Chern maps for \(U\)-theory
Let \({\mathcal A}\) be an exact \(k\)-linear category with a duality functor. In this paper the authors define the homologies and their variants for \({\mathcal A}\) using a Quillen's \(Q\)-construction and show that such a construction yields a generalization of the Dennis trace map and the Jones-Goodwillie Chern map into the \(U\)-theory of \({\mathcal A}\). This is done as follows. They first define the Dennis trace map from \(K_*({\mathcal A})\) to \(HH_*({\mathcal A})\) and the Jones-Goodwillie Chern map from \(K_*({\mathcal A})\) to \(HN_*({\mathcal A})\), which are equivalent to the maps defined by \textit{R. McCarthy} using the \(S\)-construction. These maps are extended to \(_\varepsilon U_*({\mathcal A})\) by composing these maps with the canonical map from \(_\varepsilon U_*({\mathcal A})\) to \(K_*({\mathcal A})\) which comes from forgetting the \(\varepsilon\)-hermitian form. It is proven that when 2 is invertible in the ground ring \(k\), these generalized maps factor through the corresponding involutive homologies. And also it is shown that sliding \(2p\) dimensions to the positive direction in particular the second one can be transformed into a map from \(_\varepsilon U_*({\mathcal A})\) to the dihedral homology \(HD_{*+2p} ({\mathcal A})\) which is the involutive cyclic homology. Furthermore the authors note that if all short exact sequences in \({\mathcal A}\) split, then the \(L\)-theory \(_\varepsilon L_*({\mathcal A})\) of \({\mathcal A}\) can be defined analogously and statements similar to the above can be made. This provides another proof of the early results of \textit{G. Cortinas} and \textit{J. Lodder}.
scientific article; zbMATH DE number 986705