[migrated from math.stackexchange: http://math.stackexchange.com/questions/727896/etale-cohomology-and-restricted-direct-product]

$\newcommand{\h}{\operatorname{H}}$
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.