The Cauchy-Kowalevski theorem for \(\mathcal E\)-modules
type: Item
instance of: scholarly article
title: The Cauchy-Kowalevski theorem for \(\mathcal E\)-modules
zbMATH Open document ID: 0906.35004
DOI: 10.1016/S0021-7824(98)80002-0
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-01-21T00:00:00Z
review text: The author proves a version of the Cauchy-Kowalevski theorem for \({\mathcal E}_X\)-modules and for holomorphic functions in the category \(D^b(Y;p_Y).\) Namely, upon considering a complex analytic manifold \(X,\) a point \(p_X\in T^*X\) and a coherent left \({\mathcal E}_X\)-module \({\mathcal M},\) he proves that the object \(R{\mathcal H}om_{{\mathcal E}_X}({\mathcal M},{\mathcal O}_X)_{p_X}\) is well defined in \(D^b(X;p_X)\) (by proving, among other properties that, for instance, it depends only on \({\mathcal M}\) up to isomorphisms, and that it is functorial with respect to \({\mathcal M}).\) He next takes a morphism \(f\colon Y\rightarrow X\) of complex manifolds, a point \(p\in Y\times_XT^*X,\) considers \(p_X=f_\pi(p)\) and \(p_Y= ^t f'(p),\) and supposes that \({\mathcal M}\) is defined in a neighborhood of \(p_X\) and non-characteristic with respect to \(f.\) Upon denoting by \(\underline{f_p}^{-1}{\mathcal M}\) the inverse image of \({\mathcal M}\) by \(f,\) he constructs the natural morphism \(f_p^{-1}R{\mathcal H}om_{{\mathcal E}_X}({\mathcal M},{\mathcal O}_X)_{p_X}\rightarrow R{\mathcal H}om_{{\mathcal E}_Y}(\underline{f_p}^{-1}{\mathcal M},{\mathcal O}_Y)_{p_Y}\) in the category \(D^b(Y;p_Y)\) and proves that it is an isomorphism. In the final section, he gives an application.
Mathematics Subject Classification ID: 35A10
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 58J15
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 1207282
zbMATH Keywords: \({\mathcal E}_X\)-modules
author: Ryuichi Ishimura
reviewed by: Alberto Parmeggiani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1207282