Uniform bounds in generalized Cohen–Macaulay rings

Abstract

We establish a uniform bound for the Castelnuovo-Mumford regularity of associated graded rings of parameter ideals in a generalized Cohen-Macaulay ring. As consequences, we obtain uniform bounds for the relation type and the postulation number. Moreover, we show that generalized Cohen-Macaulay rings can be characterized by the existence of such uniform bounds.

Castelnuovo-Mumford Regularity and Hilbert Coefficients of Parameter Ideals

“…Later, this result was extended on the class of m-primary ideals (see [2,10,11]). In [12], Linh-Trung gave a uniform bound for the regularity of associated graded ring with respect to parameter ideal in generalized Cohen-Macaulay ring. Goto-Ozeki [6] used the result of Linh-Trung [12] to establish a uniform bound for Hilbert coefficients of parameter ideal in generalized Cohen-Macaulay ring.…”

“…In [12], Linh-Trung gave a uniform bound for the regularity of associated graded ring with respect to parameter ideal in generalized Cohen-Macaulay ring. Goto-Ozeki [6] used the result of Linh-Trung [12] to establish a uniform bound for Hilbert coefficients of parameter ideal in generalized Cohen-Macaulay ring. They also proved that a ring is generalized Cohen-Macaulay if and only if Hilbert coefficients e i (Q) of parameter ideals are finite, for i = 1, .…”