Flat modules and exactness

Question

Assume N is flat. i want to conclude that $Tor_n(M,N)=0 \forall n>0$ and for any module $M$. To prove this I took a projective resolution $P_{.}$ of M and tensored with N. By assumption N is flat so how can i conclude that $P_{.} \otimes N$ is also exact? Is it true that if $A \to B \to C$ is exact and N is flat implies $A \otimes N \to B \otimes N \to C\otimes N$ is also exact. Please help me.

Answer 1

Take $0\to K\to P\to M\to 0$, with $P$ projective. Then the long exact sequence tells you $$ \operatorname{Tor}_{n}(P,N)\to \operatorname{Tor}_{n}(M,N)\to \operatorname{Tor}_{n-1}(K,N) $$ is exact for all $n>1$. Since $P$ is projective, $\operatorname{Tor}_{n}(P,N)=0$ for all $n\ge1$. Moreover $\operatorname{Tor}_{n-1}(K,N)=0$ by inductive hypothesis. The base case comes from exactness of $$ \operatorname{Tor}_{1}(P,N)\to \operatorname{Tor}_{1}(M,N)\to K\otimes N\to P\otimes N $$ and exactness of $0\to K\otimes N\to P\otimes N$.