A homotopical Skolem--Noether theorem
Ajneet Dhillon, P\'al Zs\'amboki

TL;DR
This paper generalizes the classical Skolem--Noether theorem to a homotopical setting, establishing a fiber sequence that relates automorphisms of perfect complexes and their endomorphism algebras in derived algebraic geometry.
Contribution
It introduces a homotopical version of the Skolem--Noether theorem applicable to presentable monoidal quasi-categories with descent, extending classical correspondences to derived and spectral algebraic contexts.
Findings
Establishes a fiber sequence linking automorphisms of perfect complexes and their endomorphism algebras.
Shows the splitting of a long exact sequence on homotopy sheaves for perfect complexes.
Applies to derived algebraic geometry, spectral algebraic geometry, and ind-coherent sheaves in characteristic zero.
Abstract
The classical Skolem--Noether Theorem [Giraud, 71] shows us (1) how we can assign to an Azumaya algebra on a scheme a cohomological Brauer class in 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 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 over a quasi-compact and quasi-separated scheme , the long exact sequence on homotopy group sheaves splits giving equalities…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory
