Adelic descent theory

A result of Andre 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 GL(n) (A) of regular matrices over the ring of adeles (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 adeles A(X)(center dot), we have an equivalence Perf (X) similar or equal to broken vertical bar Perf (A(X)(center dot))broken vertical bar between perfect complexes on X and cartesian perfect complexes for A(X)(center dot). Using the Tannakian formalism for symmetric monoidal 1-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adeles. 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.