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I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction given in the Stacks Project tag 01LH:

Let $S$ be a scheme. Let $\mathfrak{B}$ be a basis for the topology of $S$. Suppose given the following data:

1) For every $U \in \mathfrak{B}$ a scheme $f_{U}: X_{U} \rightarrow U$ over U.

2) For every pair $U,V \in \mathfrak{B}$ such that $ V \subset U$ a morphism $\rho^{U}_{V}:X_{V} \rightarrow X_{U}$.

Assume that

a) each $ρ^{U}_{V}$ induces an isomorphism $X_{V} \rightarrow f^{−1}_{U}(V)$ of schemes over $V$,

b) whenever $W,V,U \in \mathfrak{B}$, with $W \subset V \subset U$ we have $\rho^{U}_{W}= \rho^{U}_{V}\circ \rho^{V}_{W}$.

Then there exists a morphism $f:X \rightarrow S$ of schemes and isomorphisms $i_{U}:f^{−1}(U) \rightarrow X_{U}$ over $U \in \mathfrak{B}$ such that for $V,U \in \mathfrak{B} $ with $V\subset U$ the composition

$$ X_{V} \stackrel{i_{V}^{-1}}{\longrightarrow} f^{-1}(V) \stackrel{\text{inclusion}}{\longrightarrow} f^{-1}(U) \stackrel{i_{U}}{\longrightarrow} X_{U} $$ is the morphism $\rho^{U}_{V}$.

For the sake of simplicity let's just take the base $\mathfrak{B}$ to be the set of all affine opens, although this really shouldn't make a difference. I am convinced by the proof there that the scheme $X$ exists. But I really have no intuitive understanding of what this scheme is. When I read the gluing data in terms of a base of open sets, it reads much like the construction of a bundle of sections in topology. Indeed we are covering the "base space" by some basic open sets, then we have some projection maps down from the "total space".

How should one think about the scheme $X$ and morphism $f: X \rightarrow S$ more explicitly? What is it as a set? Is there some way to think of it as sections of a total space?

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  • $\begingroup$ Perhaps it might be easier to relate this construction with vector bundles first --- in this case for (sufficiently small) affine opens $U$, $X_U = \mathbb{A}^n \times U$, and you're gluing them together to form a possibly non-trivial vector bundle. $\endgroup$ – loch Oct 11 '18 at 10:30

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