Topological \(p\)-adic vector spaces and index theory
type: Item
instance of: scholarly article
title: Topological \(p\)-adic vector spaces and index theory
zbMATH Open document ID: 0838.47054
author: Zoghman Mebkhout
author: Gilles Christol
published in: Annales Mathématiques Blaise Pascal
publication date: 1996-05-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79133
review text: Suppose \(u\) is a linear transformation of a vector space \(E\) over a complete ultrametric field \(k\). This \(u\) is said to have an index \(\chi (u)\) if both kernel and cokernel of \(u\) are finite-dimensional. We set \(\chi (u)= \chi(u, E)= \dim(\ker u)- \dim(\text{coker } u)\). The basic question is to prove the existence of the index when \(u\) is an ordinary differential operator on certain power series rings. The main theorem of this paper says that if \(r_n\) is an increasing sequence of positive reals tending to \(r\) and \(P\) is a differential operator in the ring \({\mathcal H}^+ (r_n)\) of Laurent series of \({1\over x} k[[{1\over x}]]\) converging in a disk \(s\leq |x|\) for some \(s< r_n\) having an index in \({\mathcal H}^+ (r_n)\) for all \(n\), then it has an index in \({\mathcal H}^+ (r)\) and \(\chi (P,{\mathcal H}^+ (r))\) is equal to the limit of \(\chi (P,{\mathcal H}^+ (r_n))\) as \(n\) tends to \(\infty\).
Mathematics Subject Classification ID: 47S10
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 13J05
zbMATH DE Number: 800318
zbMATH Keywords: complete ultrametric field
zbMATH Keywords: ordinary differential operator
zbMATH Keywords: Laurent series
reviewed by: N. Sankaran
MaRDI profile type: Publication
OpenAlex ID: W2323150169
cites work: On the index theorem for \(p\)-adic differential equations. I
cites work: On the index theorem for \(p\)-adic differential equations. II
cites work: Éléments de géométrie algébrique. I: Le langage des schémas. II: Étude globale élémentaire de quelques classe de morphismes. III: Étude cohomologique des faisceaux cohérents (première partie)
cites work: Q5833442
cites work: Duality of projective limit spaces and inductive limit spaces over a nonspherically complete nonarchimedean field
cites work: Q3717175
DOI: 10.5802/AMBP.22
description: scientific article; zbMATH DE number 800318