On the algebraic \(K\)-theory of \(R[X,Y,Z]/(X^2+Y^2+Z^2-1)\)
The authors calculate the Brouwer degree of a certain polynomial map. It has been already proven by \textit{R. G. Swan} [Ann. Math. Stud. 113, 432--522 (1987; Zbl 0708.55014)] that \(\tilde{K_0}\) of the coordinate ring \(A=R[X,Y,Z]/(X^2+Y^2+Z^2-1)\) of the algebraic 2-sphere \(S^2(R)\) over a field \(R\) is isomorphic to the integers \({\mathbb Z}\), whenever \(R\) is of characteristic not two and contains the squared root of \(-1\). The authors use explicit idempotent matrices \(\gamma^p\) of order 2, that correspond to the integer \(p\) in the aforementioned isomorphism, and represent classes in \(\tilde{K_0}(A)\) to calculate the Brouwer degree of a polynomial map \(S^2(R)\rightarrow S^2(R)\), where \(R\) is a field of characteristic zero. They then proceed to apply the algebraic de Rham cohomology of Kähler differentials to define Brouwer degree for polynomial homomorphisms of \(A\) to itself, and relate the problem of finding hermitian representatives for \(R = K(\sqrt{-1})\), where \(K\) does not contain \(\sqrt{-1}\), to some problems of representing Brouwer degrees by polynomial maps. In particular they show that if \({\mathbb Q}[i] \subset R\), then there is a polynomial map from \(S^2(R)\) to itself of Brouwer degree \(p\) for every \(p\) in \({\mathbb Z}\). On the other hand, if \(R\) contains all algebraic real numbers, then there is a polynomial map from \(S^2(R)\) to itself of Brouwer degree \(p\) for any odd \(p\).
scientific article; zbMATH DE number 5996087
scientific article