Cox rings of algebraic stacks (Q6541857)

In this well-written article, the authors give proper definitions of the multilplicative structures on Cox rings of invertible sheaves on a general algebraic stack and on Cox rings of rank-one reflexive sheaves on a normal and excellent algebraic stack. This article builds upon previous work [\textit{A. Hochenegger} and \textit{E. Martinengo}, Math. Z. 280, No. 3--4, 1185--1202 (2015; Zbl 1329.14010); J. Pure Appl. Algebra 222, No. 6, 1287--1305 (2018; Zbl 1390.14009)] of the two first-named authors on the constructions of Mori dream stacks and their Cox rings. Here the authors prove that Cox rings in this generality always exist and show their (non-)uniqueness in terms of a particular Ext-group. At the end of the article an interesting applications to the theory of Mori dream spaces is given. It is also interesting to note that much of the machinery developed by the authors here works in even greater generality than that of algebraic stacks, namely for categories fibred in groupoids having an fpqc-atlas. For example, this allows for applications to the infinite root stack associated to a logarithmic scheme.