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. (more…)