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

This article is cited in 3 scientific papers (total in 3 papers)

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

Full text: PDF file (996 kB)
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English version:
Izvestiya: Mathematics, 2008, 72:6, 1187–1252

Bibliographic databases:

Document Type: Article
UDC: 512.73
MSC: 14C15, 14F10, 19E08, 19E20
Received: 04.07.2007

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

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathscinet  zmath  isi
    2. R. Ya. Budylin, “An adelic construction of Chern classes”, Sb. Math., 202:11 (2011), 1637–1659  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Braunling O., “Geometric Two-Dimensional Ideles With Cycle Module Coefficients”, Math. Nachr., 287:17-18 (2014), 1954–1971  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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