Example 66.41.6. Consider the algebraic space $X$ constructed in Spaces, Example 64.14.2. Recall that it is Galois twist of the affine line with zero doubled. The Galois twist is with respect to a degree two Galois extension $k'/k$ of fields. As such it comes with a morphism

which is quasi-compact. We claim that $\pi $ is universally closed. Namely, after base change by $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$ the morphism $\pi $ is identified with the morphism

which is universally closed (some details omitted). Since the morphism $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$ is universally closed and surjective, a diagram chase shows that $\pi $ is universally closed. On the other hand, consider the diagram

Since the unique point of $X$ above $0 \in \mathbf{A}^1_ k$ corresponds to a monomorphism $\mathop{\mathrm{Spec}}(k') \to X$ it is clear there cannot exist a dotted arrow! This shows that a finite separable field extension is needed in general.

## Comments (2)

Comment #2099 by Matthew Emerton on

Comment #2127 by Johan on