I was trying to solve exercise 1.24 part b) of Eisenbud's book on Schemes and to deduce from it an example of locally ringed space which is not an affine scheme. However I am having many problems in one computation
Consider the affine scheme (Spec($\mathbb C[x,y],O_{Spec\mathbb C[x,y]})$ and set $U=D(x)\cup D(y)$ and say $O$ is the restriction sheaf to $U$. I want to prove that $(U,O)$ is not an affine scheme. To do so, I would like to compute $O(U)$ which should be $\mathbb C[x,y]$ according to the text. How can I prove this? How can I deduce from this that the pair $(U,O)$ is not an affine scheme?
I mean, an element of $O(U)$ is a function from $D(x)\cup D(y)$ into the disjoint union of $\mathbb C[x,y]_{p}$ where $p$ is a prime ideal not containing $x,y$. But from this I really don't see an isomorphism to $\mathbb C[x,y]$. I tried to use the sheaf axioms but I failed.
Thank you for your help
Sections of $D(x)$ are elements of the localized rings: $\frac{p(x,y)}{x^n}$ for some $p \in \mathbb{C}[x,y]$ and $n \in \mathbb{N}$. Your sections of $U$ are rational functions that are sections of $D(x)$ and $D(y)$ at the same time... this is exactly $\mathbb{C}[x,y]$.
Now you use the correspondence between affine schemes and rings. The ring $\mathbb{C}[x,y]$ is already taken by $\mathbb{A}^2$, so there's no room for $D(x) \cup D(y) = \mathbb{A}^2 \backslash \{0\}$ to be affine.