Plan of proof about derived functors in general abelian category

Question

I have to write a report about the derived functors of the inverse limit $\lim$ functor defined from the category of inverse systems (of modules, or maybe in some cases of cochain complexes).

Now, the only reference that I have ( $\text{Strong Shape and Homology}$ by Mardesic, chapter 11) conducts proofs that are both long and tedious and far from general as they might be.

My plan is to prove the results I need for some general kind of abelian categories rather than inverse systems alone. I need help to be sure that I am not missing some crucial passage and I would appreciate any references or comments in order to clarify the proof of some lemmas that I need.

1. Inverse systems are an instance of functors, so by this result The functor category $A^J$ is abelian category if $A$ is abelian they form an abelian category when they take values in another abelian category. Either here or in what follows I might need the fact that epis and monos are characterized by being such pointwise. Here a discussion: https://mathoverflow.net/questions/17953/can-epi-mono-for-natural-transformations-be-checked-pointwise

2. I need of course that my category of inverse systems has enough projectives and injectives. That holds for modules. Can I again traslate this property to any category $A^J$ of functors from a fixed category that take values in a category with enough projectives and injectives?

3. Then I would need just to prove that whenever $A$ is an abelian category with enough injective objects and $S: A \longrightarrow \operatorname{Mod}$ is an additive left exact functor, the usual procedure of homological algebra yields right derived functors. I cannot find a reference that conducts the proof in all generality, though.

Answer 1

All three points are covered in the second chapter of Weibel‘s Book „An introduction to homological Algebra“ in the context of abelian categories.

As a short Answer:

1. Try getting more comfortable with the notions around abelian categories. Then it will be easy to verify that $A^I$ is abelian for any abelian category A and any small category I (i.e. the objects of I form a set). The thing about epis and monos can be formulated concisely as follows: Given a sequence $F’\rightarrow F\rightarrow F’’$ in $A^I$, then this sequence is exact in $A^I$ if and only if for any $i\in I$ we have that the sequence $F’(i)\rightarrow F(i)\rightarrow F’’(i)$ is exact in $A$.

2. Yes this works but you have to add another assumption: If $A$ is an abelian category with enough projectives (enough invectives) and is cocomplete (complete) which means that arbitrary direct sums (direct products) exist, them $A^I$ has enough projectives (enough invectives). Details of a proof can be found on page 43 of Weibel’s book.

3. This can be done also with $S$ having values in an arbitrary abelian category. For details see the book.

Maybe it is convenient to know the Freyd-Mitchel embedding theorem for abelian categories when working with these. It says the following:

Given a small Abelian category $A$ there exists a ring $R$ and a full exact embedding $F:A\rightarrow R-mod$. This is helpful when trying to prove exactness properties in commutative diagrams. Keep in mind that $F$ may not necessarily respect infinite direct sums and products! Also notice that if $A$ is an arbitrary abelian category with a choice of kernels, cokernels and finite direct sums, and if $C\subset A$ is a SET of objects then there exists a small abelian subcategory $A’\subset A$ with $C\subset A’$.