Related papers: Length, multiplicity, and multiplier ideals

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} =…

Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two…

This article is concerned with the number of generators of perfect ideals J in regular local rings (R,m). If J is sufficiently large modulo $m^n$, a bound is established depending only on n and the projective dimension of J. More ambitious…

Let $I$ be an arbitrary nonzero squarefree monomial ideal of dimension $d$ in a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. Let $\mu$ be the number of associated primes of $S/I$ of dimension $d$. We prove that the multiplicity of…

Let $\mathcal A$ be a simple, $\sigma$-unital, non-unital, non-elementary C*-algebra and let $I_{min}$ be the intersection of all the ideals of $\mathcal M(\mathcal A)$ that properly contain $\mathcal A$. $I_{min}$ coincides with the ideal…

A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every…

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $\omega_R$ and the property that the socle…

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$.…

We show that Serre's Intersection Multiplicity Conjecture holds for a formal power series ring A over a complete, two-dimensional regular local ring R. From this, we deduce the corresponding result for the local rings of any scheme X which…

In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the…

This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V)xGL(W)-invariant ideal in the symmetric algebra S of the tensor product of V with the dual of W, where V and W…

A zero-dimensional polynomial ideal may have a lot of complex zeros. But sometimes, only some of them are needed. In this paper, for a zero-dimensional ideal $I$, we study its complex zeros that locate in another variety $\textbf{V}(J)$…

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$.…

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…

In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a…

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s to positive characteristic such that the action of the Frobenius morphism on the top…

Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax…

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal of degree $d\leq 2$. We show that $(I^{k+1}:I)=I^k$ for all $k\geq 1$ and we disprove a motivation question that was appeared in…

By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study…

Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper answers to the question when mixed multiplicities of $M$ are positive and…