Related papers: Stability results for projective modules over Rees…

Let $R$ be a commutative noetherian ring, and $\mathcal{Z}$ a stable under specialization subset of $\Spec(R)$. We introduce a notion of $\mathcal{Z}$-cofiniteness and study its main properties. In the case $\dim(\mathcal{Z})\leq 1$, or…

In this paper we begin the classification of coherent systems $(E,V)$ on the projective line which are stable with respect to some value of a parameter $\alpha$. In particular we show that the moduli spaces, if non-empty, are always smooth…

We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but…

In this paper, we prove stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category $\mathbf{OrI}(R)$ and prove a Noetherianity theorem for the…

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}_{> 0} \cup \{\infty\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}_{>…

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules and by $\operatorname{D^b}(R)$ the bounded derived category of $\operatorname{mod} R$. In this paper, we first…

Let $G$ be a $p$-adic reductive group and $R$ be a noetherian Jacobson $\mathbb{Z}[1/p]$-algebra. In this note, we show that every smooth irreducible $R$-linear representation of $G$ is admissible using the finiteness result of Dat, Helm,…

We generalize known results on summands of completely decomposable and separable torsion-free abelian groups to modules over h-local Pr\"ufer domains. Over such domains summands of completely decomposable torsion-free modules are again…

Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or…

An unital C*-algebra A is said to have cancellation of projections if the semigroup D(A) of Murray-von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies…

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and…

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of…

Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an aleph_1-free R-module G of rank aleph_1 with endomorphism algebra End_RG=A . Clearly…

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.

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q…

It is proved that if $R$ is a valuation domain with maximal ideal $P$ and if $R_L$ is countably generated for each prime ideal $L$, then $R^R$ is separable if and only $R_J$ is maximal, where $J=\cap_{n\in\mathbb{N}}P^n$.

Let $R$ be a commutative noetherian ring. We prove that the class of modules of projective dimension bounded by $k$ is of finite type if and only if $R$ satisfies Serre's condition $(S_k)$. In particular, this answers positively a question…

We introduce the notion of the $\infty$-category of (complete) derived $G$-graded modules over a $G$-graded ring $R$ for a torsion-free abelian group $G$, and we study its foundational properties. Moreover, we prove a categorical…

In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the…

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules…