Castelnuovo's regularity and multiplicity
The main result connects several invariants of an equidimensional locally Cohen-Macaulay graded \(K\)-algebra of dimension \(d\geq 1\) and an ideal \(\mathfrak q\) generated by a system of parameters of degree \(1\): \[ \text{length}(A/{\mathfrak q})-e_ 0({\mathfrak q};A)-\text{reg}(A/{\mathfrak q})\leq I(A)- \text{reg}(A), \tag{*} \] where \(e_ 0\) denotes multiplicity, \(I(A)=\sum^{d-1}_{j=0}\binom{d-1}{j}\text{length}(H^ j_{{\mathfrak m}}(A))\) (\(\mathfrak m\) the irrelevant maximal ideal), and \(\text{reg}(A)\) is the Castelnuovo regularity, i.e. the smallest number \(m\) such that the \(j\)-th syzygy module of \(\mathfrak a\) is generated by elements of degree \(m+j\), \(A\) being given in the form \(S/\mathfrak a\), \(S=K[X_ 0,\ldots,X_ n]\). Moreover it is shown that one has equality above if \(A\) is Buchsbaum, and that equality for every \(\mathfrak q\) implies that \(A/H^ 0_{{\mathfrak m}}(A)\) is Buchsbaum. As an application the authors give a bound on \(\text{reg}(A)\) (and thus on the degrees of the generators of the syzygies of \(A\)) in the following manner: Let \(A=S/\mathfrak a\) as above and \([\mathfrak a]_ i=0\) in degrees \(i\leq t\); then \[ \text{reg}(A)\leq \deg(\mathfrak a)-\binom{t+\text{codim}(\mathfrak a)}{ t}+t+I(A). \] Under the slightly stronger hypothesis that \(A\) is arithmetically Cohen-Macaulay, this result has been previously proved by \textit{P. Maroscia} and the authors [Math. Ann. 277, 53--65 (1987; Zbl 0634.14029)]. Furthermore for \(\dim(A)=1\) a precise condition on \(H^ 0_{\mathfrak m}(A)\) is given under which one has equality in (*) for all \(\mathfrak q\). The paper concludes with several examples which show for example that equality in (*) for some \(\mathfrak q\) does not force \(A\) to be Buchsbaum and that \(A/H^ 0_{\mathfrak m}(A)\) being Buchsbaum does not imply equality in (*) for all \(\mathfrak q\).
invariants of an equidimensional locally Cohen-Macaulay graded algebra
scientific article; zbMATH DE number 4089691