Principal homogeneous spaces and group scheme extensions

Source abstract

Suppose G G is a finite commutative group scheme over a ring R R . Using Hopf-algebraic techniques, S. U. Chase has shown that the group of principal homogeneous spaces for G G is isomorphic to Ext ⁡ ( G ′ , G m ) \operatorname {Ext} (G’,{G_m}) , where G ′ G’ is the Cartier dual to G G and the Ext is in a specially-chosen Grothendieck topology. The present paper proves that the sheaf Ext ⁡ ( G ′ , G m ) \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.