Constructions and dévissage in Hermitian K-theory
The author first defines the Hermitian \(K\)-theory of an exact category \(\mathcal C\) with duality. The definition extends the known case of additive categories in a highly nontrivial way, since the Hermitian category associated to the exact category is in general not exact. The author constructs a certain category \(\mathcal L(\mathcal C)\) inspired by an approach of \textit{C. H. Giffen} [J. Pure Appl. Algebra 52, 1-30 (1988; Zbl 0655.18008)], and defines the Hermitian \(K\)-theory \(K^h_*\) of \(\mathcal C\) as the homotopy groups of the classifying space of \(\mathcal L(\mathcal C)\). In the second part of the paper the author proves dévissage for the Hermitian \(K\)-theory of a Dedekind ring \(A\), i.e., he shows that there are isomorphisms \[ K_n^h(\mathcal T_S) \cong \oplus K_n^h(P(A/\mathcal P)) \] between the Hermitian \(K\)-groups of the category \(\mathcal T_S\) of torsion modules over \(A\) and the direct sum of the Hermitian \(K\)-groups of the categories of finitely-generated projective modules over the residue fields \(A/\mathcal P\). The same result holds for \(\mathcal U\)-theory. In combination with a localization theorem of Hornbostel-Schlichting this produces a long exact localization sequence, conjectured by \textit{M. Karoubi} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 359-404 (1974; Zbl 0325.18011)]. For finite fields the \(\mathcal U\)-theory groups are calculated based on the corresponding calculations of \textit{E. M. Friedlander} for Hermitian \(K\)-groups of finite fields [Topology 15, 87-109 (1976; Zbl 0401.18007)]. This leads to a more or less explicit description of the relation between the Hermitian \(K\)-theory of the ring of integers \(A\) in a number field \(F\) and the Hermitian \(K\)-theory of \(F\).
scientific article; zbMATH DE number 1879623