Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0408351

Some asymptotic properties of the Rees powers of a module

Ana L. Branco Correia·Santiago Zarzuela
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let R be a commutative ring and let G be a free R-module with positive rank e. For any R-submodule E of G we may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components E_n of the image, that we call the n-th Rees powers of E. In this work we prove some asymptotic properties of the modules E_n, which extend well known similar ones for the case of ideals, among them Burch's inequality for the analytic spread.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ana L. Branco Correia, Santiago Zarzuela. 2004-08-25. Some asymptotic properties of the Rees powers of a module. https://arxiv.org/abs/math/0408351

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Buchsbaumness, Macaulayfication and Castelnuovo-Mumford regularity of monomial curves

Projective monomial curves are associated with rings generated by monomials of equal degree in two variables. In this paper, we give an infinite class of non-smooth, non Cohen-Macaulay $k$-Buchsbaum projective monomial curves for any $k\geq 1$ and find the monomial generators for the respective Macaulayfication. More generally, we demonstrate a method to find the Macaulayfication of a $k$-Buchsbaum monomial curve for any $k\geq 1$. We also discuss Castelnuovo-Mumford regularity of certain curves in terms of $k$-Buchsbaumness.

math.AC↗
arXiv2026

Castelnuovo-Mumford regularity of toric varieties with at most one singular point

We establish upper bounds for the Castelnuovo--Mumford regularity of the coordinate ring of a simplicial projective toric variety with at most one singular point. In the smooth case, our results recover the bound of Herzog and Hibi [Proc. Amer. Math. Soc. 131 (2003), 2641--2647], and therefore the Eisenbud--Goto bound. Furthermore, when the variety has exactly one singular point and dimension at least $3$, we prove that its regularity also satisfies the Eisenbud--Goto bound. The proof combines combinatorial and homological methods: we study the asymptotic behavior of the sumsets associated to the toric variety and relate it to Castelnuovo--Mumford regularity via a Hochster-like formula.

math.AC↗
arXiv2026

Generalized divisor topology of commutative rings

Let $R$ be a commutative ring with nonzero identity and let $R^\#$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^\#)$. Its basic open sets are \[ B_a=\{[b]\in EC(R^\#): b\mid a^n \text{ for some }n\geq 1\}. \] The relation \[ [b]\in B_a \quad\Longleftrightarrow\quad \sqrt{aR}\subseteq\sqrt{bR} \] shows that $GD(R)$ records radical divisibility among principal ideals. We prove that $GD(R)$ is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the $T_0$ and discrete properties and of the equality $GD(R)=D(R)$. We also determine the isolated points of $GD(R)$. Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain $R$, $GD(R)$ is compact if and only if $R$ is a $G$-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of $GD(R)$ with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.

math.AC↗