# Descent (Tuesday, October 26)

> Missed first few boards, find notes / see phone pics.

:::{.remark}
If $G\in \smooth\Aff\Grp\Sch\slice X$ and $T\slice X$ is a [[locally fppf trivial]] $G\dash$torsor, then $T$ is [[etale locally trivial]] by a slicing argument.

Setup: let $X\in \Var\slice k$ and $G\in\Grp\Sch\slice k$ with $T\slice X\in \torsors{G}$.
Partition $X(k)$ as 
\[
X(k) = \Disjoint_{\tau \in H^1(k; G)} \ts{x\in X(k) \st T_x \cong \tau }
.\]
Note that $H^1(k; G)\in \torsors{G}\slice k$.
Let $\tau\in H^1$ be such a torsor and $G_\tau$ be the corresponding **inner form**: note that $G$ acts on itself by conjugation, so inner forms are in the image of the induced map on $H^1$:


\begin{tikzcd}
	{\im \mathrm{conj}^*} && {\ts{\text{forms of } G}} \\
	\\
	{H^1(k; G)} && { H^1(k; \Aut G)}
	\arrow["{\mathrm{conj}^*}", from=3-1, to=3-3]
	\arrow["{\mathrm{conj}^*}", from=3-1, to=1-1]
	\arrow[from=1-1, to=1-3]
	\arrow["\cong"', tail reversed, from=3-3, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJIXjEoazsgRykiXSxbMiwyLCJIXjEoazsgXFxBdXQgRykiXSxbMiwwLCJcXHRze1xcdGV4dHtmb3JtcyBvZiB9IEd9Il0sWzAsMCwiXFxpbSBcXGNvbmpeKiJdLFswLDEsIlxcY29ual4qIl0sWzAsMywiXFxjb25qXioiXSxbMywyXSxbMSwyLCJcXGNvbmciLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJhcnJvd2hlYWQifX19XV0=)

:::

:::{.exercise title="important: on what descent means and how to compute with it"}
Prove the following claim: $\tau\in H^1(k; G)$ is a left $G\dash$torsor, and thus $\tau$ is naturally a *right* $G\dash$torsor.
:::

:::{.remark}
Let $G$ be a discrete group and $T\in \torsorsleft{G}$, or equivalently a group scheme over an algebraically closed field (in characteristic zero).
Write $T^r \da \ts{T, T\times G\to T: tg\da g\inv t}$ to exchange left and right actions, and apply this construction at the level of points.
Hint: you'll need to use [[descent]] to check this.
:::

:::{.remark}
On twisted [[torsors]] : define $T_\tau \da \leftquotient{G}{T\fiberprod{X}\tau}$, and observe that $T_\tau$ is a right $G_\tau$ torsor via an action on $\tau$.
:::

:::{.proposition title="?"}
There is an equality
\[
\ts{x\in X(k) \st T_x \cong \tau} = \im\qty{T_\tau(k) \to X(k) }
.\]
:::

:::{.proof title="?"}
Note that $x\in \im\qty{T_\tau(k) \to X(k)} \iff \ro{T_\tau}{x}$ is a trivial $G_\tau\dash$torsor, since having a rational point implies being a trivial torsor since that point can be used to translate.
This happens iff $\leftquotient{G} T_x \fiberprod{X} \tau$ iff $T_x\cong \tau$.

Conversely, left a rational point and check that you get the graph of an isomorphisms (after base change).
:::

:::{.corollary title="?"}
One can partition
\[
X(k) = \Disjoint_{\tau \in H^1(k; G)} \im\qty{T_\tau(k) \to X(k)}
.\]
:::

:::{.remark}
Everything up until now works in $\Alg\Spaces \geq \Sch$, so it's a mostly formal construction thus far.
:::

:::{.remark}
We showed the following proposition in the case of $\PGL_n$: assignments of points to Brauer classes were locally constant.
:::

:::{.proposition title="Local constancy of evaluation"}
Let $k\in \Local\Field$ and $X\in \Sch\slice k$ proper.
[^properness]

For $G\in \Grp\Sch\slice k^\et$ and $T\in\torsors{G}\slice X$, then the following map is locally constant:
\[
X(k) &\to H^1(k; G)\\
x &\to T_x
.\]


[^properness]: 
Possibly not needed here, but included in Poonen's statement.

:::

:::{.remark}
How did this proof go before?
We showed something didn't deform, i.e. an argument in cohomology of the tangent bundle, and used Artin approximation.
:::

:::{.proof title="?"}
The point: étale sheaves don't deform, i.e. $H^1(\TT_{T\slice X}) = 0$ for $\TT$ the relative cotangent bundle (i.e. $H^1$ with coefficients in the tangent sheaf).
See Poonen for a proof use Krasner's lemma.
A word on this proof: for a constant group scheme, this is a constant Galois cover and the field extensions don't change under small perturbations (i.e. of the coefficients of the polynomial).
:::

:::{.corollary title="?"}
The image $\im \qty{X(k) \to H^1(k; G)}$ is finite.
:::

:::{.proof title="?"}
Use that the map is proper and $X(k)$ is compact.
:::

:::{.remark}
<!--See [[Selmer sets]], [[regular representation]].-->
:::

## Selmer Sets

:::{.definition title="Selmer Sets"}
Let $k\in \Number\Field$, $X\in \Var\slice k, G\in \smooth\Aff\Alg\Grp\slice k, T\in \torsors{G}\slice X$.
Define
\[
\Sel_T(k; G) \da \ts{
\tau \in H^1(k; G) \st
\tau_{\kv} \in \im\qty{X(\kv) \to H^1(\kv; G) }
\text{for all }v\in \places{k}
}
.\]
:::

:::{.example title="?"}
If $A\slice k \in \Ab\Var$ and $G \da A[n]$ with $T: A \mapsvia{[n]} A$, then $\Sel_T(k; G)$ is the [[Selmer group]] of $A$.
:::

:::{.remark}
Unpacking this definition, we can write this as
\[
\Sel_T(k; G) = \ts{\tau \in H^1 \st T_\tau(\kv) \neq \emptyset} \supseteq \ts{\tau \st T_\tau(k) \neq \emptyset}
.\]
Then 
\[
X(k) = \Disjoint_{\tau\in \Sel_T(k; G)} \im\qty{T_\tau(k) \to X(k)}
.\]
:::