Integrality of the Chern character in small codimension
type: Item
instance of: scholarly article
title: Integrality of the Chern character in small codimension
zbMATH Open document ID: 1260.14013
author: Olivier Haution
published in: Advances in Mathematics
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.4084
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 14C15
zbMATH DE Number: 6084085
zbMATH Keywords: Steenrod operations
zbMATH Keywords: Chow groups
zbMATH Keywords: Chern character
zbMATH Keywords: Grothendieck-Riemann-Roch theorem
MaRDI profile type: Publication
OpenAlex ID: W2020657431
arXiv ID: 1103.4084
cites work: Q3286843
cites work: Vector fields on spheres
cites work: K-THEORY AND THE HOPF INVARIANT
cites work: POWER OPERATIONS IN K-THEORY
cites work: Riemann-Roch for singular varieties
cites work: Steenrod operations in Chow theory
cites work: Q3516073
cites work: Q4342000
cites work: Q5718764
cites work: Intersection theory using Adams operations
cites work: Degree formula for the Euler characteristic
cites work: On the first Steenrod square for Chow groups
cites work: Reduced Steenrod operations and resolution of singularities
cites work: Q3096594
cites work: Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic $0$
cites work: Q3069692
cites work: Steenrod operations, degree formulas and algebraic cobordism
cites work: Steenrod operations and degree formulas
cites work: The Steenrod algebra and its dual
cites work: Q3159970
cites work: Motivic decomposition of a compactification of a Merkurjev-Suslin variety
cites work: Opérations En K-Théorie Algébrique
cites work: Algebraic \(K\)-theory.
cites work: Reduced power operations in motivic cohomology
cites work: Degree formula for connective \(K\)-theory
DOI: 10.1016/J.AIM.2012.04.030
review text: Let \(X\) be an algebraic variety, \(K_{0}^{\prime}(X)\) the Grothendieck group of coherent sheaves, \(\mathrm{CH}_{i}(X)_{ Q}\) the Chow group of \(X\) of algebraic cycles of dimension \(i\) and \(\mathrm{ch}_{i}: K_{0}^{\prime}(X) \rightarrow \mathrm{CH}_{i}(X)_{Q}\) the homological Chern character. One can state the following conjecture:NEWLINENEWLINE Conjecture. Let \(X\) be a variety, and \(n\) a positive integer. Then NEWLINE\[NEWLINEp^{[n/(p-1)]} \cdot \mathrm{ch}_{\dim X-n} [{\mathcal O}_{X}]\in \mathrm{im} (\mathrm{CH}_{\dim X-n}(X)_{Z_{(p)}}\rightarrow \mathrm{CH}_{\dim X-n}(X)_{Q_{(p)}}NEWLINE\]NEWLINE The topological analog of this was formulated and proved by \textit{J. F. Adams} [Proc. Camb. Philos. Soc. 57, 189-199 (1961; Zbl 0103.16001)]. For another easier proof of this but with more restrictive assumptions [\textit{M. F. Atiyah}, Q. J. Math., Oxf. II. Ser. 17, 165--193 (1966; Zbl 0144.44901)]. The author gives a proof of this conjecture for any \(p\) and \(n<p(p-1).\) Using Gabber's result on existence of regular alterations the author can prove the conjecture for all \(n\) in the case when the characteristic of the base field is not equal to \(p.\) The consequences of this conjecture especially to study of correspondences between algebraic varieties are also discussed. Assuming that the conjecture holds true for \(n\leq m(p-1),\) given any (possibly singular) variety \(X\) the author constructs for any \(0<i \leq m ,\) a morphism of graded abelian groups: NEWLINE\[NEWLINET_{i}: \mathrm{CH}_{\bullet}(X)\otimes {\mathbb Z}/{p} \rightarrow {\widetilde{\mathrm{CH}}}_{{\bullet}-i(p-1)}(X)\otimes {\mathbb Z}/{p}NEWLINE\]NEWLINE \({\widetilde{\mathrm{CH}}}_{{\bullet}}\) denotes here the Chow group modulo torsion. The operations \(T_{i}\) for \(1<i \leq (p-1) \) are constructed unconditionally and for all \(i,\) when the characteristics of the base field is different from \(p.\) The operations \(T_{i}\) for \(i\leq p\) correspond up to sign to the \(i\)-t…
description: scientific article; zbMATH DE number 6084085