Witt rings and K-theory
The following result about the Witt ring W(A) of a ring A is proved: If A is a commutative ring and \(X_ A\) denotes its real spectrum [see \textit{M. Coste} and \textit{M.-F. Roy}, Contemp. Math. 8, 27-59 (1982; Zbl 0485.14007)] then there is a natural ring homomorphism \(W(A)\to KO(X_ A)\) such that both kernel and cokernel are 2-torsion groups. KO(X\({}_ A)\) is an analogoue of the usual K-theory of a topological space. A large part of the paper deals with the definition and with alternative descriptions of KO(X), where X is a constructible subset of \(X_ A\). One such description is an isomorphism KO(X)\(\cong W(C(X))\). C(X) denotes the ''ring of abstract semi-algebraic functions on X''. The paper contains a few remarks about this ring. A more extensive discussion may be found in the reviewer's Habilitationsschrift: ''Real closed spaces'' (München 1984) [for an abstract, see Rocky Mountain J. Math. 14, 971- 972 (1984; see the preceding review) and \textit{H. Delfs}, J. Reine Angew. Math. 355, 108-128 (1985; Zbl 0543.55004)]. The natural ring homomorphism \(A\to C(X_ A)\) provides the homomorphism \(W(A)\to W(C(X_ A))\cong KO(X_ A)\) of the theorem. A comparision of these Witt rings yields the theorem.
ring of abstract semi-algebraic functions
scientific article; zbMATH DE number 3920651