Linkage by Generically Gorenstein Cohen-Macaulay Ideals.

Abstract

In a Gorenstein local ring R, two ideals A and B are said to be linked by an ideal I if the two relations A = (I: B) and B = (I: A) hold. In the case that I is a complete intersection, or a Gorenstein ideal, it is known that linkage preserves the Cohen-Macaulay property. That is, if A is a Cohen-Macaulay ideal, then so is B. However, if I is allowed to be a generically Gorenstein, Cohen-Macaulay ideal, easy examples show that this type of linkage does not preserve the Cohen-Macaulay property. The primary purpose of this work is to investigate how much of the Cohen-Macaulay property this more general kind of linkage does preserve. By associating to I an auxiliary ideal J, for which J/I is isomorphic to the canonical module $K\sb{R/I}$ of R/I, we are able to give complete conditions for various types of Cohen-Macaulay conditions that B possesses, when B is linked by I to a Cohen-Macaulay ideal A. In particular, we give a criterion for B to be a Cohen-Macaulay ideal, and when it is not, for R/B to have high depth. We also give a description in some cases of the non-Cohen-Macaulay locus of R/B, including a calculation of its dimension. In these cases, there is an interesting relationship between the depth of R/B and the dimension of the non-Cohen-Macaulay locus. Finally, we give some remarks on a construction of a free resolution of R/B from given resolutions.