Canonical extensions, irregularities and the Swan conductor
type: Item
instance of: scholarly article
title: Canonical extensions, irregularities and the Swan conductor
zbMATH Open document ID: 0969.14012
author: Richard Crew
published in: Mathematische Annalen
publication date: 2000-03-30T00:00:00Z
review text: \(k\) is a perfect field of characteristic \(p>0\) and \(X/k\) is an affine smooth, absolutely irreducible curve with smooth compactification \(\overline{X}\). One fixes a finite field extension \(K\) of the field of fractions of \(W(k)\) and an action of Frobenius on \(K\) which has fixed field \(K_0\). According to N.~Tsuzuki, the category of the continuous representations of \(\pi _1(X)\) in finite dimensional vector spaces over \(K_0\) with finite local monodromy at the points of \(\overline{X}-X\) is equivalent to the category of the overconvergent unit-root \(F\)-isocrystals on \(X\) over \(K\). Moreover N.~Tsuzuki proved that the Swan conductor of the local monodromy representation at a point \(z\) of \(\overline{X}- X\) is equal to the irregularity of the connection of the corresponding crystal at \(z\). In this paper, a geometric proof of the last statement is presented. The basic tool is Katz's theory of the canonical extension of a local Galois representation which leads to an \(\ell\)-adic construction of the Swan representation. Let a representation of \(\pi _1(X)\), as above, be given. Consider a point \(z\in \overline{X}-X\). Let \(\eta _{\infty}\) be the field \(k((T^{-1}))\), equal to the completion of the function field of \(X\) at the place \(z\). The finite group \(G\) is the local monodromy group of the representation at \(z\) and gives rise to an étale covering \(Y\rightarrow X\). Let \(\eta\supset \eta _{\infty}\) denote the induced Galois extension with group \(G\). Then the \(p\)-adic version of Katz's construction states that the Swan representation is isomorphic to \(H^1_c({\mathbb{P}}^1- \{0\},Rj_*\text{Reg}_\eta)\), where \(j:{\mathbb{G}}_m\rightarrow {\mathbb{P}}^1-\{0\}\) is the inclusion and \(\text{Reg}_\eta =f_*K\) is a unit-root \(F\)-isocrystal endowed with a \(G\)-action. Here \(K\) denotes the trivial \(F\)-isocrystal on the étale covering \(Y\) of \(X\). The final statement which completes the proof is that for any overconverge…
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 1406569
zbMATH Keywords: unit-root F-isocrystal
zbMATH Keywords: Swan representation
zbMATH Keywords: irregularity of an F-crystal
MaRDI profile type: Publication
full work available at URL: s002080050002
OpenAlex ID: W2084124662
DOI: 10.1007/S002080050002
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description: scientific article; zbMATH DE number 1406569