Hilbert functions, residual intersections, and residually \(S_2\) ideals (Q2711523)

Let \(R\) be a polynomial ring over a field, let \(I\subseteq R\) be a homogeneous ideal, and let \({\mathfrak a}\subseteq I\) be a homogeneous ideal generated by \(s\) forms with \(\text{codim}({\mathfrak a}:I)\geq\) s. In this paper, the authors obtain conditions for when the Hilbert function of \(R/{\mathfrak a}\) or of \(R/({\mathfrak a}:I)\) is determined by \(I\) and the degrees of the forms. This is the case for perfect ideals of codimension \(2\) or Gorenstein of codimension \(3\). The notion of ``(weakly) \(s\)-residually \({S_2}\)'' is introduced and it is shown that if \(I\) satisfies \(G_s\), i.e. for each prime ideal \(\mathfrak p\) containing \(I\) with \(\text{codim } {\mathfrak p}\leq s-1\), the minimal number of generators of \(I_{\mathfrak p}\) is at most codim \({\mathfrak p}\) , and it is weakly \((s-1)\)-residually \(S_2\), then the Hilbert function of \(R/{\mathfrak a}\) or of \(R/({\mathfrak a}:I)\) is determined, up to \(R\)-equivalences, by \(I\) and the degrees of the homogeneous generators of \(\mathfrak a\). NEWLINENEWLINENEWLINEThe main result of the paper establishes that the vanishing of local cohomology implies the residually \(S_2\)-property. At the end the authors show that general projection can create ideals that are \(s\)-residually \(S_2\) and they obtain examples that illustrate the theory.