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

and the $k$-algebra maps

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 $\mathop{\mathrm{Spec}}(C)$:

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.

Why is $X$ separated?

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

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

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

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