We shall now approach the proof of the Cartan vanishing theorem. First, however, it will be necessary to describe a spectral sequence between Cech cohomology and derived functor cohomology. For now, the reason is that there isn’t any obvious way for us to compute derived functor cohomology, because injective sheaves are big and scary, while Cech cohomology is nice and concrete. And indeed, all we’ve done so far is compute various Cech cohomologies.

I should mention that I don’t know a standard reference for the material in this post. I didn’t find Godement’s treatment in Theorie des faisceaux to be terribly enlightening, but after a fair bit of googling I found a sketch in James Milne’s online notes on étale cohomology. Fortunately, enough details are given to enable one to work it out more fully for oneself.

Let ${X}$ be a topological space covered by an open cover ${\mathfrak{A} = \left\{U_i\right\}_{i \in I}}$, and consider the category ${\mathfrak{C}}$ of presheaves of abelian groups on ${X}$. Let ${\mathfrak{C}'}$ be the subcategory of sheaves. The spectral sequence will be the Grothendieck spectral sequence of the composite of functors

$\displaystyle \mathfrak{C}' \stackrel{F}{\rightarrow} \mathfrak{C} \stackrel{G}{\rightarrow} \mathbf{Ab}.$

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…)