The Smith normal form of an integer matrix $A\in\text{Mat}_{m \times n}(\mathbb{Z})$ is a factorization $A=UDV$ for:
- $D\in\text{Mat}_{m\times n}(\mathbb{Z})$
- Each diagonal entry of $D$ divides the next, i.e. $d_{i,i}|d_{i+1,i+1}$ (elementary divisors of $A$)
- $U\in\text{Mat}_{m\times m}(\mathbb{Z})$, $V\in\text{Mat}_{n\times n}(\mathbb{Z})$ are invertible over $\mathbb{Z}$.
Denote by $a_1,\dots, a_r$ the non-zero elementary divisors of $A$ for $r=\text{rank}(A)$. Then given a complex $\mathbb{Z}^n\overset{A}\to\mathbb{Z}^m\overset{B}\to\mathbb{Z}^k$ (where $BA=0$), the homology at the middle is given by $\ker(B)/\text{im}(A)\cong\bigoplus_{i=1}^r\mathbb{Z}/a_i\oplus\mathbb{Z}^{m-r-s}$ for $s=\text{rank}(B)$.
Is there a similar result where we instead consider homology over the field $\mathbb{Q}$ using Gaussian elimination on $A$?