Let $R = \prod_{n\in\mathbf{N}}R_n$ be an infinite direct product of discrete valuation rings $R_n$. Why is $\mathrm{Pic}(R) = 0$?
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We aim to show that every $\mathbb{G}_m$torsor over $R$ is trivial. By descent, such a torsor is represented by an affine $R$scheme $X$. Due to affineness, $X(R) = \prod_n X(R_n)$, so it remains to (invoke the axiom of countable choice and) note that each $X(R_n)$ is nonempty because the pullback of $X$ to $\mathrm{Spec} R_n$ is the trivial $\mathbb{G}_m$torsor as $R_n$ is local. For the argument to work, it suffices that $\mathrm{Pic}(R_n) = 0$ for every $n$. Also, a similar argument (coupled with limit formalism) shows that there are no nontrivial vector bundles on $\mathrm{Spec } \mathbb{A}_K$ where $\mathbb{A}_K$ is the adele ring of a global field $K$. 

