So now we have a proof (in chapter 3) that category of modules over a commutative ring has enough injectives. Actually, two proofs. One is the standard dualization argument that appears in most textbooks. The other is a variant of the “small object argument” in homotopy theory and uses a bit more set-theoretic machinery. The latter has the advantage that it can be used to show that large classes of abelian categories have enough injectives (as Grothendieck does in his Tohoku paper). In my commutative algebra class, the teacher hinted that one could prove the theorem this way.

The idea is somewhat explained in this blog post, but not very well, and some of the technical points (e.g. filtered ordinals) are obscured there. Thanks to Johan de Jong for pointing this out.

Also, the formatting has changed a little. The chapter and section titles are not simply the defaults.