Related papers: Tight Hilbert Polynomial and F-rational local ring…

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain…

An equidimensional local ring is F-rational if and only if one ideal generated by a system of parameters is tightly closed. The question of whether a non-equidimensional local ring can have a tightly closed ideal generated by a system of…

Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a…

We prove that if the first tight Hilbert coefficient vanishes then ring is $F$-rational provided it is a Buchsbaum local ring satisfying the $(S_2)$ condition.

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and…

Let $(R, \mathfrak m)$ be an unmixed Noetherian local ring, Q a parameter ideal and $K$ an $\mathfrak m$-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and…

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ with prime characteristic…

Let $(R,m)$ be a Noetherian local ring of dimension $d$ and $K,Q$ be $m$-primary ideals in $R.$ In this paper we study the finiteness properties of the sets $\Lambda_i^K(R):=\{g_i^K(Q): Q$ is a parameter ideal of $R\},$ where $g_i^K(Q)$…

We had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $\ell(R/I) <\infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK…

Let $(R, \mathfrak{m}, k)$ be an excellent equidimensional local ring of characteristic $p>0$. The aim of this paper is to show that $\ell_R(\mathfrak{q}^*/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$…

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p >…

The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many…

Let $R$ be a commutative $F$-algebra, where $F$ is a field of characteristic 0, satisfying the following conditions: $R$ is equidimensional of dimension $n$, every residual field with respect to a maximal ideal is an algebraic extension of…

In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ ( with respect to $\mathfrak{m}$)…

Let $M$ be a finitely generated module of dimention d over a Noetherian local ring (A,m) and I an m-primary ideal. Let be a pair of good I-filtrations F and F' of M. We show that the Hilbert coefficients e_i(F) are bounded below and above…

Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G.…

Given ideals $I,J$ of a noetherian local ring $(R, \mathfrak m)$ such that $I+J$ is $\mathfrak m$-primary and a finitely generated $R$-module $M$, we associate an invariant of $(M,R,I,J)$ called the $h$-function. Our results on…

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…

For a Noetherian local ring $(\RR, \m)$, the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an $\m$-primary ideal $I$ are known to code for properties of $\RR$, of the blowup of $\spec(\RR)$ along $V(I)$, and…