Contents

# Contents

## Idea

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

## Construction

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

$\vert \sum_{n\gg -\infty} [a_{n}]p^{n}\vert_{r}=\sup_{n}\vert a_{n}\vert p^{-r n}$

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

Then the Fargues-Fontaine curve $X_{E}$ is defined to be

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

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

## References

Last revised on April 24, 2021 at 01:53:23. See the history of this page for a list of all contributions to it.