82 research outputs found
Separably closed fields and contractive Ore modules
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
Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces
Let be a non-archimedean local field, a smooth and proper -scheme,
and fix a pluricanonical form on . For every finite extension of ,
the pluricanonical form induces a measure on the -analytic manifold
. We prove that, when runs through all finite tame extensions of
, suitable normalizations of the pushforwards of these measures to the
Berkovich analytification of converge to a Lebesgue-type measure on the
temperate part of the Kontsevich--Soibelman skeleton, assuming the existence of
a strict normal crossings model for . We also prove a similar result for all
finite extensions under the assumption that has a log smooth model.
This is a non-archimedean counterpart of analogous results for volume forms on
degenerating complex Calabi--Yau manifolds by Boucksom and the first-named
author. Along the way, we develop a general theory of Lebesgue measures on
Berkovich skeleta over discretely valued fields
K-theory and topological cyclic homology of henselian pairs
Given a henselian pair of commutative rings, we show that the
relative -theory and relative topological cyclic homology with finite
coefficients are identified via the cyclotomic trace . This
yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity
theorem (for mod coefficients, with invertible in ) and McCarthy's
theorem on relative -theory (when is nilpotent).
We deduce that the cyclotomic trace is an equivalence in large degrees
between -adic -theory and topological cyclic homology for a large class
of -adic rings. In addition, we show that -theory with finite
coefficients satisfies continuity for complete noetherian rings which are
-finite modulo . Our main new ingredient is a basic finiteness property
of with finite coefficients.Comment: 59 pages, revised and final versio
Contracting Endomorphisms of Valued Fields
We prove that the class of separably algebraically closed valued fields
equipped with a distinguished Frobenius endomorphism is
decidable, uniformly in . The result is a simultaneous generalization of the
work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and
the work of the first author and Hrushovski (in the case where the fields are
algebraically closed).
The logical setting for the proof is a model completeness result for valued
fields equipped with an endomorphism which is locally infinitely
contracting and fails to be onto. Namely we prove the existence of a model
complete theory amalgamating the theories
and introduced in [4] and [9],
respectively. In characteristic zero, we also prove that
is NTP and classify the stationary types: they
are precisely those orthogonal to the fixed field and the valuation group
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