On the Cayley-Bacharach Property

TL;DR

This paper extends the Cayley-Bacharach property to 0-dimensional affine algebras over arbitrary fields, providing characterizations and algorithms for verification, and links it to Gorenstein rings and Hilbert functions.

Contribution

It introduces a generalized framework for the Cayley-Bacharach property applicable to broader algebraic structures and offers explicit algorithms for its assessment.

Findings

Characterization of the Cayley-Bacharach property via the canonical module.

Algorithms for checking the property directly and through Gorenstein conditions.

Linking the property to the symmetry of affine Hilbert functions and Gorenstein rings.

Abstract

The Cayley-Bacharach property, 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 Cayley-Bacharach property 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 Cayley-Bacharach property and the symmetry of their affine Hilbert function, as well as by the strict Cayley-Bacharach property and the last difference of their affine Hilbert function.