Castelnuovo-Mumford regularity and degrees of generators of graded submodules (Q1409638)

Let \(A\) be a homogeneous graded ring over an Artinian ring and \(M\) a finitely generated graded \(A\)-module. The Castelnuovo-Mumford regularity \(\operatorname {reg} M\) of \(M\) is an important invariant of~\(M\). For example \(\operatorname {reg} M\) is an upper bound of the generating degree \(d(M)\), that is, the largest degree of a minimal homogeneous generator of~\(M\), if \(A\) is a polynomial ring over a field. In the present paper, the author gives an upper bound of \(\operatorname {reg} M\) by using \(d(M)\). Let \(M\) be a finitely generated graded \(A\)-module and \(L\) a graded submodule of~\(M\). The author gives a polynomial \(P(x)\), which depends only on the Hilbert polynomial of~\(M\), such that \(\operatorname {reg} L \leq P(\max\{d(L), \operatorname {reg} M + 1\})\).