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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.…

For an exact category we provide two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel--Quillen embedding. For the first construction we describe a pre-aisle and its…

The monomorphism category $\mathcal S_n(\mathcal X)$ is introduced, where $\mathcal X$ is a full subcategory of the module category $A$-mod of Artin algebra $A$. The key result is a reciprocity of the monomorphism operator $\mathcal S_n$…

We prove that an object $U$ in a triangulated category with coproducts is silting if and only if it is a (weak) generator of the category, the orthogonal class $U^{\perp_{>0}}$ contains $U$, and $U^{\perp_{>0}}$ is closed under direct sums.…

Let $R$ be a finite dimensional algebra of finite global dimension over a field $k$. In this paper, we will characterize a $k$-linear abelian category $\mathscr C$ such that $\mathscr C\cong \operatorname {tails} A$ for some graded right…

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring which is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also compatible…

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over complex numbers. The gradation of such a Lie superalgebra $\ggg$ naturally arises, with the zero component $\ggg_0$ being a reductive Lie…

In this paper, we examine the class of cofibrant modules over a group algebra $kG$, that were defined by Benson in [2]. We show that this class is always the left-hand side of a complete hereditary cotorsion pair in the category of…

Let {\cal T} be a triangulated category, {\cal A} a full subcategory of {\cal T} and {\cal X} a functorially finite subcategory of {\cal A}. If {\cal A} has the properties that any {\cal X}-monomorphism of {\cal A} has a cone and any {\cal…

For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions.

The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that…

Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free…

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study complete cohomology of objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ by applying…

Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite…

Every Serre subcategory of an abelian category is assigned a unique type. The type of a Serre subcategory of a Grothendieck category is in the list: $$(0, 0), \ (0, -1), \ (1, -1), \ (0, -2), \ (1, -2), \ (2, -1), \ (+\infty, -\infty);$$…

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of…

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an…

The main result of this paper may be stated as a construction of "almost representations" for the canonical presheaves of object extensions of length n on the C-systems defined by locally cartesian closed universe categories with binary…

Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is called (1) $\mathcal C$-$closed$ if $X$ is closed in every topological semigroup $Y\in\mathcal C$ containing $X$ as a discrete subsemigroup, (2) $ideally$ $\mathcal…

We invesigate the relation between projective and anomalous representations of categories, and show how to any anomaly $J\colon \mathcal{C}\to 2\mathrm{Vect}$ one can associate an extension $\mathcal{C}^J$ of $\mathcal{C}$ and a subcategory…