Cokernel of a faithfully flat homomorphism

Question

Let $f:A\to B$ be faithfully flat ring homomorphism and $N=\operatorname{Coker}(f)$ the cokernel of $f$. Let $I$ be an ideal of $A$. How can I use the fact that if $B$ is a faithfully flat $A$-algebra, then $IB\cap A=I$ for every ideal $I$ of $A$ to show that $I\otimes_AN\to IN$ is injective? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.19(b).)

Answer 1

We have an exact sequence of $A$-modules $A\stackrel f\to B\to N\to 0$.
Then $I\otimes_A A\to I\otimes_AB\to I\otimes_AN\to 0$ is also exact.
Since $B$ is $A$-flat we have $I\otimes_AB\simeq IB$. We also know that $I\otimes_AA\simeq I$.
Moreover, $IN=(IB+f(A))/f(A)\simeq IB/IB\cap f(A)$, so the kernel of the canonical projection $IB\to IN\to0$ is $IB\cap f(A)=f(I)$. (Here one uses the exercise 2.6(b).)
Now one can use the Snake Lemma and find $0\to\ker(I\otimes_AN\to IN)\to0$ since the cokernel of the map $I\stackrel{f}\to f(I)$ is zero.