Fargues-Fontaine curve



Let EE be a perfectoid field (for example p\mathbb{C}_{p}). The Fargues-Fontaine curve X EX_{E} is a complete algebraic curve whose closed points parametrize the untilts of EE. Such an untilt may be recovered as the residue field of the corresponding point.


Let EE be a perfectoid field, let 𝒪 E\mathcal{O}_{E} be its ring of integers, and let W(𝒪 E)W(\mathcal{O}_{E}) be the Witt vectors of 𝒪 E\mathcal{O}_{E}. Let ϕ:EE\phi:E\to E be the Frobenius morphism. We define a norm || r\vert\cdot\vert_{r} on W(𝒪 E)[1/p]W(\mathcal{O}_{E})[1/p] as follows:

| n[a n]p n| r=sup n|a n|p rn\vert \sum_{n\gg -\infty} [a_{n}]p^{n}\vert_{r}=\sup_{n}\vert a_{n}\vert p^{-r n}

Let B EB_{E} be the Frechet completion of W(𝒪 E)[1/p]W(\mathcal{O}_{E})[1/p] with respect to all the norms || r\vert\cdot\vert_{r} for every positive rr.

Then the Fargues-Fontaine curve X EX_{E} is defined to be

X E=Proj( nB E ϕ=p n).X_{E}=\mathrm{Proj}(\oplus_{n\in\mathbb{N}} B_{E}^{\phi=p^{n}}).

The Fargues-Fontaine curve may also be constructed as an adic space.