Vanishing cycles and non-classical parabolic cohomology
type: Item
instance of: scholarly article
title: Vanishing cycles and non-classical parabolic cohomology
zbMATH Open document ID: 0870.14013
author: Anthony J. Scholl
published in: Inventiones Mathematicae
publication date: 1996-05-29T00:00:00Z
review text: The paper deals with the formalism of vanishing cycles and some applications to non-classical parabolic cohomology. In particular, one constructs non-zero elements in motivic cohomology (i.e. algebraic \(K\)-groups) classifying extensions in the category of mixed motives over some number field \(K\), associated to subgroups of \(\text{SL}_2({\mathbb Z})\) which are not congruence subgroups. Let \(S\) be the spectrum of a strictly henselian discrete valuation ring with generic point \(\eta\) and closed point \(s\). Write \(I=\text{Gal}(\overline{\eta}/\eta)\), and let \(\ell\) be a prime different from the residue characteristic of \(S\). Consider a smooth curve \(f:X\rightarrow S\) with generic point \(\xi\). Assume \(f\) has two sections with images \(Z_1\), \(Z_2\) in \(X\) such that \(Z_1\), \(Z_2\) intersect in \(x\in X_s\). Write \(Z=Z_1\cup Z_2\) and \(q:Z\hookrightarrow X\) and \(g:U=X-Z\hookrightarrow X\) for the inclusions. Assume there is a \({\mathbb Q}_{\ell}\)-sheaf \({\mathcal F}\) on \(U\), lisse on the intersection of \(U\) with some neighbourhood of \(x\). Finally, assume the local monodromy representation of \(\pi_1(\overline{U},\overline{\xi})\) on \({\mathcal F}_{\overline{\xi}}\) is unipotent and admits a factorization \[ \pi_1(\overline{U},\overline{\xi})\longrightarrow\bigoplus_{r=1,2} \pi_1^{(\ell)}(\overline{X}_x-\overline{Z}_r,\overline{\xi}) {\buildrel\sim\over\longrightarrow}{\mathbb Z}_{\ell}(1)^2{\buildrel \chi\over\longrightarrow}\text{Aut}{\mathcal F}_{\overline{\xi}}, \] for some \(\chi\), where \(\overline{X}_x\) is the strict henselization of \(X\) at \(x\), \(\overline{U}=\overline{X}_x\times_XU\), and \(\overline{Z}_r=\overline{X}_x\times_XZ_r\). One has an action of \(I\) on \({\mathcal F}_{\overline{\xi}}\). For any \({\mathbb Q}_{\ell}\)-sheaf \({\mathcal G}\) on \(X\) denote the sheaves of vanishing cycles by \(R^i\Phi{\mathcal G}\). The main result of the paper now says that there is an \(I\)-equivariant isomorphism \((R^1\P…
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14A20
zbMATH DE Number: 866100
zbMATH Keywords: vanishing cycles
zbMATH Keywords: non-classical parabolic cohomology
zbMATH Keywords: motivic cohomology
zbMATH Keywords: spectrum of a strictly henselian discrete valuation ring
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
full work available at URL: s002220050061
OpenAlex ID: W2067875078
DOI: 10.1007/S002220050061
description: scientific article; zbMATH DE number 866100