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Résumé |
The Brauer group ${\rm Br}(X)$ of an algebraic variety $X$ is defined as the group of Azumaya algebras on $X$ up to Morita equivalence. There is an injective map (the Brauer map) ${\rm Br}(X) \hookrightarrow {\rm H}^2_{\mbox{ét}} (X,{\mathbb G}_m)$. Understanding the image of this map constitutes the so-called Brauer problem.
Toën introduced the notion of derived Azumaya algebra, later also developed by Lurie. Derived Azumaya algebras modulo Morita equivalence form the derived Brauer group dBr(X), which contains Br(X) and admits a map $\phi : {\rm Br}(X) \hookrightarrow {\rm H}^2_{\mbox{ét}} (X,{\mathbb G}_m)$ extending the classical Brauer map. Unlike that, however, $\phi$ is an isomorphism, and thus offers a natural way to describe those cohomology classes not contained in the image of the Brauer map.
With Michele Pernice (KTH Stockholm) we gave a more concrete description of $\phi$ and its inverse, by using the interpretation of ${\rm H}^2_{\mbox{ét}} (X,{\mathbb G}_m)$ via ${\mathbb G}_m$-gerbes and by implementing the notion of twisted sheaves in the derived setting.
I will explain this result and give some perspectives on ongoing work regarding the interaction of the derived Brauer group with Beilinson’s theory of adèles, in the case of a curve.
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