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? 



Does every scheme admit an open atlas consisting only of morphisms representable by an affine?
A morphism $X \to Y$ is representable by an affine iff it is an affine morphism (preimages of open affines are open affine), since the latter is stable under base change. Now an open immersion $U \to X$ is affine iff $U \cap V$ is open affine in $X$ for every open affine $V$ in $X$. Thus $X$ has an atlas consisting of such maps iff every two open affines intersect in an open affine, i.e. iff the diagonal $\Delta : X \to X \times\_{\mathbb{Z}} X$ is affine (which is the case when $X$ is separated).


