TITLE: Sheaf cohomology and injective resolutions
QUESTION [55 upvotes]: In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come into play? It is very confusing and awkward to me that why taking injective stuff into consideration would allow you to "extend" a left exact functor.

REPLY [22 votes]: This is an interesting discussion to someone raised in the 60's.  It illustrates how lack of motivation creeps into books unnoticed.  Back when Hartshorne was being written everyone was steeped in the then standard derived functor formalism of Cartan Eilenberg and Grothendieck, axiomatizing constructions of cohomology via complexes.  So the pattern of this formalism explained so nicely by Anton and Andrew above was taken for granted. As the subject evolved, it was understood that acyclic resolutions could replace the more categorically natural injective ones.  This is the gist of Evan's answer. Perhaps also there is a tradition in mathematics books of giving definitions without historical background.  I have always had difficulty with any unmotivated definitions, such as abstract "residues" and modern Riemann Roch theorems, so I keep stressing to my students the value of learning the original version of Riemann, even if their goal is to understand cohomological and arithmetic versions.  Anyway, excellent question.