Bulletin of the Belgian Mathematical Society - Simon Stevin

On the algebraic K-theory of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$

Marek Golasiński and Francisco Gómez Ruiz

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Abstract

It is well known after R. Swan that $\tilde K_0(R[X,Y,Z]/(X^2+Y^2+Z^2-1))$ is isomorphic to the integers $\mathbb Z$, whenever $R$ is a field of characteristic not two which contains the squared root of $-1$. \par First, we give explicit idempotent matrices $\gamma^p$ of order two, corresponding to the integer $p,$ in the isomorphism above, if $R$ is a field of characteristic zero. Then, we use the algebraic de Rham cohomology of Kähler differentials to define Brouwer degree for polynomial homomorphisms of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$ to itself, and relate the problem of finding hermitian representatives for $R=K(i),$ $K$ a field not containing $i,$ to some unsolved problems of representing Brouwer degrees by polynomial maps.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 18, Number 5 (2011), 849-860.

Dates
First available in Project Euclid: 13 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1323787172

Digital Object Identifier
doi:10.36045/bbms/1323787172

Mathematical Reviews number (MathSciNet)
MR2918651

Zentralblatt MATH identifier
1234.30025

Subjects
Primary: 14P25: Topology of real algebraic varieties 19A49: $K_0$ of other rings
Secondary: 55R50: Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX}

Keywords
algebraic $2$-sphere idempotent matrix $\tilde K_0$ polynomial map

Citation

Golasiński, Marek; Ruiz, Francisco Gómez. On the algebraic K-theory of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$. Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 5, 849--860. doi:10.36045/bbms/1323787172. https://projecteuclid.org/euclid.bbms/1323787172


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