Related papers: FI- and OI-modules with varying coefficients

Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded…

We call a semigroup $S$ f-noetherian if every right congruence of finite index on $S$ is finitely generated. We prove that every finitely generated semigroup is f-noetherian, and investigate whether the properties of being f-noetherian and…

The groups $\Gamma_{n,s}$ are defined in terms of homotopy equivalences of certain graphs, and are natural generalisations of $\mbox{Out}(F_n)$ and $\mbox{Aut}(F_n)$. They have appeared frequently in the study of free group automorphisms,…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, and let $M$ be a finitely generated $R$-module. For a non-negative integer $t$, we prove that $H_{\fa}^t(M)$ is $\fa$-cofinite whenever $H_{\fa}^t(M)$ is Artinian and…

The purpose of this paper is to give information on the FI-homology of the standard injective cogenerators of the category of FI-modules, where FI is the category of finite sets and injections. Working over a field k of characteristic zero,…

We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We…

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially ``sewn…

Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a…

For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$.…

Let A be a noetherian commutative ring, and let I be an ideal in A. We study questions of flatness and I-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and I-adically free A-module.

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated…

Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a graded $K[t]$-module, we introduce and study "$N$-fiber-full up to $h$" $A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and only if…

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in…

In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the…

Let $R$ be a strong $n$-coherent ring such that each finitely $n$-presented $R$-module has finite projective dimension. We consider $\mathcal{FP}_{n}(R)$ the full subcategory of $R$-Mod of finitely $n$-presented modules. We prove that…

We find the model completion of the theory modules over $A$, where $A$ is a finitely generated commutative algebra over a field $K$. This is done in a context where the field $K$ and the module are represented by sorts in the theory, so…

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…

In this paper we show that for a torsion-free abelian group $G$, $\operatorname{rank}_\mathbb{Z}G<\infty$ if and only if there exists a Noetherian $G$-graded ring $R$ such that the set $\{R_g \neq 0\}$ generates the group $G$. For every $G$…

In this paper, we inspect a relatively unexplored notion of finite generation in semirings, namely semirings in which all congruences are finitely generated. Such semirings are dubbed Congruence Noetherian. After developing sufficient…

Let $R$ be a commutative noetherian ring and $I$ an ideal of $R$. Assume that for all integers $i$ the local cohomology module $H_I^i(R)$ is $I$-cofinite. Suppose that $R_\mathfrak{p}$ is a regular local ring for all prime ideals…