The Stacks project

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

\[ \pi : X \longrightarrow S = \mathbf{A}^1_ k \]

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

\[ \text{affine line with zero doubled} \longrightarrow \text{affine line} \]

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

\[ \xymatrix{ \mathop{\mathrm{Spec}}(k((x))) \ar[r] \ar[d] & X \ar[d]^\pi \\ \mathop{\mathrm{Spec}}(k[[x]]) \ar[r] \ar@{..>}[ru] & \mathbf{A}^1_ k } \]

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

At the end of line two, in the phrase ``in a Galois twisted relative'', the language is slightly garbled.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03KI. Beware of the difference between the letter 'O' and the digit '0'.