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We relate the annihilators of graded components of the canonical module of agraded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximalideal. In particular, we connect them to the core of the maximal ideal. Anapplication of our results characterizes Cayley-Bacharach sets of points interms of the structure of the core of the maximal ideal of their homogeneouscoordinate ring. In particular, we show that a scheme is Cayley-Bacharach ifand only if the core is a power of the maximal ideal.

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