Algebraic \(K\)-theory of special groups (Q2576921)

Special groups were introduced by the authors [``Special groups. Boolean-theoretic methods in the theory of quadratic forms'', Mem. Am. Math. Soc. 689 (2000; Zbl 1052.11027)] in order to axiomatize the algebraic theory of quadratic forms over fields in terms of isometry relation on binary forms. This approach is equivalent to other axiomatic approaches proposed and studied in the 1980s. The authors introduced also to the context of special groups the formalism of Milnor's \(K\)-theory modulo \(2\) in [Algebra Colloq. 10, 149--176 (2003; Zbl 1053.11036)]. In the present paper they compute the \(K\)-theory of special groups arising as Boolean algebras, inductive limits, finite products or extensions by a group of exponent \(2\), and also in some other situations. A second aim is the study of a property SMC which is the analog of the injectivity of the multiplication by \(\ell(-1)\) map in the Milnor's \(K\)-theory modulo \(2\). It is proved that SMC is preserved in the constructions on special groups mentioned above.