相关论文: Homotopy categories of unbounded complexes of proj…

Let $A$ be an artin algebra. We show that the bounded homotopy category of finitely generated right $A$-modules has Auslander-Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [H2];…

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov-Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a…

We prove that for all noetherian rings, the level of any homologically bounded complex $M$ with respect to the collection of projective or injective modules is bounded above by the projective dimension of $\bigoplus H(M)$ plus one or the…

It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue…

Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the…

For a tensor ring $T_R(M)$, we obtain sufficient and necessary conditions to describe all complete projective resolutions and all Gorenstein projective modules. As a consequence, we provide a method for constructing Gorenstein projective…

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…

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. In this paper, we make use of three diferent techniques addressing intrinsic and homological properties of several classes of relative Gorenstein…

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize projectively coresolved Gorenstein flat modules over $T_R(M)$, showing that a $T_R(M)$ module $(X,u)$ is…

This article investigates the splitting problem for finitely generated projective modules $P$ over affine algebras over algebraically closed fields and their polynomial extensions. We then address an open question due to M. Roitman on monic…

Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\V$-schemes. Then we build a category…

Let $T$ be a tilting module. In this paper, Gorenstein $\pi[T]$-projective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein…

The stable module category has been realized as a subcategory of the unbounded homotopy category of projective modules by Kato. We construct the triangulated hull of this subcategory inside the homotopy category. This can also be used to…

We prove that the projective model structure on the category of unbounded cochain complexes extends naturally to the category of contractions. The proof is completely elementary and we do not assume familiarity with model categories.

We consider a finite acyclic quiver $\mathcal{Q}$ and a quasi-Frobenius ring $R$. We endow the category of quiver representations over $R$ with a model structure, whose homotopy category is equivalent to the stable category of…

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we…

Let $T=\left( \begin{array}{cc} R & M 0 & S \end{array} \right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of…

We formulate a notion of "geometric reductivity" in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…

We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders…