Generalizations of the stacked bases theorem

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.