Related papers: Blowups of hypersurfaces

Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in…

The aim of this paper is to extend Cohen structure theorem beyond local rings. Both Cohen structure theorem and Nagata's generalization of it are special cases of our results. We investigate for which rings $R$ there exists a maximal ideal…

Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well-known that if the associated graded ring $\gr_I(R)$ is Cohen-Macaulay (Gorenstein), then so is $R$, but the converse is not true in general. In this paper we investigate…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). We consider singularities of the normalization of the blow-up Proj R[It] of I. A theorem of Lipman implies that the ideal I has a unique factorization as…

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…

The aim of this paper is to define the notion of the Cohen-Macaulay cone of a Noetherian local domain R and to present its application to the theory of Hilbert-Kunz functions. It has been shown in Kurano's paper "Numerical equivalence…

The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are…

This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which…

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…

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…

For a Cohen-Macaulay local ring $(R,\mathfrak{m})$ with canonical module, we study how relations between $\text{index}(R)$ and $\text{g}\ell\ell(R)$ and between $\text{index}(R)$ and $e(R)$ are preserved when factoring out regular sequences…

In this paper, we prove "prismatic Kunz's theorem" which states that a complete Noetherian local ring $R$ of residue characteristic $p$ is a regular local ring if and only if the Frobenius lift on a prismatic complex of (a derived…

Let $R$ be a Cohen-Macaulay local ring of dimension one with a canonical module $\rm{K_R}$. Let $I$ be a faithful ideal of $R$. We explore the problem of when $I \otimes_RI^{\vee}$ is torsionfree, where $I^{\vee} = \operatorname{Hom_R(I,…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists…

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined…

Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let $\varphi$ denote its Hilbert-Burch matrix. When $\varphi$ has linear entries, the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood under…

Let $E$ be a module of projective dimension one over a Noetherian ring $R$ and consider its Rees algebra $\mathcal{R}(E)$. We study this ring as a quotient of the symmetric algebra $\mathcal{S}(E)$ and consider the ideal $\mathcal{A}$…

In this work we describe the local cohomology of reflexive modules of rank one over normal semigroup rings with respect to monomial ideals. Using our description we show that the problem of classifying maximal Cohen-Macaulay modules of rank…

An ideal I of a local Cohen-Macaulay ring R is called a cohomologically complete intersection if H^i_I(R) = 0 for all i \neq c = height(I). Here H^i_I(R), i \in Z denotes the local cohomology of R with respect to I. For instance, a…

Let $\mathcal I=\{I_n\}$ be a divisorial filtration on a two dimensional normal excellent local ring $(R,m_R)$. Let $R[\mathcal I]=\oplus_{n\ge 0}I_n$ be the Rees algebra of $\mathcal I$ and $\tau:\mbox{Proj}R[\mathcal I])\rightarrow…