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A generalization of central simple algebras over arbitrary commutative rings, or over a scheme

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$\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?

Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection …