I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd of the determinants of all the $m\times m$ minors of $A$ is $1$.
I know that for there to be surjectivity between $\mathbb{Z}^n$ and $\mathbb{Z}^m$ $n$ must be greater than or equal to $ m$ and for there to even be $m \times m$ minors $n$ again must be greater than or equal to $ m$, so I just assume this throughout.
I sort of have one direction $\Leftarrow$
i) Greatest Common Divisor =1 implies surjectivity: First observe that the if $ | \mathbb{Z}^m/ Im(M)| < \infty$ then $|\det M| = | \mathbb{Z}^m/ Im(M) |$ otherwise $\det(M) = 0$ where $M$ is an $m\times m$ matrix. We can consider the $n$ columns of $A$ as column vectors $v_1, v_2, \ldots, v_n$. These $n$ column vectors live in $\mathbb{Z}^m$. Let $S'' = \{ v_i\}$ and then let $S'$ be subsets of $S''$ of cardinality $m$ and lastly let $S$ be the elements of $S'$ such that when the $m$ $v_i$ vectors are considered as $m\times m$ matrices, the determinant is not zero, thus $S$ consists of all $m\times m$ minors of $A$ with non-zero determinant (we ignore zeroes since they do not affect gcd). For each $s\in S$ define a map $i_s: \mathbb{Z}^m \rightarrow \mathbb{Z}^n$ that maps the standard basis of
$\mathbb{Z}^m$ to the basis elements $e_k \mathbb{Z}^n$ such that $v_k \in s$. That is, $\phi \circ i_s$ gives the matrix created by the column vectors of
$s$. Let $\Lambda$ be the lattice Im$\phi \supset \sum_{s\in S}$ Im
$\phi\circ i_s =\sum_{s \in S} \Lambda_s$. Thus $\forall s \in S$,
$\Lambda_s \subset \Lambda \subset \mathbb{Z}^m$. Thinking in terms of group theory, we have that $\Lambda$ is a subgroup of $\mathbb{Z}^m$ and all the
$\Lambda_s$ are subgroups of $\Lambda$. Thus by Lagrange's Theorem, we have
$|\mathbb{Z}/\Lambda| \Big\vert |\mathbb{Z}^m/\Lambda_s|$ Since $|\mathbb{Z}^m/\Lambda_s|$ are the determininants of the $m\times m$ minors and the definition of the common divisor of several integers is the greatest positive integer dividing all of them. Thus by hypothesis $|\mathbb{Z}/\Lambda| \leq 1$ and so $|\mathbb{Z}/\Lambda| =1$ and we have that Im$A=\Lambda = \mathbb{Z}^m$ so the map is surjective.
I was hoping to get a more elementary proof that doesn't rely on the observation that the if $ | \mathbb{Z}^m/ Im(M)| < \infty$ then $|\det M| = | \mathbb{Z}^m/ Im(M) |$ otherwise $\det(M) = 0$ where $M$ is an $m\times m$ matrix or normal forms.
Thanks!
Best Answer
Proof (which I learned from Darij Grinberg): Assume that $f$ is surjective. Then there is some $n \times m$ matrix $B$ with $AB=1_m$. Let $Z_1,\dotsc,Z_v$ denote the $m \times m$ submatrices of $B$. Then the Cauchy-Binet formula (which has a nice graph theoretic proof) implies $1=\mathrm{det}(AB)=\sum_{s=1}^{v} \mathrm{det}(Z_s) \mathrm{det}(Y_s)$.
Conversely, assume $\sum_s \lambda_s \mathrm{det}(Y_s)=1$ for some $\lambda_s \in R$. Let $B_s$ denote the $n \times m$ matrix, which is built up out of $\mathrm{adj}(Y_s)$ and with zero columns which were deleted in $A \mapsto Y_s$. Let $B = \sum_{s=1}^{v} \lambda_s B_s$. Then we have
$$AB = \sum_s \lambda_s A B_s = \sum_s \lambda_s Y_s \mathrm{adj}(Y_s) = \sum_s \lambda_s \mathrm{det}(Y_s) 1 = 1.$$
Remark: One can show that $f$ is injective iff $(\mathrm{det}(Y_1),\dotsc,\mathrm{det}(Y_s))$ is a regular ideal.