G-torsors over a Dedekind scheme

We prove the equivalence of three “points of view” on the notion of a G-torsor when the base scheme is a Dedekind scheme, generalizing known results when the base is a field. The two main tools that we generalize are Chevalley’s theorem on semi-invariants (cf. Borel (1991)[1, II.5.1], ) and a Tannakian description of G-torsors given by Nori and Saavedra (cf. Nori (1976)[8, Sec. 2], and Saavedra Rivano (1972) [11, II.4.2]). As an application, we show that the fibered category of G-torsors on a regular proper curve over a field k is an Artin stack locally of finite presentation over k.