Residual intersections in Cohen-Macaulay rings

Residual intersections generalize the notion of linkage. Let R be a Noetherian ring and s be an integer. Two proper ideals I and J of R are linked if J = a : I and I = a : J where a is generated by a regular sequence contained in I ∩ J. An R-ideal J is an s-residual intersection of I if there exists an s-generated ideal a ⊂ I such that J = a : I and ht J ≥ s. Two central questions in residual intersection theory are: for a residual intersection J of I, when is R/J Cohen-Macaulay, and what is the canonical module for R/J .;Artin and Nagata introduced the concept of residual intersections in 1972. Peskine and Szpiro outlined when two linked ideals are Cohen-Macaulay in 1974, but it is not clear when residual intersections are Cohen-Macaulay. Huneke and Ulrich studied residual intersection theory in the case where R is Gorenstein in 1988. We obtain some answers to two above questions in the case where R is Cohen-Macaulay.;In proving these results, we develop for a finite R-module M the following concepts: M-s-residually S2, M-ANs, M-sliding depth, and M-strongly Cohen-Macaulay. In our first theorem, we relate the vanishing of local cohomology modules to the M-s-residually S2 property. Our main theorem gives answers to the two central questions of residual intersection theory. We apply the main result to determine properties of minimal reductions and reduction numbers. To conclude, let a be a minimal reduction of an ideal I, and assume that a can be minimally generated by ℓ elements. We prove that the extended Rees ring of a is isomorphic to B/J where B is the polynomial ring R[T1,..., Tℓ, U] and J is a residual intersection of I. Then we prove B/J is Cohen-Macaulay and its canonical module has an "expected form."...