Related papers: Triangulated Matlis equivalence

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…

Let $\mathcal{C}$ be a small category. In this paper, we mainly study the category of modules $\mathfrak{M}\mbox{od-}\mathfrak{R}$ on ringed sites $(\mathbf{C},\mathfrak{R})$. We firstly reprove the Theorem A of the paper (M. Wu and F. Xu.…

Mutation of compact silting objects is a fundamental operation in the representation theory of finite-dimensional algebras due to its connections to cluster theory and to the lattice of torsion pairs in module or derived categories. In this…

We study aisles in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence,…

In the stable category of bounded below $\mathcal{A}(1)$--modules, every module is determined by an extension between a module with trivial $Q_0$-Margolis homology and a module with trivial $Q_1$-Margolis homology. We show that all bounded…

It is shown that any localisation of triangulated categories induces (up to an equivalence) a localisation of abelian categories when one passes to their abelianisations. From this one obtains for any enlargement of Grothendieck universes…

This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category…

In this paper, we show that the homotopy category of N-complexes of projective R-modules is triangle equivalent to the homotopy category of projective T_{N-1}(R)- modules where T_{N-1}(R) is the ring of triangular matrices of order N-1 with…

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is…

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules…

This paper is based on the material of Section 4 and Appendix C in arXiv:1503.05523v6, which was excluded from the subsequent versions of arXiv:1503.05523. We present the definition of a dedualizing complex of bicomodules over a pair of…

Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for…

Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. It is shown that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a…

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Let $T$ be a tilting $A$-module and $B={\rm End}_A\ T$ be the endomorphism algebra of $T$. In this paper, we consider the correspondence between the tilting…

Let $R$ by a right coherent ring and $R$-Mod denote the category of left $R$-modules. We show that there is an abelian model structure on $R$-Mod whose cofibrant objects are precisely the Gorenstein flat modules. Employing a new method for…

Given a noetherian abelian category $\mathcal Z$ of homological dimension two with a tilting object $T$, the abelian category $\mathcal Z$ and the abelian category of modules over $\text{End} (T)^{\textit{op}}$ are related by a sequence of…

Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the…

Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings is a simple-minded system if…

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define…

In this paper, we introduced the concept of crossed module for Hom-Lie antialgebras. It is proved that the category of crossed modules for Hom-Lie antialgebras and the category of $Cat^1$-Hom-Lie antialgebras are equivalent to each other.…