Annihilators of Graded Components of the Canonical Module, and the Core of Standard Graded Algebras

TL;DR

This paper explores the relationship between the annihilators of graded components of the canonical module and colon ideals of powers of the maximal ideal in graded Cohen-Macaulay rings, linking these to the core of the ideal and characterizing Cayley-Bacharach sets.

Contribution

It establishes a novel connection between the canonical module's graded components and the core of the maximal ideal, providing a new characterization of Cayley-Bacharach schemes.

Findings

Annihilators of canonical module components relate to colon ideals of powers of the maximal ideal.

The core of the maximal ideal characterizes Cayley-Bacharach schemes.

A scheme is Cayley-Bacharach iff its core is a power of the maximal ideal.

Abstract

We relate the annihilators of graded components of the canonical module of a graded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An application of our results characterizes Cayley-Bacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. In particular, we show that a scheme is Cayley-Bacharach if and only if the core is a power of the maximal ideal.