In the last two and and a half weeks I’ve updated the material on derived categories and derived functors. You can now find this material in a new chapter entitled Derived Categories.

The original exposition defined the bounded below derived category as the homotopy category of bounded below complexes of injectives. This is actually a very good way to think about derived categories if you are mainly interested in computing cohomology of sheaves on spaces and/or sites. On the other hand, it does not tell you which problem derived functors really solve. Let’s discuss this a bit more in the setting of sheaves of modules on a ringed space (X, O_X). I will assume you know how to define cohomology of sheaves by injective resolutions, left derived functors by projective resolutions, you have heard that D(A) is complexes up to quasi-isomorphism, but you don’t yet know exactly why one makes this choice.

Let F : Mod(O_X) —> A be a right exact functor from the abelian category of O_X-modules into an abelian category A. The category Mod(O_X) usually does not have enough projectives. Hence it wouldn’t work to define the bounded above derived category in terms of bounded above complexes of projectives. You could still make this definition but there wouldn’t be a functor from the category of modules into it and hence it wouldn’t suffice to compute left derived functors of F. In fact, what should be the “left derived functors” of F in this setting? Grothendieck, Verdier, and Deligne’s solution is the following: Let M be an O_X-module. Consider the category of all resolutions

… —> K^{-1} —> K^0 —> M —> 0

where K^i is an arbitrary O_X-module. For any such resolution we can consider the complex

F(K^*) = ( … —> F(K^{-1}) —> F(K^0) —> 0 )

in the abelian category A. We say that LF is defined at M if and only if the system of all F(K^*) is essentially constant up to quasi-isomorphism, i.e., essentially constant in the bounded above derived category D^-(A). If one can choose K^* so that F(K^*) is actually equal to this essentially constant value, then one says that K^* computes LF(M). These definitions are motivated by the case where there do exist enough projectives: in that case one shows that given a projective resolutions P^* there always exists a map P^* —> K^*, hence the system is essentially constant with value F(P^*). We say an object M is left acyclic for F if M computes LF. Note that this makes sense without knowing that LF is everywhere defined! It turns out that LF is defined for any M which has a resolution K^* where all K^n are left acyclic for F and that in this case F(K^*) is the value of RF(M) in D^-(A). For example, why is one allowed to use bounded above flat resolutions to compute tors? The reason is that flat modules are left acyclic for tensoring with a sheaf (this is not a triviality — it is something you have to prove; hint: use Lemma Tag 05T9).

I started rewriting the material on derived categories because I gave 2 lectures about derived categories and derived functors in my graduate student seminar, and I wanted to understand the details. Let me know if you find any typos, errors, or lack of clarity. Also, there is still quite a bit missing, for example a discussion of derived categories of dg-modules would be cool.