I am having trouble understanding the Tor functor as presented in Dummit and Foote.

Given $\dotsb\to P\_n\to P\_{n-1}\to\dotsb\to P\_0\to B\to 0$ as a projective resolution with homomorphisms $d\_i:P\_n\to P\_{n-1}$.

The book defines projective resolution as an exact sequence as above with each $P\_i$ being a projective module.

An exact sequence is defined as a sequence such that for each $P\_i$ in our sequence, $\ker(d\_n)={\rm im}(d\_{n+1})$.

We then tensor each module with the module $A$, giving $\dotsb\to A\otimes P\_n\to\dotsb\to A\otimes P\_0\to A\otimes B\to 0$ and related homomorphisms, $1\otimes d\_n:A\otimes P\_n\to A\otimes P\_{n-1}$.

The definition in Dummit and Foote says $\operatorname{Tor}\_n^R(A,B)=\ker(1\otimes d\_n)/{\rm im}(1\otimes d\_{n+1})$. 

Is $\dotsb\to A\otimes P\_n\to\dotsb\to A\otimes P\_0\to A\otimes B\to 0$ always exact? It feels like it is not, since if it was exact, it would be fairly uninteresting.

Assuming it is not exact, what does $\operatorname{Tor}\_n^R(A,B)$ look like? In other words, how are $\ker(1\otimes d\_n)$ and ${\rm im}(1\otimes d\_{n+1})$ different?

Specifically I am trying to figure out this problem ([Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}\_n^R(A,B)$ is also torsion.](https://math.stackexchange.com/questions/704470/given-a-torsion-r-module-a-where-r-is-an-integral-domain-mathrmtor-n)), but I clearly am having trouble understanding the definitions, so recalling them is not really helping me.

Edit: Let's also assume that I will want to solve other problems involving the $\operatorname{Tor}$ functor, so I want to understand it in general and not just well enough to answer the linked question.