There are couple parts in the proof of Corollary II 4.3.2 Hartshorne which is not clear to me and I would appreciate clarifications.
We have a morphism of schemes $f: X \to Y$, $P \in X$, $U$ open affine so that $f(U)$ is contained in V, an open affine of $Y$.
What I would like to understand is how do we know that this $U \times_V U$ is open affine neighbourhood of $\Delta (P)$? where $\Delta$ is the diagonal morphism $X \rightarrow X \times_Y X$. More specifically,
1) how do we know $\Delta(P) \in U \times_V U$?
2)how do we know $U \times_V U$ is an open subset of $X \times_Y X$?
3) I understand that $U \times_V U$ is an affine scheme, because $U$ and $V$ are affine, but how do we know that this structure is the one obtained by restricting $X \times_Y X$ to $U \times_V U$?
Thank you...
Consider projections $p_1,p_2:X\times_YX\rightarrow X$. Then you may show that
$W = p_1^{-1}(U)\cap p_2^{-1}(U)$ with restrictions ${p_1}_{\mid W}, {p_2}_{\mid W}$ is a fiber-product of $U$ over $Y$ with itself (the proof consists of verification that it admits the required universal property). Hence $U\times_YU$ can be identified with $W$ and you can actually show that it is identified with $W$ by (fiber) product of open immersions $U\hookrightarrow X$ (employ the universal property of $W$).
$U\times_YU$ is canonically isomorphic with $U\times_VU$, because $f(U)\subseteq V$. Again just try to verify that they share the same universal property.
Since $p_i\circ \Delta = 1_{X}$ for $i=1,2$ and $P\in U$, we derive that $\Delta(P) \in p_1^{-1}(U)\cap p_2^{-1}(U)$.
Now three properties above give answers to your questions.