On the Cayley-Bacharach Property

Abstract

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.

Keywords:

• Affine Hilbert function
• canonical module
• Cayley-Bacharach Property
• complete intersection
• Gorenstein ring
• separator

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13H10
• Secondary: 13P99
• 14M05
• 14Q99

Acknowledgments

The third author thanks the University of Passau for its hospitality and support during part of the preparation of this paper.