Derived Functors and nice Resolutions

Question

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following:

"Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor between two abelian categories and $\mathscr A$ has enough projective objects. The left derived functors $L_i$$\mathscr F$ for each object $A$ in $\mathscr A$ is defined as $H_i(\mathscr F(P))$, where P is a projective resolution of $A$ in $\mathscr A$."

Then, he continues with a remark on $\mathscr F$-Acyclic objects, for which functors $L_i$$\mathscr F$ is zero for every positive integer $i$, and mentions that one could compute the left derived functors of a right exact functor $\mathscr F$ using the $\mathscr F$-acyclic resolutions of an object A in $\mathscr A$.

It has always seemed to me that the idea of derived funtors and using different resolutions of an object is to understand the given object better based on the structure of some better-understood objects of the category (e.g. projectives, injective, flat, etc.). Namely, we wish to resolve the complexity of a given object by some objects we know better.

For a given functor $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ (let's say right exact) between two abelian categories, we can define the $\mathscr F$-Acyclic objects, and projectives are always $\mathscr F$-Acyclic, if $\mathscr A$ has enough projectives.

These are the question that came to my mind:

• I am looking for a pair of abelian categories $\mathscr A$and $\mathscr B$ and a right exact functor $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ such that $\mathscr A$ does not have enough projectives, but we can explicitly introduce a class of $\mathscr F$-Acyclic objects of the category $\mathscr A$.
• Since we vastly use $Tor_i(-)$, as the left derived functors of the tensor product, we know that every projtive is flat and seems that projectives are the "best" $Tensor$-Acyclic objects. So, I was wondering if for every given right exact functor (similarly left) $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$, there is a "best" type of $\mathscr F$-Acyclic objects which could be compared to the other classes of $\mathscr F$-Acyclic objects in the same way we compare projectives and flats.

Thanks for you help in advance (and sorry for the long question).

Answer 1

1. Take any abelian category $\mathscr A$ with insufficiently many projectives, and let $\mathscr F$ be the zero functor.

2. Flat modules are precisely those which are acyclic for Tor (in either variable). In fact, $M$ is flat if and only if the functor $\text{Tor}_1(-, M)$ vanishes, if and only if the functor $\text{Tor}_i(-, M)$ vanishes for every $i>0$. As for projectives being the best acyclics - this is true, in the sense that a module $M$ is projective if and only if $L_iF(M)=0$ for every $i>0$ and every right-exact functor $F$.