$Tor(X,Y)=0$ for every R-module Y implies $Tor_n(X,Y)$?

Question

I have to prove that for any R-module X,

the following statements are equivalent :

1. $Tor_1(X,Y)=0$ for every R-module Y
2. $Tor_n(X,Y)=0$ for every n>0 and every R-module Y

i don’t know how to prove 1->2 ...

Answer 1

We have $\operatorname{Tor}_n(X,Y) = \operatorname{Tor}_{n-1}(X, K)$ where $K$ is a syzygy module in a projective presentation for $Y$.

Answer 2

Both are equivalent to $X$ being flat. The proof of $(1)\implies(2)$ is "dimension-shifting". Suppose $\text{Tor}_n^R(X,-)$ is identically zero for some $n\ge1$. For each module $Y$ there is an exact sequence $$0\to Y'\to F\to Y\to0$$ with $F$ free. The long exact sequence for Tor implies $$\text{Tor}_{n+1}^R(X,Y)\cong\text{Tor}_{n}^R(X,Y')=0.$$