Tor of submodule

Question

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between:

$Tor_{A^e}$ and $Tor_{B^e}$?

Answer 1

For the inclusion $i : A \to B$ you have an induced map $i^* : A^e \to B^e$ and you may regard every $B^e$ module as an $A^e$ module, so you have an induced map $i_0^* : M \otimes_{B^e} N \to M \otimes_{A^e} N$ for every right $B^e$ module $M$ and left $B^e$ module $N$. Here you must note that tensor product is contravariant with respecto to morphisms of algebras (or rings) and not just for inclusions.

Now you have induced natural maps $i_n^* : Tor_n^{B^e}(M,N) \to Tor_n^{A^e}(M,N)$ for every $n\geq 1$. Then you have a morphism of functors $i^* : Tor^{B^e} \to Tor^{A^e}$.

I'm pretty sure you were looking for an answer like " $Tor^{A^e} \ is \ a \ subfunctor \ of \ Tor^{B^e}$". I think there must be a counterexample, but at this level it is very difficult to find one.