Related papers: Totally acyclic complexes over noetherian schemes

The category of coherent sheaves over a noetherian scheme is very important for studying the properties of a given scheme. For noetherian schemes it is a well-known fact that the topology can be fully recovered from the corresponding…

We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations.…

We show that Bloch's complex of relative zero-cycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over…

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…

For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all…

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…

Recently, several authors have adopted new alternative approaches in the study of some classical notions of modules. Among them, we find the notion of subprojectivity which was introduced to measure in a way the degree of projectivity of…

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably…

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods…

A correspondence between quasicoherent sheaves on toric schemes and graded modules over some homogeneous coordinate ring is presented, and the behaviour of several finiteness properties under this correspondence is investigated.

Let $X$ be a scheme, proper over a commutative noetherian ring $A$. We introduce the concept of an ample filter of invertible sheaves on $X$ and generalize the most important equivalent criteria for ampleness of an invertible sheaf. We also…

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed…

We classify indecomposable non-projective Gorenstein-projective modules over a monomial algebra via the notion of perfect paths. We apply this classification to a quadratic monomial algebra and describe explicitly the stable category of its…

A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C) \simeq A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call…

Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in…

Quasi-projective dimension of modules over associative rings is generalized in this paper to the one of complexes of modules. Basic properties of this dimension are established, including a comparison result with projective dimension and a…

We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein projective modules to…

The study of rings and modules with homological criteria is a cornerstone of commutative algebra. Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. In this paper, a relative…

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring $(R,m)$ with $m^3=0$. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes,…

Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of…