Tame Loci of Certain Local Cohomology Modules

TL;DR

This paper investigates the tame loci of local cohomology modules of graded modules over Noetherian rings, focusing on primes of height up to three, to understand their structural properties in algebraic geometry.

Contribution

It introduces the concept of tame loci at level i in codimension ≤ 3 for local cohomology modules, providing new insights into their behavior over graded rings.

Findings

Characterization of tame loci in codimension ≤ 3.

Identification of conditions for tameness of local cohomology modules.

Structural properties of local cohomology modules in specific codimensions.

Abstract

Let $M$ be a finitely generated graded module over a Noetherian homogeneous ring $R = \bigoplus_{n \in \mathbb{N}_0}R_n$. For each $i \in \mathbb{N}_0$ let $H^i_{R_{+}}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+ = \bigoplus_{n > 0} R_n$ of $R$, furnished with its natural grading. We study the tame loci $\ft^i(M)^{\leq 3}$ at level $i \in \mathbb{N}_0$ in codimension $\leq 3$ of $M$, that is the sets of all primes $\fp_0 \subset R_0$ of height $\leq 3$ such that the graded $R_{\fp_0}$-modules $H^i_{R_{+}}(M)_{\fp_0}$ are tame.