Exercise 3.15 [Atiyah/Macdonald] I have a question regarding a claim in Atiyah, Macdonald. A is a commutative ring with $1$, $F$ is the free $A$-module $A^n$. Assume that $A$ is local with residue field $k = A/\mathfrak m$, and assume we are given a surjective map $\phi: F\to F$ with kernel $N$. Then why is the following true?

Since $F$ is a flat $A$-module, the exact sequence $0\to N \to F\overset\phi\to F\to 0$ gives an exact sequence $0\to k\otimes N \to k\otimes F \overset{1\otimes \phi}\to k\otimes F \to 0$.

I can see that $F$ is a free $A$-module, and that the first sequence is exact. But how does flatness of $F$ tell me something about the second sequence?
Thanks!

From the long exact sequence of Tor the second short exact sequence looks like 
$$\text{Tor}_1^A(k,F) \rightarrow k\otimes N \rightarrow k\otimes F \rightarrow k\otimes F \rightarrow 0$$
But $\text{Tor}_1^A(k,F) = 0$ since $F$ is flat. Look at Chapter 2 exercise 24.
