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Local cohomology and Segre products

Jiamin Li, Wenliang Zhang

TL;DR

The paper develops a K"unneth-type framework for local cohomology under Segre products of standard graded rings, establishing an exact sequence for $k=0,1$ and a direct sum of three terms for $k\ge2$ in $H^k_{I\#J}(M\#N)$, with Saturation components $M^{sat}_I$ and $N^{sat}_J$ and a detailed construction via $E^{\bullet}\#F^{\bullet}$. It further derives a sharp lower bound on depth for Segre products and proves its optimality in key cases, while also connecting to Eulerian graded $\mathcal{D}$-modules to obtain precise cohomological dimension formulas, notably $\mathrm{cd}_{I\#J}(R\#S)=\mathrm{cd}_I(R)+\mathrm{cd}_J(S)-1$ in the polynomial setting. The results extend to finite Segre products and provide structural insights into saturation behavior under Segre, enriching the study of local cohomology and depth in non-regular contexts. Altogether, the work unifies homological techniques and $\mathcal{D}$-module methods to advance understanding of cohomological dimensions and depth for Segre-constructed rings.

Abstract

We prove a Künneth formula for local cohomology of a Segré product of graded modules supported in a Segré product of ideals. In order to apply our formula to the study of cohomological dimension, we also investigate asymptotic behaviors of Eulerian graded $\scr{D}$-modules.

Local cohomology and Segre products

TL;DR

The paper develops a K"unneth-type framework for local cohomology under Segre products of standard graded rings, establishing an exact sequence for and a direct sum of three terms for in , with Saturation components and and a detailed construction via . It further derives a sharp lower bound on depth for Segre products and proves its optimality in key cases, while also connecting to Eulerian graded -modules to obtain precise cohomological dimension formulas, notably in the polynomial setting. The results extend to finite Segre products and provide structural insights into saturation behavior under Segre, enriching the study of local cohomology and depth in non-regular contexts. Altogether, the work unifies homological techniques and -module methods to advance understanding of cohomological dimensions and depth for Segre-constructed rings.

Abstract

We prove a Künneth formula for local cohomology of a Segré product of graded modules supported in a Segré product of ideals. In order to apply our formula to the study of cohomological dimension, we also investigate asymptotic behaviors of Eulerian graded -modules.
Paper Structure (5 sections, 19 theorems, 44 equations)

This paper contains 5 sections, 19 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on local cohomology and graded rings
  3. A Künneth formula for local cohomology
  4. A sharp lower bound on depth
  5. Results on graded $\mathcal{D}$-modules with an application to cohomological dimension

Key Result

Theorem 1.1

Let $R,S$ be Noetherian $\mathbb{N}$-graded rings with $R_0=S_0=\Bbbk$ and let $M$ (and $N$) be a $\mathbb{Z}$-graded $R$-module (a $\mathbb{Z}$-graded $S$-module, respectively). Denote by $\mathfrak{m}_R$, $\mathfrak{m}_S$ and $\mathfrak{m}_{R\#S}$ the homogeneous maximal ideals in $R$, $S$, and $R

Theorems & Definitions (42)

  • Theorem 1.1: Goto-Watanabe
  • Theorem 1.2: =Theorem \ref{['Kunneth formula: two components']}
  • Theorem 1.3: =Theorem \ref{['nonzero in negative degree']}
  • Theorem 1.4: = Theorem \ref{['thm: optimal in polynomial case']}
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 32 more