On extending \(\mathcal A\) -modules through the coefficients
type: Item
instance of: scholarly article
title: On extending \(\mathcal A\) -modules through the coefficients
zbMATH Open document ID: 1269.18007
author: Anastasios Mallios
author: Patrice Ntumba
published in: Mediterranean Journal of Mathematics
publication date: 2013-03-20T00:00:00Z
review text: Ambient space in this paper is the category of free \(\mathcal A\)-modules, where \(\mathcal A\) is an \(\mathbb R\)-algebra sheaf. The authors model their results on free \(\mathcal A\)-modules on the results in the classical case. Complexification \(\mathcal A_\mathbb C\) of free \(\mathcal A\)-modules is the process of obtaining new free \(\mathcal A\)-modules by enlarging the \(\mathbb R\)-algebra sheaf \(\mathcal A\) to a \(\mathbb C\)-algebra sheaf \(\mathcal A_\mathbb C=\mathbb C\otimes_\mathbb R \mathcal A\). For an \(\mathcal A\)-module \(\mathcal E\) , the complete presheaf of sections of \(\mathcal E\) (denoted by \(\Gamma(\mathcal E)\)) is a complete \(\Gamma(\mathcal A)\)-presheaf. Some of the results are as follows: Theorem 1.1: Let \(\mathcal A\) be an algebra sheaf, \(\mathcal B\) a subalgebra sheaf in \(\mathcal A\), \(\mathcal E\) a free \(\mathcal B\)-module and let \(\vartheta\) be the canonical \(\Gamma(\mathcal B)\)-morphism \(\vartheta=1\otimes id_{\Gamma(\mathcal E)}\). Then, the ordered pair \((\Gamma(\mathcal E_\mathcal A), \vartheta)\) has the following universal property: For every pair \((\Gamma(\mathcal F),\nu)\), where \(\mathcal F\) is an \(\mathcal A\)-module and \(\nu\) is a \(\Gamma(\mathcal B)\) morphism of the \(\Gamma(\mathcal B)\) presheaf \(\Gamma(\mathcal E)\) into the underlying \(\Gamma(\mathcal B)\)-presheaf of \(\Gamma(\mathcal F)\), there is a uniquely defined \(\Gamma(\mathcal A)\) morphism \(\tilde\nu\) sending \(\Gamma(\mathcal E_\mathcal A)\) into \(\Gamma(\mathcal F)\), such that \(\nu=\tilde\nu\circ\vartheta\). Theorem 2.1: Let \(\mathcal A\) be an ordered nonzero-nilpotent free \(\mathbb R\)-algebra sheaf, and \(\mathcal E\) a free \(\mathcal E\)-module. Then, \(\mathcal E \) carries an \(\mathcal A_\mathbb C\)-module structure if and only if, there exists a \(J\in\)End\(_\mathcal A\mathcal E\), such that \(J^2=-1\). Theorem 2.2: The sheaf \(\mathcal S\) of sets of almost complex structures on \((_\mathbb R\mathca…
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 08A35
Mathematics Subject Classification ID: 18D99
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 6147173
zbMATH Keywords: subalgebra sheaf
zbMATH Keywords: \(\Gamma(\mathcal A)\)-presheaf
zbMATH Keywords: almost complex structure
zbMATH Keywords: Kronecker gauge
zbMATH Keywords: complexification
zbMATH Keywords: nonzero-nilpotent \(\mathcal A\)-module
zbMATH Keywords: complex conjugation
zbMATH Keywords: \(\mathcal A\)-automorphism
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: s00009-011-0170-4
OpenAlex ID: W1990638701
cites work: Q4520910
cites work: Q4040889
cites work: Q5580324
cites work: Q3919827
cites work: Complex geometry. An introduction
cites work: Q4393464
cites work: Complexification in \(A\)-modules
DOI: 10.1007/S00009-011-0170-4
description: scientific article; zbMATH DE number 6147173