The Stacks project

Example 26.21.8. Let $k$ be a field. Consider the structure morphism $p : \mathbf{P}^1_ k \to \mathop{\mathrm{Spec}}(k)$ of the projective line over $k$, see Example 26.14.4. Let us use the lemma above to prove that $p$ is separated. By construction $\mathbf{P}^1_ k$ is covered by two affine opens $U = \mathop{\mathrm{Spec}}(k[x])$ and $V = \mathop{\mathrm{Spec}}(k[y])$ with intersection $U \cap V = \mathop{\mathrm{Spec}}(k[x, y]/(xy - 1))$ (using obvious notation). Thus it suffices to check that conditions (2)(a) and (2)(b) of Lemma 26.21.7 hold for the pairs of affine opens $(U, U)$, $(U, V)$, $(V, U)$ and $(V, V)$. For the pairs $(U, U)$ and $(V, V)$ this is trivial. For the pair $(U, V)$ this amounts to proving that $U \cap V$ is affine, which is true, and that the ring map

\[ k[x] \otimes _{\mathbf{Z}} k[y] \longrightarrow k[x, y]/(xy - 1) \]

is surjective. This is clear because any element in the right hand side can be written as a sum of a polynomial in $x$ and a polynomial in $y$.

Comments (0)

There are also:

  • 18 comment(s) on Section 26.21: Separation axioms

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Your email address will not be published.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 01KQ. Beware of the difference between the letter 'O' and the digit '0'.