arXiv subjects

Christian Serpé

Publications and source records attributed to Christian Serpé.

6 recordsLinked to original sources

arXiv2018

On the Spectrum of Nonstandard Dedekind Rings

The methods of nonstandard analysis are applied to algebra and number theory. We study nonstandard Dedekind rings, for example an ultraproduct of the ring of integers of a number field. Such rings possess a rich structure and have interesting relations to standard Dedekind rings and their completions. We use lattice theory to classify the ideals of nonstandard Dedekind rings. The nonzero prime ideals are contained in exactly one maximal ideal and can be described using valuation theory. The localisation at a prime ideal gives a valuation ring and we determine the value group and the residue field. The spectrum of a nonstandard Dedekind ring is described using lattices and value groups. Furthermore, we investigate the Riemann-Zariski space and the valuation spectrum of nonstandard Dedekind rings and their quotient fields.

math.NT↗

arXiv2009

Uniform bounds and ultraproducts of cycles

This paper is about the question whether a cycle in the l-adic cohomology of a smooth projective variety over the rational numbers, which is algebraic over almost all finite fields, is also algebraic over the rationals. We use ultraproducts respectively nonstandard techniques in the sense of A. Robinson, which the authors applied systematically to algebraic geometry. We give a reformulation of the question in form of uniform bounds for the complexity of algebraic cycles over finite fields.

math.AG↗

arXiv2008

Nonstandard model categories and homotopy theory

In order to apply nonstandard methods to questions of algebraic geometry we continue our investigation from "Enlargements of categories" (Theory Appl. Categ. 14 (2005), No. 16, 357--398) and show how important homotopical constructions behave under enlargements.

math.CT↗

arXiv2008

Etale and motivic cohomology and ultraproducts of schemes

This paper is a continuation of the authors article "Enlargements of schemes" (Log. Anal.1 (2007), no. 1, 1-60) We mainly study the behaviour of etale cohomology, algebraic cycles and motives under ultraproducts respectively enlargements. The main motivation for that is to find methods to transfer statements about etale cohomology and algebraic cycles from characteristic zero to positive characteristic and vice versa. We give one application to the independence of $l$ of Betti numbers in etale cohomology and applications to the complexity of algebraic cycles.

math.AG↗

arXiv2005

On a spectral sequence for equivariant K-theory

We apply the machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin-Graham) and an Atiyah-Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch-Lichtenbaum and Friedlander-Suslin.

math.KT↗

arXiv2004

Nonstandard Etale Cohomology

A lot of good properties of etale cohomology only hold for torsion coefficients. We use "enlargement of categories" as developed in http://arxiv.org/abs/math.CT/0408177 to define a cohomology theory that inherits the important properties of etale cohomology while allowing greater flexibility with the coefficients. In particular, choosing coefficients *Z/P (for P an infinite prime and *Z the enlargement of Z) gives a Weil cohomology, and choosing *Z/l^h (for l a finite prime and h an infinite number) allows comparison with ordinary l-adic cohomology. More generally, for every N in *Z, we get a category of *Z/N-constructible sheaves with good properties.

math.AG↗