A homotopical Skolem--Noether theorem

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 $\mathscr C^\otimes$ 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 $\pi_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Perf}}} E,\mathrm{id}_E)=\pi_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Alg}}\mathop{\mathrm{Perf}}}\mathop{\mathbf R\mathrm{End} E},\mathrm{id}_{\mathop{\mathbf R\mathrm{End} E}})$ for $i\ge1$. 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.