Abstract
In this paper we study graded ideals I in a polynomial ring S such that the numerical function k↦depth(S/Ik) 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 homogeneous generators of I is a direct summand of S, then depth(S/Ik) is constant. All the ideals with constant depth-function discovered by Herzog and Vladoiu in [11] satisfy the criterion given above. In the contest of square-free monomial ideals, there is a chance that a converse of the previous fact holds true.
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