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Adelic descent theory

Part of: (Co)homology theory Families, fibrations

Published online by Cambridge University Press:  31 May 2017

Michael Groechenig
Michael Groechenig*
Affiliation:
Department of Mathematics and Computer Science, Freie Universität Berlin, D-14195 Berlin, Germany email [email protected]
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Abstract

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

Keywords

adèlesvector bundlesperfect complexesmoduli stacks

MSC classification

Primary: 14D23: Stacks and moduli problems 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 8 , August 2017 , pp. 1706 - 1746
Copyright
© The Author 2017 

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