We give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from Joyal.

We give an affirmative answer to a conjecture proposed by Tevelev [16] in characteristic 0 case: any variety contains a schön very affine open subvariety. Also we show that any fan supported on the tropicalization of a schön very affine variety produces a schön compactification. Using toric schemes over a discrete valuation ring, we extend tropical compatifications to the non-constant coefficie...

For a prime number p > 2, we give a direct proof of Breuil’s classification of finite flat group schemes killed by p over the valuation ring of a p-adic field with perfect residue field. As application we establish a correspondence between finite flat group schemes and Faltings’s strict modules which respects associated Galois modules via the Fontaine-Wintenberger field-of-norms functor

We prove, for an arithmetic scheme X/S over a discrete valuation ring whose special fiber is a strict normal crossings divisor in X, that the Swan conductor of X/S is equal to the Euler characteristic of the torsion in the logarithmic de Rham complex of X/S. This is a precise logarithmic analog of a theorem by Bloch [1].

Contents Introduction 2 1. Homology theory, cycle map, and Kato complex 6 2. Vanishing theorem 10 3. Bertini theorem over a discrete valuation ring 14 4. Surjectivity of cycle map 17 5. Blowup formula and moving lemma 19 6. Proof of main theorem 21 7. Applications of main theorem 23 References 25 1 2 SHUJI SAITO AND KANETOMO SATO

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.

Contents Introduction 2 1. Homology theory and cycle map 6 2. Kato homology 12 3. Vanishing theorem 16 4. Bertini theorem over a discrete valuation ring 20 5. Surjectivity of cycle map 23 6. Blow-up formula 25 7. A moving lemma 28 8. Proof of main theorem 30 9. Applications of main theorem 33 Appendix A.

We give axiomatizations and quantifier eliminations for first-order theories of finitely ramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from Joyal.

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

The aim of this paper is to report on recent work on liftings of groups of au-tomorphisms of a formal power series ring over a eld k of characteristic p to characteristic 0, where they are realised as groups of automorphisms of a formal power series ring over a suitable valuation ring R dominating the Witt vectors W(k): We show that the lifting requirement for a group of automorphisms places se...