  In this paper we study graded ideals I in a polynomial ring S such that the
numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees
algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not
larger than the projective dimension of S/I and (iii) the K-algebra generated
by some generators of I is a direct summand of S, then f(k) is constant. When I
is a square-free monomial ideal, the above criterion includes as special cases
all the results of a recent paper by Herzog and Vladoiu. In this combinatorial
setting there is a chance that the converse of the above fact holds true.
