Stable K-theory, Topological Hochschild Homology, Generalized free products, S-Algebras, Localization K-theory and generalized free products of S-Algebras: Localization methods Roland Schwänzl Schwänzl Roland Ross E. Staffeldt Staffeldt Ross E. Friedhelm Waldhausen Waldhausen Friedhelm Diskrete Strukturen in der Mathematik, P 99, SFB 343 Bielefeld"

K-theory and generalized free products of S-algebras: Localization methods

Roland Schwänzl, Ross E. Staffeldt, Friedhelm Waldhausen

Preprint series: Diskrete Strukturen in der Mathematik, P 99, SFB 343 Bielefeld

MSC 2000
19D10 Algebraic $K$-theory of spaces

Abstract
A generalized free product diagram of S-algebras is a generalization and stabilization of the diagram of group rings arising from a Seifert-van Kampen situation. Our eventual goal is to obtain a description of the algebraic K-theory of the ``large'' algebra in a generalized free product diagram in terms of the K-theories of the three smaller algebras. We first provide foundational material on generalized free product diagrams of S-algebras and associated categories of Mayer-Vietoris presentations. We show that the categories of Mayer-Vietoris presentations are categories with cofibrations, weak equivalences, and mapping cylinders. In particular, the hypotheses of the ``generic fibration theorem'' of Waldhausen (Algebraic K-theory of spaces, Lecture Notes in Math. 1126(1985), 318-419) are satisfied for two fundamental notions of weak equivalence, and there is, therefore, a three-term fibration sequence up to homotopy in which the K-theory of Mayer-Vietoris presentations with respect to the fine notion of equivalence is compared with the K-theory with respect to the coarse notion. The rest of the paper is concerned with interpreting these K-theories in terms of the K-theories of the algebras in the generalized free product diagram.

The first interpretative result of the paper uses the additivity theorem to identify the K-theory of Mayer-Vietoris presentations with respect to fine equivalences with the product of the K-theories of the three smaller algebras in a given generalized free product diagram.

The second interpretative result uses the approximation theorem to identify the K-theory of Mayer-Vietoris presentations with respect to coarse equivalences with the K-theory of the generalized free product algebra. In order to confirm the hypotheses of the approximation theorem, we develop a localization tool for Mayer-Vietoris presentations which resembles Bousfield's theory of localization of spectra with respect to generalized homology theories.

Investigation of the third term in the fibration sequence will be the subject of another work.

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