Related papers: On functors that detect $S_n$

We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0)…

In this paper, we first study $fs$-modules, i.e., modules with finitely many small submodules. We show that every $fs$-module with finite hollow dimension is Noetherian. Also, we prove that an $R$-module $M$ with finite Goldie dimension, is…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \Vert f(z) \Vert=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $\mu$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We…

Let (A,m_A) -> (B,m_B) be a local morphism of local noetherian rings and M a finitely generated B-module. Then it follows from Tor^A_1(M,A/m_A) = 0 that M is a flat A-module. This is usually called the "local criterion of flatness". We give…

Let $R,S$ be rings, $\mathcal{X}\subseteq \text{mod}$-$R$ a covariantly finite subcategory, $\mathcal{C}$ the smallest definable subcategory of $\text{Mod}$-$R$ containing $\mathcal{X}$ and $\mathcal{D}$ a definable subcategory of…

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…

Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that…

Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}\Lambda$, where $\Lambda$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(\Lambda)$, the submodule category of…

Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ($u$-$S$-Noetherian for abbreviation) ring provided there exists an element $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq…

Let R be a Noetherian ring, I an ideal of R and M a ZD-module. Let S be a Melkersson subcategory with respect to I such that M/IM doesn't belong to S. We show that all maximal S-sequences on M in I, have equal length. If this common length…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and…

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of…

We prove that any noetherian quasi-excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms. We show that as an easy formal consequence of this result one obtains…

Given a non-positive DG-ring $A$, associated to it are the reduction and coreduction functors $F(-) = \mathrm{H}^0(A)\otimes^{\mathrm{L}}_A -$ and $G(-) = \mathrm{R}\operatorname{Hom}_A(\mathrm{H}^0(A),-)$, considered as functors…

Given a homomorphism of commutative noetherian rings $\phi: R \to S$, Daniel Quillen conjectured in 1970 that if the Andre-Quillen homology functors $D_n(S|R,-)$ vanish for all $n \gg 0$, then they vanish for all $n \ge 3$. We prove the…

Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first…

Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i) $I_n\supseteq…

In this survey, we summarize some results in the literature involving the mesh category, which is a combinatorial representation of the category of modules over a finite-dimensional associative algebra. We discuss Riedtmann's well-behaved…