The finiteness dimension of local cohomology modules and its dual notion

Let aa be an ideal of a commutative Noetherian ring RR and MM a finitely generated RR-module. We explore the behavior of the two notions fa(M)fa(M), the finiteness dimension of MM with respect to aa, and, its dual notion qa(M)qa(M), the Artinianness dimension of MM with respect to aa. When (R,m)(R,m) is local and r≔fa(M)r≔fa(M) is less than fam(M), the mm-finiteness dimension of MM relative to aa, we prove that Har(M) is not Artinian, and so the filter depth of aa on MM does not exceed fa(M)fa(M). Also, we show that if MM has finite dimension and Hai(M) is Artinian for all i>ti>t, where tt is a given positive integer, then Hat(M)/aHat(M) is Artinian. This immediately implies that if q≔qa(M)>0q≔qa(M)>0, then Haq(M) is not finitely generated, and so fa(M)≤qa(M)fa(M)≤qa(M).