Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings (Q1891216)

Recently, the problem of finding better upper bounds for the so-called Castelnuovo-Mumford regularity of projective varieties and graded modules over graded rings has been investigated by many authors. The authors of this paper consider here the problem for generalized Cohen-Macaulay modules. Let \(R = K[R_1] = \bigoplus_{n \geq 0} R_n\) be a noetherian graded ring over an infinite field \(K\). For a \(d\)-dimensional finitely generated graded \(R\)-module \(M = \bigoplus_{n \in \mathbb{Z}} M_n\), put \(a_i (M) = \max \{n \mid [H_m^i (M)]_n \neq 0\}\), and \(\text{reg}_n (M) = \max \{i + a_i (M) \mid n \leq i \leq d\}\), where \(m\) is the maximal homogeneous ideal of \(R\). We say that \(M\) is a \(k\)-Buchsbaum module if \(m^k H_m^i (M) = 0\) for any \(i < d\) and that \(m^k\) is a standard \(M\)-ideal if any homogeneous system of parameters \(x_1, \ldots, x_d\) for \(M\) contained in \(m^k\) is standard, i.e., \((x_1, \ldots, x_d) H_m^i (M/(x_1, \ldots, x_j) M) = 0\), \(i + j < d\). Under these assumptions on \(M\), the main theorem of this paper gives, using properties of standard system of parameters, new upper bounds for \(\text{reg}_n (M)\) in terms of \(a_d (M)\) and \(k\). For example, if \(R\) is a Buchsbaum ring with \(\dim (R) = d\), then we have \(\text{reg}_0 (R) \leq a_d (R) + d - 1\). As corollaries, upper bounds for \(\text{reg}_n (R)\) in terms of the multiplicity of \(R\) or the reduction number of \(m\) are given.