  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.
