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 1Let be a noetherian topological space, and an inductive system of abelian groups on . Then for each .

The idea is to use the Grothendieck spectral sequence. First, consider the abelian category of -indexed inductive systems of sheaves on . This is a functor category of (a poset is a category!) in an abelian category (namely, the category of sheaves) so is an abelian category. Then the maps are functors on this category. We first study the first functor. (more…)