Classification of derived Azumaya Algebras over derived smooth manifolds via derived Brauer Groups

2026/07/05 by Yimu Mao, Christopher Tropp
#math.GM

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Abstract

For a derived smooth manifold (X,O_X) in the sense of Spivak, we pull back Toën's categorical derived Brauer stack along the forgetful functor from simplicial C^∞-rings to connective simplicial commutative rings and then stackify on the open site of X. The resulting categorical Brauer stack has the homotopy type \mathfrakBr_X ≃ K(\underline\mathbb Z,1) × B²GL₁(O_X). Consequently its group of stackified Brauer classes is dBr(X):=π₀Γ(X,\mathfrakBr_X) ≅ H¹(X,\underline\mathbb Z) × π₀Γ(X,B²GL₁(O_X)), where GL₁(O_X) is the sheaf of derived units. The usual formula with H²(X,O_X^×) 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.

Citations

• A Serre-Swan theorem for gerbe modules on étale Lie groupoids
• Higher Topos Theory
• The homotopy theory of dg-categories and derived Morita theory

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• GAGA problems for the Brauer group via derived geometry
• A homotopical Skolem--Noether theorem
• Derived Azumaya algebras and generators for twisted derived categories
• The derived Picard group of an affine Azumaya algebra
• A note on secondary K-theory