Related papers: On the core of ideals

Let $F$ be a field and $M_n(F)$ the ring of $n \times n$ matrices over $F$. Given a subset $S$ of $M_n(F)$, the null ideal of $S$ is the set of all polynomials $f$ with coefficients from $M_n(F)$ such that $f(A) = 0$ for all $A \in S$. We…

Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the…

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. The asymptotic behavior of the quotient modules M/I^n M of M is an actively studied subject in commutative algebra. The main result of this paper…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ \wrt \ $I$ and let $\R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with…

The classical "generalized principal ideal theorems" of Macaulay, Eagon-Northcott, and others give sharp bounds on the heights of determinantal ideals in arbitrary rings. But in regular local rings (or graded polynomial rings) these are far…

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We…

Let $S$ be a polynomial ring over a field $K$, with a monomial order $\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $\delta_I$ and the footprint…

Let I be an ideal of height two in R=k[x_0,x_1] generated by forms of the same degree, and let K be the ideal of defining equations of the Rees algebra of I. Suppose that the second largest column degree in the syzygy matrix of I is e. We…

In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and…

Let $R$ be a standard graded polynomial ring that is finitely generated over a field, and let $I$ be a homogenous prime ideal of $R$. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of $R/I^t$, as $t$ grows…

When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the…

Let $(R, \mathfrak{m}) $ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a Cohen-Macaulay ring of dimension $d$. There is given a complete answer to the question…

Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $\deg A = 0$ and $\deg X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$…

In this paper, we present an algorithm for computing the minimal reductions of $\mathfrak{m}$-primary ideals of Cohen--Macaulay local rings. Using this algorithm, we are able to compute the Hilbert--Samuel multiplicities and solve the…

Let $(A,\mathfrak{m})$ be an excellent normal local ring of dimension $d \geq 2$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal. Then the following assertions are equivalent: (i) The extended Rees algebra $A[It,…

For a one dimensional analytically unramified Cohen-Macaulay local ring $R$, the blowup algebra of the canonical ideal is a module finite birational extension. The conductor of this extension always contains the conductor of $R$. We study…

We propose an effective method for primary decomposition of symmetric ideals. Let $K[X]=K[x_1,\ldots,x_n]$ be the $n$-valuables polynomial ring over a field $K$ and $\mathfrak{S}_n$ the symmetric group of order $n$. We consider the…

Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$ and let $I$ be an $\mathfrak{m}$-primary ideal. We show that there is a non-negative integer $r_I$ (depending only on $I$) such that if $M$ is any non-free maximal…

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such…

Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose that $B$ contains the rationals, $B/M$ is uncountable and $|B| = |B/M|$. Let the minimal prime ideals of $B$ be partitioned into $m \geq 1$ subcollections $C_1,…