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In topology, we can construct $\mathbb{CP}^\infty$ as the direct limit of $\cdots\rightarrow \mathbb{CP}^n \rightarrow \mathbb{CP}^{n+1}\rightarrow \cdots$ with the embedding given by $[x_0: x_1: x_2: x_3: \cdots: x_n] \mapsto [x_0: x_1, x_2: x_3: \cdots: x_n: 0]$.

Is there a similar construction of an infinite-dimensional version of projective space as a scheme?

One natural choice would be $\text{Proj } \mathbb{C}[x_0, x_1, \ldots]$, but this doesn't quite work. This ring is the direct limit of $\mathbb{C}[x_0, x_1, \ldots, x_n] \rightarrow \mathbb{C}[x_0, x_1, \ldots, x_{n+1}]$ with the natural inclusions, so the corresponding projective scheme should be the inverse limit of the $\mathbb{CP}^n$. (This isn't quite true because the maps aren't actually maps of graded rings, and only induce rational maps on the projective spaces, but it at least works in the affine category: $\text{Spec } \mathbb{C}[x_0, x_1, \ldots]$ is a variety over $\mathbb{C}$ with points given by arbitrary infinite sequences of numbers in $\mathbb{C}$, which is the inverse rather than direct limit of the affine spaces).

What happens if we take $\text{Proj } R$ (or if it's easier, $\text{Spec } R$), where $R$ is the inverse limit of $\mathbb{C}[x_1, \ldots, x_{n+1}] \rightarrow \mathbb{C}[x_1, \ldots, x_n]$ given by $x_{n+1} \mapsto 0$?

The inverse system corresponds to the direct system of $\mathbb{CP}^n$ given above, but perhaps the limit does not give quite the right thing.

The MO post here discusses this problem, mentioning why $\text{Proj } \mathbb{C}[x_0, \ldots]$ doesn't work, but it does not say what goes wrong when you take $\text{Proj } R$

EDIT I believe $R$ can be described as follows: elements of $R$ are infinite sums of monomials of finite degree in the $x_i$, such that for any $n$, the set of non-zero terms involving $x_1, \ldots, x_n$ is finite.

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In topology $\mathbb{CP}^{\infty}$ is important because it classifies line bundles. The analogous object in algebraic geometry is the classifying stack $B \mathbb{G}_m$, which is not a scheme. I think if you make precise what you've written down (for starters, it will be an ind-scheme, so also not a scheme), you'll write down an object classifying pairs consisting of a line bundle and some sections that generate them, or something like that?

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    $\begingroup$ I don't know anything really about ind-schemes, but what is the difference between the ind-scheme which is the direct limit of $\cdots \rightarrow \mathbb{A}^n \rightarrow \mathbb{A}^{n+1} \rightarrow \cdots$ not just $\text{Spec }R$? $\endgroup$ – Dorebell May 18 '16 at 0:41

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