The minimal free resolutions of the Huneke—Ulrich deviation two gorenstein ideals

Abstract

Structure theorems exist for Gorenstein ideals of small grade. A different parameter one can use for organizing ideals is deviation (or complete intersection defect). The deviation of the ideal I is the minimal number of generators of I minus the grade of I. A deviation zero ideal is a complete intersection. Kunz [S] proved that there are no deviation one Gorenstein ideals. Presently people are trying to find all deviation two Gorenstein ideals. See, for example, [3]. A hypersurface section of a grade g 1 deviation two Gorenstein ideal is a grade g deviation two Gorenstein ideal. We consider this to be a trivial example. Very few non-trivial examples of deviation two Gorenstein ideals are known. The Buchsbaum-Eisenbud structure theorem [2] describes all grade three Gorenstein ideals. In particular, every deviation two grade three Gorenstein ideal is generated by the maximal order pfaflians of a 5 x 5 alternating matrix. No non-trivial deviation two Gorenstein ideals of even grade are known. Herzog and Miller [3] have taken a step toward proving that all grade four deviation two Gorenstein ideals are trivial. Let I be a grade four deviation two Gorenstein ideal. If I is a generic complete intersection and I/I* is CohenMacaulay, then I is a hypersurface section. The only known non-trivial deviation 2 Gorenstein ideals of grade at least four are the ideals of Huneke and Ulrich [4], which are the ideals studied in this paper. For each integer s, let y’“’ be a 1 x s matrix of indeterminates, X’“’ be an s x s alternating matrix of indeterminates, g’“’ be the product y(“X(‘), and ct.‘) be the sequence gp),..., g’,“! , . Let A, be the ideal of