Suppose I have some scheme $(X, O_X)$. Suppose I have two open subsets of X, $U$ and $V$. I was wondering about the following:
1) Is $\Gamma (U \cup V, O_X) \cong \Gamma (U , O_X) \times_{\Gamma (U \cap V, O_X)} \Gamma (V , O_X)$? (here the product here denotes a fiber product)
1') Would it be $=$ or $\cong$ in 1)?
2) If the answer to 1) is yes, then this looks a lot like a section of a sheaf when you glue two schemes together. I guess is it possible to interpret this as a section of scheme (made by gluing together $U$ and $V$ along $U \cap V$ via the identity map?)
Thanks!