Universal extension crystals of 1-motives and applications
type: Item
instance of: scholarly article
title: Universal extension crystals of 1-motives and applications
zbMATH Open document ID: 1225.14014
author: Fabrizio Andreatta
author: Alessandra Bertapelle
published in: Journal of Pure and Applied Algebra
publication date: 2011-05-10T00:00:00Z
review text: This paper contains the construction of the Marvin map in the context of 1-motives, so providing realizations of the Manin map in the de Rham context, in the context of mixed Hodge structures (MHS) and in the crystalline context. The Manin map is a homomorphism \[ {\mathbf A}(S)\to \text{Ext}^1_S({\mathbf H}^1_{dR}({\mathbf A}/S,\nabla_{{\mathbf A}}),({\mathcal O}_S, d)), \] where \(S\) is a smooth connected curve and \({\mathbf A}\) is an abelian scheme. Here \(\text{Ext}^1_S\) is the group of isomorphism classes of\({\mathcal O}_S\)-modules endowed with an integral connection, extension of \({\mathbf H}^1_{dR}({\mathbf A}/S)\), with its Gauss-Marvin connection \(\nabla_{{\mathbf A}}\), by \({\mathcal O}_S\), with the usual derivation \(d\). A 1-motive \({\mathbf M}\) over the scheme \(S\) is the datum of an \(S\)-group scheme \({\mathbf X}\) locally defined, for the étale topology, by a finite \({\mathbf Z}\)-module, an \(S\)-group scheme \({\mathbf G}\), which is an extension over \(S\) of an abelian scheme \({\mathbf A}\) by a torus \({\mathbf T}\) and a homomorphism of \(S\)-group schemes \(u:{\mathbf X}\to{\mathbf G}\). For a 1-motive \({\mathbf M}\) one can define a Cartier dual \({\mathbf M}^\vee\) and the \(S\)-valued points \({\mathbf M}(S)\) which are the elements \(x\in\Hom({\mathbf Z},{\mathbf M})\) in the bounded derived category of sheaves on the fppf site of \(S\). Then there is an isomorphism of abelian groups \[ {\mathcal M}_{{\mathbf M}}:{\mathbf M}(S)\to \text{Ext}^1_1\text{-mot}({\mathbf M}^\vee,{\mathbf G}_m) \] which is functorial in \({\mathbf M}\) and \(S\). The authors construct a Gauss-Manin connection \(\nabla_{{\mathbf M}}\) on the locally free \({\mathcal O}_S\)-module \({\mathbf T}_{dR}({\mathbf M})\), the de Rham realization of \({\mathbf M}\). This connection is functorial in \({\mathbf M}\) and preserves the weight filtration on \({\mathbf T}_{dR}({\mathbf M})\). Composing \({\mathcal M}_{{\mathbf M}}\) with the de Rham realization …
reviewed by: Claudio Pedrini
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 5886350
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.11.004
OpenAlex ID: W2109493996
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DOI: 10.1016/J.JPAA.2010.11.004
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description: scientific article; zbMATH DE number 5886350