Derived Azumaya Algebras on Compact Derived Smooth Manifolds
Published in arXiv, 2026
We extend Toën’s derived Brauer theory to the open site of Spivak’s derived smooth manifolds. For a compact derived smooth manifold $X$, we define a spectralized structure sheaf $\mathbb O_X$ and prove that every Picard twist can be represented by a derived Azumaya algebra, whose Morita equivalence class is uniquely determined by the twist. In the open locally Morita trivial setting, this gives a natural bijection
\[\Phi_X:\mathbf{dBr}^{\mathrm{open}}(X) \xrightarrow{\cong} H^1(|X|;\underline{\mathbb Z}) \times \pi_0\Gamma\bigl(|X|,\mathrm B^2GL_1(\mathbb O_X)\bigr).\]If we consider $C^\infty_X(\mathbb C)$ on ordinary smooth manifolds, the second factor is $H^3(\lvert X\rvert;\underline{\mathbb Z})$, which recovers the Dixmier–Douady invariant. If we consider $C^\infty_X(\mathbb R)$, the second factor is $H^2(\lvert X\rvert;\underline{\mathbb Z/2})$, which recovers the corresponding invariant of Donovan and Karoubi.
Recommended citation: Y.Mao, C.Tropp. (2026). "Derived Azumaya Algebras on Compact Derived Smooth Manifolds"
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