Multiplicative Structure on Resolutions of Algebras Defined by Herzog Ideals

Abstract

There is only a small collection of algebras whose finite free resolutions are known to admit the structure of an associative, differential, graded, commutative algebra. Avramov and Hinič have shown that there are algebras (even Gorenstein ones) whose resolutions admit no such structure. We enlarge the affirmative list by showing that Herzog's algebras k0(g) defined by ‘a sequence and a matrix’ have resolutions that do support associative DGC structure. In fact, we do this for the versal deformations k(g). The proof, which is valid in arbitrary codimension g ⩾ 4, rests on our earlier ‘big from small’ construction and involves combining two Koszul resolutions (exterior algebras). As corollaries we prove that the k(g) are factorial, with normal, rigid, generic completions. We use the DGC structure to identify the singular locus; it follows that perfect specializations of k(g) in affine space of dimension at most g + 6 have smooth deformations.