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1. Picard sheaves, local Brauer groups, and topological modular forms

   https://doi.org/10.1112/topo.12333  

   Antieau, Benjamin; Meier, Lennart; Stojanoska, Vesna (May 2024, Journal of Topology)

   Abstract We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group. 

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2. The Cohomological Brauer Group of a Torsion $${\mathbb{G}}_{m}$$-Gerbe

   https://doi.org/10.1093/imrn/rnz235  

   Shin, Minseon (November 2019, International Mathematics Research Notices)

   Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes. 

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3. Vanishing of Brauer classes on K3 surfaces under reduction

   https://doi.org/10.1112/jlms.70051  

   Maulik, Davesh; Tayou, Salim (January 2025, Journal of the London Mathematical Society)

   Abstract Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many reductions where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei–Hassett–Várilly‐Alvarado. 

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4. The nil-Brauer category

   https://doi.org/10.5802/art.2  

   Brundan, Jonathan; Wang, Weiqiang; Webster, Benjamin (January 2024, Annals of Representation Theory)

   We introduce the nil-Brauer category and prove a basis theorem for its morphism spaces. This basis theorem is an essential ingredient required to prove that nil-Brauer categorifies the split \imath-quantum group of rank one. As this \imath-quantum group is a basic building block for \imath-quantum groups of higher rank, we expect that the nil-Brauer category will play a central role in future developments related to the categorification of quantum symmetric pairs. 

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5. A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

   https://doi.org/10.4153/CJM-2018-030-8  

   Brundan, Jonathan; Comes, Jonathan; Kujawa, Jonathan Robert (October 2019, Canadian Journal of Mathematics)

   Abstract We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q. 

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