 Elementary decomposition of a finite free chain complex

Over the principal ideal domain $\mathbb{Z}$, Smith normal form decomposes a chain complex of finitely generated free modules into direct sums of one-term complexes $\mathbb{Z}$ and two-term complexesThe one-term summands record free homology, the summands with $m>1$ record cyclic torsion, and those with $m=1$ are acyclic.