Algebra structures on some canonical resolutions

Abstract

The study of algebra structures on finite free resolutions of cyclic modules begins with Buchsbaum and Eisenbud [B-El]. The classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field [Gu], complete intersections (Koszul complex), modules of homological dimension atmost 3 [B-El], Gorenstein of codimension four [K-M], and Herzog algebras [K-M23. Avramov [Al] gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure. Let R be a noetherian local ring with maximal ideal m. In this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules R/I, namely, when Z is of the form Jk, where J is an ideal generated by a regular sequence, and when Z is the ideal of maximal minors of a generic n x m matrix, provided R contains the rationals. In [Al] Avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non-zero obstructions. The general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module. When Z is an ideal generated by a regular sequence in R and M= R/Z” for any positive integer k, then these obstructions are all zero. Therefore, Avramov and Schlessinger asked whether the minimal resolutions of R/Z’ admit an algebra structure. Corollary 3.6 of this paper provides an affirmative answer to this question.