The finiteness dimension of local cohomology modules and its dual notion

TL;DR

This paper investigates the properties of finiteness and Artinianess dimensions of local cohomology modules over Noetherian rings, establishing new relationships and conditions under which these modules exhibit specific finiteness or Artinian properties.

Contribution

It introduces dual notions of finiteness and Artinianess dimensions in local cohomology, providing new results on their behavior and interrelations in Noetherian rings.

Findings

H^r_{a}(M) is not Artinian when f_{a}(M)<f_{a}^{m}(M)

If H^i_{a}(M) is Artinian for all i>t, then H^t_{a}(M)/a H^t_{a}(M) is Artinian

q_{a}(M)>0 implies H^q_{a}(M) is not finitely generated

Abstract

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has finite dimension and H^i_{\fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{\fa}(M)/\fa H^t_{\fa}(M) is Artinian. It immediately implies that if q:=q_{\fa}(M)>0, then H^q_{\fa}(M) is not finitely generated, and so f_{\fa}(M)\leq q_{\fa}(M).