Classification of derived Azumaya Algebras over derived smooth manifolds via derived Brauer Groups
Published in arXiv, 2026
For a derived smooth manifold $(X,\mathcal O_X)$ in the sense of Spivak, we pull back Toën’s categorical derived Brauer stack along the forgetful functor from simplicial $C^\infty$-rings to connective simplicial commutative rings and then stackify on the open site of $X$. The resulting categorical Brauer stack has the homotopy type
\[\mathfrak{Br}_X \simeq K(\underline{\mathbb Z},1) \times B^2GL_1(\mathcal O_X).\]Consequently its group of stackified Brauer classes is
\[dBr(X):=\pi_0\Gamma(X,\mathfrak{Br}_X) \cong H^1(X,\underline{\mathbb Z}) \times \pi_0\Gamma(X,B^2GL_1(\mathcal O_X)),\]where $GL_1(\mathcal O_X)$ is the sheaf of derived units. The usual formula with $H^2(X,\mathcal O_X^\times)$ is recovered when the structure sheaf is discrete. This shows that the pullback categorical Brauer invariant is governed by the full homotopy type of the derived unit sheaf.
Recommended citation: Y.Mao, C.Tropp. (2026). "Classification of derived Azumaya Algebras over derived smooth manifolds via derived Brauer Groups"
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