Let $R = \prod\_{n\in\mathbf{N}}R\_n$ be an *infinite* direct product of discrete valuation rings $R\_n$. Why is $\mathrm{Pic}(R) = 0$?



Picard group of infinite direct product of DVRs trivial
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 $K$.


