Related papers: Homology of perfect complexes
We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the…
The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present…
In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological…
We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this…
We make use of the concepts of Tor-rigid and rigid-test modules, among others, to investigate the interplay between cohomology vanishing and the finiteness of several homological dimensions such as projective, injective and Gorenstein…
Let G be a finite group scheme over an algebraically closed field of positive characteristic. Assume further that the connected component of G is unipotent. It is shown that the projectivity of a rational G-module can be detected on a…
We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived)…
A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which…
Let $R$ be a local ring and $M$ a finitely generated $R$-module. The complete intersection dimension of $M$--defined by Avramov, Gasharov and Peeva, and denoted $\cidim_R(M)$--is a homological invariant whose finiteness implies that $M$ is…
Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…
In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several…
Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized…
It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that $\operatorname{Tor}_i^R(M, {}^{f^{e}}\!R)=0$ for $t\le i\le t+\dim R$ and…
We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that,…
We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion $\widehat R^{\mathfrak a}$ of a commutative noetherian ring $R$ with respect to a proper ideal…
We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by…
Let $\Lambda$ be an artinian ring. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for finitely generated $\Lambda$-modules.
In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…
The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite…
It is now very known how the subprojectivity of modules provides a fruitful new unified framework of the classical projectivity and flatness. In this paper, we extend this fact to the category of complexes by generalizing and unifying…