Generalized divisors and total reflexivity

Abstract

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent O{script}X-modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent {script}X-modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature. © Taylor & Francis Group, LLC.