The finiteness dimension of local cohomology modules and its dual notion

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).