The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture

2015 ◽  
pp. 97-120 ◽  
Author(s):  
Manfred Kolster
Keyword(s):  
Residue Theorem ◽  
Norm Residue

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue

scholarly journals Two families of pro-𝑝 groups that are not absolute Galois groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Claudio Quadrelli
Keyword(s):  
Galois Cohomology ◽  
Galois Groups ◽  
Residue Theorem ◽  
Smoothness Property ◽  
Massey Products ◽  
Norm Residue

Abstract Let 𝑝 be a prime. We produce two new families of pro-𝑝 groups which are not realizable as absolute Galois groups of fields. To prove this, we use the 1-smoothness property of absolute Galois pro-𝑝 groups. Moreover, we show in these families, one has several pro-𝑝 groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of the norm residue theorem), or the vanishing of Massey products in Galois cohomology.

A general reciprocity law for symbols on arbitrary vector spaces

2018 ◽  
Vol 17 (06) ◽  
pp. 1850101
Author(s):  
Fernando Pablos Romo
Keyword(s):  
General Theory ◽  
General Expression ◽  
Vector Spaces ◽  
Residue Theorem ◽  
Arbitrary Vector ◽  
The Past ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Norm Residue ◽  
Hilbert Norm

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov — M. Kerr and the author — together with D. Hernández Serrano — can also be deduced from this general expression.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Large Scale ◽  
Residue Theorem ◽  
Geometric Constructions ◽  
Complete Proof ◽  
Motivic Cohomology ◽  
Symmetric Powers ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Many Sources ◽  
First Time

This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

On the generalized fractional Laplacian

2021 ◽  
Vol 24 (6) ◽  
pp. 1797-1830
Author(s):  
Chenkuan Li
Keyword(s):  
Fractional Laplacian ◽  
Special Functions ◽  
Computational Techniques ◽  
Residue Theorem ◽  
End Points ◽  
Proper Subspace ◽  
Classical Sense ◽  
First Time ◽  
Cauchy’S Residue Theorem ◽  
Integral Identities

Abstract The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u(x) over the space Ck (Rn ) (which contains S(Rn ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

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