Artin exercise on free modules homomorphism

Question

2.3 Let $A$ be the matrix of a homomorphism $\varphi:\mathbb Z^n\to\mathbb Z^m$ of free $\mathbb Z$-modules.

(a) Prove that $\varphi$ is injective if and only if the rank of $A$, as a real matrix, is $n$.

(b) Prove that $\varphi$ is surjective if and only if the greatest common divisor of the determinants of the $m\times m$ minors of $A$ is $1$.

This is from Chapter 12 (1st Edition) of Artin. I believe it is Ch 14 in the 2nd Edition. I saw this question here earlier, but it appears to have been deleted. After coming across it in Artin, I tried it myself and am stuck, particularly on (b). I thought to extend $\varphi$ to a $\mathbb Q$-homomorphism $\mathbb Q^n\to\mathbb Q^m$ and use Smith normal form, but I am having trouble making this work and not sure if it will. Does anyone have a direction that may be helpful?

Also, is there a generalization of this exercise? For instance, a homomorphism $\varphi:R^n\to R^m$ with $R$ a Euclidean domain?

Answer 1

(b) $\varphi$ is surjective iff its SNF is surjective. This means that the SNF of $\varphi$ has only $1$'s on the main diagonal. But how one can get the invariant factors of $\varphi$? By using minors, for example. More precisely, the $i$th invariant factor of $\varphi$, say $d_i$, is $\Delta_i/\Delta_{i-1}$, where $\Delta_i$ is the $\gcd$ of all $i$th minors of $\varphi$. If $\Delta_m=1$, then $\Delta_i=1$ for all $i\le m$. Conversely, if $d_i=1$ for all $i=1,\dots,m$, then $\Delta_i=1$ for all $i=1,\dots,m$. (On the way one needs $n\ge m$, but this follows easily by extending $\varphi$ to $\mathbb Q$ and noticing that the extension remains surjective.)