A homotopical Skolem--Noether theorem

Abstract

The classical Skolem--Noether Theorem [Giraud, 71] shows us (1) how we can assign to an Azumaya algebra $A$ on a scheme $X$ a cohomological Brauer class in $H^2(X,{\mathbf{G}}_m)$ and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem [Lieblich, 09] generalizes this result to weak algebras in the derived 1-category locally quasi-isomorphic to derived endomorphism algebras of perfect complexes. We show that in general for a co-family ${\mathcal{C}}^{\otimes }$ of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving in particular the above correspondences. For a totally supported perfect complex $E$ over a quasi-compact and quasi-separated scheme $X$, the long exact sequence on homotopy group sheaves splits giving equalities ${\pi }_i({\mathrm{Aut}}_{\mathrm{Perf}}E,{\mathrm{id}}_E)={\pi }_i({\mathrm{Aut}}_{\mathrm{Alg}\mathrm{Perf}}\mathrm{REndE},{\mathrm{id}}_{\mathrm{REndE}})$ for $i\ge 1$. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algeraic Geometry in characteristic 0.

56 pages. Minor changes

References in corpus (2)

• Brauer groups and Galois cohomology of commutative ring spectra
• On the $\infty$-stack of complexes over a scheme