The Stacks project

Exercise 111.52.8. Let $k$ be a field. Consider the rings

\begin{align*} A & = k[x, y]/(xy) \\ B & = k[u, v]/(uv) \\ C & = k[t, t^{-1}] \times k[s, s^{-1}] \end{align*}

and the $k$-algebra maps

\[ \begin{matrix} A \longrightarrow C, & x \mapsto (t, 0), & y \mapsto (0, s) \\ B \longrightarrow C, & u \mapsto (t^{-1}, 0), & v \mapsto (0, s^{-1}) \end{matrix} \]

It is a true fact that these maps induce isomorphisms $A_{x + y} \to C$ and $B_{u + v} \to C$. Hence the maps $A \to C$ and $B \to C$ identify $\mathop{\mathrm{Spec}}(C)$ with open subsets of $\mathop{\mathrm{Spec}}(A)$ and $\mathop{\mathrm{Spec}}(B)$. Let $X$ be the scheme obtained by glueing $\mathop{\mathrm{Spec}}(A)$ and $\mathop{\mathrm{Spec}}(B)$ along $\mathop{\mathrm{Spec}}(C)$:

\[ X = \mathop{\mathrm{Spec}}(A) \amalg _{\mathop{\mathrm{Spec}}(C)} \mathop{\mathrm{Spec}}(B). \]

As we saw in the course such a scheme exists and there are affine opens $\mathop{\mathrm{Spec}}(A) \subset X$ and $\mathop{\mathrm{Spec}}(B) \subset X$ whose overlap is exactly $\mathop{\mathrm{Spec}}(C)$ identified with an open of each of these using the maps above.

  1. Why is $X$ separated?

  2. Why is $X$ of finite type over $k$?

  3. Compute $H^1(X, \mathcal{O}_ X)$, or what is its dimension?

  4. What is a more geometric way to describe $X$?

Comments (0)

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Your email address will not be published.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 02B3. Beware of the difference between the letter 'O' and the digit '0'.



View Exercise 111.52.8 as pdf