--- date: 2021-07-01 title: Fargues-Fontaine Reading Notes created: 2023-05-17T18:12 updated: 2023-05-17T22:49 --- - **Tags:** - #projects/notes/reading - #todo - #web/quick-notes - #web/blog - #arithmetic-geometry/p-adic-hodge-theory - **Refs:** - [Projects/2021 Fargues Fontaine MOC](Projects/2021%20Fargues%20Fontaine%20MOC.md) # Fargues-Fontaine Reading Notes ## Useful references - [https://www.mathi.uni-heidelberg.de/fg-sga/docs/Programm_la-courbe.pdf](https://www.mathi.uni-heidelberg.de/fg-sga/docs/Programm_la-courbe.pdf) Seminar outline: - - - [http://www.bourbaki.ens.fr/TEXTES/Exp1150-Morrow.pdf](http://www.bourbaki.ens.fr/TEXTES/Exp1150-Morrow.pdf) - - ## Todos Motivate the following: - [Perfectoid spaces](Perfectoid%20spaces) - [Unsorted/tilting](Unsorted/tilting.md) and [untilts](untilt) - [Galois representations](Galois%20representations.md) (might have notes elsewhere..?) - Langlands, period! - [absolute Galois group](absolute%20Galois%20group.md), particularly $G(\Qbar/\QQ)$ (see notes elsewhere) - Relations to [stable homotopy](stable%20homotopy.md) : an isomorphism of Lubin-Tate, and the [Drinfeld tower](Drinfeld%20tower)? What is tilting? How is it different from tilting in a derived category? What are the coherent sheaves on the FF curve $X_{FF}$? What are stability conditions on $\Coh(X_{FF})$? The FF curve has sheaves like $\OO(n)$ for $n\in \QQ$, e.g. for an extension of a [Severi-Brauer curve](Severi-Brauer%20variety) of degree $k$ you can get $\OO(1/k)$. How do sheaves on $X_{FF}$ capture the info of sheaves on related Severi-Brauer curves? How do you construct the curve? ## Motivation / Summary ![](attachments/Pasted%20image%2020220503105503.png) Some notes on the Fargues-Fontaine curve $\Xff$, "the fundamental curve of [p-adic Hodge theory](p-adic%20Hodge%20theory.md)". What's the point? There's supposed to be a "curve" $\Xff$ over $\QQpadic$ where [local Langlands](local%20Langlands) for $\QQpadic$ should be encoded as [geometric Langlands](geometric%20Langlands.md) on $\Xff$, which glues together important [period rings](period%20rings) from [p-adic Hodge theory](p-adic%20Hodge%20theory.md). Stems from conjectures of Grothendieck wanting to related de Rham cohomology to [Unsorted/etale cohomology](Unsorted/etale%20cohomology), and a similar theorem proved by Faltings in the 80s. **Why care**: this is a hot topic right now because of a conjecture related to [local Langlands](local%20Langlands) : supposed to give a way to go from the Galois side to the [automorphic](automorphic) side. **Important object**: For $G$ a [reductive algebraic group](reductive%20algebraic%20group) over a [local field](local%20field), $\Bun_G$ [the moduli stack of $G\dash$torsors](Unsorted/BunG.md) over a family of $\Xff$ curves. A useful overall analogy: it's like the Riemann sphere $\CP^1 \da \PP^1(\CC)$, and in fact the adic version is a $p\dash$adic Riemann surface. The full ring of meromorphic functions on $\PP^1(\CC)$ is $\CC\qty{z}$, but $\CC[z]$ captures most of the data away from $\infty$. $\CC[z]$ as a $\CC\dash$algebra consists of regular (polynomial) functions on $\PP^1(\CC)$ with a pole at $\infty$ of order equal to the degree of the polynomial. View the $\ZZ\dash$algebra $\ZZ$ as the regular functions on $P$ the set of primes (finite [places](place)), with a point at $\infty$ (infinite place) given by the usual valuation $\abs{\wait}$. Make this more algebro-geometric by replacing $\ZZ$ with either $\ZZpadic$ or $\QQpadic$ (so $p\dash$adic things) and looking at $\QQpadic\dash$algebras $B$ as replacements for regular functions. $\Xff$ is also supposed to "geometrize" period rings from [p-adic Hodge theory](p-adic%20Hodge%20theory.md). One can also geometrize [class field theory](class%20field%20theory.md), and realize $\Gal(\bar\QQ/\QQ)$ as a fundamental group. $\Xff$ is also roughly a moduli of [untilts](Unsorted/tilting.md), whisch allow passing between $\FF_p$ and $\QQpadic$. A major goal is to go from characteristic zero to characteristic $p$ (relatively easy) and then to go back to characteristic zero (relatively hard). The curve is useful because many linear algebraic objects of $p\dash$adic theory can be translated into vector bundles over $\Xff$, and there is [Grothendieck splitting](splitting%20principle.md) type of theorem for those. ### Definitions - *Curve*: over $k=\kbar$, an integral separated scheme of finite type over $k$ of dimension 1. - *Finite type*: on affines, when $\spec A\to \spec B$ induces $B\to A$ making $A \in \modsleft{B}$ finitely generated. - [algebraic curve](algebraic%20curve.md) : $X$ of pure [Krull dimension](Krull%20dimension.md) 1, or equivalently $X$ has an affine open cover $\ts{\spec R_i}_{i\in I} \covers X$ where each $\spec R_i$ is Krull dimension 1. - For (irreducible) varieties over $k=\kbar$, just an algebraic variety of dimension 1. - *Complete curve*: for algebraic varieties $X$, universally closed: the projections $X\cross (\wait)\to(\wait)$ are closed maps when evaluated on any variety. - Equivalently, $X\to k$ is a [proper](Unsorted/proper%20morphism.md) morphism (separated, finite type, and universally closed) - For topological spaces, $X$ is compact iff $X$ is complete. - Complete (smooth) varieties of dimension 1 are always projective. - *Completely valued field*: a field $k$ equipped with a valuation $v$, where $k$ is complete as a metric space with respect to $v$. - *Valuation*: a group morphism $(K, \cdot) \to (\RR_{\geq 0}, +)$ with $v(a) = \infty \iff a =0$ satisfying an [ultrametric triangle inequality](ultrametric%20triangle%20inequality). - Almost a metric. Supposed to capture the multiplicity of zeros/poles of a function. - *Valuation ring*: defined as $\OO_K \da \ts{x\in K \st v(x) \leq 1}$, the unit disc with respect to the valuation. - Has a unique maximal ideal, the interior of the disc: $\mfm \da \OO_K\sm \ts{v(x) = 1}$. - *Dedekind scheme*: a Noetherian integral scheme of dimension 1 where every local ring is regular. - Example: $\spec R$ for $R$ a Dedekind domain. - Slogan: any non-generic point is a closed point. - *Tilts*: for $k$ a field, denoted \[ k\tilt \da \inverselim (k\mapsvia{x\mapsto x^p} k \mapsvia{x\mapsto x^p} k \cdots) ,\] realized as sequences $\ts{x_n}$ where $x_{n+1}^p = x_n$, made into a characteristic $p$ field with pointwise multiplication and a $p\dash$twisted addition law involving limits. - Idea: an inverse limit of applying Frobenius. Yields a perfect $\FF_p\dash$algebra. Equipped with a valuation and a valuation ring. - *Untilts*: For $k$ a field, a pair $(K, \iota)$ where $\iota: k\mapsvia{\sim} K\tilt$ is an isomorphism of fields, plus a condition on valuation rings $\OO_k$ and $\OO_K\tilt$. - [formal scheme](formal%20scheme) : topologically ringed spaces $(X, \OO_X)$ where $\OO_X$ is a sheaf of topological rings, which is locally a [formal spectrum](formal%20spectrum.md) of a Noetherian ring. - *Formal spectrum*: $\Spf R \da \Spf_I R = \colim_{n} \Spec R/I^n$, where $I^n$ is a system of ideals forming neighborhoods of zero in the sense that if $U\ni 0$ then $I^n \subseteq U$ for some $n$. Take the colimit in $\Top\Ringedspace$. The structure sheaf of $\Spf_I R$ is $\colim_{n} \OO_{\spec R/I^n}$ - Meant to accommodate formal power series as regular functions. - Examples: $I\normal R$, take the $I\dash$adic topology with basic open sets $r + I^n$. Then $\realize{\Spf A} = \realize{\Spec A/I}$, so the underlying point-set spaces are the same. - *Formal disk*: an infinitesimal thickening of a point. - *Punctured formal disk*: remove the unique global point from a formal disk. - *adic space*: for $X$ a variety over a [nonarchimedean field](nonarchimedean%20field), e.g. $\QQpadic$, the associated analytic space $X^\an$. - [Periods](Periods) : the results of integrating an [algebraic differential form](algebraic%20differential%20form) in $H_\dR^*$ over a cycle in singular cohomology $H_{\sing}^*$. Just a number in $\CC$! - [Archive/AWS2019/Witt vectors](Archive/AWS2019/Witt%20vectors.md) : complicated construction, similar to $p\dash$adic integers. Uniquely characterized as a lift $W(K)$ of a perfect $\FF_p\dash$algebra $K$ to $\ZZpadic\dash$algebra, which becomes $p\dash$adically complete and $p\dash$torsionfree. Also lifts Frobenius. - [Perfectoid space](Perfectoid%20space) : ? Frobenius is an isomorphism? Plus other conditions? - For rings, there is a natural map in $p\dash$adic Hodge theory $\AA_\inf(S) \to S$, and if $S$ is perfectoid this is supposed to look like a 1-parameter deformation ("pro-infinitesimal"). - If $\characteristic S = p$, then $S$ is perfectoid iff $S$ is perfect, i.e. Frobenius is an automorphism. - If $K$ is a perfectoid field, then $G(\bar K/K) \cong G(\bar{K\tilt}/K\tilt)$, so we can study absolute Galois groups here (result of "almost purity" theorems). - [Unsorted/mixed characteristic](Unsorted/mixed%20characteristic.md) : a ring $R$ with an ideal $I\normal R$ with $\characteristic R = 0$ but $\characteristic R/I = p > 0$. The motivating examples: $\ZZ$, or $\OO_K$ for $K$ a number field, $\ZZpadic$. ### Main Results - Why care about tilts/untilts: for $k$ a field and $X\da \spec \QQ$ as a scheme, characteristic zero tilts of $k$ are a good replacement for $X(k)$ which makes $X\cross X$ nontrivial. :::{.theorem title="Main Theorem"} If $k = \kbar$ is a completely valued characteristic $p$ field, then there exists a Dedekind scheme $X \to \spec \QQpadic$ whose closed points $x$ correspond to (isomorphism classes) of characteristic 0 untilts of $k$, modulo the action of the Frobenius $\varphi(x) = x^p$. ::: - So moduli of untilts of a perfectoid field, behaves like open disc in $\CC$. The unique characteristic $p$ untilt is like $z=0 \in \CC$. - Why the FF "curve" isn't a curve: it's a scheme over $\QQpadic$, but not finite-type: specifically, the structure morphism $X\to \spec \QQpadic$ is not finite type. - Why not finite type? Since if $x\in X$ is a closed point then the residue field $\kappa(x) \slice\QQpadic$ is not a finite extension. - Incidentally, $\kappa(x)$ is exactly the untilt of $k$ corresponding to $x$. - Nice properties of $\Xff$ coinciding with curves: - $\sum \deg_f(x) = 0$ for every rational function $f$ on $\Xff$, similar to being projective/complete as a scheme/variety. - Note: very similar to a theorem from complex analysis: sum of orders of poles and zeros equals zero for a meromorphic function on $\PP^1(\CC)$. - Line bundles are classified by degree - $H^1(C, \OO_C) = 0$, similar to having genus $g=0$. - Usually it's like $2g = \rank H^1$ - The structure morphism $X\to \spec \QQpadic$ has simply connected fibers - Useful heuristics: ![](attachments/2021-07-02_01-42-27.png) ## Constructing the curve Notation: - $B_\eps \da B^{\varphi = 1}_\crys$. - $\AA_\inf(K) \da W(K\tilt)$, i.e. just [Archive/AWS2019/Witt vectors](Archive/AWS2019/Witt%20vectors.md) of the tilt. - Supposed to interpolate between the $\characteristic 0$ geometry of $K$ and $\characteristic p$ geometry of $K\tilt$. - Importantly, even if $K$ doesn't have Frobenius, $K\tilt$ does! This will yield an action of Frobenius on cohomology -- this is probably used to set up trace formulas. ### Method 1: Schematically, Proj Construction - Punchline: \[ \Xff \approx \Proj\qty{\bigoplus_{n\geq 0} B^{\varphi = p^n}} \in \Sch .\] - $B^{\varphi = p^n}$ are the elements in $B$ where $\varphi(x) = p^n x$ where $\varphi$ is the Frobenius. - $B \da \colim_{I\subseteq [0, 1] \subset \RR} B_{I}$ is a $p\dash$adic Frechet space - Supposed to look like holomorphic functions of $p$. - For $I\da [a, b]$, $B_{[a, b]}$ is the completion of $W(\OO_C\tilt)\adjoin{{1\over p}, {1\over \abs \pi }}$ with respect to Gauss norms, where $W$ denotes the Witt vectors, and we've localized at $p$ and $\pi$ a pseudo-uniformizer. - Alternatively using the Riemann sphere analogy: $\PP^1(\CC)$ can be recovered as a proj. Write $\Fil_k \CC[z] \da \ts{f\in \CC[z] \st \deg f \leq k}$ to be the $k$th filtered piece using a filtration by degree. - The claim is that $\PP_1(\CC) \cong \Proj\qty{\bigoplus_{k\geq 0} \Fil_k \CC[z]}$. - Somehow this uses that $\CC[z_0, z_1] \mapsvia{\sim} \bigoplus_{k\geq 0} \Fil_k \CC[z]$, no clue. - By analogy, maybe $\Xff = \Proj\qty{ \bigoplus_{k\geq 0} \Fil_k B_\eps}$ where $\Fil_k B_\eps \da \ts{b\in B_\eps \st v(b) \geq k}$, where we take a valuation $v$ coming from $B_\dR$. - There is a subalgebra $B_\crys^+ \subseteq B_\crys$ where $B_\crys^+\invert{t} \cong B_{\crys}$ for some $t$, plus some other properties. - Can then obtain \[ \Xff = \Proj\qty{\bigoplus _{k\geq 0} \qty{B_\crys^+}^{\varphi = p^k}} .\] #### Informal Description Glue together [period rings](period%20rings) : an affine scheme to a [formal disk](formal%20disk.md) along a formal punctured disc, so like \[ \Xff = \spec B_{\crys}^{\varphi = 1} \glue{\spec B_{\dR }} \spec B_{\dR}^+ .\] - $\spec B_{\crys}^{\varphi = 1}$ is an affine scheme. - $\spec B_{\dR^+}$ is a formal disk. - $\spec B_{\dR}$ is a formal punctured disk. - $B_\dR$ is a [period ring](period%20ring.md), cooked up to relate $H^*_\dR(Y)$ to $H^*_{\et}(Y_{\bar K}; \QQpadic)$ when $Y$ is the [generic fiber](generic%20fiber) of a scheme over $W(k)$ and $Y_{\bar K} \da Y \fiberprod{\spec K} \spec \bar K$ (see [Crystalline comparison conjecture](Crystalline%20comparison%20conjecture)). - $B_{\dR}$ is a complete DVR, a field, and a (very large) $\QQpadic\dash$algebra with [residue characteristic](residue%20characteristic) zero, and $B_{\crys} \leq B_{\dR}$ is a subring. - $B_\dR^+$ is its [valuation ring](valuation%20ring.md) (or [place](place)), i.e. a subring such that for every element $x$ of $\ff(B_{\dR})$, either $x \in B_{\dR}^+$ or $x\inv \in B_{\dR}^+$. - This is supposed to be like a [ring of integers](ring%20of%20integers.md) $\OO_K$ for a number field $K$. It is non-canonically isomorphic to the $p\dash$adic Laurent field $\CC_p((t))$. - So it's like the one-point compactification of $\spec$ of the period ring of $p\dash$adic Hodge theory. - This is supposed to look like building the Riemann sphere $\PP^1(\CC)$ as $\CC[z]$ (regular away from $\infty)$ glued to $\CC\qty{1\over z}$ (germs of meromorphic functions at $\infty$). - $B_{\crys}^{\varphi = 1}$ is supposed to look like $\CC[z]$. Classically, $\CC\qty{z}$ is the full field of meromorphic functions on $\CP^1$, and $\CC\adjoin{z}$ are regular functions on $\CP^1\sm\ts{\infty}$. - And $B_\dR$ is like $\CC\qty{1\over z}$, germs of meromorphic functions at infinity. - Taking $\CP^1$ and puncturing at a point yields a disc, so $\CP^1\sm\ts{\infty}, \CP^1\sm\ts{0}$ are discs. Their overlap is $\CP^1\sm\ts{\infty, 0} \cong \DD\sm\ts{0}$, a disc punctured at $z=0$. ![](attachments/2021-07-03_21-14-26.png) ### Method 2: As an [adic space](adic%20space) ?? ### Method 3: As a [diamond](diamond) ?? ## Other Random Notes - Much like $\PP^1(\CC)$, we have a [Grothendieck splitting principle](Grothendieck%20splitting%20principle) : every [vector bundle](vector%20bundle) $\mce \to \Xff$ splits uniquely as $\mce \cong \bigoplus_{k=1}^m \OO_{\Xff}(\lambda _k)$ where $\lambda_k \in \QQ$ are weakly decreasing and $\OO_{\Xff}(\lambda)$ is a (somewhat complicated) rational twist. - Proof uses [p-divisible groups](p-divisible%20groups), can be used to answer questions in [p-adic Hodge theory](p-adic%20Hodge%20theory.md) (e.g. about [Galois representations](Galois%20representations.md)). - Needs Scholze's theory of [diamonds](diamonds): Weil's proof of RH for curves uses the surface $X\fiberprod{\FF_q} X$ for $X$ a curve (this surface is also used in shtuka theory in [geometric Langlands](geometric%20Langlands.md)), need a similar object in arithmetic geometry that should morally look like $\ZZ \tensor_{\FF_1} \ZZ$. - $H^1(\Xff; \OO_{\Xff}) = \QQpadic$, which can apparently be computed using Čech cohomology and the fundamental exact sequence: \[ 0 \longrightarrow \mathbb{Q}_{p} \longrightarrow B_{e} \longrightarrow B_{\mathrm{dR}} / B_{\mathrm{dR}}^{+} \longrightarrow 0 .\]