Lemma 1.3.4(b) in Bruns and Herzog

My question refers to the proof of the second of the following lemma given in Cohen-Macaulay rings by Bruns and Herzog.

Lemma 1.3.4 (Bruns and Herzog): Let $(R,\mathfrak m,k)$ be a local ring, and $\phi:F \rightarrow G$ a homomorphism of finite $R$-modules. Suppose that $F$ is free, and let $M$ be an $R$-module with $m \in \operatorname{Ass}(M)$. Suppose that $\phi \otimes M$ is injective. Then
(a) $\phi \otimes k$ is injective
(b) if $G$ is a free $R$-module, then $\phi$ is injective, and $\phi(F)$ is a free direct summand of $G$.

I can see why (a) is true. But for (b), Bruns and Herzog write: "one notes that its conclusion is equivalent to the injectivity of $\phi \otimes k$. This is an easy consequence of Nakayama's lemma."

Question 1: I find this statement confusing, since if the conclusion of (b) is equivalent to the injectivity of $\phi \otimes k$, which we already proved in (a), then why do we need the extra assumption that $G$ is free?

Question 2: What would be a proof of (b)?

Solution 1:

Let $(R,\mathfrak m,k)$ be a local ring, and $\phi:F \rightarrow G$ a homomorphism of finite free $R$-modules. Then $\phi \otimes k$ is injective iff $\phi$ is injective, and $\phi(F)$ is a free direct summand of $G$.

"$\Leftarrow$" This is not difficult and I leave it to you.

"$\Rightarrow$" Let $\{e_1,\dots,e_m\}\subset F$ be an $R$-basis. (In the following we denote $\phi \otimes k$ by $\overline{\phi}$, and $G\otimes k$ by $\overline G$.) Then $\overline{\phi}(\overline e_1),\dots,\overline{\phi}(\overline e_m)$ are linearly independent over $k$ and we can find $\{\overline g_{m+1},\dots,\overline g_n\}\subset\overline G$ such that $\{\overline{\phi}(\overline e_1),\dots,\overline{\phi}(\overline e_m),\overline g_{m+1},\dots,\overline g_n\}$ is a $k$-basis of $\overline G$. Nakayama's Lemma shows that $\{\phi(e_1),\dots,\phi(e_m),g_{m+1},\dots,g_n\}$ is a minimal system of generators for $G$. Since $\hbox{rank}\ G=\dim_k\overline G=n$, it follows that $\{\phi(e_1),\dots,\phi(e_m),g_{m+1},\dots,g_n\}$ is an $R$-basis of $G$. This shows that $G=\phi(F)\oplus\langle g_{m+1},\dots,g_n\rangle$ and $\phi $ injective.