On the ubiquity of Gorenstein rings

Introduction APIARY [23~, and subsequently GORENSTEIN Eg] and SAMUEL [2(r proved 9 3-~ t that if (2 is a point on a plane curve then no=2d Q, where dQ=dlmw0/ 9 Q and nQ = dim C'Q/~Q, 9 being the local ring of Q, CQ its normalization, and ~Q the conductor. This condition has received attention from a variety of algebraic geometers, and recently ROQUETTE [19J and BEI~GER [52 have put it i n a rather general algebraic setting. Furthermore, ROSENLICHT [24~ (see also [211) proved that ~ o = 2 d o if, and only if, f2b is a free 9 of rank one, where f2b is the module of differentials regular at Q (in the sense of ROSENLICHT). If V is a non singular variety of dimension n and.C2 V the sheaf of differential forms of degree n on V, then .0 V is a locally free sheaf of rank one. Let W be a subvariety of codimension q at all of its points. If W is non singular Dw is defined, and GROTHENDIECK has shown that