Generalizations of the stacked bases theorem

Abstract

Let H H be a subgroup of the free abelian group G G . In order for there to exist a basis { x i } i ∈ I {\{ {x_i}\} _{i \in I}} of G G for which H = ⊕ i ∈ I ⟨ n i x i ⟩ H = { \oplus _{i \in I}}\langle {n_i}{x_i}\rangle for suitable nonnegative integers n i {n_i} , it is obviously necessary for G / H 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 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.