Eisenbud-Goto inequality for Stanley-Reisner rings (Q2717188)

Let \(A\) be a polynomial ring in \(n\) variables, each of degree one, over a field \(k\). To each simplicial complex \(\Delta\), with \(n\) vertices, there is associated an ideal \(I_{\Delta}\) and a Stanley-Reisner ring \(k[\Delta]=A/I_{\Delta}\). The ideal \(I_{\Delta}\) is generated by the square-free monomials whose corresponding vertex sets do not constitute a face of \(\Delta\).NEWLINENEWLINENEWLINEThe main result of this paper is that if \(\Delta\) is pure and strongly connected, then the Castelnuovo-Mumford regularity of \(I_{\Delta}\) is at most equal to \(\deg k[\Delta]-\operatorname {codim} k[\Delta]+1\). The proof uses Hochster's theorem from 1977 which calculates the graded Betti numbers of the \(A\)-module \(k[\Delta]\) in terms of the reduced simplicial homology of various subcomplexes of \(\Delta\). The proof also uses a recent result by the author which shows that the regularity of \(I_{\Delta}\) is equal to the projective dimension of the Stanley-Reisner ring for the Alexander dual of \(\Delta\).NEWLINENEWLINENEWLINEOne application of the above result is to prove the exact same inequality for the prime ideal \(P\) of \(A\) in the case that the initial ideal of \(P\), with respect to some term order, defines a reduced ring. Eisenbud and Goto conjectured that this inequality holds for prime ideals.NEWLINENEWLINENEWLINEThe final result in the paper is a classification of the simplicial complexes for which the above inequality becomes an equality.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].