# Proving that the higher direct images of quasi-compact, quasi-separated morphisms preserve quasi-coherence

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?

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Use the Cech to derived functor spectral sequence to bootstrap from septd qc to qsqc, analogous to bootstrapping from affine to septd qc. Namely, if $\mathfrak{U}$ is a finite open cover of $X$ by septd qc opens (e.g., affine) then all overlaps among them are also septd qc, and we have spectral sequence ${\rm{H}}^i(\mathfrak{U},\underline{\rm{H}}^j(F)) \Rightarrow {\rm{H}}^{i+j}(X,F)$, and similarly on $X_f$. So for qcoh $F$, to see that $f$-localization commutes with cohomology on $X$ we reduce (by finiteness of $\mathfrak{U}$) to analogue for overlaps among members of $\mathfrak{U}$. QED –  BCnrd Nov 8 '10 at 21:51
As a separated remark, this kind of spectral sequence argument is an extremely important method for bootstrapping from schemes to algebraic spaces (using that fiber products of schemes over an algebraic space is a scheme) and from algebraic spaces to Artin stacks (using that fiber products of algebraic spaces over an Artin stack is an algebraic space). In that sense, going from septd qc schemes to qcqs schemes is a toy model for the more general version of this kind of problem. –  BCnrd Nov 8 '10 at 21:54
Akhil:Have you looked at George Kempf's paper titled Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves. It is in Rocky Mountain J of mathematics vol 10 no 3 1980. –  Mohan Ramachandran Nov 8 '10 at 22:05
@BCnrd: Thanks! This is what I was looking for. –  Akhil Mathew Nov 8 '10 at 23:15
@Mohan: I've seen the paper; it's a separated (ha!) argument though. –  Akhil Mathew Nov 8 '10 at 23:15