TITLE: Why are injective modules more complicated than projective modules?
QUESTION [22 upvotes]: For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ring, projective resolution can be taken as resolution by free modules of finite rank, but I don't see how one can easily write down injective resolutions.
I'm wondering if there is a deep reason behind this. What makes injective modules so complicated?

REPLY [12 votes]: I'll give you my more pedestrian reason. The proofs of existence of injective  resolutions require the axiom of choice, in one form or another. Translation:  these proofs  are not constructive, so there are no general algorithms for producing such objects.  This becomes a painful issue in concrete  situations.     This  has similarities with another famous existence result, the Hahn-Banach theorem  which postulates the existence of  continuous linear functionals with certain properties. It is particularly useful for existence theorems for PDE's. Unfortunately it gives you no guide for finding those solutions.