Coherence of certain overconvergent isocrystals without Frobenius structures on curves
type: Item
instance of: scholarly article
title: Coherence of certain overconvergent isocrystals without Frobenius structures on curves
zbMATH Open document ID: 1264.12005
author: Tomoyuki Abe
published in: Mathematische Annalen
publication date: 2011-08-16T00:00:00Z
review text: This paper proves coherence as \(\mathcal D^\dagger\)-modules of certain overconvergent isocrystals on curves, even at the absence of Frobenius structures. Let \(R\) be a complete discrete valuation ring of mixed characteristic, with perfect residue field \(k\). Let \(K\) denote its field of fractions. Let \(\mathcal X\) be a smooth formal curve over \(R\), \(\mathcal U\) a dense open subscheme, and \(S: = \mathcal X - \mathcal U\) (which we may assume to be a finite set of \(k\)-points). Let \(M\) be a convergent isocrystal on \(\mathcal U\) overconvergent along \(S\). The author makes the following assumption: at each point \(s \in S\), the \(0\)-slope part (tame part) of the differential modules at \(s\) is trivial; here the slope refers to applying the deep theorem of Christol-Mebkhout to the base change of \(M\) to the formal completion of \(\mathcal X\) at \(s\), so that we obtain a decomposition of the differential module into a direct sum of ones with pure slopes (which is a measure of irregularity in the \(p\)-adic setting). If we use \(\mathrm{sp}: \mathcal X_K \to \mathcal X\) to denote the specialization map, then the main theorem of this paper states that \(\mathrm{sp}_*(M)\) is a coherent \(\mathcal D^\dagger_{\mathcal X, \mathbb Q}\)-module. Note that there is no assumption on the existence of Frobenius. The proof of the theorem is to construct a finite presentation of \(\mathrm{sp}_*(M)\) as a \(\mathcal D^\dagger_{\mathcal X, \mathbb Q}\)-module directly. For this, one may work locally. If \(e_1, \dots, e_\mu\) form a basis of \(M\), the idea is to construct differential operators \(P^{(i)}_{k,j}\) such that \(\sum_j P^{(i)}_{k,j}e_j = x^{-k} e_i\), where \(x\) is a local parameter at a point \(s \in S\). The technical core lies in, when \(M\) is a differential module pure of some positive slope near \(s\), one can write down such differential operators \(P^{(i)}_{k,j}\) satisfying certain nice bounds on their norms. The paper is very technical. Alt…
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 5939568
zbMATH Keywords: overconvergent isocrystals
zbMATH Keywords: coherence of arithmetic D-modules
zbMATH Keywords: p-adic differential equations
zbMATH Keywords: slope decomposition
MaRDI profile type: Publication
full work available at URL: s00208-010-0573-8
OpenAlex ID: W2053630891
cites work: Q3325851
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cites work: On exponents of p-adic differential modules.
cites work: Explicit Frobenius descent for \({\mathcal D}^\dagger\)-modules
cites work: Descent of isocrystals and \({\mathcal D}^\dag\)-modules
cites work: Q3579590
cites work: Radii of Convergence and Index for p-Adic Differential Operators
DOI: 10.1007/S00208-010-0573-8
description: scientific article; zbMATH DE number 5939568