A general resolution for grade four Gorenstein ideals

Gorenstein rings occur in a multitude of different guises: as rings of invariants, as coordinate rings of certain determinantal varieties and symmetric semigroup curves, as representatives of linkage classes, and so on. In an attempt to unify this motley collection of examples (at least for finite projective dimension) one seeks a generic free resolution whose specializations yield all examples of given embedding codimension. The present paper describes a resolution for codimension four, not generic, but general enough to encompass many diverse examples. The structure of this resolution is intimately related to the differential, graded, commutative algebra that it supports, and to the deformation theory of codimension four Gorenstein algebras. These ideas are brought together in the determination of the singular locus of certain codimension four Gorenstein varieties. More generally they suggest a classification of codimension four Gorenstein rings that begins to impose some order on the examples.