Resolutions of almost complete intersections

Abstract

All rings we consider in this paper are commutative, Noetherian, and with identity. Let R be either a local ring or an N-graded ring whose degree 0 piece is a field (in this case all ideals and elements are homogeneous). Let & denote the maximal ideal in the local case and the irrelevant maximal ideal in the graded case. We study minimal resolutions of ideals Z of R satisfying the following conditions: (i) Z= (a,, . . . . a, ~ 1, a,) and a,, . . . . a, _, is a regular sequence, (ii) Z is not a complete intersection, and (iii) there is b,~R-(u~ ,..., a,-,) such that (a, ,..., u,~,):u,,=(u ,,..., a,-,,b,) and (a I, .-., unp,):b,=(u, )...) a,-,, a,). Condition (i) is merely a definition for almost complete intersection. In the language of [PSI, an ideal satisfying conditions (i) and (ii) is an almost complete intersection which is linked to an almost complete intersection.) We will prove: