Exterior powers of torsion-free Abelian groups.
type: Item
instance of: scholarly article
title: Exterior powers of torsion-free Abelian groups.
zbMATH Open document ID: 0616.20025
DOI: 10.1007/BF01189860
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: Some results concerning exterior products of torsionfree abelian groups of finite rank are given. Firstly if \(\Lambda^ nG\) is the n-th-exterior product of the torsionfree abelian group G of rank r and F is a free abelian subgroup of maximal rank \(\left( \begin{matrix} r\\ n\end{matrix} \right)\) in \(\Lambda^ nG\), then the structure of \(\Lambda^ nG/F\) is described. Using the natural correspondence between \(\Lambda^ nG\) and its associated projective space of dimension \(\left( \begin{matrix} r\\ n\end{matrix} \right)\) the author shows that the elements of \(\Lambda^ nG\), which can be written as exterior powers \(g_ 1\wedge g_ 2\wedge...\wedge g_ n\) with \(g_ i\in G\) correspond to the points of the Grassmannian manifold \(G_{r-1,n-1}\) in the projective space. (Satz 2.2). This characterization is used to determine the torsionfree abelian groups of rank \(\left( \begin{matrix} r\\ n\end{matrix} \right)\) that may arise as exterior powers. Finally, it is shown that the group \(\Lambda^ nG\) and the nth- homology group \(H_ n(G,Z)\) are isomorphic.
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 3998090
zbMATH Keywords: exterior products of torsionfree abelian groups of finite rank
zbMATH Keywords: Grassmannian manifold
zbMATH Keywords: projective space
zbMATH Keywords: homology group
author: Huberta Lausch
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01189860
OpenAlex ID: W2043252520
description: scientific article; zbMATH DE number 3998090