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On the Parshin-Beilinson adeles for schemes. (English) Zbl 0763.14006
The classical theory of adeles in algebraic number theory, which works also for curves in algebraic geometry, has been generalized over the past few years, first to smooth proper surfaces over a perfect field, and then to all noetherian schemes \(X\). A construction of adeles in the latter case was given by A. A. Bejlinson [in Funct. Anal. Appl. 14, 34-35 (1980; translation from Funkts. Anal. Prilozh. 14, No. 1, 44-45 (1980; Zbl 0509.14018)]. This construction associates to a quasi-coherent sheaf \({\mathcal F}\) on \(X\) a complex (rather than a single ring as in the classical case) of adeles \(\mathbb{A}^*(X,{\mathcal F})\), functorially in \({\mathcal F}\), and it has several applications, principally because of the following theorem: The cohomology of the complex \(\mathbb{A}^*(X,{\mathcal F})\) is isomorphic to the cohomology of \({\mathcal F}\).
As Bejlinson’s article is very short and concise, the present paper aims mainly at providing a more accessible exposition of this construction with proofs in the language of commutative algebra. The author also gives a new construction of rational (as opposed to Bejlinson’s analytic) adeles and shows that the complex of rational adeles also computes the cohomology of \({\mathcal F}\).
Reviewer: B.Singh (Bombay)

14F20 Étale and other Grothendieck topologies and (co)homologies
11R56 Adèle rings and groups
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