproj R is atriangulated category [17]. By a fundamental result of Buchweitz, there exists a fully faithful tri-angle functor F : Gproj R → D sg ( R ), which sends every Gorenstein projective module to the stalkcomplex concentrated in degree zero. Furthermore, F is a triangle-equivalence provided that R isGorenstein (that is, the left and right self-injective dimensions of R are finite). Consequently, Im F is a triangulated subcategory of D sg ( R ). Following [8], the Verdier quotient D def ( R ) := D sg ( R ) / Im F is called the Gorenstein defect category of R . Lemma 2.4. (see [25, Theorem A.5]) We have the following exact commutative diagram: / / Gproj R F / / (cid:15) (cid:15) D sg ( R ) / / D def ( R ) (cid:15) (cid:15) / / / / Gperf( R ) / K b (proj R ) / / D sg ( R ) / / D b (mod R ) / Gperf( R ) / / with all vertical functors triangle-equivalences. Let T , T ′ and T ′′ be triangulated categories. A recollement [5] of T relative to T ′ and T ′′ is adiagram of triangulated categories and triangle functors T ′ i ∗ / / T i ! a a i ∗ z z j ∗ / / T ′′ j ∗ a a j ! | | (2 . i ∗ , i ∗ ), ( i ∗ , i ! ), ( j ! , j ∗ ) and ( j ∗ , j ∗ ) are adjoint pairs;(R2) i ∗ , j ! and j ∗ are fully faithful;(R3) Im i ∗ = Ker j ∗ . If T , T ′ and T ′′ in diagram (2.1) are abelian categories, and the six functors involving are additivefunctors. Then we call diagram (2.1) a recollement of abelian categories , see [18, 31, 32] for details. Definition 2.5. (see [33]) Let T , T ′ and T ′′ be triangulated categories (resp. abelian categories). A of T relative to T ′ and T ′′ is given by a diagram T ′ i ? H H i ∗ ) ) T j ? G G i ! j j i ∗ (cid:7) (cid:7) j ∗ * * T ′′ j ∗ i i j ! (cid:7) (cid:7) (2 . Lemma 2.6. (compare [27]) Let (2.1) be a recollement of triangulated categories. Assume that N , N ′ and N ′′ are thick subcategories of T , T ′ and T ′′ respectively. The following statements are equivalent:(1) (2.1) restricts to the following recollement: N ′ i ∗ / / N i ! a a i ∗ z z j ∗ / / N ′′ . j ∗ b b j ! z z (2 . (2) (2.1) induces the following recollement: T ′ / N ′ i ∗ / / T / N i ! g g i ∗ z z j ∗ / / T ′′ / N ′′ , j ∗ h h j ! y y (2 . where these six functors are induced by those in (2.1).(3) i ∗ ( N ) ⊆ N ′ , i ∗ ( N ′ ) ⊆ N , j ∗ ( N ) ⊆ N ′′ and j ∗ ( N ′′ ) ⊆ N .Proof. (1) ⇒ (3) and (2) ⇒ (3) are trivial. Now assume that conditions in (3) are satisfied, we claim i ∗ ( N ) = N ′ and j ∗ ( N ) = N ′′ . To do this, let X ′ ∈ N ′ . Put X = i ∗ X ′ , it follows that X ∈ N .Hence X ′ ∼ = i ∗ i ∗ ( X ′ ) ∼ = i ∗ ( X ) and then X ′ ∈ i ∗ ( N ). Thus i ∗ ( N ) = N ′ as desired. Similarly, onecould obtain j ∗ ( N ) = N ′′ . Therefore, the claim follows. Now we infer (3) ⇒ (1) and (3) ⇒ (2)from [27, Remark 2.4 and Proposition 2.5].Let R be an Artin algebra and e ∈ R an idempotent. Recall from [16, Chapter 6] that the Schurfunctor S e : mod R → mod eRe associative to e is defined to be S e ( X ) = eX for any X ∈ mod R .Clearly, S e admits a fully faithful left adjoint Re ⊗ eRe − : mod eRe → mod R and a fully faithful rightadjoint Hom eRe ( eR, − ) : mod eRe → mod R. Denote by i − e : mod R/ReR → mod R the canonicalinclusion functor induced by the natural homomorphism R → R/ReR . We have the following
Example 2.7.
Let R be an Artin algebra and e ∈ R an idempotent.(1) (see [31, 32]) We have the following recollement of module categories: mod R/ReR i − e / / mod R Hom R ( R/ReR, − ) k k R/ReR ⊗ R − v v S e / / mod eRe. Hom eRe ( eR, − ) j j Re ⊗ eRe − w w (2 . (2) (see [15, 29]) We have the following recollement of bounded derived categories: D b (mod R/ReR ) D b ( i − e ) / / D b (mod R ) R Hom R ( R/ReR, − ) l l R/ReR ⊗ L R − u u D b ( S e ) / / D b (mod eRe ) R Hom eRe ( eR, − ) l l Re ⊗ L eRe − u u (2 . such that all functors are the derived versions of those in (2.5) if and only if the follow-ing conditions are satisfied: (i) Ext nR ( R/ReR, R/ReR ) = 0 for every integer n ≥ ; (ii) pd R R/ReR < ∞ ; (iii) pd R/ReR R < ∞ . In this section, T = (cid:18) A M B (cid:19) is a triangular matrix algebra with its corner algebras A and B Artinian and A M B an A - B -bimodule. Liu-Lu [27] and Zhang [34] gave sufficient conditions for theexistence of a recollement of D sg ( T ) relative to D sg ( A ) and D sg ( B ). In this section, we provide nec-essary and sufficient conditions for the existence of such recollement. Besides, we also give equivalentcharacterizations when there is a 2-recollement of D sg ( T ) relative to D sg ( A ) and D sg ( B ). We firstrecall some basic definitions needed in the sequel.Recall that a left T -module is identified with a triple (cid:18) XY (cid:19) φ , where X ∈ mod A , Y ∈ mod B and φ : M ⊗ B Y → X ia an A -morphism. If there is no possible confusion, we shall omit the morphism φ and write (cid:18) XY (cid:19) for short. Analogously, a left T -module (cid:18) XY (cid:19) φ is also identified with the triple (cid:18) XY (cid:19) e φ , where e φ : Y → Hom A ( M, X ) is a B -morphism defined by e φ ( y )( m ) = φ ( m ⊗ y ) for any m ∈ M and y ∈ Y .A T -morphism (cid:18) XY (cid:19) φ → (cid:18) X ′ Y ′ (cid:19) φ ′ will be identified with a pair (cid:18) fg (cid:19) , where f ∈ Hom A ( X, X ′ )and g ∈ Hom B ( Y, Y ′ ), such that the following diagram M ⊗ B Y φ / / ⊗ g (cid:15) (cid:15) X f (cid:15) (cid:15) M ⊗ B Y ′ φ ′ / / X ′ is commutative.A sequence 0 → (cid:18) X Y (cid:19) φ f g −−−−−−→ (cid:18) X Y (cid:19) φ f g −−−−−−→ (cid:18) X Y (cid:19) φ → T is exact ifand only if 0 → X f −→ X f −→ X → → Y g −→ Y g −→ Y → A and mod B ,respectively. A T -module (cid:18) XY (cid:19) φ is projective if and only if Y ∈ proj B and φ : M ⊗ B Y → X isan injective A -morphism with Coker φ ∈ proj A . We refer the reader to [3, Section III.2] for moredetails.Let e A = (cid:18) (cid:19) and e B = (cid:18) (cid:19) be idempotents of T . It is known that A ∼ = e A T e A ∼ = T /T e B T and B ∼ = e B T e B ∼ = T /T e A T as algebras. As a consequence of Example 2.7 (1), we have thefollowing observation. 7 emma 3.1. We have the following 2-recollement of module categories: mod A Hom A ( e A T, − ) A A i eA * * mod T Hom T ( B, − ) @ @ S eA j j A ⊗ T − (cid:0) (cid:0) S eB + + mod B, i eB j j T e B ⊗ B − (cid:0) (cid:0) (3 . where A ⊗ T (cid:18) XY (cid:19) φ ∼ = Coker φ , i e A ( X ) ∼ = (cid:18) X (cid:19) and Hom A ( e A T, X ) ∼ = (cid:18) X Hom A ( M, X ) (cid:19) e id ;while T e B ⊗ B Y ∼ = (cid:18) M ⊗ B YY (cid:19) , i e B ( Y ) ∼ = (cid:18) Y (cid:19) and Hom T ( B, (cid:18) XY (cid:19) φ ) ∼ = Ker e φ .Proof. To take R = T and e = e B (resp. e = e A ) as in Example 2.7 (1), we have the following tworecollements of module categories:mod A i eA / / mod T S eA j j A ⊗ T − w w S eB / / mod B, i eB j j T e B ⊗ B − w w mod B i eB / / mod T Hom T ( B, − ) j j S eB w w S eA / / mod A. Hom A ( e A T, − ) j j i eA w w To glue them together, we get the diagram (3.1). We will use the adjoint functors to get the expressionsof the functors in (3.1). Indeed, let X ∈ mod A and (cid:18) X ′ Y ′ (cid:19) φ ′ ∈ mod T . We haveHom T ( (cid:18) X ′ Y ′ (cid:19) φ ′ , Hom A ( e A T, X )) ∼ = Hom A ( S e A ( (cid:18) X ′ Y ′ (cid:19) φ ′ ) , X ) ∼ = Hom A ( X ′ , X ) ∼ = Hom T ( (cid:18) X ′ Y ′ (cid:19) e φ ′ , (cid:18) X Hom A ( M, X ) (cid:19) e id ).By Yoneda Lemma, we get Hom A ( e A T, X ) ∼ = (cid:18) X Hom A ( M, X ) (cid:19) e id . Similarly, one could obtain theexpressions of another functors.Since the module category (over an arbitrary algebra) is a subcategory of its bounded derivedcategory, it is natural to ask whether and when the 2-recollement (3.1) lifts to a 2-recollement oftheir bounded derived categories. We provide the following necessary and sufficient conditions forthis question. Lemma 3.2.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B an A - B -bimodule.
1) (compare [12]) We have the following recollement of bounded derived categories: D b (mod A ) D b ( i eA ) / / D b (mod T ) D b ( S eA ) k k A ⊗ L T − v v D b ( S eB ) / / D b (mod B ) , D b ( i eB ) l l T e B ⊗ L B − u u (3 . such that these six functors are the derived versions of those in (3.1) if and only if pd M B < ∞ .(2) We have the following recollement of bounded derived categories: D b (mod B ) D b ( i eB ) / / D b (mod T ) R Hom T ( B, − ) k k D b ( S eB ) v v D b ( S eA ) / / D b (mod A ) , R Hom A ( e A T, − ) k k D b ( i eA ) u u (3 . such that these six functors are the derived versions of those in (3.1) if and only if pd A M < ∞ .(3) We have the following 2-recollement of bounded derived categories: D b (mod A ) R Hom A ( e A T, − ) > > D b ( i eA ) , , D b (mod T ) R Hom T ( B, − ) > > D b ( S eA ) l l A ⊗ L T − } } D b ( S eB ) , , D b (mod B ) D b ( i eB ) l l T e B ⊗ L B − } } (3 . if and only if pd A M < ∞ and pd M B < ∞ .Proof. We only prove (2), (1) could be obtained by a similar argument (see also [12, Theorem 2] forthe proof of the “if” part) and (3) is a consequence of (1) and (2). We proceed by letting R = T and e = e A as in Example 2.7 (2). Note that T B ∼ = (cid:18) B (cid:19) as left T -modules and B T ∼ = e B T as right T -modules. It follows that B T is projective and condition (iii) in Example 2.7 (2) follows. Notice thatExt nT ( B, B ) ∼ = Ext nT ( (cid:18) B (cid:19) , (cid:18) B (cid:19) ) ∼ = Ext nB ( B, B ) = 0 for any n ≥
1, where the last isomorphismcould be easily checked by choosing a projective resolution of (cid:18) B (cid:19) . So condition (i) in Example2.7 (2) follows. From [25, Lemma 2.4], we know pd T B < ∞ if and only if pd A M < ∞ . FollowingExample 2.7 (2), we have the recollement (3.3) if and only if pd A M < ∞ . Proof of Theorem 1.1.
The proof of Theorem 1.1 is divided into the following two parts,including Proposition 3.3 and Proposition 3.5.
Proposition 3.3.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B an A - B -bimodule.Assume that pd M B < ∞ . Then the following statements are equivalent.
1) We have the following recollement of perfect complexes K b (proj A ) D b ( i eA ) / / K b (proj T ) D b ( S eA ) k k A ⊗ L T − v v D b ( S eB ) / / K b (proj B ) D b ( i eB ) k k T e B ⊗ L B − v v , (3 . where these six functors are the restrictions of those in (3.2).(2) We have the following recollement of singularity categories D sg ( A ) D b ( i eA ) / / D sg ( T ) D sg ( S eA ) j j A ⊗ L T − w w D sg ( S eB ) / / D sg ( B ) D b ( i eB ) j j T e B ⊗ L B − w w , (3 . where these six functors are induced by those in (3.2).(3) pd A M < ∞ .Proof. Following Lemma 3.2 (1), we have the recollement (3.2) since pd M B < ∞ . (3) ⇒ (2) couldbe found in [27, Theorem 3.2] and the equivalence of (1) and (2) follows directly from Lemma 2.6.(1) ⇒ (3) Assume that we have the recollement (3.5), then D b ( i e B )( B ) ∼ = i e B ( B ) ∈ K b (proj T ).Since i e B ( B ) ∼ = (cid:18) B (cid:19) by Lemma 3.1, it follows that pd T (cid:18) B (cid:19) < ∞ . Now consider the followingexact sequence of T -modules: 0 → (cid:18) M (cid:19) → (cid:18) MB (cid:19) → (cid:18) B (cid:19) → . ( ex (cid:18) MB (cid:19) ∈ proj T and pd T (cid:18) B (cid:19) < ∞ , we have pd T (cid:18) M (cid:19) < ∞ . By [25, Lemma2.3], we get pd A M < ∞ . Remark 3.4.
Assume that pd M B < ∞ . Liu and Lu showed in [27] that if pd A M < ∞ then wehave the recollement (3.6) of singularity categories. Whereas, by the equivalence of (2) and (3) inProposition 3.3, we know that “pd A M < ∞ ” is also a necessary condition for the existence of therecollement (3.6).Similarly, we get the following Proposition 3.5.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B an A - B -bimodule.Assume that pd A M < ∞ and M ∈ ⊥ A . Then the following statements are equivalent.(1) We have the following recollement of perfect complexes K b (proj B ) D b ( i eB ) / / K b (proj T ) R Hom T ( B, − ) k k D b ( S eB ) v v D b ( S eA ) / / K b (proj A ) R Hom A ( e A T, − ) k k D b ( i eA ) v v , (3 . where these six functors are the restrictions of those in (3.3).
2) We have the following recollement of singularity categories D sg ( B ) D b ( i eB ) / / D sg ( T ) R Hom T ( B, − ) j j D b ( S eB ) w w D b ( S eA ) / / D sg ( A ) R Hom A ( e A T, − ) j j D b ( i eA ) w w , (3 . where these six functors are induced by those in (3.3).(3) pd B Hom A ( M, A ) < ∞ .Proof. Following Lemma 3.2 (2), we have the recollement (3.3) since pd A M < ∞ . In view of Lemma2.6, it suffices to show pd B Hom A ( M, A ) < ∞ if and only if the four functors D b ( S e B ), D b ( i e B ), D b ( S e A ) and R Hom A ( e A T, − ) preserve prefect complexes.Assume these four functors preserve prefect complexes. Then we have that R Hom A ( e A T, A ) ∈ K b (proj T ). Since M ∈ ⊥ A by assumption and e A T ∼ = A ⊕ M as A -modules, it follows thatExt nA ( e A T, A ) = 0 for any n ≥
1. Hence Hom A ( e A T, A ) ∼ = R Hom A ( e A T, A ) ∈ K b (proj T ) andthen pd T Hom A ( e A T, A ) < ∞ . Notice that Hom A ( e A T, A ) ∼ = (cid:18) A Hom A ( M, A ) (cid:19) e id by Lemma 3.1, weconsider the following exact sequence of T -modules:0 → (cid:18) A (cid:19) → (cid:18) A Hom A ( M, A ) (cid:19) e id → (cid:18) A ( M, A ) (cid:19) → . ( ex (cid:18) A (cid:19) is projective, we infer that pd T (cid:18) A ( M, A ) (cid:19) < ∞ since pd T (cid:18) A Hom A ( M, A ) (cid:19) e id < ∞ . Thus we get pd B Hom A ( M, A ) < ∞ from [25, Lemma 2.3].Conversely, Assume pd B Hom A ( M, A ) < ∞ . Since S e B ( T ) ∼ = B , one has D b ( S e B )( K b (proj T )) ⊆ K b (proj B ). By Lemma 3.1, we have that i e B ( B ) ∼ = (cid:18) B (cid:19) and S e A ( T ) ∼ = S e A ( (cid:18) A (cid:19) ⊕ (cid:18) MB (cid:19) ) ∼ = A ⊕ M . We infer that S e A ( T ) ∈ K b (proj A ) since pd A M < ∞ , and hence D b ( S e A )( K b (proj T )) ⊆ K b (proj A ). Notice that (cid:18) MB (cid:19) ∈ proj T and pd T (cid:18) M (cid:19) < ∞ from [25, Lemma 2.3], we inferpd T i e B ( B ) < ∞ from the exactness of the sequence ( ex i e B ( B ) ∈ K b (proj T )and therefore D b ( i e B )( K b (proj B )) ⊆ K b (proj T ).Finally, we will show R Hom A ( e A T, − ) preserves perfect complexes to complete the proof. Since M ∈ ⊥ A by assumption, it follows that Ext nA ( e A T, A ) = 0 for any n ≥
1. Thus R Hom A ( e A T, A ) ∼ =Hom A ( e A T, A ) ∼ = (cid:18) A Hom A ( M, A ) (cid:19) e id . As pd B Hom A ( M, A ) < ∞ , we infer pd T (cid:18) A ( M, A ) (cid:19) < ∞ from [25, Lemma 2.3]. Notice that (cid:18) A (cid:19) ∈ proj T , we conclude pd T (cid:18) A Hom A ( M, A ) (cid:19) e id < ∞ bythe exact sequence ( ex R Hom A ( e A T, A ) ∈ K b (proj T ) and then we have R Hom A ( e A T, − )preserves perfect complexes as desired. In this section, we take T = (cid:18) A M B (cid:19) the triangular matrix algebra as that in Section 3. As asuccessor, we will study 2-recollement of the Gorenstein defect category D def ( T ) relative to D def ( A )11nd D def ( B ).To prove Theorem 1.2, we need some preparations. Firstly, we should know the concrete form ofGorenstein projective modules over T . Lemma 4.1. (see [34, Theorem 1.4]) Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B compatible. Then (cid:18) XY (cid:19) φ ∈ Gproj T if and only if Y ∈ Gproj B and φ : M ⊗ B Y → X is aninjective A -morphism with Coker φ ∈ Gproj A . We also need the following fact.
Lemma 4.2. (see [26, Corollary 4.2]) Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B compatible. The following statements hold true.(1) Gpd T (cid:18) X (cid:19) = Gpd A X .(2) Assume that M is right Gorenstein singular. Then Gpd T (cid:18) Y (cid:19) < ∞ if and only if Gpd B Y < ∞ . Note that the 2-recollement (1.2) in Theorem 1.2 is consisted of two recollements. We will giveequivalent characterizations for the existence of each one.
Proposition 4.3. (compare [25, Theorem 1.2] and [28, Theorem 3.12]) Let T = (cid:18) A M B (cid:19) bea triangular matrix algebra with A M B compatible. Assume that pd M B < ∞ . Then the followingstatements are equivalent.(1) We have the following recollement of Gorenstein perfect complexes Gperf( A ) D b ( i eA ) / / Gperf( T ) D b ( S eA ) k k A ⊗ L T − v v D b ( S eB ) / / Gperf( B ) D b ( i eB ) k k T e B ⊗ L B − v v , (4 . where these six functors are the restrictions of those in (3.2).(2) We have the following recollement of Gorenstein defect categories D def ( A ) ^D b ( i eA ) / / D def ( T ) ^D b ( S eA ) k k ^ A ⊗ L T − w w ^D b ( S eB ) / / D def ( B ) ^D b ( i eB ) k k ^ T e B ⊗ L B − w w , (4 . where these six functors are induced by those in (3.2).(3) M is right Gorenstein singular. roof. Following Lemma 3.2 (1), we have the recollement (3.2) since pd M B < ∞ . The equivalenceof (1) and (2) follows from Lemmas 2.4 and 2.6, and (3) ⇒ (2) could be found in [25, Theorem 1.2].(1) ⇒ (3) Assume that we have the recollement (4.1). Take any Y ∈ Gproj B , we obtain D b ( i e B )( Y ) ∼ = i e B ( Y ) is Gorenstein perfect. Notice that i e B ( Y ) ∼ = (cid:18) Y (cid:19) , it follows that Gpd T (cid:18) Y (cid:19) < ∞ . Consider the following exact sequence of T -modules:0 → (cid:18) M ⊗ B Y (cid:19) → (cid:18) M ⊗ B YY (cid:19) → (cid:18) Y (cid:19) → . Since (cid:18) M ⊗ B YY (cid:19) ∈ Gproj T by Lemma 4.1 and Gpd T (cid:18) Y (cid:19) < ∞ , we get Gpd T (cid:18) M ⊗ B Y (cid:19) < ∞ . Hence we conclude Gpd A M ⊗ B Y < ∞ from Lemma 4.2, this means M is right Gorensteinsingular.Viewing the stable category of Gorenstein projective modules as a triangulated subcategory ofthe singularity category, we get the following Corollary 4.4.
Suppose that A M B has finite projective dimension both as a left A - and right B -module. Then we have the following recollement of stable categories of Gorenstein projective modules Gproj A / / Gproj T i i v v / / Gproj B j j v v (4 . such that all these functors are the restrictions of those in (3.6) if and only if M is right Gorensteinsingular.Proof. Since A M B has finite projective dimension both as a left A - and right B -module, it is nothard to see A M B is compatible. By Proposition 3.3, we get the recollement (3.6).For the “if” part, assume M is right Gorenstein singular. From Proposition 4.3, we get therecollement (4.2), which is also induced by the recollement (3.6) since their functors are initially from(3.2). Therefore, we have the recollement (4.3) from Lemma 2.6.Conversely, assume that we have the recollement (4.3). From Lemma 2.6, we have the recollement(4.2). Then we infer that M is right Gorenstein singular from Proposition 4.3. Remark 4.5.
The recollement (4.3) of stable categories of Gorenstein projective modules has beenconsidered by Zhang [34], where he proves that if T is Gorenstein and A M is projective then we havethe recollement (4.3) (see [34, Theorem 3.5]). Whereas, if T is Gorenstein and A M is projective, it isnot hard to see (combine [34, Theorem 2.2]) all the conditions in Corollary 4.4 are satisfied and M isright Gorenstein singular. Therefore, our result generalizes Zhang’s to a more general case. Besides,our proofs are quite different. Example 4.6.
Let k be a field and Q the following quiver:1 α ( ( α ′ h h γ O O β ( ( δ O O θ / / β ′ h h . Consider the k -algebra T = kQ/I , where I is generated by α ′ α , αα ′ , β ′ β , ββ ′ , θβ , αγ − δβ and α ′ δ − γβ ′ . Let e i be the idempotent corresponding to the vertex i and put e = e + e . Denote13y A = eT e and B = (1 − e ) T (1 − e ). It follows that T = (cid:18) A M B (cid:19) with M = eT (1 − e ). Itis easy to check A is self-injective, then D sg ( A ) ≃ Gproj A ≃ mod A and hence D def ( A ) vanishes.Since B is of radical square zero but not self-injective, we infer from [14] that B is CM-free (that isGproj B = proj B ). Hence we obtain Gproj B vanishes and D def ( B ) = D sg ( B ). Notice that A M and M B are projective, and M is right Gorenstein singular since A is self-injective. Following Propositions3.3, 4.3 and Corollary 4.4, we get the following recollement of singularity categories D sg ( A ) / / D sg ( T ) h h v v / / D sg ( B ) h h v v and triangle-equivalences D def ( T ) ≃ D def ( B ) = D sg ( B ) and Gproj T ≃ Gproj A ≃ D sg ( A ) ≃ mod A .Let R be an Artin algebra and X • a complex of R -modules. The length l ( X • ) of X • is definedto be the cardinal of the set { X i = 0 | i ∈ Z } . Let n ∈ Z , denote by X • > n the complex with the i thcomponent equal to X i whenever i > n and to 0 elsewhere. Proposition 4.7.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B compatible. Assumethat pd A M < ∞ and M ∈ ⊥ Gproj A . Then the following statements are equivalent.(1) We have the following recollement of Gorenstein perfect complexes Gperf( B ) D b ( i eB ) / / Gperf( T ) R Hom T ( B, − ) k k D b ( S eB ) v v D b ( S eA ) / / Gperf( A ) R Hom A ( e A T, − ) k k D b ( i eA ) v v , (4 . where these six functors are the restrictions of those in (3.3).(2) We have the following recollement of Gorenstein defect categories D def ( B ) ^D b ( i eB ) / / D def ( T ) ^R Hom T ( B, − ) k k ^D b ( S eB ) w w ^D b ( S eA ) / / D def ( A ) ^R Hom A ( e A T, − ) k k ^D b ( i eA ) w w , (4 . where these six functors are induced by those in (3.3).(3) M is Gorenstein singular.Proof. Following Lemma 3.2 (2), we have the recollement (3.3) since pd A M < ∞ . In view ofLemmas 2.4 and 2.6, it suffices to show M is Gorenstein singular if and only if the four functors D b ( S e B ), D b ( i e B ), D b ( S e A ) and R Hom A ( e A T, − ) preserve Gorenstein perfect complexes.Assume these four functors preserve Gorenstein perfect complexes. Take any Y ∈ Gproj B , fromLemma 4.1 we know (cid:18) M ⊗ B YY (cid:19) ∈ Gproj T and then it is Gorenstein perfect. It follows that D b ( S e A )( (cid:18) M ⊗ B YY (cid:19) ) ∼ = S e A ( (cid:18) M ⊗ B YY (cid:19) ) ∼ = M ⊗ B Y ∈ Gperf( A ) , A M ⊗ B Y < ∞ . Hence M is right Gorenstein singular. Notice that e A T ∼ = A ⊕ M ,we infer e A T ∈ ⊥ Gproj A since M ∈ ⊥ Gproj A . For any X ∈ Gproj A , it follows that R Hom A ( e A T, X ) ∼ = Hom A ( e A T, X ) ∼ = (cid:18) X Hom A ( M, X ) (cid:19) e id . Since R Hom A ( e A T, − ) preserves Gorenstein perfect complexes, we get Gpd T (cid:18) X Hom A ( M, X ) (cid:19) e id < ∞ . Consider the following exact sequence of T -modules:0 → (cid:18) X (cid:19) → (cid:18) X Hom A ( M, X ) (cid:19) e id → (cid:18) A ( M, X ) (cid:19) → . ( ex (cid:18) X (cid:19) ∈ Gproj T by Lemma 4.1, one has Gpd T (cid:18) A ( M, X ) (cid:19) < ∞ . By Lemma4.2 (2), we obtain Gpd B Hom A ( M, X ) < ∞ and then M is left Gorenstein singular. To sum up, weget that M is Gorenstein singular.Conversely, assume that M is Gorenstein singular. Since S e B preserves Gorenstein projectivemodules by Lemma 3.1, it is easy to check that D b ( S e B ) preserves Gorenstein perfect complexes. Forany Y ∈ Gproj B , by Lemma 3.1, we obtain i e B ( Y ) ∼ = (cid:18) Y (cid:19) . Since M is right Gorenstein singular,we infer Gpd B i e B ( Y ) < ∞ from Lemma 4.2 (2). Let Y • be a bounded complex of Gorenstein B -modules. Since i e B is exact, we have D b ( i e B )( Y • ) ∼ = i e B ( Y • ), it is a bounded complex with eachdegree being of finite Gorenstein projective dimension. It follows from Lemma 2.3 that D b ( i e B )( Y • ) ∈ Gperf( T ). Now for any (cid:18) FG (cid:19) φ ∈ Gproj T , by Lemma 3.1 we have S e A ( (cid:18) FG (cid:19) φ ) ∼ = F . FollowingLemma 4.1, we have the following exact sequence of A -modules0 → M ⊗ B G → F → Coker φ → G ∈ Gproj B and Coker φ ∈ Gproj A . Then Gpd A M ⊗ B G < ∞ since M is right Gorensteinsingular and hence Gpd A F < ∞ . Similarly as above, we conclude that D b ( S e A ) preserves Gorensteinperfect complexes.Finally, it remains to show R Hom A ( e A T, − ) preserves Gorenstein perfect complexes to completeour proof. To do this, let X • be a bounded complex of Gorenstein projective A -modules. We proceedby induction on length l ( X • ) of X • . If l ( X • ) = 1, we may suppose X • = X is the stalk complexconcentrated in degree 0. Since M ∈ ⊥ Gproj A , it is clear to see R Hom A ( e A T, X ) ∼ = Hom A ( e A T, X ) ∼ = (cid:18) X Hom A ( M, X ) (cid:19) e id . As X ∈ Gproj A , one has (cid:18) X (cid:19) ∈ Gproj T . Besides, we have Gpd B Hom A ( M, X ) < ∞ since M is left Gorenstein singular. From Lemma 4.2 (2), we know Gpd T (cid:18) A ( M, X ) (cid:19) < ∞ .Hence from the exactness of the sequence ( ex T (cid:18) X Hom A ( M, X ) (cid:19) e id < ∞ and then R Hom A ( e A T, X ) ∈ Gperf( T ).Now suppose l ( X • ) = n ≥ n . We mayassume X • = 0 → X → X → · · · → X n − → . It induces a triangle X [ − → X •≥ → X • → X D b (mod A ). Apply the functor R Hom A ( e A T, − ) to it, we get the following triangle R Hom A ( e A T, X )[ − → R Hom A ( e A T, X •≥ ) → R Hom A ( e A T, X • ) → R Hom A ( e A T, X )in D b (mod T ). By the induction hypothesis, we have both R Hom A ( e A T, X ) and R Hom A ( e A T, X •≥ )are Gorenstein perfect. Note that Gperf( T ) is a thick subcategory of D b (mod T ). Hence we obtain R Hom A ( e A T, X • ) ∈ Gperf( T ), that is, R Hom A ( e A T, − ) preserves Gorenstein perfect complexes.Combine Proposition 4.7 with Proposition 3.5, we get the following equivalent characterizationsfor the existence of a recollement of Gproj T relative to Gproj B and Gproj A . Corollary 4.8.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B compatible. Assumethat pd A M < ∞ , pd B Hom A ( M, A ) < ∞ and M ∈ ⊥ Gproj A . We have the following recollement ofstable category of Gorenstein projective modules Gproj B / / Gproj T j j v v / / Gproj A i i v v (4 . such that all these six functors are the restrictions of those in (3.8) if and only if M is Gorensteinsingular.Proof. The proof is similar as that in Corollary 4.4, we omit it.
Proof of Theorem 1.2.
This follows directly from Propositions 4.3 and 4.7.Consequently, we have the following commutative diagram such that all the functors involving inthe recollements and 2-recollements are mentioned above.
Corollary 4.9.
Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B compatible.(1) Assume that pd M B < ∞ . Then we have the following commutative diagram of recollements (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Gproj A (cid:15) (cid:15) / / Gproj T (cid:15) (cid:15) j j v v / / Gproj B (cid:15) (cid:15) j j v v D sg ( A ) (cid:15) (cid:15) / / D sg ( T ) (cid:15) (cid:15) i i v v / / D sg ( B ) (cid:15) (cid:15) i i v v D def ( A ) (cid:15) (cid:15) / / D def ( T ) (cid:15) (cid:15) i i v v / / D def ( B ) (cid:15) (cid:15) i i v v if and only if pd A M < ∞ and M is right Gorenstein singular.(2) Assume that pd A M < ∞ and M ∈ ⊥ Gproj A . Then we have the following commutative iagram of recollements (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Gproj B (cid:15) (cid:15) / / Gproj T (cid:15) (cid:15) j j v v / / Gproj A (cid:15) (cid:15) j j v v D sg ( B ) (cid:15) (cid:15) / / D sg ( T ) (cid:15) (cid:15) i i v v / / D sg ( A ) (cid:15) (cid:15) i i v v D def ( B ) (cid:15) (cid:15) / / D def ( T ) (cid:15) (cid:15) i i v v / / D def ( A ) (cid:15) (cid:15) i i v v if and only if pd B Hom A ( M, A ) < ∞ and M is Gorenstein singular.(3) Assume that pd A M < ∞ , M ∈ ⊥ Gproj A and pd M B < ∞ . Then we have the followingcommutative diagram of 2-recollements (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Gproj A (cid:15) (cid:15) - - Gproj T (cid:15) (cid:15) m m w w . . Gproj B (cid:15) (cid:15) m m w w D sg ( A ) (cid:15) (cid:15) - - D sg ( T ) (cid:15) (cid:15) m m w w - - D sg ( B ) (cid:15) (cid:15) m m w w D def ( A ) (cid:15) (cid:15) - - D def ( T ) (cid:15) (cid:15) n n w w . . D def ( B ) (cid:15) (cid:15) m m w w if and only if pd B Hom A ( M, A ) < ∞ and M is Gorenstein singular. Recall that an A - B -bimodule A M B is called a Frobenius bimodule if it is projective as a left A -and right B -module, and there is an A - B -bimodule isomorphism B Hom B op ( M, B ) A ≃ B Hom A ( M, A ) A . Meanwhile, an extension A ⊆ B of algebras is called a Frobenius extension if B is projective as an A -module and B ∼ = Hom A ( A B, A ) as a B - A -bimodule. In this case, both A B B and B B A are Frobeniusbimodules. We refer to Kadison [23] for more details on this matter. Proposition 4.10. (1) Let T = (cid:18) A M B (cid:19) be a triangular matrix algebra with A M B a Frobenius imodule. Then we have the following commutative diagram of 2-recollements (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Gproj A (cid:15) (cid:15) - - Gproj T (cid:15) (cid:15) m m w w . . Gproj B (cid:15) (cid:15) m m w w D sg ( A ) (cid:15) (cid:15) - - D sg ( T ) (cid:15) (cid:15) m m w w - - D sg ( B ) (cid:15) (cid:15) m m w w D def ( A ) (cid:15) (cid:15) - - D def ( T ) (cid:15) (cid:15) n n w w . . D def ( B ) (cid:15) (cid:15) m m w w . (2) Let A ⊆ B be a Frobenius extension of algebras. Assume T ′ = (cid:18) A B B (cid:19) , then we have thefollowing commutative diagram of 2-recollements (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Gproj A (cid:15) (cid:15) . . Gproj T ′ (cid:15) (cid:15) m m w w - - Gproj B (cid:15) (cid:15) n n v v D sg ( A ) (cid:15) (cid:15) - - D sg ( T ′ ) (cid:15) (cid:15) m m v v - - D sg ( B ) (cid:15) (cid:15) m m v v D def ( A ) (cid:15) (cid:15) . . D def ( T ′ ) (cid:15) (cid:15) n n v v . . D def ( B ) (cid:15) (cid:15) n n v v . (3)Let A ⊆ B be a Frobenius extension of algebras. Assume T ′′ = (cid:18) B B A (cid:19) , then we have thefollowing commutative diagram of 2-recollements (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Gproj B (cid:15) (cid:15) . . Gproj T ′′ (cid:15) (cid:15) m m w w - - Gproj A (cid:15) (cid:15) n n v v D sg ( B ) (cid:15) (cid:15) - - D sg ( T ′′ ) (cid:15) (cid:15) m m v v - - D sg ( A ) (cid:15) (cid:15) m m v v D def ( B ) (cid:15) (cid:15) . . D def ( T ′′ ) (cid:15) (cid:15) n n v v - - D def ( A ) (cid:15) (cid:15) n n v v . Proof.
Since A ⊆ B is a Frobenius extension, it follows that both A B B and B B A are Frobeniusbimodules. We only prove (1), because (2) and (3) are its consequences. As M is Frobenius, it is easy18o see pd B Hom A ( M, A ) < ∞ . Following [21, Theorem 3.4], we know that M ⊗ B − and Hom A ( M, − )preserve Gorenstein projective modules. Hence M is Gorenstein singular and then we get the desired2-recollement diagram by Corollary 4.9. Acknowledgements
This research was partially supported by NSFC (Grant No. 11626179, 11671069, 11701455,11771212), Qing Lan Project of Jiangsu Province, Jiangsu Government Scholarship for OverseasStudies (JS-2019-328), Shaanxi Province Basic Research Program of Natural Science (Grant No.2017JQ1012, 2020JM-178) and Fundamental Research Funds for the Central Universities (Grant No.JB160703).
References [1] L. Angeleri H¨ugel, S. Koenig, Q.H. Liu, D. Yang,
Ladders and simplicity of derived modulecategories , J. Algebra (2017), 15–66.[2] M. Auslander, M. Bridger,
Stable module theory , Memoirs Amer. Math. Soc. , Amer. Math.Soc., Providence, RI, 1969.[3] M. Auslander, I. Reiten, S.O. Smalø, Representation Theory of Artin Algebras , CambridgeUniversity Press, 1995.[4] L.L. Avramov, A. Martsinkovsky,
Absolute, relative, and Tate cohomology of modules of finiteGorenstein dimension , Proc. Lond. Math. Soc. (2002), 393–440.[5] A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers , Ast´erisque (1982).[6] A.A. Beilinson, V.A. Ginsburg, V.V. Schechtman,
Koszul duality , J. Geom. Phys. (3) (1998),317–350.[7] A. Beligiannis, The homological theory of contravariantly finite subcategories: Gorenstein cate-gories, Auslander-Buchweitz contexts and (co-)stabilization , Comm. Algebra (2000), 4547–4596.[8] P.A. Bergh, D.A. Jørgensen, S. Oppermann, The Gorenstein defect category , Quart. J. Math. (2) (2015), 459–471.[9] R.O. Buchweitz, Maximal Cohen-Macaulay Modules and Tate Cohomology over GorensteinRings , Unpublished manuscript, 1986.[10] H.X. Chen, C.C. Xi,
Good tilting modules and recollements of derived module categories , Proc.London Math. Soc. (3) (2012), 959–996[11] H.X. Chen, C.C. Xi,
Recollements of derived categories, III: Finitistic dimensions , J. LondonMath. Soc. (2) (2017), 633–658.[12] Q.H. Chen, Y.N. Lin, Recollements of extension of algebras , Sci. China Ser. A (4) (2003),530–537. 1913] X.W. Chen, Singularity Categories, Schur functors and triangular matrix tings , Algebra Rep-resent. Theor. (2009), 181–191.[14] X.W. Chen, Algebras with radical square zero are either self-injective or CM-free , Proc. Amer.Math. Soc. (1) (2012), 93–98.[15] E. Cline, B. Parshall, L. Scott,
Finite dimensional algebras and hight weightest categories , J.Reine. Angew. Math. (1988), 85–99.[16] J.A. Green,
Polynomial Representations of GL n , Lecture Notes in Math, vol. 830. Springer,New York (1980).[17] D. Happel, Triangulated Categories in Representation Theory of Finite Dimensional Algebras ,Lond. Math. Soc. Lect. Notes Ser. , Cambridge Univ. Press, Cambridge, 1988.[18] D.Happel,
Partial tilting modules and recollement , in: Proceedings of the International Con-ference on Algebra, Part2, Novosibirsk, 1989, in: Contemp. Math., vol.131, Amer. Math. Soc.,Providence, RI, 1992, pp.345–361.[19] D. Happel,
On Gorenstein Algebras , in: Representation theory of finite groups and finite-dimensional algebras (Proc. Conf. at Bielefeld, 1991), Progress in Math. , Birkh¨auser, Basel,1991, pp.389–404.[20] H. Holm, Gorenstein homological dimensions , J. Pure Appl. Algebra (2004), 167–193.[21] J.S. Hu, H.H. Li, Y.X. Geng, D.D. Zhang,
Frobenius functors and Gorenstein flat dimensions ,Comm. Algebra (3) (2020), 1257–1265.[22] P. Jørgensen, Reflecting recollements , Osaka J. Math. (2010), 209C-213.[23] L. Kadison, New Examples of Frobenius Extensions, Univ. Lecture Ser., vol. 14, Amer. Math.Soc., Providence, RI, 1999.[24] F. Kong, P. Zhang, From CM-finite to CM-free , J. Pure Appl. Algebra (2) (2016), 782–801.[25] H.H. Li, Y.F. Zheng, J.S. Hu, H.Y. Zhu,
Gorenstein projective modules and recollements overtriangular matrix rings , Comm. Algebra (to appear), arXiv:1910.02626.[26] H.H. Li, J.S. Hu, Y.F. Zheng,
When the Schur functor induces a triangle-equivalence betweenGorenstein defect categories , preprint, arXiv:2003.06782.[27] P. Liu, M. Lu,
Recollements of singularity categories and monomorphism categories , Comm.Algebra (2015), 2443–2456.[28] M. Lu, Gorenstein defect categories of triangular matrix algebras , J. Algebra (2017), 346–367.[29] J.I. Miyachi,
Recollement and idempotent ideals , Tsukuba J. Math. (2) (1992), 545–550.[30] D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models ,Proc. Steklov Inst. Math. (2004), 227–248.2031] C. Psaroudakis,
Homological theory of recollements of abelian categories , J. Algebra (2014),63–110.[32] C. Psaroudakis, O. Skartsaterhagen, O. Solberg,
Gorenstein categories, singular equivalencesand finite generation of cohomology rings in recollements , Trans. Amer. Math. Soc. Ser. B (2014), 45–95.[33] Y.Y. Qin, Y. Han, Reducing homological conjectures by n-recollements , Algebra Represent.Theor. (2016), 377–395.[34] P. Zhang, Gorenstein-projective modules and symmetric recollements , J. Algebra (2013),65–80.[35] P. Zhang, Y.H. Zhang, G.D. Zhou, L. Zhu,
Unbounded ladders induced by Gorenstein algebras ,Colloq. Math. (1) (2018), 37–56.[36] Y.F. Zheng, Z.Y. Huang,
Triangulated equivalences involving Gorenstein projective modules ,Canad. Math. Bull.60