Characteristic cycle and wild ramification for nearby cycles of étale sheaves
type: Item
instance of: scholarly article
title: Characteristic cycle and wild ramification for nearby cycles of étale sheaves
zbMATH Open document ID: 1478.14037
author: H. Y. Hu
author: Jean-Baptiste Teyssier
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2021-08-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1811.02860
review text: Let \(S\) be a henselian trait with a perfect residue field of characteristic \(p>0\). Let \(\eta\) and \(\tilde{\eta}\) be respectively the generic and geometric generic point of \(S\), \(G\) the Galois group of \(\tilde{\eta}\) over \(\eta\), \(X\) a scheme and \(f:X\rightarrow S\) a morphism of finite type. Let \(\ell \neq p\) be a prime number, \(\Lambda\) a finite field of characteristic \(\ell\) and \(\mathcal{K}\) a bounded below complex of étale sheaves of \(\Lambda\)-modules on \(X\). Introduced by Grothendieck in the 1960s, the nearby cycles complex \(R\Psi (\mathcal{K},f)\) and the vanishing cycles complex \(R\Phi (\mathcal{K},f)\) of \(\mathcal{K}\) relative to \(f\) are complexes on the geometric special fiber of \(f\) endowed with a \(G\)-action, called the monodromy action. The authors give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes' and Saito's logarithmic ramification filtration. This provides a positive answer to a conjecture of Leal for smooth morphisms in equal characteristic. The ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait is also studied.
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 14D05
zbMATH DE Number: 7377460
zbMATH Keywords: henselian trait
zbMATH Keywords: étale sheaf
zbMATH Keywords: abelian scheme
zbMATH Keywords: monodromy
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
OpenAlex ID: W3136322455
arXiv ID: 1811.02860
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DOI: 10.1515/CRELLE-2021-0009
description: scientific article