A generalized Hochster's formula for local cohomologies of monomial ideals

相对Hochster—Takayama公式与Cohen—Macaulay单项理想商理想

Relative Hochster--Takayama formula and Cohen--Macaulay monomial ideal quotients

本文提出相对Hochster—Takayama公式，用于研究单项理想商理想，给出Cohen-Macaulay性判别准则及符号幂滤相关模的性质。

Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals

We give a generalization of Hochster's formula for local cohomologies of square-free monomial ideals to monomial ideals, which are not necessarily square-free. Using this formula, we give combinatorial characterizations of generalized Cohen-Macaulay monomial ideals. We also give other applications of the generalized Hochster's formula.

An equivariant Hochster's formula for $\mathfrak S_n$-invariant monomial ideals

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $\mathfra…

Extremal Betti Numbers and Applications to Monomial Ideals

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alte…

The Monomial Conjecture and Order Ideals

In this article first we prove that a special case of the order ideal conjecture, originating from the work of Evans and Griffith in equicharacteristic, implies the monomial conjecture due to M. Hochster. We derive a necessary and sufficient condition for the validity of this special case in terms certain syzygis of canonical modules of normal domains possessing free summands. We also prove some special cases of this observation.

走向由simplicial poset给出的理想局部余维理论

Toward the theory on local cohomologies at the ideals given by simplicial posets

本文研究了由simplicial poset给出的理想局部余维理论，通过构造和分析渐进注射包络来建立局部余维的理论基础。