This will be a gentle introduction into Brauer groups and twisted sheaves. The emphasis will be on geometric aspects and eventually on moduli spaces of twisted sheaves on K3 surfaces. We will study the different ways to think about Brauer groups as groups of Azumaya algebras, Brauer-Severi varieties, twisted sheaves, 𝔾m-gerbes... How to translate from one to the other, how to define Chern classes, how to split Brauer classes, etc.
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