Etale cohomology and restricted direct product

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$\newcommand{\h}{\operatorname{H}}$ Let $k$ be a global field, $A$ an abelian variety over $k$, and $\mathbf A$ the ring of adeles of $k$. One defines the Tate-Shafarevich group of $A$ by $$Ш(A) = \ker\left(\h^1(k,A) \to \prod_v \h^1(k_v,A)\right)$$ where $\h^1(-)$ is etale cohomology, and we interpret $A$ as a sheaf on the etale site of $k$. In fact, the image of $\h^1(k,A)$ lands inside of the smaller group $$\prod_v' (\h^1(k_v,A), \h^1(\mathfrak o_v,A))$$ (restricted direct product). This makes sense because for almost all $v$, $A$ has good reduction at $v$.

My question is, do we have $$\h^1(\mathbf A_k,A) \simeq \prod_v' (\h^1(k_v,A), \h^1(\mathfrak o_v,A))?$$

In that case, we could define $Ш(A) = \ker(\h^1(k,A) \to \h^1(\mathbf A_k,A))$. I'm a little worried because $\mathbf A_k$ is pretty horrible when considered as a commutative ring, so its etale site will probably not be easy to manage.

Edit: Just so its clear, this was inspired by Poonen's lectures at the most recent AWS.

• Would it help to phrase this geometrically? Presumably, the homomorphism you want takes an $A$-torsor over $\mathbf{A}_k$ and base-changes to an $A$-torsor over each $k_v$. Mar 28, 2014 at 21:09

You don't define what is meant by ${\rm{H}}^1(\mathfrak{o}_v, A)$ (it is an abuse of notation, as $A$ lives over $k$ or $k_v$, not $\mathfrak{o}_v$), but the only definition which comes to mind at a good place (namely, cohomology over $\mathfrak{o}_v$ with coefficients in the abelian scheme Neron model at $v$) vanishes due to Lang's theorem over the finite residue field and the smoothness of torsors over $\mathfrak{o}_v$ for a smooth group scheme.
In other words, the proposed alternative definition is nothing more or less than the usual definition, in view of the fact that for any finite type smooth connected commutative group scheme $A$ over a global field $k$ and any class $\xi \in {\rm{H}}^1(k,A)$, for all but finitely many $v$ the local restriction $\xi_v \in {\rm{H}}^1(k_v,A)$ vanishes. (The proof of such vanishing at almost all $v$ is a pleasant exercise with the torsor interpretation of such H$^1$'s and standard "spreading out" arguments.)
Since $\mathbf{A}_k = \varinjlim_S (k_S \times \prod_{v \not\in S} \mathscr{O}_v)$ where $S$ varies through the finite sets of places of $k$ (containing the archimedean places) and $k_S := \prod_{v \in S} k_v$, the compatibility of etale cohomology with respect to arbitrary directed systems of rings gives that $${\rm{H}}^1(\mathbf{A}_k,A) = \varinjlim_S (\prod_{v \in S} {\rm{H}}^1(k_v,A) \times {\rm{H}}^1(\prod_{v \not\in S} \mathscr{O}_v, A_S))$$ where $A_S$ is the abelian scheme Neron model of $A$ over the ring $\mathscr{O}_{k,S}$ of $S$-integers for sufficiently big $S$.
So the content of the question then reduces to showing that if $R = \prod_{v\not\in S} \mathscr{O}_v$ and $G$ is a smooth finite type $R$-group scheme with connected fibers then ${\rm{H}}^1(R, G) = 1$. The elements in this H$^1$ are represented by algebraic spaces over $R$ that are fppf $G$-torsors (hence $R$-smooth with geometrically connected fibers). Using this geometric interpretation and a concrete description of finite etale $R$-algebras (and finiteness of the residue fields of the local factor rings $\mathscr{O}_v$), the vanishing is proved on pages 613--614 (and top of page 615) of the paper "Finiteness theorems for algebraic groups over function fields" in Compositio 148 (2012).