Statement in stack project on Relative Gluing

Question

This is a screenshot of a statement from stacksproject.

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I have a number of confusion of this statement:

1. What is a scheme $f_U:X_U \rightarrow U$ over $U$?

Does this means that $X_U$ is represented by a locally ringed space $(U, O'_U)$, $O'_U$ being a contravariant functor.

2. How does $\rho^U_V$ induce an isomorphism $X_V \rightarrow f^{-1}_U(V)$ of schemes over $V$.

I have two confusion, again, what does it mean "over", and secondly, what exactly is $f^{-1}_U(V)$?

Answer 1

What is a scheme $f_U: X_U→U$ over $U$?

Quite generally, a scheme $X$ over $S$ for any schemes $X, S$ is just the scheme $X$, together with a distinguished morphism $X \to S$. This notion allows us to consider the category of schemes over $S$, $(Sch/S)$:

• Elements of $(Sch/S)$ are tuples $(X, f: X \to S)$ of a scheme $X$ with a distinguished morphism $f$
• Morphisms of $(Sch/S)$ between $(X,f)$ and (Y,g) morphisms of schemes $\varphi: X \to Y$, such that $f = g \circ \varphi$, i.e. morphisms that commute with the maps to $S$.

This so called relative treatment of schemes might seem a bit special at first sight, but notice that doing things in such a situation is actually more general, than treating schemes alone, because the category of scheme is just the category of schemes over its final object, $\text{Spec }\mathbb{Z}$.

How does $ρ^U_V$ induce an isomorphism $X_V→f^{−1}_U(V)$ of schemes over V?

That is just a condition. $f^{-1}_U(V)$ is the preimage of $V$ under the morphism $f_U$. Then it is easy, that $\rho^U_V$ maps $X_V$ to $f_U^{-1}(V)$, because it should be a morphism over $U$ (actually, this is not given in the statement, though I don't see any other reason why $\rho^U_V$ should end up in $f_U^{-1}(V)$). Now we just demand that $\rho^U_V: X_V \to f_U^{-1}(V)$ is an isomorphism.