The higher \(K\)-theory of real curves
type: Item
instance of: scholarly article
title: The higher \(K\)-theory of real curves
zbMATH Open document ID: 1026.19004
DOI: 10.1023/A:1020865906645
author: C. Weibel
author: Claudio Pedrini
published in: \(K\)-Theory
publication date: 2003-04-03T00:00:00Z
review text: Let \(X\) be a smooth real curve, i.e., a smooth one-dimensional reduced quasi-projective scheme over \(\mathbb R\), which is not defined over \(\mathbb C\). It is a classical fact that \(K_{0}(X)\) is closely related to topological invariants of the spaces \(X(\mathbb R)\) and \(X(\mathbb C)\) of real and complex points of \(X\). In this paper, the authors relate the higher \(K\)-groups of \(X\) to such topological invariants. They show that for \(n\neq 0\) the group \(K_n(X)\) is the direct sum of a divisible group and an elementary abelian \(2\)-group. The torsion subgroup of \(K_n(X)\) is a sum of copies of \(\mathbb Q/\mathbb Z\) and of \(\mathbb Z_2\), with the number of summands explicitly described in terms of topological invariants of \(X(\mathbb R)\) and \(X(\mathbb C)\). The results are slightly different depending on whether \(X\) is projective of affine as well as whether \(X(\mathbb R)\) is empty or non-empty. In particular, the torsion subgroups are periodic with period \(8\) if \(X(\mathbb R)\neq\emptyset\) and with period \(4\) is \(X(\mathbb R)=\emptyset\). One important ingredient of the proof is a transfer argument based on Comessatti's theorem. On the other hand, the authors use a spectral sequence converging to \(K\)-theory, whose \(E_2\) term is given by motivic cohomology of \(X\). The relevant motivic cohomology groups are computed using results of Suslin and Voevodsky.
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 1892282
zbMATH Keywords: real curve
zbMATH Keywords: higher algebraic \(K\)-theory
zbMATH Keywords: motivic cohomology
reviewed by: Andreas Cap
MaRDI profile type: Publication
full work available at URL: a:1020865906645
OpenAlex ID: W1991005111
description: scientific article; zbMATH DE number 1892282