Properties of abelian categories via recollements

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 example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

Recollements of Singularity Categories and Monomorphism Categories

We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules.     

商范畴的Recollement

证明了三角范畴的recollement可以自然诱导其商范畴的recollement.特别地,得到类似于群同态第二基本定理的结果,即若U是三角范畴D的局部化(或余局部化)子范畴,V是U的三角满子范畴,则U/V是D/V的局部化(或余局部化)子范畴,并且有三角等价(D/V)/(U/V)≌D/U.同理,对Abel范畴的recollement也有相应的结果.     

三角范畴的recollement与AR-三角

设{D＇，D，D＇＇；i^＊，i＊=i!，i^!，j!，j^＊=j^!，j＊）是一个recollement，本文证明了当D有AR-三角时，D＇，D＇＇也有AR-三角，并且它们的AR-三角完全可由D中AR-三角诱导．     

正合范畴中的复形、余挠对及粘合

作者在弱幂等完备的正合范畴(A,ε)中引入了复形的新的定义,并且证明了ε-正合复形的同伦范畴κ_ex(ε)是同伦范畴κ_ε(A)的厚子范畴.给定(A,ε)中的余挠对(x,y),定义了正合范畴(c_ε(A),C(ε))中的两个余挠对(x^~_ε,dgy^~_ε)和(dgx^~_ε,y^~_ε),并且证明了当A是可数完备时,c_ε(A)中任意无界复形的dgx^~_ε,y^~_ε-分解存在.作为应用,建立了相对于范畴κ_ex(ε)和D_ε(A)的范畴κ_ε(A)的左粘合,给出了R-模范畴的粘合的例子.     

Construction of CPs-Stratified Algebras

The results of [7 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representations of Algebras and Related Topics, London Math. Society Lecture Note Series 168:200–224 . [Google Scholar]] and [2 Ágoston , I. , Dlab , V. , Lukács , E. ( 2011 ). Constructions of stratified algebras . Comm. Algebra 39 : 2545 – 2553 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3 Ágoston , I. , Lukács , E. Stratifying pairs of subcategories for CPS-stratified algebras . To appear in Journal of Algebra and Its Applications , p. 11 . [Google Scholar]], we generalize these earlier results to construct recursively all CPS-stratified algebras.     

Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules

The aim of this paper is two-fold. Given a recollement (T′, T, T″, i*, i_*, i^!, j_!, j*, j_*), where T′, T, T″ are triangulated categories with small coproducts and T is compactly generated. First, the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i_* preserves compact objects. As a con-sequence, given a ladder (T′, T, T″, T, T′) of height 2, then the certain BBD-induction of compactly generated t-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement (D(B-Mod),D(A-Mod),D(C-Mod), i*, i_*, i^!, j_!, j*, j_*) induced by a homological ring epimorphism, the last aim of this work is to show that if A is Gorenstein, _A B has finite projective dimension and j_! restricts to D ^b (C-mod), then this recollement induces an unbounded ladder (B-Gproj,A-Gproj, C-Gproj) of stable categories of finitely generated Gorenstein-projective modules. Some examples are described.     

Derived Equivalences of Upper Triangular Differential Graded Algebras

This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form (R, S, M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R, S, M) and (S, M′, R′), where the DG-bimodule M′ is obtained from M and X and R′ is the endomorphism differential graded algebra of a K-projective resolution of X.