# 10/20/2020 Lecture 18

## Comparison for $\pi_1^\et$

:::{.remark}
Last time: we defined $\pi_1^\et X$ for $X$ a normal variety over a field $k$, defined in terms of the site $X_{\fet}$ and a geometric point $\bar{x}$.
Note that the geometric point need not be $\spec \kbar$, one could take an algebraic closure of the resiude field at the generic point $\eta$ of $X$.
For $X = \spec k$ a field, this recovers the absolute Galois group, and $\bar{x}$ is a choice of algebraic closure $\kbar$.
We also have a comparison theorem for $\pi_1^\et X$ and $\pi_1^\Top(X)\procomplete$ for varieties $X$ over $\CC$, which we'll explain today.
There was also an example in characteristic $p$, where $\pi_1^\et \AA^1$ was not topologically finitely-generated.
:::

:::{.theorem title="SGA1"}
There is an equivalence of categories induced by
\[
F_{\bar x}: X_{\fet} &\to G\dash\Fin\Set, \qquad G \da\pi_1^\et X \\
Y/X &\mapsto Y_{\bar x}
,\]
where all actions in sight are continuous.
:::

:::{.corollary title="?"}
If $\characteristic k > 0$ and $X\da \AA^1\slice k$, then $\pi_1^\et( X, \bar x)$ is not topologically finitely-generated for any $\bar x$.
:::

:::{.proof title="?"}
This will follow from the fact that $H^1(X_\et; \FF_p)$ is not finitely-generated.
The claim is that there is a natural map 
\[
\Hom_\Top(\pi_1^\et (X, \bar x), \FF_p) \iso H^1(X, \FF_p)
.\]

Note that $\FF_p\actson \FF_p$ by addition, thus a map from $G\da \pi_1^\et(X, \bar x)$ to $\FF_p$ gives an action $G\actson \FF_p$.
This assembles to a map $\Hom_\Top(G, \FF_p) \to S$, the set of finite continuous $G\dash$sets such that the action $\FF_p\selfmap$ factors through a map $G\to \FF_p$.
By the theorem, $S\cong \torsors{\FF_p}$.
:::

:::{.corollary title="?"}
For any two geometric points $\bar x_1, \bar x_2$, there is an isomorphism
\[
\pi_1^\et(X, \bar x_1) \iso \pi_1^\et(X, \bar x_2)
.\]
:::

:::{.proof title="?"}
Note that this uses the assumption that $X$ is a connected variety.
If $X$ is disconnected, $\pi_1^\et$ will depend on which connected component $\bar x$ is in.

**Step 1**: let $G_i = \pi_1^\et(X, \bar x_i)$; there is an equivalence of categories \({}_{G_1}\Set \iso {}_{G_2}\Set\), since they are both equivalent to $X_{\fet}$.

**Step 2**: the category determines the abstract group.
:::

:::{.remark}
In fact, more is true: there is not just an abstract isomorphism, but rather one defined up to inner conjugation.
This is similar to the situation in topology: choosing paths $x_1\to x_2$ and $x_2\to x_1$, defining a loop $\gamma: x_1\to x_2\to x_1$ and conjugating by $\gamma$ induces an isomorphism $f_\gamma: \pi_1(X, x_1)\iso \pi_1(X, x_2)$.
Choosing an isomorphism for $\pi_1^\et$ can be done by choosing a sequence of specializations and generizations, which is the scheme-theoretic analog of a path:

\begin{tikzcd}
	& {y_1} && {y_3} && {y_n} \\
	{\bar x_1} && {y_2} && \cdots && {\bar x_2}
	\arrow[squiggly, from=1-2, to=2-1]
	\arrow[squiggly, from=1-2, to=2-3]
	\arrow[squiggly, from=1-4, to=2-3]
	\arrow[squiggly, from=1-4, to=2-5]
	\arrow[squiggly, from=1-6, to=2-5]
	\arrow[squiggly, from=1-6, to=2-7]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=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)


Here the $y_i$ are points in the scheme $X$, and $a\to b$ means $b \in \cl_X(a)$.
Note that one can also genericize once to the generic point and then specialize, if one choose an algebraic closure of the function field.
However, there is not a canonical way to choose such an algebraic closure.

The claim now is that if $\bar{x}$ specializes to $\bar{y}$, one gets a natural transformation:

\begin{tikzcd}
	{X_{\fet}} \\
	\\
	\Set
	\arrow[""{name=0, anchor=center, inner sep=0}, "{F_{\bar x}}"', curve={height=30pt}, from=1-1, to=3-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{F_{\bar y}}", curve={height=-30pt}, from=1-1, to=3-1]
	\arrow[shorten <=12pt, shorten >=12pt, Rightarrow, from=0, to=1, "\eta"]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsMixbMCwwLCJYX3tcXGZldH0iXSxbMCwyLCJcXFNldCJdLFswLDEsIkZfe1xcYmFyIHh9IiwyLHsiY3VydmUiOjV9XSxbMCwxLCJGX3tcXGJhciB5fSIsMCx7ImN1cnZlIjotNX1dLFsyLDMsIiIsMix7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XV0=)

Here $\eta(Y/X)(z) = \cl_X(z) \intersect Y$:

![](figures/2023-02-13_16-15-08.png)

:::

:::{.theorem title="Comparison"}
If $X$ is normal over $\CC$, there is a map $\pi_1^\et(X, \bar x) \from \pi_1^\Top(X^\an, \bar{x}^\an)$ which induces an isomorphism 
\[
\pi_1^\Top(X^\an, \bar{x}^\an)\procomplete \iso \pi_1^\et(X, \bar x)
.\]
:::

:::{.proof title="?"}
Idea: given a finite $\pi_1^\et\dash$set, cook up a finite $\pi_1^\Top\dash$set and vice-versa.
One can define $\pi_1^\Top(X^\an, \bar{x}^\an) \da \Aut(F_{\bar x}^\an)$ where $F_{\bar x}^\an: \Cov(X)\to \Set$ is a functor out of the category of (not necessarily finite) covering spaces of $X$.
It's ETS the following diagram commutes: 

\begin{tikzcd}
	{X_{\et}} && {\Cov(X)} \\
	\\
	& \Set
	\arrow["{F_{\bar x}}"', from=1-1, to=3-2]
	\arrow["{F_{\bar x}^\an}", from=1-3, to=3-2]
	\arrow["\an", from=1-1, to=1-3]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJYX3tcXGV0fSJdLFsyLDAsIlxcQ292KFgpIl0sWzEsMiwiXFxTZXQiXSxbMCwyLCJGX3tcXGJhciB4fSIsMl0sWzEsMiwiRl97XFxiYXIgeH1eXFxhbiJdLFswLDEsIlxcYW4iXV0=)

and $\an$ restricts to an equivalence $X_{\fet}\to \Fin\Cov(X)$.
This equivalence exists by the Riemann existence theorem, and is in fact equivalent to the statement that there is such an isomorphism.
This is a nontrivial theorem in analytic geometry!

Riemann existence says that a finite etale cover, i.e. a finite continuous $\pi_1^\et\dash$set, is the same as a finite analytic cover, i.e. a finite $\pi_1^\Top\dash$set.
Thus $\pi_1^\et$ is the universal object through which an action $\pi_1^\Top\actson H$ factors for $H\in \Fin\Grp$, which is another definition of profinite completion.
:::

:::{.corollary title="?"}
If $X$ is a smooth proper curve, then 
\[
\pi_1^\et(X, \bar x) = \gens{a_1,b_1,\cdots, a_g, b_g \st \prod [a_i b_i] = 1}\procomplete
.\]
The proof amounts to doing the topological computation and taking the profinite completion, and there is no other known proof (and in particular, this has no purely algebraic proof).
:::

:::{.theorem title="The SES in $\pi_1^\et$"}
Let $X\in \Var\slice k$, so there are maps $X_{\bar k}\to X\to \spec k$.
This induces a SES 
\[
\pi_1^\et(X_{\bar k}, \bar{x}_{\bar k})\injects \pi_1^\et(X, \bar x) \surjects \Gal(\kbar/k)
.\]

:::

:::{.remark}
We won't cover the proof, but e.g. the content of surjectivity here is that if one has a connected cover of $k$, i.e. a separable extension of $k$, it remains connected after base changing to $X$ since $X$ is a variety and thus geometrically connected.
Exactness in the middle: given an etale cover of $X$, it becomes disconnected when base changing to $X_{\bar k}$ iff it has a piece pulled back from $\Gal(\kbar/k)$, so this characterizes exactly which covers are geometrically connected.
The proof is nontrivial!
:::

## Specialization

:::{.theorem title="?"}
Setup: $X$ is a proper and flat over a complete DVR $R$ with geometrically connected fibers, dropping the assumption that $X$ is a variety over a field.
Let $K = \ff(R)$ with residue field $k = R/\mfm_R$.
Given $\bar{x} \to X_k$ a point on the special fiber, the natural map
\[
\pi_1^\et(X_k, \bar x)\iso \pi_1^\et(X, \bar x)
\]
induced by inclusion of the special fiber is an isomorphism.
:::

:::{.proof title="?"}
This is a very nontrivial theorem!
One needs to show that the restriction functor induced by base change to $k$, the restriction $X_{\fet}\to (X_k)_\fet$, is an equivalence of categories.
The picture to keep in mind: a family over a DVR is like a family of varieties over a complex disc $\DD$, and for small enough discs, the family deformation retracts on the central fiber and induces a homotopy equivalence.

Essential surjectivity: 

**Step 1**: Given $Y\to X_k$ a finite etale cover, construct a finite etale cover $Y\to X\hat{{}^\mfm}$, the completion of $X$ at the maximal ideal of the base.
A digression on deformation theory: the idea is to lift $Y$ to $X/\mfm^n$ one step at a time.
This amounts to filling in the following diagram: 

\begin{tikzcd}
	Y && {?} \\
	\\
	{X_k} && {X\tensor R/\mfm^2}
	\arrow[from=1-3, to=3-3]
	\arrow[hook, from=3-1, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow[dashed, from=1-1, to=1-3]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJZIl0sWzAsMiwiWF9rIl0sWzIsMiwiWFxcdGVuc29yIFIvXFxtZm1eMiJdLFsyLDAsIj8iXSxbMywyXSxbMSwyLCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFswLDFdLFswLDMsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

We'll need existence and uniqueness properties of $(?)$, and more generally, the same for the following diagrams for each $n$: 

\begin{tikzcd}
	{Y_n} && {?} \\
	\\
	{X\tensor R/\mfm^n} && {X\tensor R/\mfm^{n+1}}
	\arrow[from=1-3, to=3-3]
	\arrow[hook, from=3-1, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow[dashed, from=1-1, to=1-3]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJZX24iXSxbMCwyLCJYXFx0ZW5zb3IgUi9cXG1mbV5uIl0sWzIsMiwiWFxcdGVuc29yIFIvXFxtZm1ee24rMX0iXSxbMiwwLCI/Il0sWzMsMl0sWzEsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMCwxXSxbMCwzLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

Let $I$ be the ideal defining this embedding.
The existence of the appropriate lift is controlled by an obstruction class in some $H^2$, and if a lift exists, the set of lifts is a torsor for $H^1$.
Here $\obs \in \Ext^2_{X_k}(\Omega^1_{Y/X_k}, I)$, and if $\obs = 0$ there exists $Y_{n+1}$ flat over $X\tensor R/\mfm^{n+1}$ making the diagram Cartesian, and $\ts{Y_{n+1}} \in \torsors{H}$ for $H\da \Ext^1_{X_k}( \Omega^1 _{Y/X_k}, I)$ in this case.
Since $Y$ is etale over $X_k$ is étale, in particular it is smooth so $\Omega^1_{Y/X_k} = 0$ and thus both of these groups are zero.


Thus lifts exist and are unique.

**Step 2**:
We have a unique $Y\to \complete{X}{m}$, now using properness one can apply formal GAGA to produce $Y\to X$.
:::

:::{.exercise title="?"}
Show $Y_{n+1}$ is etale over $X\tensor R/\mfm^{n+1}$.
This can be checked using the conormal exact sequence, or that one can check if a map is etale by checking it on the special fiber for flat maps.
:::

:::{.remark}
A word on where the obstructions come from: these theorems are proved using Čech cohomology.
One needs to show that locally on $X$, one can find lifts, which is a variant of the definition of formal smoothness.
This follows because $Y_n\to X\tensor R/\mfm^n$ is smooth, so locally looks like $\AA^N$, and one can lift affine space using the structure theorem for smooth morphisms.
To glue lifts together: locally the lifts are unique, so pick isomorphisms between them.
Typically these won't satisfy the cocycle condition, and a nontrivial computation shows that the failure of this is measured by an element in $\Hc^2$, and in fact will represent a class in $\Ext^2$.
The gluing data will be acted on by Čech cocycles for $\Ext^1$.
:::

:::{.corollary title="?"}
Given $X$ as in the theorem and $\bar\eta\to X_{K}$ a geometric point specializing to some $\bar\xi\to X_k$ in the special fiber, there is a canonical map 
\[
\mathrm{sp}: \pi_1^\et(X_K, \bar \eta) \to \pi_1^\et(X_k, \bar\xi)
.\]
:::

:::{.proof title="?"}
This is really a definition: there is a map $(X_K, \bar\eta) \to (X, \bar\eta)$ which induces 
\[
\pi_1^\et(X_K, \bar\eta) \to\pi_1^\et(X, \bar\eta) \isovia_{\text{specialization}} \pi_1^\et(X, \bar\xi)\isovia{\text{thm}} \pi_1^\et(X_k; \bar \xi)
,\]
and the map in the corollary is this composition.
So this is essentially induces by inclusion of the generic fiber and using that $\pi_1(X)$ is the same as $\pi_1$ of the special fiber.
:::

:::{.theorem title="?"}
If $X$ is normal, $\mathrm{sp}$ is surjective.
:::

:::{.proof title="?"}
The essential content: given $Y\to X$ finite etale with $Y$ connected, $Y_K$ is also connected.
This follows from Zariski-Nagata purity since $Y$ is the normalization of $X$ in the function field of $Y$, which is a domain.
:::

:::{.corollary title="?"}
Let $X$ be normal, flat, proper over $R$, and let $\xi$ as above. 
Then there is a surjection 
\[
\pi_1^\et(X_{\bar K}, \bar\eta) \surjects \pi_1^\et(X_{\bar k}, \bar \xi)
\]
where the form is the geometric generic fiber and the latter the geometric special fiber.
:::

:::{.theorem title="?"}
If $X\in \Var\slice k$ and $k=\kbar$ with $\characteristic k = 0$ and $L/k$ is an extension with $L = \bar L$, there is an induced map 
\[
\pi_1^\et(X_L)\iso \pi_1^\et(X)
.\]
:::

:::{.proof title="Idea"}
Galois descent.
One has to show that any etale cover of $X_L$ is pulled back from an etale cover of $X$, and this follows from the rigidity of etale covers in characteristic zero.
This is highly nontrivial, and false in characteristic $p$!
:::

:::{.example title="?"}
Let $X$ be a smooth proper curve over $k=\kbar$ with $\characteristic k = p > 0$.
Then $\pi_1^\et(X, \bar x)$ is topologically generated by at most $2g$ elements.

**Step 1**: Lift to characteristic zero, using that obstructions are in $\Ext^2$ with coefficients in some vector bundles, which will always vanish by Grothendieck vanishing.
Then one algebraizes using formal GAGA.

**Step 2**: One then gets a surjective specialization map $\mathrm{sp}: \pi_1^\et(X_{\bar k}) \surjects \pi_1^\et(X)$ from the geometric generic fiber of the lift, and the LHS can be computed over $\CC$.

The situation is that we're over an algebraically closed field, e.g. the algebraic closure of Witt vectors, but $X$ is always defined over some algebraically closed finitely generated subfield which can embed in $\CC$.
The theorem guarantees that $\pi_1^\et$ doesn't change under this descent and embedding.
We can then compute using the comparison theorem, and conclude by surjectivity.
This is essentially the only technique we know of to compute $\pi_1^\et$ for varieties in characteristic $p$!
:::

:::{.theorem title="?"}
For $X$ as above, the map $\pi_1(X_{\bar K})\to \pi_1^\et(X_{\bar k})$ from geoemtric generic to special fibers induces an isomorphism on prime-to-$p$ completions for $p\da \characteristic k$.
:::

:::{.remark}
For the proof, see SGA1, it's not that bad.
This gives finer information about these groups, namely a presentation of the prime-to-$p$ completion, and we only know how to prove this using analytic means.

There are analogous theorems for varieties with an SNC compactification which are substantially harder, see Grothendieck-Mure for results on the *tame* fundamental group.
One can replace this information about the prime-to-$p$ completion with a surjection onto $\pi_1^{\et, \tame}$.
:::

:::{.proposition title="?"}
There is an equivalence of cateogires between lcc sheaves on $X_\et$ and continuous $\pi_1^\et(X)\dash$modules.
From right to left, such a $G\dash$module $M$ is in particular a $G\dash$set and thus induces a covering space and thus an lcc sheaf $F_M$.
Conversely, lcc sheaves are represented by finite etale covers, and more generally constructible sheaves are represented by quasifinite morphisms.
:::

:::{.remark}
There is a canonical map 
\[
H_\cts^i(\pi_1^\et(X, \bar x), M)\to H^i(X_\et; F_M)
\]
which induces an isomorphism for $i=0, 1$.
:::

:::{.proof title="?"}
Morally the same as the topological situation: a semilocally simply connected space $X$ has a covering space which is a $\pi_1(X)\dash$torsor, so there is a map $X\to \B \pi_1(X)$, and this is the pullback on cohomology.

In this situation, there is a morphism of sites $f: X_\et\to X_{\fet}$ induced by inclusion.
The claim is that $\Sh(X_\fet) = \pi_1^\et(X)\dash\Set$ and $F_M = f^* M$ with $\RR^{1}f_* F_M = 0$.
The content of the last statement is that given a cohomology class with coefficients in an lcc sheaf, it can be killed by a finite etale cover.
Such a cohomology class is a torsor, so it kills itself.
:::