In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring A and its idempotent ideal J, we show that an A/J-module T is a silting A-module if and only if T is a silting A/J-module. For the finite dimensional k-algebras, with k a field, we show that the tensor products of silting modules are still silting. We also show that the (partial) Gorenstein silting properties can be glued by the recollements of module categories of Noetherian rings. As a consequence, we glue the Gorenstein silting modules of an upper triangular matrix Gorenstein ring by those of the involved rings.
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a…
(Partial) Gorenstein silting modules are introduced and investigated. It is shown that for finite dimensional algebras of finite CM-type, partial Gorenstein silting modules are in bijection with τ_G-rigid modules; Gorenstein silting modules are the module-theoretic counterpart of 2-term Gorenstein silting complexes; and the relation between 2-term Gorenstein silting complexes, t-structures and torsion pairs in module categories. Furthermore, the corresponding version of the classical Brenner-Butler theorem in this setting are characterised; and the upper bound of the global dimension of endomo…
In the paper, we investigate the lifting of recollements with respect to Gorenstein-projective modules. Specifically, a homological ring epimorphism can induce a lifting of the recollement of the stable category of finitely generated Gorenstein-projective modules; the recollement of the bounded Gorenstein derived categories of some upper triangular matrix algebras can be lifted to the homotopy category of Gorenstein-projective modules. As a byproduct, we give a sufficient and necessary condition on the upper triangular matrix algebra T_{n}(A) to be of finite CM-type for an algebra A of finite …
It is shown that a 4-recollement of derived categories of CM-finite algebras induces a 2-recollement of the corresponding Cohen-Macaulay Auslander algebras, which generalises the main theorem of Pan [S. Y. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012), 355{363]. Moreover, both Auslander- Reiten conjecture and Gorenstein projective conjecture are shown invariant under 3 (or 4)-recollement of unbounded derived categories of algebras.
Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{\rm{b}}({\rm{mod\mbox{-}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a $\mathbb{D}^-({\rm{Mod\mbox{-}}} )$ level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we…
Relations between Gorenstein derived categories, Gorenstein defect categories and Gorenstein stable categories are established. Using these, the Gorensteinness of an algebra $A$ and invariants with respect to recollements of the bounded Gorenstein derived category $D^{b}_{gp}(A\mbox{-}{\rm mod})$ of $A$ are investigated. Specifically, the Gorensteinness of $A$ is characterized in three ways: the existence of Auslander-Reiten triangles in $D^{b}_{gp}(A\mbox{-}{\rm mod})$; recollements of $D^{b}_{gp}(A\mbox{-}{\rm mod})$; and also Gorenstein derived equivalences. It is shown that the finiteness …
We apply the technique of recollement to study the Gorenstein defect categories of triangular matrix algebras. First, we construct a left recollement of Gorenstein defect categories for a triangular matrix algebra under some conditions, using it, we give a categorical interpretation of the Gorenstein properties of the triangular matrix algebra obtained by X-W. Chen, B. L. Xiong and P. Zhang respectively. Second, under some additional conditions, a recollement of Gorenstein defect categories for a triangular matrix algebra is constructed. As an application, for a special kind of triangular matr…
The relation between the $n$-recollements of stable categories of Gorenstein projective modules and the virtual Gorensteinness of algebras are investigated. Let $A,B$, and $C$ be finite dimensional algebras. We prove that if the stable category of Gorenstein projective $A$-modules admits a $n$-recollement relative to that of $B$ and $C$ with $n\geq 2$, then $A$ is virtually Gorenstein if and only if so are $B$ and $C$.
Invariants with respect to recollements of the stable category of Gorenstein projective A-modules over an algebra A and stable equivalences are investigated. Specifically, the Gorenstein rigidity dimension is introduced. It is shown that the Gorenstein rigidity dimension is invariant with respect to both Morita equivalences and the stable equivalences of Gorenstein projective modules. As a consequence, the Gorenstein rigidity dimension is shown the invariant of derived equivalences. The Gorenstein rigidity dimension is compared along the recollements of the stable category of Gorenstein projec…
In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to Calabi-Yau reduction, silting reduction and $τ$-tilting reduction. More precisely, if an abelian category $\mathcal{A}$ admits a recollement relative to abelian categories $\mathcal{A}'$ and $\mathcal{A}"$, diagrammatically expressed by $$\xymatrix@!C=2pc{ \mathcal{A'} \ar@{>->}[rr]|{i_{*}} && \mathcal{A} \ar@<-4.0mm>@{->>}[ll]_{i^{*}} \ar@{->>}[rr]|{j^{*}} \ar@{->>}@<4.0mm>[ll]^{i^{!}}&& \mathcal{A''} \ar@{>->}@<-4.0mm>[ll]_{j_{!}} \ar@{>->}@<4.0mm>[ll]^{j_{*}} },$$ then the assignment $…
In this paper we show that the (un)bounded derived categories$\colon$(i) of the monomorphism category, (ii) of the morphism category and (iii) of the double morphism category, admit a periodic infinite ladder of recollements. These results are based on a characterization that we provide for a recollement of (compactly generated) triangulated categories to admit a ladder of some height going either upwards or downwards. Moreover, we introduce and study the singularity category of the monomorphism category over an Artin algebra $Λ$ and show that there is a periodic infinite ladder that connects …
Let $T=(A,M,0,B)$ be a triangular matrix algebra with its corner algebras $A$ and $B$ Artinian and $_AM_B$ an $A$-$B$-bimodule. The 2-recollement structures for singularity categories and Gorenstein defect categories over $T$ are studied. Under mild assumptions, we provide necessary and sufficient conditions for the existences of 2-recollements of singularity categories and Gorenstein defect categories over $T$ relative to those of $A$ and $B$. Parts of our results strengthen and unify the corresponding work in [27,28,34].
We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let $ω$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $ω$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{\mathcal{G}_ω}$, where $\mathcal{G}_ω$ is the subcategory of all $ω$-Gorenstein objects, is a triangulated category and it is, under some conditions, triangle equivalent …
n-recollements of triangulated categories and n-derived-simple algebras are introduced. The relations between the n-recollements of derived categories of algebras and the Cartan determinants, homological smoothness and Gorensteinness of algebras respectively are clarified. As applications, the Cartan determinant conjecture is reduced to 1-derived-simple algebras, and the Gorenstein symmetry conjecture is reduced to 2-derived-simple algebras.
Given an artin algebra $Λ$ with an idempotent element $a$ we compare the algebras $Λ$ and $aΛa$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element $a$ which ensure that $Λ$ is Gorenstein if and only if $aΛa$ is Gorenstein, that the singularity categories of $Λ$ and $aΛa$ are equivalent and that Fg holds for $Λ$ if and only if Fg holds for $aΛa$. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorenstei…
We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwa…
For a recollement $(\mathcal{D}B,\mathcal{D}A,\mathcal{D}C)$ of derived categories of algebras, we investigate when the functor $j^*:\mathcal{D}A\rightarrow\mathcal{D}C$ is an eventually homological isomorphism. In this context, we compare the algebras $A$ and $C$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. The results are applied to stratifying ideals, triangular matrix algebras and derived discrete algebras.
Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.
In this paper, we consider the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ for a Gorenstein monomial algebra $A$. Firstly, for a positively graded $1$-Gorenstein algebra, we prove that its ${\mathbb Z}$-graded singularity category admits silting objects. Secondly, for $A=KQ/I$ being a Gorenstein monomial algebra, we prove that $D_{sg}(\mod^{\mathbb Z} A)$ has tilting objects. As a consequence, $D_{sg}(\mod^{\mathbb Z}A)$ is triangulated equivalent to the derived category of a hereditary algebra $H$ which is of finite repre…
The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some int…
We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complxes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay $\opn{T}_2(R)$-modules.
Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently. In this paper, we prove that the Gabriel-Zisman localization $\mathcal B/({\rm thick}\mathcal W)$ of an extriangulated category $\mathcal B$ with respect to a presilting subcategory $\mathcal W$ satisfying certain condition can be realized as a subfactor category of $\mathcal B$. This generalizes the result by Iyama-Yang for silting reduction on triangulated categories. Then we discuss the relation between silting subcategories and tilting subcategories in extriangulated categori…
For a $Z$-graded Gorenstein ring $R$, we study the stable category $CM^ZR$ of $Z$-graded maximal Cohen-Macaulay $R$-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory given as the stable category of $CM_0^ZR$ is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. In the case $dim R=1$, we prove that the stable category of $CM_0^ZR$ always admits a silting object, and that it admits a tilting object if and only if either $R$ is regular or the $a$-invariant of $R$ …
The monomorphism category $\mathscr{S}(A, M, B)$ induced by a bimodule $_AM_B$ is the subcategory of $Λ$-mod consisting of $\left[\begin{smallmatrix} X\\ Y\end{smallmatrix}\right]_ϕ$ such that $ϕ: M\otimes_B Y\rightarrow X$ is a monic $A$-map, where $Λ=\left[\begin{smallmatrix} A&M\\0&B \end{smallmatrix}\right]$. In general, it is not the monomorphism categories induced by quivers. It could describe the Gorenstein-projective $\m$-modules. This monomorphism category is a resolving subcategory of $\modcatΛ$ if and only if $M_B$ is projective. In this case, it has enough injective objects and Aus…
