Let (R,m) be a commutative Noetherian complete local ring, a an ideal of R, and A an Artinian R-module with N-dimA = d. We prove that if d > 0, then Cosupp(Ha d−1(A)) is finite and if d ≤ 3, then the set Coass(Ha i (A)) is finite for all i. Moreover, if either d ≤ 2 or the cohomological dimension cd(a) = 1 then Ha i (A) is a-coartinian for all i; that is, TorRj (R/a,Ha i (A)) is Artinian for all i, j. We also show that if Ha i (A) is a-coartinian for all i < n, then TorRj (R/a,Ha n (A)) is Artinian for j = 0,1. In particular, the set Coass(Ha n (A)) is finite
