Corrigendum to N\'eron models, Lie algebras, and reduction of curves of genus one [LLR1] and The Brauer group of a surface [LLR2]

TL;DR

This corrigendum corrects a key formula used in previous work on the Brauer group of surfaces over finite fields, ensuring the validity of earlier results about its order being a perfect square.

Contribution

It provides a corrected version of Gordon's formula, enabling the original proofs about the Brauer group's properties to remain valid.

Findings

Corrected Gordon's formula for the key mathematical relation

Validated the original result that the Brauer group's order is a perfect square

Ensured the mathematical accuracy of previous proofs

Abstract

Let X be a proper smooth and connected surface over a finite field. We proved in [LLR2] that the order of the Brauer group Br(X) of X is a perfect square if it is finite. Our proof is based in part on a result of Gordon [Gor], which we used in [LLR1] to establish a key formula. Thomas Geisser noted that the formula in [LLR1] is incorrect, due to an omission in [Gor], and provides a corrected formula. We explain in this corrigendum how to modify the work of Gordon to establish a correct formula. The corrected formula can be used to prove the result in [LLR2] without further modifications.