Cox rings of algebraic stacks

Abstract

We give a proper definition of the multiplicative structure of the fol lowing rings: the Cox ring of invertible sheaves on a general algebraic stack; an the Cox ring of rank-one reflexive sheaves on a normal and excellent algebrai stack. We show that such Cox rings always exist and establish their (non-)unique ness in terms of an Ext-group. Moreover, we compare our definition with th classical construction of a Cox ring on a variety. Finally, we give an applicatio to the theory of Mori dream stacks.