Generalized Divisors and Biliaison

Abstract

We extend the theory of generalized divisors so as to work on any scheme $X$ satisfying the condition $S_2$ of Serre. We define a generalized notion of Gorenstein biliaison for schemes in projective space. With this we give a new proof in a stronger form of the theorem of Gaeta, that standard determinantal schemes are in the Gorenstein biliaison class of a complete intersection. We also show, for schemes of codimension three in ${\mathbb P}^n$, that the relation of Gorenstein biliaison is equivalent to the relation of even strict Gorenstein liaison.