Today, I will discuss the proof of a technical lemma in Hartshorne on sheaf cohomology, which he uses to prove Grothendieck’s vanishing theorem by reducing to finitely generated sheaves.

Prerequisite: Some familiarity with the Grothendieck spectral sequence. I’m probably going to blog about this (and spectral sequences more generally) soon, though I need to polish what I have written so far. Note that this has already been blogged about over at A Mind for Madness.

Proposition 1 Let ${X}$ be a noetherian topological space, and ${\mathcal{F}_\alpha, \alpha \in A}$ an inductive system of abelian groups on ${X}$. Then ${H^i( X, \varinjlim \mathcal{F}_\alpha) \simeq\varinjlim H^i(X, \mathcal{F}_\alpha)}$ for each ${i}$.

The idea is to use the Grothendieck spectral sequence. First, consider the abelian category ${\mathfrak{C}}$ of ${A}$-indexed inductive systems of sheaves on ${X}$. This is a functor category of ${A}$ (a poset is a category!) in an abelian category (namely, the category of sheaves) so ${\mathfrak{C}}$ is an abelian category. Then the maps ${\left\{\mathcal{F}_\alpha\right\} \rightarrow H^i( X, \varinjlim \mathcal{F}_\alpha) ,\varinjlim H^i(X, \mathcal{F}_\alpha)}$ are functors on this category. We first study the first functor.

Let ${F}$ be the functor ${\mathfrak{C} \rightarrow \mathbf{Sh}(X)}$ sending ${\left\{\mathcal{F}_{\alpha}\right\} \rightarrow \varinjlim \mathcal{F}_\alpha}$ and ${G: \mathbf{Sh}(X) \rightarrow \mathbf{Ab}}$ the global section functor. For the first functor ${\mathfrak{C} \rightarrow \mathbf{Ab}}$ (i.e. direct limit, then cohomology), we are looking at the composites ${R^i G \circ F}$.

Lemma 2 The category ${\mathfrak{C}}$ has enough injectives.

Indeed, we shall obtain a generator for ${\mathfrak{C}}$. Let ${U}$ be a generator for ${\mathbf{Sh}(X)}$–for instance, a suitable sum of constant sheaves over open subsets of ${X}$. For ${\alpha \in A}$, let ${U_\alpha \in \mathfrak{C}}$ be the diagram scheme containing ${U}$ in the ${\alpha}$ spot and ${0}$ everywhere else. To hom out of ${U_\alpha}$ into another diagram scheme is the same as homming from ${U}$ into the ${\alpha}$-th spot into the other diagram scheme. From this it is easy to see that ${\bigoplus U_\alpha}$ is a generator for ${\mathfrak{C}}$.

Moreover, since ${\mathbf{Sh}(X)}$ satisfies the condition (A) of the previous post, it follows that ${\mathfrak{C}}$ does (the condition (A) is preserved under functor categories, as remarked there). So by the theorem of the previous post, ${\mathfrak{C}}$ admits enough injectives.

We now want to apply the Grothendieck spectral sequence to ${G, F}$. It needs to be checked first that ${F}$ maps injectives in ${\mathfrak{C}}$ to ${G}$-acyclic objects. Any injective object in ${\mathfrak{C}}$ must be a diagram consisting only of injective sheaves; this follows by testing exactness when homming out of diagram schemes containing precisely one object, and is a straightforward exercise.

Now we have

Lemma 3 The direct limit of flasque sheaves on a noetherian space is flasque.

Since (cf. Hartshorne) every injective is flasque, it follows that direct limits (i.e. ${F}$) maps injective objects in ${\mathfrak{C}}$ into flasque (hence acyclic) sheaves. So the lemma will allow us to apply the Grothendieck spectral sequence.

The lemma, in turn, is a direct corollary of the fact that ${(\varinjlim \mathcal{F}_\alpha)(U) = \varinjlim (\mathcal{F}_\alpha(U))}$ on a noetherian space. This is an exercise in Hartshorne II.1. In fact, if ${\mathcal{F}_\alpha(U) \rightarrow \mathcal{F}_\alpha(V)}$ is surjective whenever ${ V \subset U}$, it follows by taking inductive limits and using the above fact that ${(\varinjlim \mathcal{F}_\alpha)(U) \rightarrow (\varinjlim \mathcal{F}_\alpha)(V)}$ is surjective, since inductive limits of abelian groups preserve surjectivity.

So, anyway, it is now clear that we can apply the Grothendieck spectral sequence to ${G \circ F}$. There is a spectral sequence ${R^i G \circ R^j F \rightarrow R^{i+j}(GF) = R^{i+j}(FG)}$. We have used the fact that ${FG = GF}$, which is just a restatement of the fact stated above that taking sections and direct limits are operators that commute for sheaves on a noetherian space. (This is a slight abuse of notation, since for instance ${F}$ is used to refer to direct limits both of sheaves and of abelian groups.)

But the above fact about inductive limits of sheaves implies that inductive limits of sheaves form an exact functor, in particular the higher derived functors of ${F: \mathfrak{A} \rightarrow \mathbf{Sh}(X)}$ are zero. So we find that this spectral sequence degenerates and ${R^{i}(GF) = R^i G \circ F}$.

Now, let’s use the spectral sequence for ${F \circ G}$. On the category of inductive systems of abelian groups, the inductive limit is an exact functor (so any object is acyclic with respect to this functor). So ${F}$ is exact and its higher derived functors are zero. In particular, the Grothendieck spectral sequence ${R^{i}(F) \circ R^{j}(G) \rightarrow R^{i+j}(FG)}$ exists and degenerates. We find that

$\displaystyle R^{i}(FG) = F \circ R^i (G).$

But ${FG = GF}$, so ${R^{i}(FG) = R^i(GF)}$. It now follows that

$\displaystyle R^i G \circ F = F \circ R^i G,$

which is the same as saying that taking global sections commutes with direct limits.
Hartshorne doesn’t actually give this argument; he prefers to avoid spectral sequences throughout the text. In the Tohoku paper, Grothendieck gives a general criterion (Proposition 3.6.2) about when the derived functors of a given functor commute with inductive limits: basically, the initial functor has to commute with inductive limits (which here was a fact about sheaves to which I referred to an exercise in Hartshorne), and the fact that inductive limits of injective objects are acyclic with respect to this functor (which was the fact above that the inductive limit of injective sheaves is flasque).  I think the same spectral sequence argument I gave here goes through.
Weirdly, Grothendieck gives an explicit argument, preferring to avoid his own spectral sequence. I’m not sure that spectral sequences make life much easier in this particular case; you only use the Grothendieck spectral sequence when one of the functors is actually exact, which is kind of degenerate, and the argument in Hartshorne is no longer than this post. Nonetheless, I’m using this as an exercise to get a better feel for spectral sequences and how they are used.