On top dimensional Lyubeznik numbers in mixed characteristic

混合特征的Lyubeznik数

Lyubeznik numbers in mixed characteristic

这份手稿定义了一个新的家庭的不变量，类似于Lyubeznik数，与任何局部环的剩余域有素特征。特别地，正如它们的命名所暗示的，这些“混合特征的Lyubeznik数”被定义为所有混合特征的局部环。这些新的不变量具有与等特征线中的性质相似的性质。值得注意的是，混合特征中的“最高”Lyubeznik数是一个定义明确的概念。虽然对于某些等特征p>0的局部环，混合特征的Lyubeznik数与等特征的Lyubeznik数是相同的，但我们也给出了它们不同的一个例子.

Lyubeznik numbers and injective dimension in mixed characteristic

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of $\mathbb{Z}$-algebras, we also exhibit an exa…

Cohomological dimension, Lyubeznik numbers, and connectedness properties in mixed characteristic

We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.

Lyubeznik numbers of local rings and linear strands of graded ideals

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for Lyubeznik numbers. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik…

局部上同调模的第二消失定理

The Second Vanishing Theorem for Local Cohomology Modules

本文证明了非分歧正则局部环局部上同调模的第二消失定理并给出素特征$p$下的新证明，将等特征局部环最高Lyubeznik数的拓扑刻画推广到混合特征，还得到了混合特征下局部上同调维数的上界。

关于Hellus--Lyubeznik--Yildirim局部上同调模猜想

On Hellus--Lyubeznik--Yildirim's conjecture of local cohomology modules

本文研究Hellus-Lyubeznik-Yildirim猜想，在深度为1、特定条件下深度为2、平方自由单项理想及混合特征离散赋值环形式幂级数环等情形下给出部分肯定答案。