One way to think of a manifold is as a family of of open subsets $U_i \subset \mathbb{R}^n$, together with distinguished subsets $V_{ij} \subset U_i$ and isomorphisms $\psi_{ij}: V_{ij} \to V_{ji}$ that satisfy the cocycle condition. This may not be useful practically, but occasionally it might be an intuitive crutch. Now, just as manifolds are obtained by gluing together subsets of $\mathbb{R}^n$, a scheme is obtained by gluing together affines. In other words, we have open affines $U_i = \mathrm{Spec} A_i$ for suitable rings $A_i$, open subsets $V_{ij} \subset U_i$, and isomorphisms $\psi_{ij}$ (of locally ringed spaces) as before. However, the open subsets $V_{ij}$ need not be themselves affine.

**Question**: Is it possible to formulate this definition such that the sets $V_{ij}$ are affine? I know this can be done if the scheme is separated (because the intersection of open affines is affine).

One of the nice things about this is that wouldn't have to worry about the isomorphisms $\psi_{ij}$ being isomorphisms of locally ringed spaces, just isomorphisms of the corresponding rings.

This is probably a bad way of thinking of schemes in general; the only reason I was interested in it was because then the fibered product could perhaps be thought of more "explicitly."

The following (related) question also occurred to me when I was thinking about this.

**Question'**: Is there an easy way to tell when the complement of $V(\mathfrak{a}) \subset \mathrm{Spec} A$ is affine? Of course, this is true when $\mathfrak{a}$ is principal. (Answered: see the comments of Matthew Emerton and David Speyer.)