Formal GAGA for gerbes

2023/05/30 by Kresch, Andrew, Mathur, Siddharth · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.19114

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/Iⁿ⁺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.

Cited by

• Extensions of Abelian Schemes and the Additive Group

Related

• GAGA problems for the Brauer group via derived geometry
• On the Brauer groups of fibrations II
• Purity for the Brauer group
• The bigger Brauer group is really big
• Brauer groups and étale homotopy type