Maximally differential ideals in regular local rings

Hilbert functions of monomial ideals containing a regular sequence

Let $M$ be an ideal in $K[x_1,...,x_n]$ ($K$ is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing $M$ satisfy the Eisenbud-Green-Harris conjecture and moreover prove that the Cohen-Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that $h$-vector of Cohen-Macaulay simplicial complex $Δ$ is the $h$-vector of Cohen-Macaulay $(a_1-1,...,a_t-1)$-balanced simplicial complex where $t$ is the height of the Stanley-Reisner ideal of $Δ$ and $(a_1,...,a_t)$ is the …

Regular local rings of dimension four and Gorenstein syzygetic prime ideals

Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.

二维正则局部环中完全理想乘积的核心与伴随理想研究

On the core and adjoint of the product of complete ideals in two-dimensional regular local rings

该研究借助完全理想的联合约化，给出二维正则局部环中完全理想乘积的核心、伴随理想表达式及其余长度计算方法，强化了Rees与Sally对Briançon-Skoda定理的推广结果。

Regular Sequences On Squares Of Monomial Ideals

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular sequences on $R/I^2$. We then obtain a criterion for when $\depth R/I^2>1$ and a lower bound on $\depth R/I^2$.

Trace ideal and annihilator of Ext and Tor of regular fractional ideals, and some applications

Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we answer in one-dimensional local analytically unramified case, a question raised by the present author and R. Takahashi. As another application, we give an alternative proof of a recent result of Ö. Esentepe that for one-dimensional analytically unramified Gorenstein local rings, the cohomology annihilator of Iyengar and Takahashi coincides with the conducto…

The core of a module and the adjoint of an ideal over a two dimensional regular local ring

The core of an module is the intersection of all its reductions. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two dimensional regular local ring is the product of the module and the adjoint of an ideal. This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules $M$ and $N$ over a two-dimensional regular local ring with $M\subset N$ and $M^{**}=N^{**}$, the core of $M$ is containe…

Frobenius test exponent for ideals generated by filter regular sequences

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 uniform bound for Frobenius test exponents of ideals generated by filter regular sequences of length at most $t$.

Adjoint ideals and Gorenstein blowups in two-dimensional regular local rings

In this article we investigate when a complete ideal in a two-dimensional regular local ring is a multiplier ideal of some ideal with an integral multiplying parameter. In particular, we show that this question is closely connected to the Gorenstein property of the blowup along the ideal.

组合学中产生的二项式理想的正则性和深度

Regularity and depth of binomial ideals arising from combinatorics

研究组合学中由相邻2-子式及有限格产生的二项式理想的正则性与深度。

The Center and Radius of the Regular Graph of Ideals

The regular graph of ideals of the commutative ring $R$, denoted by ${Γ_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is proved that the radius of $Γ_{reg}(R)$ equals $3$. The central vertices of $Γ_{reg}(R)$ are determined, too.

Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein-Lazarsfeld-Smith and Hochster-Huneke.

Bounds on the Stanley depth and Stanley regularity of edge ideals of clutters

Let $I$ be the edge ideal of a clutter $\mathcal{C}$ in a polynomial ring $S$. In this paper, we present estimations of the Stanley depth of $I$ as well as the Stanley regularity of $S/I$, in terms of combinatorial data from the clutter $\mathcal{C}$.

Bounds for the regularity of product of edge ideals

Let $I$ and $J$ be edge ideals in a polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$. In this paper, we obtain a general upper and lower bound for the Castelnuovo-Mumford regularity of $IJ$ in terms of certain invariants associated with $I$ and $J$. Using these results, we explicitly compute the regularity of $IJ$ for several classes of edge ideals. Let $J_1,\ldots,J_d$ be edge ideals in a polynomial ring $R$ with $J_1 \subseteq \cdots \subseteq J_d$. Finally, we compute the precise expression for the regularity of $J_1 J_2\cdots J_d$ when $d \in \{3,4\}$ and $J_d$ is the …

Regularity of S_n-invariant monomial ideals

For a polynomial ring S in n variables, we consider the natural action of the symmetric group S_n on S by permuting the variables. For an S_n-invariant monomial ideal I in S and j >= 0, we give an explicit recipe for computing the modules Ext^j(S/I,S), and use this to describe the projective dimension and regularity of I. We classify the S_n-invariant monomial ideals that have a linear free resolution, and also characterize those which are Cohen-Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and anoth…

正则局部环中有限支撑理想的一个消失定理

A vanishing theorem for finitely supported ideals in regular local rings

本文证明了正则局部环中有限支撑理想的上同调消失性质，推导了其伴随的阶、与严格变换的可交换性及正规理想的约化数等结果，得到了混合特征下未知的Briancon-Skoda型结论。

Hilbert functions of d-regular ideals

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let $p \geq 0$ and $d>0$ be integers. If the base field is a field of characteristic 0 and there is a graded ideal $I$ whose projective dimension $\mathrm{proj\ dim}(I)$ is smaller than or equal to $p$ and whose regularity $\mathrm{reg}(I)$ is smaller than or equal to $d$, then there exists a monomia…

Integrally closed ideals in two-dimensional regular local rings are multiplier ideals

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field, there is in fact no difference between "multiplier" and "integrally closed" (or "complete.") However, among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals) the only simple complete one…

Jumping numbers of a simple complete ideal in a two-dimensional regular local ring

In this manuscript we shall give an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, we obtain a formula for the jumping numbers of an analytically irreducible plane curve. We then show that the jumping numbers determine the equisingularity class of the curve.

Regularity and algebraic properties of certain lattice ideals

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. The…