Isomorphism with affine scheme

Question

Suppose $X\rightarrow {\rm Spec}(A)$ is a morphism between a scheme and an affine scheme that

1) is a bijection topologically and

2) each scheme theoretic fiber is a single reduced point

3) both $X$ and $A$ are reduced

Then is $X$ affine?

Answer 1

Here is one example where $X$ is not affine. A similar idea would work if $A$ is reduced with infinitely many points in $\operatorname{Spec}A$, and you took the same definition for $X$ below using residue fields of points in $\operatorname{Spec}A$. A more interesting question might be if your question is true assuming $X$ is connected.

Let $k$ be a field, and consider the morphism $$X := \coprod_{x \in \mathbf{A}^1_k} \operatorname{Spec} \kappa(x) \longrightarrow \mathbf{A}^1_k$$ where $\kappa(x)$ denotes the residue field at a point $x \in \mathbf{A}^1_k$, and the map is defined by mapping the unique point in $\operatorname{Spec} \kappa(x)$ to $x$. This is a bijection on topological spaces by construction, each scheme theoretic fiber is a spectrum of a field, hence is a single reduced point, and $X$ is reduced. On the other hand, $X$ is not quasicompact, hence $X$ cannot be affine.