Castelnuovo–Mumford Regularity and Powers

Abstract

This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo–Mumford regularity for standard graded rings \(R=\bigoplus _{i\in {\mathbb N}} R_i\) over general base rings R0. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers Iv of an homogeneous ideal I of R is eventually a linear function in v. Finally we show how the flexibility of the definition of the Castelnuovo–Mumford regularity over general base rings can be used to give a simple proof of a result proved by the authors in “Maximal minors and linear powers”.