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arXiv · 2305.19114

Formal GAGA for gerbes

Andrew Kresch·Siddharth Mathur
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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 $X_n$ 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.

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Andrew Kresch, Siddharth Mathur. 2023-05-30. Formal GAGA for gerbes. https://arxiv.org/abs/2305.19114

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