On arithmetic class invariants

Let F be a number field with ring of integers OF , and let S be a finite set of places of F . Assume that S contains the set S∞ of archimedean places of F , and write Sf for the set of finite places contained in S. Let OS (or O, when there is no danger of confusion) denote the ring of Sf -integers of F . Write F for an algebraic closure of F . Let Y be any scheme over Spec(O). Suppose thatG is a finite, flat commutative group scheme over Y of exponent N , and let G denote the Cartier dual of G. Let π : X → Y be aG-torsor, and write π0 : G → Y for the trivialG-torsor. ThenOX is anOG-comodule, and so it is also anOGD -module (see [12]). As an OGD -module,OX is locally free of rank one, and it therefore gives a line bundle Mπ over G. Set Lπ := Mπ ⊗M−1 π0 . Then the map ψ : H (Y,G) → Pic(G) ; [π ] → [Lπ ]