**Speaker:** Pal Zsamboki (Renyi Institute)

"A homotopical Skolem-Noether theorem"

Joint work with Ajneet Dhillon. See arXiv:2007.14327 [math.AG]. The classical Skolem--Noether Theorem by Giraud shows us (1) how we can assign to an Azumaya algebra A on a scheme X a cohomological Brauer class in H^2(X,G_m) and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem by Lieblich 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 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(Aut Perf E,id_E)=pi_i(Aut_Alg Perf REnd E, id_REnd E) for i >= 1. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algebraic Geometry in characteristic 0.