Invariants, exponents and formal group laws
type: Item
instance of: scholarly article
title: Invariants, exponents and formal group laws
zbMATH Open document ID: 1303.14053
author: Changlong Zhong
author: J. Malagón-López
author: Kirill Zainoulline
published in: Journal of Algebra
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.1880
reviewed by: Xiao Xiao
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 17B22
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 55N22
zbMATH DE Number: 6335796
zbMATH Keywords: oriented cohomology
zbMATH Keywords: flag variety
zbMATH Keywords: linear algebraic group
zbMATH Keywords: algebraic cycles
zbMATH Keywords: formal group law
zbMATH Keywords: algebraic cobordism
zbMATH Keywords: gamma filtration
MaRDI profile type: Publication
OpenAlex ID: W2963331844
arXiv ID: 1207.1880
cites work: Basic polynomial invariants, fundamental representations and the Chern class map
cites work: On the torsion of Chow groups of twisted spin-flags
cites work: Invariants, torsion indices and oriented cohomology of complete flags
cites work: Invariants symétriques entiers des groupes de Weyl et torsion
cites work: Désingularisation des variétés de Schubert généralisées
cites work: Q4342000
cites work: The \(\gamma\)-filtration and the Rost invariant
cites work: Topological methods in algebraic geometry. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel.
cites work: Q3998721
cites work: Algebraic Cobordism
cites work: Q3947818
cites work: Oriented cohomology theories of algebraic varieties
cites work: The Arithmetic of Elliptic Curves
cites work: On the \({\gamma}\)-filtration of oriented cohomology of complete spin-flags
DOI: 10.1016/J.JALGEBRA.2014.03.026
review text: The author extends the notion of an exponent \(\tau_d\) to a more general setting, namely, in the context of algebraic oriented cohomology theories and the associated formula group laws. Then the authors use these generalized exponents to estimate the torsion part of an arbitrary algebraic oriented cohomology theories of some flag varieties. \vskip 0.1in More specifically, if \(W\) is the Weyl group of a crystallographic root system which acts by means of simple reflections on the respective weight lattice \(\Lambda\). Let \(F\) be a formal group law over a coefficient ring \(R\). The formal group algebra \(R\Lambda_F\) is the quotient of \(R[[x_{\Lambda}]]\) by the ideal \(I_F\), where \(R[[x_{\Lambda}]]\) is the \(\ker(\epsilon)\)-adic completion of \(R[x_{\Lambda}]\) and \(I_F\) is the closure of the ideal of \(R[[x_{\Lambda}]]\) generated by \(x_0\) and elements of the form \(x_{\lambda_1+\lambda_2} - (x_{\lambda_1}+_F x_{\lambda_2})\). Let \(R\Lambda_F^W\) be the subring of invariants of \(R\Lambda_F\), and \(I_F^W\) the associated ideal. Given two formal group laws \(F\) and \(F'\), there is an isomorphism \(\Phi: R\Gamma_F \to R\Gamma_{F'}\). The generalized exponent \(\tau_d^{F, F'}\) from \(F\) to \(F'\) is defined to be the exponent of the image \(\Phi(I_F^W)\) in \(I_{F'}^W\), after passing to the \(d\)-th subsequent quotients of the \(I_F-\) and \(I_{F'}-\) filtrations. When \(F\) is the multiplicative and \(F'\) is the additive formal group laws, the exponent coincide with the original definition in [\textit{S. Baek} et al., Doc. Math., J. DMV 17, 135--150 (2012; Zbl 1244.13004)].NEWLINENEWLINE The main result of the paper is the following:NEWLINENEWLINETheorem. Let \(F\) and \(F'\) be formal group laws over a ring \(R\) of characteristic different from \(2\). For the root system of type \(B_n\) with \(n \geq 3\) (resp. \(D_n\) with \(n \geq 4\)), we have \(\tau_d^{F,F'} \; | \; 2^d\) for \(d \geq 1\) (resp. \(n>d\geq 1\)). In partic…
description: scientific article; zbMATH DE number 6335796