Step in the construction of the global spec of a sheaf of algebras

Question

I'm working my way through the construction of the global spec of a sheaf of algebras. Here is the setup. Let $ Y $ be a scheme. Let $ \mathscr{A } $ be a quasi coherent sheaf of $ \mathcal{O}_Y$algebras. For open affine subsets $V \subset U \subset Y$ we define $ X_{V}=\text{Spec}(\mathscr{A}(V)) $ and $ X_{U}=\text{Spec}(\mathscr{A}(U)) $ with the map $ {X}_{V}\to{X}_{U}$ induced by the restriction $\mathscr{A}(U)\to\mathscr{A}(V)$. My question is, why is $X_{V}\to{X}_{U}$ an open immersion?

I tried to use basic affines but I'm losing it with the multi-level application of functors in this construction.

Answer 1

It's hard to not get hung up on the criterion for representability, but I think the ingredients here are simpler.

If $f\colon \operatorname{Spec} A \to \operatorname{Spec} B$ and $N$ is a $B$-module then $f^*\widetilde{N} \simeq \widetilde{N \otimes_B A}$.

In your situation the morphism is just the inclusion map $f\colon V \to U$. So $\mathcal{A}(V) = \mathcal{A}(U) \otimes_{\mathcal{O}(U)} \mathcal{O}(V)$. This is exactly the ring corresponding to $X_U \times_U V$. So the morphism $X_V \to X_U$ is the base change of $V \to U$ by $X_U \to U$, and we know that open immersions are stable under base change.