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Let \fa be an ideal of a commutative Noetherian ring R and M a finitelygenerated R-module. We explore the behavior of the two notions f_{\fa}(M), thefiniteness 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 andr:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of Mrelative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filterdepth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M hasfinite dimension and H^i_{\fa}(M) is Artinian for all i>t, where t is a givenpositive integer, then H^t_{\fa}(M)/\fa H^t_{\fa}(M) is Artinian. Itimmediately implies that if q:=q_{\fa}(M)>0, then H^q_{\fa}(M) is not finitelygenerated, and so f_{\fa}(M)\leq q_{\fa}(M).

The finiteness dimension of local cohomology modules and its dual notion