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 $X$?

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