Let OK be a complete discrete valuation ring of residue characteristic p > 0, and G be a finite flat group scheme over OK of order a power of p. We prove in this paper that the Abbes-Saito filtration of G is bounded by a linear function of the degree of G. Assume OK has generic characteristic 0 and the residue field of OK is perfect. Fargues constructed the higher level canonical subgroups for ...

The fabrication of ultrathin-ring electrodes with a diameter of 2 mm and a thickness of 100 nm is established. The ultrathin-ring electrodes provide a large density of pseudo-steady-state currents, and realize pseudo-steady-state amperometry under quiescent conditions without a Faraday cage. Under the limiting current conditions, the current response at the ultrathin-ring electrode can be well ...

CARL FAITH All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective . SISI rings include Noetherian rings, Morita rings, and almost maximal valuation rings ([Vil) . In [F3] we raised the question of whether a polynomial ring R[-1 over a SISI ring R is again SISI . In this paper we show this is not...

We establish a trace formula for rigid varieties X over a complete discretely valued field of equicharacteristic zero, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the analytic germ of f at x. We develop a theory of motivic integration for formal schem...

In this paper, on a non-standard extension (∗X ,∗d) of a metric space (X ,d), we construct a chain of new non-standard topologies in terms of convex subrings of ∗R, its minimal element is the Stopology and its maximal is the Q-topology. Next, we construct X̂ , the F -asymptotic hull of X , and we prove that such space is metrizable and complete when F is generated by an asymptotic scale. Finally...

This paper is part of the program of reverse mathematics. We assume the reader is familiar with this program as well as with RCA,, and WKL,, the two weak subsystems of second-order arithmetic we are going to work with here. (If not, a good place to start is [2].) In [2], [3], [4], many well-known theorems about countable rings, countable fields, etc. were studied in the context of reverse mathe...

We show that an algebraic immediate valuation ring extension of characteristic $$p>0$$ is a filtered union complete intersection algebras finite type.

We develop a valuation theory for generalized polygons similar to the existing theory for dense near polygons. This valuation theory has applications for the study and classification of generalized polygons that have full subpolygons as subgeometries.

In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring A. We give two versions of the construction: the first, ...