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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