Related papers: Generalized local cohomology and the Intersection …

The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional…

In this paper we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over…

A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the…

We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far…

Let $R$ denote a Noetherian ring and an ideal $J \subset R$ with $U = \operatorname{Spec R} \setminus V(J)$. For an $R$-module $M$ there is an isomorphism $\Gamma(U, \tilde{M}) \cong \varinjlim \operatorname{Hom}_R(J^n,M)$ known as…

Let $R$ be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if $\mathfrak{a}$ is an ideal of $R$ which is linked by the ideal $I$, then $cd(\mathfrak{a},R) \in \{ grad \mathfrak{a},…

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation $Q\to R$ by a…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the…

In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds…

Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded $k$-algebra; these notions are justified by examples from noncommutative invariant theory.

We prove that if $f:R \rightarrow S$ is a local homomorphism of noetherian local rings, and $M$ is a non-zero finitely generated or artinian $S$-module whose injective dimension over $R$ is bounded by the difference of the embedding…

Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that for each $i>0$, $\mathrm{Tor}^R_i(M,N)=0$. We prove that the cohomological support of $M\otimes_R N$ (in the sense of Avramov-Buchweitz) is equal to…

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…

A cohomological support, Supp_A(M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the…

Extending a notion defined for surjective maps by Blanco, Majadas, and Rodicio, we introduce and study a class of homomorphisms of commutative noetherian rings, which strictly contains the class of locally complete intersection…

Let $f\colon (\mathbb C^n,S)\to (\mathbb C^p,0)$ be a finite map-germ with $n<p$ and $Y_\delta$ the image of a small perturbation $f_\delta$. We show that the reduced cohomology of $Y_\delta$ is concentrated in a range of degrees determined…

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely…

Let $R$ be a commutative Noetherian ring. Denote by $\textrm{mod}R$ the category of finitely generated $R$-modules. In this paper, a contravariantly infinite subcategory of $\textrm{mod}R$ is defined as a full subcategory $\mathscr{X}$ of…

We deal with the complete-intersection property of maximally differential ideals. Also, we connect the Gorenstein homology of derivations to the Gorenstein property of the base rings. These equipped with some applications.