Generalized Divisors on DMH Stacks

Hartshorne developed a theory of generalized divisors on Gorenstein schemes to characterize codimension-one closed subschemes without embedded points. Generalized divisors can be viewed as a generalization of Weil divisors to non-normal schemes. The purpose of this paper is to extend generalized divisors on schemes to DMH stacks, where DMH stacks are Deligne-Mumford stacks satisfying specific conditions. We provide a detailed proof of the properties of total quotient sheaves under \'etale morphisms, thereby demonstrating the difficulty of directly defining generalized divisors on DMH stacks through fractional ideals of the total quotient sheaf. Instead, we propose to define generalized divisors on DMH stacks using reflexive coherent sheaves that are locally free of rank one at generic points. Furthermore, we rigorously establish the rationality of this definition on stacks.