The story of how I ended up writing this post is a bit roundabout. I was trying to figure out whether a technical lemma in Hartshorne on the cohomology of inductive limits of sheaves could be proved using spectral sequences, and I thought I had it, but I then realized that I had no justification for asserting that the category in question had enough injectives (which is necessary to apply the Grothendieck spectral sequence). So I tracked down the original reference to the result, which was–conveniently enough–in Grothendieck’s famous Tohoku paper (available openly). It turns out that there is a way to see that the category I was interested in had enough injectives, but it is a fairly interesting and involved result. I will explain what I learned from reading the section of the paper today. Next time, I will explain my thoughts that led me here. Also, before I proceed, here’s a PDF of the post.
It is known that the category of modules over a ring has enough injectives, i.e. any object can be imbedded as a subobject of an injective object. This is one of the first things one learns about injective modules, though it is a nontrivial fact and requires some work. Similarly, when introducing sheaf cohomology, one has to show that the category of sheaves on a given topological space has enough injectives, which is a not-too-difficult corollary of the first fact.
However, it is more difficult to see that, for instance, the category of inductive diagrams of sheaves (within a fixed inductive system) has enough injectives. For this a general result that states that wide classes of abelian categories (satisfying minor categorical conditions) have enough injectives is handy and convenient. This result is in Tohoku, but Grothendieck claims it is not his.
I will assume familiarity with basic diagram-chasing in abelian categories (e.g. the notion of the inverse image of a subobject). (more…)