Regularity stabilization for the powers of graded $\operatorname {\frak {M}}$-primary ideals

Abstract:

This note provides first a generalization of the stabilization result of Eisenbud and Ulrich for the regularity of powers of an $\mathfrak {m}$-primary ideal to the case of ideals that are not generated in a single degree (see Theorem 1.5). We then partially extend our previous results expressing this stabilization degree in terms of the regularity of a specific graded strand of the Rees ring: The natural extension of the statement holds at least if the stabilization index or the regularity is greater than the number of variables. In any case, a precise comparison is given (see Theorem 4.1).

For simplicity, we do not introduce a graded module as in the work of Eisenbud and Ulrich. It can be done along the same lines, but makes the statements less transparent (see Remark 1.6 where we derive the key point for such an extension).