The Stacks project

Lemma 27.18.5. Let $U \to V$ be an open immersion of quasi-affine schemes. Then

\[ \xymatrix{ U \ar[d] \ar[rr]_-j & & \mathop{\mathrm{Spec}}(\Gamma (U, \mathcal{O}_ U)) \ar[d] \\ U \ar[r] & V \ar[r]^-{j'} & \mathop{\mathrm{Spec}}(\Gamma (V, \mathcal{O}_ V)) } \]

is cartesian.

Proof. The diagram is commutative by Schemes, Lemma 25.6.4. Write $A = \Gamma (U, \mathcal{O}_ U)$ and $B = \Gamma (V, \mathcal{O}_ V)$. Let $g \in B$ be such that $V_ g$ is affine and contained in $U$. This means that if $f$ is the image of $g$ in $A$, then $U_ f = V_ g$. By Lemma 27.18.3 we see that $j'$ induces an isomorphism of $V_ g$ with the standard open $D(g)$ of $\mathop{\mathrm{Spec}}(B)$. Thus $V_ g \times _{\mathop{\mathrm{Spec}}(B)} \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A)$ is an isomorphism onto $D(f) \subset \mathop{\mathrm{Spec}}(A)$. By Lemma 27.18.3 again $j$ maps $U_ f$ isomorphically to $D(f)$. Thus we see that $U_ f = U_ f \times _{\mathop{\mathrm{Spec}}(B)} \mathop{\mathrm{Spec}}(A)$. Since by Lemma 27.18.4 we can cover $U$ by $V_ g = U_ f$ as above, we see that $U \to U \times _{\mathop{\mathrm{Spec}}(B)} \mathop{\mathrm{Spec}}(A)$ is an isomorphism. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 27.18: Quasi-affine schemes

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Your email address will not be published.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