It is well known that for a Noetherian ring R, an ideal I of R, and M a finitely generated R-module, the local cohomology modules HiI(M) are not always finitely generated. On the other hand, if R is local and m is its maximal ideal, then Him(M) are Artinian modules, which is equivalent to the following two properties:

(i) SuppR(Him(M)) ⊆{m};

(ii) the vector space HomR(k, Him(M)) has finite dimension over k, where k = R/m.

Taking these facts into account, Grothendieck [9] made the following conjecture.