Tameness and Artinianness of Graded Generalized Local Cohomology Modules

TL;DR

This paper investigates the properties of graded generalized local cohomology modules, including their vanishing behavior, bounds on their degrees, conditions for tameness, and Artinian properties of certain submodules, advancing understanding in graded cohomology theory.

Contribution

It provides new conditions for vanishing, bounds, tameness, and Artinian properties of graded generalized local cohomology modules, extending existing theoretical frameworks.

Findings

Homogeneous components of cohomology modules vanish for large degrees.

Established bounds for the supremum of end degrees of cohomology modules.

Proposed sufficient conditions for tameness and Artinian properties.

Abstract

Let $R=\bigoplus_{n\geq 0}R_n$, $\fa\supseteq \bigoplus_{n> 0}R_n$ and $M$ and $N$ be a standard graded ring, an ideal of $R$ and two finitely generated graded $R$-modules, respectively. This paper studies the homogeneous components of graded generalized local cohomology modules. First of all, we show that for all $i\geq 0$, $H^i_{\fa}(M, N)_n$, the $n$-th graded component of the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\fa$, vanishes for all $n\gg 0$. Furthermore, some sufficient conditions are proposed to satisfy the equality $\sup\{\en(H^i_{\fa}(M, N))| i\geq 0\}= \sup\{\en(H^i_{R_+}(M, N))| i\geq 0\}$. Some sufficient conditions are also proposed for tameness of $H^i_{\fa}(M, N)$ such that $i= f_{\fa}^{R_+}(M, N)$ or $i= \cd_{\fa}(M, N)$, where $f_{\fa}^{R_+}(M, N)$ and $\cd_{\fa}(M, N)$ denote the $R_+$-finiteness dimension and the cohomological dimension of $M$ and $N$ with respect to $\fa$, respectively. We finally consider the Artinian property of some submodules and quotient modules of $H^j_{\fa}(M, N)$, where $j$ is the first or last non-minimax level of $H^i_{\fa}(M, N)$.