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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Generalizations of the stacked bases theorem


Authors: Paul Hill and Charles Megibben
Journal: Trans. Amer. Math. Soc. 312 (1989), 377-402
MSC: Primary 20K21
DOI: https://doi.org/10.1090/S0002-9947-1989-0937245-8
MathSciNet review: 937245
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Abstract: Let $ H$ be a subgroup of the free abelian group $ G$. In order for there to exist a basis $ {\{ {x_i}\} _{i \in I}}$ of $ G$ for which $ H = { \oplus _{i \in I}}\langle {n_i}{x_i}\rangle $ for suitable nonnegative integers $ {n_i}$, it is obviously necessary for $ G/H$ to be a direct sum of cyclic groups. In the 1950's, Kaplansky raised the question of whether this condition on $ G/H$ is sufficient for the existence of such a basis. J. Cohen and H. Gluck demonstrated in 1970 that the answer is "yes"; their result is known as the stacked bases theorem, and it extends the classical and well-known invariant factor theorem for finitely generated abelian groups. In this paper, we develop a theory that contains and, in fact, generalizes in several directions the stacked bases theorem. Our work includes a complete classification, using numerical invariants, of the various free resolutions of any abelian group.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0937245-8
Article copyright: © Copyright 1989 American Mathematical Society