We show that the algebraic K -theory of generalized archimedean valuation rings occurring in Durov’s compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces.We also show that the “residue field at infinity” is badly behaved from a K -theoretic point of view.

Let F be a local non-archimedean field and let G be the group of F valued points of a connected reductive algebraic group over F. In this paper we compute the Ext-groups of generalized Steinberg representations in the category of smooth G-representations with coefficients in a certain self-injective ring. AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 11F70.

Let F be a non-Archimedean locally compact field, let G be a split connected reductive group over F . For a parabolic subgroup Q ⊂ G and a ring L we consider the G-representation on the L-module (∗) C∞(G/Q, L)/ ∑

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring A∞ = ∏ ν|∞Kν , provides the right theory to obtain, using the λ-operations, Serre’s archimedean local factors of the complex Lfunction of X as regularized determinants.

In this paper, we generalize the main result of [Mzk2] (to the effect that very general noetherian log schemes may be reconstructed from naturally associated categories) to the case of log schemes locally of finite type over Zariski localizations of the ring of rational integers which are, moreover, equipped with certain “archimedean structures”.

It is well known [4] that the non-Archimedean residue class fields K of the ring of continuous real valued functions on a space are realclosed and 7/i-sets. It does not appear to be known that the exponential function in the reals induces an exponential function in K (definitions to follow) ; thus K is exponentially closed. The property of being exponentially closed is a new invariant which wil...

Berarducci (2000) studied irreducible elements of the ring k((G<0))⊕Z, which is an integer part of the power series field k((G)) where G is an ordered divisible abelian group and k is an ordered field. Pitteloud (2001) proved that some of the irreducible elements constructed by Berarducci are actually prime. Both authors mainly concentrated on the case of archimedean G. In this paper, we study ...

Let K be a non-archimedean field with residue field k, and suppose that k is not an algebraic extension of a finite field. We prove two results concerning wandering domains of rational functions φ ∈ K(z) and Rivera-Letelier’s notion of nontrivial reduction. First, if φ has nontrivial reduction, then assuming some simple hypotheses, we show that the Fatou set of φ has wandering components by any...