Related papers: Perfect complexes and completion

Cohen proved that the infinite variable polynomial ring $R=k[x_1,x_2,\ldots]$ is noetherian with respect to the action of the infinite symmetric group $\mathfrak{S}$. The first two authors began a program to understand the…

We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the…

We prove that if $R$ is a G-ring then every fully dualizable $R$-linear cocomplete category is equivalent to a twist by a $\mathbb{G}_m$-gerbe of the category of modules over a finite \'etale $R$-algebra. We also show that this holds more…

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$…

Let $I$ be an ideal of a commutative Noetherian complete local ring $R$. In the present paper, we establish the equality $\dim R/(I+\Ann_R M)=\dim M$ for all $I$-cofinite $R$-modules $M$.

A finitely generated module $M$ over a commutative Noetherian ring $R$ is called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M), \] where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure…

Among the finitely generated modules over a Noetherian ring R, the semidualizing modules have been singled out due to their particularly nice duality properties. When R is a normal domain, we exhibit a natural inclusion of the set of…

Given any commutative Noetherian ring $R$ and an element $x$ in $R$, we consider the full subcategory $\C(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by $x$ is zero. Our…

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a…

Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{F} = \{ J_n \}_{n \geq 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F}) = \bigoplus_{n \geq 0} J_n$ is a finitely generated…

We compute the Balmer spectrum of the category of perfect complexes on an algebraic stack admitting a finite locally free cover by an affine scheme and identify it with the homogeneous spectrum of the cohomology ring.

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the…

We prove two results about the derived functor of $a$-adic completion: (1) Let $K$ be a commutative noetherian ring, let $A$ be a flat noetherian $K$-algebra which is $a$-adically complete with respect to some ideal $a\subseteq A$, such…

Let $T$ be a local (Noetherian) ring and let $Q_1$ and $Q_2$ be prime ideals of $T$. We find sufficient conditions for there to exist a quasi-excellent local subring $B$ of $T$ satisfying the following conditions: (1) the completion of $B$…

We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for…

We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy…

We study categories of dualizable torsion and complete objects for compactly-rigidly generated tensor-triangulated categories T with a Noetherian central action of a graded commutative Noetherian ring R. We show that they always admit a…

Let $R$ be a commutative local uniserial ring of length $n$, $p$ a generator of the maximal ideal, and $k$ the radical factor field. The pairs $(B,A)$ where $B$ is a finitely generated $R$-module and $A\subset B$ a submodule of $B$ such…

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…

We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a…