I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following:
I wonder how we get that $\mathcal{O}_{X_i} (X_{ij}) = \mathcal{O}_{X_j}(X_{ji})$ implies there is such a canonical isomorphism, which I shall call $f_{ij}$. I know that $\textrm{Hom} (X,Y) = \textrm{Hom} (\mathcal{O}_Y (Y), \mathcal{O} _X (X))$ and it seems to be related except those are global sections.
Next, there are additional conditions to check when glueing, namely
$f_{ij} (X_{ij} \cap X_{ik}) = X_{ji} \cap X_{jk}$ and $f_{ij} = f_{jk} \circ f_{ij}$ on $X_{ij} \cap X_{ik}$.
How can we verify these conditions?