The definition of the left derived functor I know is from Rotman's Advanced Modern Algebra II, where he gave the hypothesis that the original functor $T$ that another functor $L_nT$ can derive from is that $T$ is from $_R\textbf{Mod}$ to $_S\textbf{Mod}$ (here I think both $R$ and $S$ are two general rings).
However, few pages later, I saw that the author defined $\text{Tor}$ to be as the red line shows. I feel weird that $T$ is a functor from $_R\textbf{Mod}$ to $\textbf{Ab}$, how can he get a left derived functor from such $-\otimes_RB$? So my question is:
- What is the definition of left-derived functor in most mathematics literature. (is the original functor defined from a module to another? Or can it be the case which is from a module to $\textbf{Ab}$?)
- Did Rotman somehow make a mistake here?
Also, the definition of derived functors he wrote:
$\mathbf{Ab}$ is the same as ${}_\mathbb{Z}\mathbf{Mod}$, so this is using the earlier definition in the case $S=\mathbb{Z}$.
You should be aware though that Rotman's definition is not the most general possible definition of derived functors. In particular, left derived functors can be defined for functors between any two abelian categories as long as the domain category has enough projectives. The definition is exactly the same, just working with abstract abelian categories instead of modules.