Etale topology

Of course this proof only works on objects (U,T,6) such that Ot has no 2-torsion. To prove it in general we argue as follows. Since there are no obstructions to deforming a principally polarized abelian scheme over an affine scheme we can take locally on T a lift of the pair (Ajj, A). Hence we are reduced to the case T = U. Locally on U in the etale topology the pair (Air, A) is the pullback of a pair (A%, A) over a scheme Z smooth over Spec(Z). (This is true since Ag Spec(Z) is smooth.) Since Oz has no 2-torsion we are done. 

Locally in the etale topology on S we can choose a rigidified line bundle (i.e. a line bundle trivialized along the origin) L on X such that A(L) = A (see [GIT] Definition 6.2). This... 

Our reference on algebraic stacks is [DM] 4. A stack will be a stack in groupoids over the category of schemes with the etale topology. Stacks will be denoted by calligraphic uppercase letters. 

Remark 1.21. — One can verify easily that the statement of Propositions 1.20 also holds in the Zariski and etale topologies. A general proof working for all three cases can be obtained using the fact that in all of them there is a notion of the small site over a smooth scheme X (Zariski, Nisnevich or 6tale) which has fiber products preserved by the base change functors for arbitrary morphisms of base schemes. 

I t G be a sheaf of groups on Sm/S)yu- Using the etale topology and the pair of adjoint functors between the simplicial homotopy categories associated with the obvious morphism of sites... 

F satisfies the second condition of the proposition the morphism of sheaves in etale topology po = (Eq Xg F), —is seen to be an epimorphism in Nisnevich... 

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