  Fix an $I$-adically complete Noetherian ring $A$ and suppose $X$ is a proper
$A$-scheme. This article concerns the relationship between the Brauer group of
$X$ and that of the various $X_n$ where $X_n$ is the fiber over $A/I^{n+1}$. In
particular, we answer a question of Grothendieck by showing that, in positive
and mixed characteristic, there are examples of $X$ with nontrivial Brauer
classes that restrict to zero on all the $X_n$. We characterize such behavior,
prove this cannot happen in characteristic zero, and deduce a formal GAGA
statement for Brauer classes.
