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…)