2020/07/28 by Ajneet Dhillon, Dhillon, Ajneet, Pál Zsámboki +1
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2007.14327
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 H2(X,\mathbf Gm) 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^⊗ 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 πi(\mathopAut_\mathopPerf E,idE)=πi(\mathopAut_\mathopAlg\mathopPerf\mathop\mathbf REnd E,id_\mathop\mathbf 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 Algeraic Geometry in characteristic 0.