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- Title
Duality and socle generators for residual intersections.
- Authors
Eisenbud, David; Ulrich, Bernd
- 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 is an ideal with s = g + t generators such that K := J : I has codimension s. Let I ¯ be the image of I in 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 is a Gorenstein ring, and that the modules I ¯ u and I ¯ t + 1 - u are dual to one another via the multiplication pairing into I ¯ ≅ t + 1 ωR ¯. In the second part of the paper we study the analogue of residue theory, and prove that, when R / K is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of I t + 1 / J I t ≅ ωR/K is generated by a Jacobian determinant.
- Subjects
GORENSTEIN rings; K-theory; ALGEBRA; MULTIPLICATION; INTERSECTION theory
- Publication
Journal für die Reine und Angewandte Mathematik, 2019, Vol 2019, Issue 766, p183
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2017-0045