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 Uspekhi Mat. Nauk, 2020, Volume 75, Issue 6(456), Pages 3–84 (Mi umn9975)

Iterated Laurent series over rings and the Contou-Carrère symbol

S. O. Gorchinskiy, D. V. Osipov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol.
The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case $n>1$, for the group of invertible elements of the algebra of $n$-iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher-dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described.
As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the $n$-iterated algebraic loop group of the Milnor $K$-group of degree $n+1$ to the above group scheme factor through the higher-dimensional Contou-Carrère symbol.
Bibliography: 67 titles.

Keywords: iterated Laurent series over rings, higher-dimensional Contou-Carrère symbol, Milnor $K$-group of a ring, higher-dimensional Witt pairing, group schemes.

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614 This research was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).

DOI: https://doi.org/10.4213/rm9975

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English version:
Russian Mathematical Surveys, 2020, 75:6, 995–1066

Bibliographic databases:

UDC: 512.71+512.666+512.747+511.22
MSC: Primary 19D45; Secondary 13J05, 14L15, 19F05, 13F25

Citation: S. O. Gorchinskiy, D. V. Osipov, “Iterated Laurent series over rings and the Contou-Carrère symbol”, Uspekhi Mat. Nauk, 75:6(456) (2020), 3–84; Russian Math. Surveys, 75:6 (2020), 995–1066

Citation in format AMSBIB
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