finitely generated and flat module with respect to exact sequences and tensor product

Question

everyone! I got stuck in an exercise and have to ask for help. I have to consider an exact sequence of $A-Mod$ $$0\rightarrow M'\rightarrow M\rightarrow M'' \rightarrow 0 $$ with $M''$ finitely generated and flat. With this assumption I should conclude that the sequence $$ 0\rightarrow M'\otimes_A N\rightarrow M\otimes_A N\rightarrow M''\otimes_A N\rightarrow 0 $$ is exact $\forall N$ in $A-Mod$. May you give me a hint, please?

Answer 1

Hint: Put $N$ in an exact sequence $0 \rightarrow K \rightarrow F \rightarrow N \rightarrow 0$ with $F$ a free module. Write out the diagram obtained by tensoring that sequence with $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$, being careful to extract all the information that you can from flatness of $M''$ and $F$. Chase the diagram.