Principal Homogeneous Spaces and Group Scheme Extensions

Abstract

Suppose $G$ is a finite commutative group scheme over a ring $R$. Using Hopf-algebraic techniques, S. U. Chase has shown that the group of principal homogeneous spaces for $G$ is isomorphic to $\operatorname {Ext} (G’,{G_m})$, where $G’$ is the Cartier dual to $G$ and the Ext is in a specially-chosen Grothendieck topology. The present paper proves that the sheaf $\operatorname {Ext} (G’,{G_m})$ vanishes, and from this derives a more general form of Chase’s theorem. Our Ext will be in the usual (fpqc) topology, and we show why this gives the same group. We also give an explicit isomorphism and indicate how it is related to the existence of a normal basis.