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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.Thus the main goal of this paper is to study, quite broadly, the interplay between annihilators of graded components of ω on the one hand and colon ideals of powers of m on the other hand.More generally, let R be a standard graded Cohen-Macaulay algebra over a field k, of dimension d > 0, and write m for its homogeneous maximal ideal and ω = ω R for its graded canonical module. By a = a(R) we denote the a-invariant of R, which is the negative of the initial degree of ω. Recall that if J is any ideal generated by a linear system of parameters, then 'J is a minimal reduction of m with reduction number r J (m) = a + d ', which simply means that m i = J i−j m j for every i ≥ j ≥ a + d, but for no smaller j. It is easy to see that J i : m j = m i−j+a+d for every i and j ≥ a + d, provided R is Gorenstein or, more generally, level, which means that ω is generated by homogeneous elements of the same degree, ω = [ω] −a R. Under these assumptions, the R-submodules [ω] t R of ω generated by the homogeneous elements of a fixed degree t are all faithful as long as t ≥ −a; this is obvious since ω = [ω] −a R is faithful and there exists a form of positive degree regular on ω. The same holds if R is a domain because a suitable shift of ω embeds into R. Without the additional assumptions on the ring, neither the statement about the colon ideals nor the one about the graded components of the canonical module are true, but there is a close relationship between the two conditions. In fact in one of our main results we express, more generally, the annihilators of graded components of ω in terms of colon ideals of powers of m,for every t and j ≥ a + d (see Theorem 2.7). Conversely, this allows one to write the colon ideals J i : m j for any i and j ≥ a + d in the formwhere m i−j+a+d is the 'expected' part and N is an ideal of height zero that is generated in degrees ≤ i − j + a + d − 1 and can be described as(see Corollaries 2.14 and 2.15). To prove these results it is useful to replace the colon ideals J i : R m j in R by the corresponding colon ideals J i ω : ω m j in ω, which we then relate to truncations [ω] ≥i−j+d of ω and to graded components of the canonical module Ω of the extended Rees ring of m. This is done in one of our main technical results. There we also use a bound on the regularity of Ω to show that Theorem 2.3 and, for related results, [10,14,11,4]).Our results imply, for instance, that [ω] −a R is faithful, equivalently, [ω] t R is faithful for every t ≥ −a, if and only if J i : m j = m i−j+a+d for every i and ...