Formal GAGA for gerbes

Abstract 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>X</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> X_{n} where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>X</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> X_{n} is the fiber over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>/</m:mo> <m:msup> <m:mi>I</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>X</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> X_{n} . We characterize such behavior, prove this cannot happen in characteristic zero, and deduce a formal GAGA statement for Brauer classes.