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: (For those that are familiar with it, or just want to see the linked tag, it is the proof that a certain functor is representable, with representing object being the relatively glued scheme)

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?

  • It's like specifying a bundle on a space by choosing a trivializing open cover and defining its value on every element in the cover (which is how one can think about the schemes X_U), as well as the compatibility data on intersections of elements of this cover. – skd Oct 11 at 4:54

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