Algebraic \(K\)-theory of the infinite place (Q897097)

While it is well-known in number theory that ``the spectrum of \(\mathbb{Z}\) should be completed with an infinite prime'', an actual technical framework for doing so has only recently been proposed by Durov (unpublished). Using that approach, the author analyses the algebraic \(K\)-theory of the local ring \(\mathbb{Z}_{(\infty)}\) at the infinite prime, and of similar rings of number-theoretic interest. The results are phrased in terms of Durov's ``generalised rings''. These are monads in the category of sets which commute with filtered colimits and satisfy a certain commutativity condition. The relevant definitions and results are recalled in \S2. In \S3, generalised valuation rings are introduced and the theory of their free and projective modules is developed. Using Waldhausen's \(\mathcal{S}_{\bullet}\)-construction, algebraic \(K\)-theory of generalised valuation rings is defined. The author shows that the resulting \(K\)-groups are isomorphic to the stable homotopy groups of the classifying space of norm-1 elements in the generalised valuation ring (Theorem~3.14). In \S4, the paper finishes with illustrating differences between the generalised and classical ring setup in the context of localisation and completion.