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 $\cc\mapsto j^*(\cc)$ defines a bijection between wide subcategories in $\mathcal{A}$ containing $i_{*}(\mathcal{A}')$ and wide subcategories in $\mathcal{A}"$. Moreover, a wide subcategory $\mathcal{C}$ of $\mathcal{A}$ containing $i_{*}(\mathcal{A}')$ admits a new recollement relative to $\mathcal{A}'$ and $j^{*}(\mathcal{C})$ which is induced from the original recollement.
Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We provide some methods to construct resolving subcategories in $(\mathcal{A},\mathcal{B},\mathcal{C})$. As applications of the Auslander-Reiten correspondence, we get the gluing of cotilting modules in a recollement of module categories for artin algebras. We also give some bounds of resolution dimensions of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$ with respect to re…
For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric subcategory $\mathscr{E}$ of a module category such that the derived category of the endomorphism ring of $T$ is a recollement of the derived categories of $\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of $\mathscr{E}$.
In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.
We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms of dg categories to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated…
Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal B,\mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal A$ and $\mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal B$ under certain conditions.
For any abelian category $\mathcal{A}$, Auslander constructed a localisation $w:\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})\to \mathcal{A}$ called the defect, which is the left adjoint to the Yoneda embedding $Y:\mathcal{A}\to\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})$. If $\mathcal{A}$ has enough projectives, then this localisation is part of a recollement called the defect recollement. We show that this recollement is an instance of the MacPherson-Vilonen construction if and only if $\mathcal{A}$ is hereditary. We also discuss several subcategories of $\mathrm{fp}(\mathcal{A}^…
A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an exam…
We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a $2n$-gon of recollements in an $(m/n)$-Calabi-Yau triangulated category. Second, we show the homotopy category $\mathsf{K}(\mathsf{Mor}_{N-1}(\mathcal{B}))$ of complexes of an additive category $\mathsf{Mor}_{N-1}(\mathcal{B})$ of $N-1$ sequences of split monomorphisms of an additive category $\mathcal{B}$ has a $2N$-gon of recollments. Third, we show the homotopy category $\mathsf{K}_{N}(\mathcal{B})$ of $N$-complexes of $\mathcal{B}$ has also a $2N$-gon of recollment…
It is shown that the silting reduction $\ct/\thick\cp$ of a triangulated category $\ct$ with respect to a presilting subcategory $\cp$ can be realized as a certain subfactor category of $\ct$, and that there is a one-to-one correspondence between the set of (pre)silting subcategories of $\ct$ containing $\cp$ and the set of (pre)silting subcategories of $\ct/\thick\cp$. This is analogous to a result for Calabi-Yau reduction. This result is applied to show that Amiot-Guo-Keller's construction of $d$-Calabi-Yau triangulated categories with $d$-cluster-tilting objects takes silting reduction to C…
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…
In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $τ$-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal{U},\mathcal{T}]$ in the lattice $\operatorname{\mathsf{tors}} \mathcal{A}$ of torsion classes in $\mathcal{A}$ such that $\mathcal{W}:=\mathcal{U}^\perp \cap \mathcal{T}$ is a wide subcategory of $\mathcal{A}$; we call these intervals wide intervals. We prove that a wide interval $[\mathcal{U},\mathcal{T}]$ is isomorphic to the lattice $\o…
Recollements of derived categories from $n$-term big tilting complexes
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；Beijing Key Laboratory of Biological Big Data and Topological Statistics, Beijing Jiaotong University（北京交通大学生物大数据与拓扑统计北京市重点实验室）；School of Science, Beijing University of Civil Engineering and Architecture（北京建筑大学理学院）
Let $Λ$ be a finite-dimensional algebra over a field $K$. We describe how Buan and Marsh's $τ$-exceptional sequences can be used to give a "brick labeling" of a certain poset of wide subcategories of finitely-generated $Λ$-modules. When $Λ$ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn our attention towards the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with…
We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated st…
