Related papers: Rees' theorem for filtrations, multiplicity functi…

Let $(R,\frak m)$ be a Noetherian local ring of prime characteristic $p>0$, and $t$ an integer such that $H_{\frak m}^j(R)/0^F_{H^j_{\frak m}(R)}$ has finite length for all $j<t$. The aim of this paper is to show that there exists an…

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…

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation…

We prove that if $I$ is a monomial ideal with linear quotients in a ring of polynomials $S$ in $n$ indeterminates and $\operatorname{depth}(S/I)=n-2$, then $\operatorname{sdepth}(S/I)=n-2$ and, if $I$ is squarefree,…

Let $S = \mathbb{C}[x_{i,j}]$ be the ring of polynomial functions on the space of $m \times n$ matrices, and consider the action of the group $\mathbf{GL} = \mathbf{GL}_m \times \mathbf{GL}_n$ via row and column operations on the matrix…

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…

Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert…

We search for principal ideals. As a sample, let $R$ be a strongly-normal, almost-factorial, and complete-intersection local ring with a prime ideal $P$ of height one. If $depth(R/ P)\geq dim R-2$, we show $P$ is principal. As an immediate…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $I$ an ideal of $R$. We study how local cohomology modules with support in $\mathfrak{m}$ change for small perturbations $J$ of $I$, that is, for ideals $J$ such that $I\equiv J\bmod…

Ananyan and Hochster proved the existence of a function $\Phi(m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}(I) \le \Phi(m,d)$. Relatedly,…

For a Noetherian commutative ring $R$, let $H^i_I(R)$ be the $ i$-th local cohomology module of $R$ with respect to $I$. In \cite{Hel-08}, Hellus posed the question of identifying rings $R$ such that $\operatorname{injdim}_R…

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $\gamma_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is…

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to…

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1}…

Given a reduced, local ring $R$ and an ideal $\mathfrak{a}$ of positive height, we give a decomposition of the test module, $\tau(\omega_T, t^{-\lambda})$, of the extended Rees algebra, $T =R[\mathfrak{a} t, t^{-1}]$. In particular, the…

Let $(R,P)$ be a commutative, local Noetherian ring, $I$, $J$ ideals, $M$ and $N$ finitely generated $R$-modules. Suppose $J + ann_R M + ann_R N$ is $P$-primary. The main result of this paper is Theorem 6, which gives necessary and…

In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let $I$ be a homogeneous ideal in a polynomial ring over a field of…

The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis…

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals…

We study multiplicities of jumping numbers of multiplier ideals in a smooth variety of arbitrary dimension. We prove that the multiplicity function is a quasi-polynomial, hence proving that the Poincar\'e series is a rational function. We…