Exterior powers of torsion-free Abelian groups.
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.
exterior products of torsionfree abelian groups of finite rank
scientific article; zbMATH DE number 3998090