Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four

Abstract

Recently Buchsbaum and Eisenbud [3] exploited the algebra structure on a finite free resolution of a Gorenstein ideal of codimension three to obtain a complete determinantal description of such an ideal. As they pointed out, the study of algebra structures on resolutions has for the most part been confined to the (generally infinite) minimal free resolution of the residue field of a local ring or the Koszul resolution of an ideal generated by a regular sequence. They proposed, however, to extend the scope of the study to all minimal free resolutions of cyclic modules. Khinich [1] furnished an example of a grade four ideal I for which the minimal resolution of R/I does not admit the structure of an associative, differential, graded commutat ive algebra (DGC algebra). Khinich's ring R/I is Cohen-Macaulay, but not Gorenstein. We conjecture that minimal finite free resolutions of Gorenstein factor rings R/a admit D G C algebra structures. In this paper we establish the conjecture for R a Gorenstein local ring in which 2 is a unit and a a Gorenstein ideal of grade (or height) four. We begin by clarifying what we mean by an algebra structure on a resolution. Let