Let $M$ be a $n \times m$ matrix with integer coefficients.

Let $$\text{Coker}(M) = \mathbb{Z}^n /\, \text{Colspace}(M).$$ Why is it the case that there exist positive intergers $a_1, \cdots, a_{\rho}$ with $a_1 \mid \cdots \mid a_{\rho}$ such that $$ \text{Coker}(M) \simeq \mathbb{Z}^{n-\rho} \oplus \bigoplus_{i=1}^{\rho} \mathbb{Z}_{a_i}.$$

Furthermore, why is it that these numbers are actually given by $a_i = \frac{g_{i}}{g_{i-1}}$, where $g_i$ denotes the gcd of the $i \times i $ minors of M (and $g_0 = 1$)?


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