Let $X$ be a scheme. An open atlas for $X$ is a jointly epimorphic family of Zariski-open immersions $\{X_i\to X\}$ where each $X_i$ is an affine scheme.

A morphism $X\to S$ of schemes is called representable by an affine if for any map $Y\to S$ where $Y$ is affine, the pullback $X\times_S Y$ is itself affine.

Then the question:

Given a scheme $S$, does there exist an open atlas for $S$ consisting only of morphisms representable by an affine?

Recursive definitionit is! $\endgroup$ – Harry Gindi Oct 11 '10 at 20:36