Duality and socle generators for residual intersections
David Eisenbud, Bernd Ulrich · Journal für die reine und angewandte Mathematik (Crelles Journal) · 2018
Abstract We prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt. Suppose that I is an ideal of codimension g in a Gorenstein ring, and J ⊂ I {J\subset I} is an ideal with s = g + t {s=g+t} generators such that K := J : I {K:=J:I} has codimension s. Let I ¯ {{\overline{I}}} be the image of I in R ¯ := R / K {{\overline{R}}:=R/K} . In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring R ¯ ⊕ I ¯ ⊕ ⋯ ⊕ I ¯ t + 1 {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules I ¯ u {{\overline{I}}{}^{u}} and I ¯ t + 1 - u {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into I ¯ ≅ t +