Let $f: X \to Y$ be a quasi-compact, quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Then the higher direct images $R^if_*(\mathcal{F})$ are quasi-coherent on $Y$. In the case of $i=0$, this is local on $Y$, and it reduces to the following fact:

Let $X$ be a quasi-compact, quasi-separated scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$ be a global section. Then $\Gamma(X_f, \mathcal{O}_X) \simeq \Gamma(X, \mathcal{O}_X)_f$ via the natural map between the right-hand side and the left-hand side coming from restriction.

This is not too difficult to prove directly (i.e. by covering $X$ by open affines whose intersections can be covered by finitely many open affines). In the case of $i>0$, it seems harder. Now one has to show

Let $X$ be a quasi-compact, quasi-separated scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$ be a global section. Then $H^i(X_f, \mathcal{O}_X) \simeq H^i(X, \mathcal{O}_X)_f$ via the natural map between the right-hand side and the left-hand side coming from restriction.

In EGA III (1.4.5), this is proved only for quasi-compact *separated* schemes, and the argument used is to show that the Cech complex localizes appropriately (which is direct). The Cech argument does not seem to apply for quasi-separated schemes, though. Nonetheless, many results that are stated for quasi-compact separated morphisms do work with similar proofs when separatedness is relaxed to quasi-separatedness.

Question:Is there a way to modify this kind of argument to work for quasi-separated morphisms?

What *is* true is that the functors $H^i(X, -)_f$ and $H^i(X_f, -)$ are universal (effaceable) $\delta$-functors on the category of sheaves of $\mathcal{O}_X$-modules over $X$, and there is a morphism at level zero. By Tohoku, there is a natural morphism between these two $\delta$-functors. We know that it is an isomorphism on the category of quasi-coherent sheaves in dimension zero. The problem is that the Tohoku argument doesn't quite apply since we don't know (I think) that the functors are effaceable on the subcategory of quasi-coherent sheaves.

Is there a way to fix this argument?

finitenessof $\mathfrak{U}$) to analogue for overlaps among members of $\mathfrak{U}$. QED – BCnrd Nov 8 '10 at 21:51