Gorenstein Algebras in Codimension 2 and Koszul Modules

Abstract

The notion of Koszul module introduced in Grassi (1996) is a generalization of the concept of complete intersection ring to arbitrary finite modules over some base ring A (cf. Def. 1.3). A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that holds as R-modules, where A is a Cohen–Macaulay local ring with dim A − dim A R = 2. Here we prove (Thm. 1.4) that every Gorenstein algebra R of codimension 2 has a resolution over A (assuming 2 is invertible in A) that is simultaneously Gorenstein-symmetric and of Koszul module type; this is a generalization to algebras of higher rank of the easy fact that, if R is a quotient of A, it has a resolution over A given by a length two Koszul complex.