# Tuesday, October 12

> Reference: FGA Explained.

:::{.proposition title="?"}
For $R$ a complete local ring with residue field $\kappa$, there is an isomorphism $\Br(R) \iso \Br(\kappa)$.
:::

:::{.remark}
We'll prove a stronger claim that there is a bijection $\SB\Sch\slice{R}/\sim \to \SB\Sch\slice k/\sim$, which requires some deformation theory.
A summary of obstruction theory for schemes:

Let $A \in \CRing$, $I\normal A$ is square zero ideal, and $X\slice{A/I}$ a smooth scheme.
Then there exists a functorial class $\obs(X) \in H^2(X; \T_X \tensor_{A/I} I)$ such that $X$ admits a flat lift to $A$ iff $\obs(X) = 0$.
If the obstruction vanishes, the set of lifts is a torsor for $H^1$, and the automorphisms of the lift are given by $H^0$.
Here $\T_X$ is the tangent sheaf, and a *flat lift* is a flat scheme $\tilde X\slice{A}$ equipped with an isomorphism $\tilde X\tensor (A/I) \iso X$.

A word on this deformation-theoretic result is proved:

- Show affine schemes lift, e.g. using Cohen structure theorem.
  Alternatively, something about being étale?

- Try to glue, which may not satisfy the cocycle condition -- failure to glue will show up in this cohomology.
  Why the tangent sheaf: the difference between two gluing data is a derivation.

Note that for vector bundles $E\to X$, the cohomology would be in $\Endo(E)$.

> See also tangent/cotangent complex.

:::

:::{.proof title="?"}
We'll try to lift a Severi-Brauer over $k$ to one over $R$.
Claim: letting $R_n \da R/\mfm^n$, given a lift to $R_n$, there exists a unique lift to $S_n \da R_{n+1}$.
We have 
\[
\obs(S_n) \in H^2(S^n; \T_{S_n} \tensor \mfm^n/\mfm^{n+1}) = H^2(S_n; \T_{S_n}) \tensor_k \mfm^n/\mfm^{n+1}
,\]
which follows from base change in cohomology using $\T_{S_n} \tensor_{R_n} k \tensor_k \mfm^n/\mfm^{n+1}$.
Here $\obs(S_n) = 0$, since 
\[
H^2(S; \T_S) \tensor_k \kbar = H^2(S_{\kbar}; \T_{S, \kbar}) = H^2(\PP^n\slice{\kbar}; \T_{\PP^n\slice k}) = 0
.\]

 See Hartshorne, this uses the Euler exact sequence.

So a lift exists for each $R_n$.

This lift is unique since lifts are torsors for $H^1(S_n; \T_{S_n} \tensor \mfm^n/\mfm^{n+1})$.

Why this lifts to $R$: formal GAGA, which gives a way of going from formal schemes to actual schemes.
See "FGA Explained", Ch. 8.
This is because giving a scheme over $R^n$ for all $n$ amounts to giving a formal scheme, since the underlying topological spaces are the same.
The input is an ample line bundle: here for $\PP^n$ we can take the dual of the dualizing sheaf $\OO_{S_n}\dual$.
:::

:::{.remark}
Formal GAGA: one of the most useful techniques!
:::

:::{.proposition title="?"}
Suppose $X\in \Var\slice k$ and let $A\in \Br(X)$ (e.g. represented by an Azumaya algebra), then 

- If $k$ is a \(p\dash \)adic field, then there is a map
\[
X(k) &\to \Br(X) \\
x &\mapsto x^*(A)
.\]

- For $k=\RR$, the map $X(\RR) \to \Br(k) = {1\over 2}\ZZ/\ZZ$ is locally constant, i.e. constant on connected components.

:::

:::{.proof title="?"}
For $x\in X(k)$, $\hat{\OO_{X, x}}$ is a complete local $k\dash$algebra with residue field $k$.
Then for $A\in \Br(X)$, we have a map $\psi: \ro{A}{\hat{ \OO_{X, x} } } \iso (A_x)\tensor_k \hat{\OO_{X, x}}$.
We want to spread $\psi$ out to a \(p\dash \)adic neighborhood of $x$.
In the analytic setting, this can be done using **Artin approximation**, which will imply there exists an étale neighborhood $U$ of $x$ and a map
\[
U &\to X \\
y &\mapsto x
,\]
which extends (?) and induces an isomorphism on complete local rings.
Now applying the implicit function theorem, there exists a \(p\dash \)adic neighborhood of $x$ in any $U(k)$.
:::

:::{.corollary title="?"}
Let $X\slice k$ for $k$ a number field and $A\in \Br(X)$.
Then

a. The following map on adeles is locally constant:
\[
A^*: X(\AA_k) &\to \QQ/\ZZ \\
x &\mapsto \sum_{v\in \places{k}} m_{v_x}(x^* A)
.\]

b. $X(\AA)^{\AA} \da (A^*)\inv(0)$ is closed and open.
c. $X(\AA)^{\Br} = \Intersect _{A\in \Br(X)} X(\AA)^{\AA}$ is closed.
d. $\bar{X(k)} \subseteq X(\AA)^{\Br}$.
e. If $X$ is proper, then $X(\AA)^{\Br} \neq X(\AA)$ and weak approximation does not hold.

:::

:::{.proof title="?"}
\envlist

a. Use the same Lang-Weil argument used previously, and that this is a sum of locally constant maps.
b. 0 is closed and open in $\QQ/\ZZ$ and $A^*$ is continuous.
c. This is an intersection of closed sets.
d. We already know $X(k)$ is contained in the RHS, and by (c) we know it's closed, so the RHS contains its closure.
e. Immediate from (d).

:::

:::{.warnings}
The adelic topology is not the product topology.
:::

:::{.definition title="Symbol Algebra"}
For $k\in \Field$ and let $\chi: \Gal(\kbar\slice k)\to C_n$ and $a\in k\units / (k\units)^n$, then recall that $(\chi, a) \da L_\chi\gens{x}_{\sigma}/\gens{x^n-a}$ where $L_\chi$ is the fixed field of $\chi$ and $L_\chi\gens{x}_\sigma$ is the twisted polynomial ring where $\ell x = x\sigma(\ell)$.
:::

:::{.example title="?"}
Take a smooth proper model of $U = \ts{y^2 + z^2 = (3-x^2)(x^2-2)}$ and the *symbol algebra* $A = (3-x^2, -1)$.


:::{.exercise title="Homework"}
Check that this has points locally!
:::

Our goal is to show that $X(\AA)^A = \emptyset$.
By Kummer theory, choosing an isomorphism $\mu_n(k) \to C_n$ induces a bijection
\[
k\units/(k\units)^n &\iso \ts{\chi: \Gal(\kbar \slice k) \to C_n} \\
a &\mapsto k[x] / \gens{x^n-a}
.\]

For $n=2$ and $\ch k \neq 2$, there is a canonical isomorphism $\ts{\pm 1} \iso \mu_2(k) \iso C_2$.
View $(\chi, a) \in H^2(k, \mu_n)$, and there is a cup product
\[
H^1(k; C_n) \times H^1(k; \mu_n) &\to H^2(l; \mu_n) \\
\chi &\mapsto [\chi] \cupprod [a]
.\]

Another point of view: if $L\slice k$ is Galois with Galois group $C_n$, it comes with a choice of generator $\sigma$ and thus a canonical element in $[ \sigma] \in H^2(L\slice k; \ZZ) \iso C_n$.
Then there is another cup product
\[
k\units = H^0(L\slice k; L\units) \mapsvia{(\wait) \cupprod [\sigma]} H^2(L\slice k; L\units) = \Br(L\slice k) = k\units/\Norm_{L\slice k} k\units
,\]
in which case $(\chi, a) = a \cupprod [\sigma]$.
:::

:::{.corollary title="?"}
$(\chi, a) =0 \iff a \in \Norm_{L\slice k} L\units$.
:::

:::{.remark}
For $n=2$, one has $(a, b) = k\adjoin{\sqrt{b}}\gens{x}_\sigma / \gens{x^n-a}$, and this splits iff $a$ is a norm from $k(\sqrt{b})$ when this is a field.
:::

:::{.exercise title="?"}
What are the equations for the Severi-Brauer arising from $(a, b)$.
:::