Derived Functors and Sheaf Cohomology

It is a note taken from Hartshorne (Chapter III) about derived functors and sheaf cohomology. With the category language, everything is very clear. We have some remarks:

1. We first construct the derived functors of a very familiar functors: global section functors, and then prove its derived functors agree with Cech's cohomology on noetherian separated schemes. By this, we also prove the Cech's cohomology does not depend on the choice of open affine covering.

2. We construct derived functors of other functors: $Hom(F,-), \mathscr{Hom}(F,-), f_*$. The derived functor of $Hom$ is $Ext^i(F,-)$, the derived functors of $\mathscr{Hom}(F,-)$ are $\mathscr{Ext}^i(F,-)$, and the derived functors of $f_*$ are $R^if_*$-the higher direct image sheaves.

3. The theorem of Grothendieck about $\delta$-universal functors turns out to be very powerful in proving some key observation. And it plays an important role in our later parts.