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 Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 6, Pages 133–202 (Mi izv2702)

An adelic resolution for homology sheaves

S. O. Gorchinskiyab

a Steklov Mathematical Institute, Russian Academy of Sciences
b Independent University of Moscow

Abstract: We propose a generalization of the ordinary idele group by constructing certain adelic complexes for sheaves of $K$-groups on schemes. Such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated with the presheaf of a homology theory with certain natural axioms satisfied, in particular, by $K$-theory. In this case it is proved that the adelic complex provides a flabby resolution for this sheaf on smooth varieties over an infinite perfect field and that the natural morphism to the Gersten complex is a quasi-isomorphism. The main advantage of the new adelic resolution is that it is contravariant and multiplicative. In particular, this enables us to reprove that the intersection in Chow groups coincides (up to a sign) with the natural product in the corresponding $K$-cohomology groups. Also, we show that the Weil pairing can be expressed as a Massey triple product in $K$-cohomology groups with certain indices.

DOI: https://doi.org/10.4213/im2702

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English version:
Izvestiya: Mathematics, 2008, 72:6, 1187–1252

Bibliographic databases:

UDC: 512.73
MSC: 14C15, 14F10, 19E08, 19E20

Citation: S. O. Gorchinskiy, “An adelic resolution for homology sheaves”, Izv. RAN. Ser. Mat., 72:6 (2008), 133–202; Izv. Math., 72:6 (2008), 1187–1252

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv2702
• https://doi.org/10.4213/im2702
• http://mi.mathnet.ru/eng/izv/v72/i6/p133

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This publication is cited in the following articles:
1. Gorchinskiy S., “Notes on the biextension of Chow groups”, Motives and algebraic cycles, Fields Inst. Commun., 56, Amer. Math. Soc., Providence, RI, 2009, 111–148
2. R. Ya. Budylin, “An adelic construction of Chern classes”, Sb. Math., 202:11 (2011), 1637–1659
3. Braunling O., “Geometric Two-Dimensional Ideles With Cycle Module Coefficients”, Math. Nachr., 287:17-18 (2014), 1954–1971
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