## Section7.3A cohomological interlude (Ricky)

Overview:

1. Introduction
2. Etale cohomology
3. Artin-Schreier covers and consequences

### Subsection7.3.1Introduction

$k = \bar k$ field of char $p\text{.}$

Let $G$ be a profinite group, and $G(p)$ the maximal closed quotient of $G$ which is pro-$p\text{.}$

##### Goal.

Let $K = k((t\inv ))\text{,}$ then there is an isomorphism

\begin{equation*} G_K(p)\xrightarrow\sim \pi _1(\aff^1, 0)(p)\text{.} \end{equation*}

### Subsection7.3.2Etale cohomology and $\pi _1^\et$ (for curves)

###### Definition7.3.2.Covers.

Say a morphism of schemes

\begin{equation*} Y \to X \end{equation*}

is a cover if it is finite etale. We say that it is Galois with group $G$ if it is locally of the form

\begin{equation*} \bigsqcup_{\sigma \in G} U_\sigma \to U \end{equation*}

with $U_\sigma \simeq U\text{.}$

Normally we study

\begin{equation*} X_{Zar} = \{ U \to X \text{ open immersion}\} \end{equation*}

where the morphisms are commuting triangles of such maps.

“Classical” sheaves on $X$ are

\begin{equation*} \sheaf F_{new} (U \to X) = \sheaf F_{old}(U)\text{.} \end{equation*}

We can create more “exotic” topologies on $X$ by changing this category

\begin{equation*} X_\et = \{ U \to X \text{ etale}\}\text{.} \end{equation*}

We have sheaves on this topology as before.

Still study sheaf cohomology in this context,

\begin{equation*} R^i \Gamma _\et \sheaf F = H_\et^i(X, \sheaf F) \end{equation*}

morally:

\begin{equation*} H^i_\et(X, \sheaf F) \end{equation*}

\begin{equation*} H^i_\sing (X(\CC)_{top}, \sheaf F)\text{.} \end{equation*}

We have as usual

\begin{equation*} H^1_\et(X, \FF_p) \simeq \Hom_{cts} (\pi _1^\et(X, \bar x), \FF_p)\text{.} \end{equation*}

Here $\pi _1^\et(X, \bar x)$ classifies connected covers of $X\text{:}$ start with

\begin{equation*} \left\{\lb (X_i,x_i) \xrightarrow{f_i} (X, \bar x)\rb _{i\in I}\text{ Galois etale covers}\right\} \end{equation*}

then

\begin{equation*} \pi _1^\et(X, \bar x) = \varprojlim _{i\in I} \Aut_{(X, \bar x)} (X_i, x_i)\text{.} \end{equation*}

Elements of

\begin{equation*} \Hom_{cts} (\pi _1^\et(X, \bar x), \FF_p) \end{equation*}

are either zero (trivial cover) or an isomorphism of a finite quotient of $\pi_1^\et(X, \bar x)$ with $\FF_p$ (a Galois cover with group $\FF_p$).

Given a SES of sheaves on $X_\et$

\begin{equation*} 0\to \sheaf F_1\to \sheaf F_2\to \sheaf F_3 \to 0 \end{equation*}

we get a LES

\begin{equation*} 0 \to H^0_\et(X, \sheaf F_1) \to H^0_\et(X, \sheaf F_2)\to H^0_\et(X, \sheaf F_3) \to H^1_\et(X, \sheaf F_1) \to H^1_\et(X, \sheaf F_2)\to H^1_\et(X, \sheaf F_3) \to \cdots\text{.} \end{equation*}

### Subsection7.3.3Artin-Schreier covers and consequences

#### Subsubsection7.3.3.1Cohomological computations

We have a map

\begin{equation*} \wp \colon \aff^1_k \to \aff^1_l \end{equation*}
\begin{equation*} t \mapsto t^p - t \end{equation*}

since

\begin{equation*} \wp ' (t ) = - 1 \end{equation*}

this is a cover.

The kernel of this map (of group varieties) is then

\begin{equation*} \Spec (k\lb t\rb /(t^p- 1)) \simeq (\Spec k)^p \end{equation*}
\begin{equation*} \simeq (\FF_p)_k\text{.} \end{equation*}

We can upgrade this to a SES of sheaves on any $X/k$

\begin{equation*} 0 \to (\FF_p)_k\to \GG_{a,k}\xrightarrow\wp \GG_{a,k}\to 0\text{.} \end{equation*}
1. Use fact from SGA 1
\begin{equation*} H_{Zar}^q(X, \sheaf O_X) = H_{\et}^q(X, \GG_a) \end{equation*}
use the LES associated to Artin-Schreier sequence
\begin{equation*} 0 \to H^0_\et(X, \FF_p) \to H^0_\et(X, \GG_a)\to H^0_\et(X,\GG_a) \to H^1_\et(X, \FF_p) \to H^1_\et(X, \GG_a)\to H^1_\et(X, \GG_a) \to \cdots \end{equation*}
swap $\et$ for $Zar\text{.}$
2. Follows from first part using
\begin{equation*} H^q_{Zar}(\Spec A, \sheaf O_{\Spec A}) = \begin{cases} A \amp \text{if } q =0\\0 \amp \text{otw}\end{cases}\text{.} \end{equation*}

For instance

\begin{equation*} H^1_\et(\aff^1_k, \FF_p ) = k\lb t\rb / \wp k\lb t\rb \end{equation*}

we have

\begin{equation*} k\lb t\rb \hookrightarrow k ((t \inv)) = K \end{equation*}

View $K$ as

\begin{equation*} k \lb t \rb \hookrightarrow K = k\lb t\rb \oplus t\inv k \lb \lb t\inv \rb \rb \end{equation*}

check that $\wp$ preserves this decomposition and using Hensel's lemma

\begin{equation*} \wp(t\inv k \lb \lb t\inv \rb \rb ) = t \inv k \lb \lb t\inv \rb \rb \end{equation*}

so we get

\begin{equation*} H_\et^1 = k\lb t\rb / \wp k\lb t\rb \text{.} \end{equation*}

Cohomological dimension of $\pi _1\text{.}$

###### Definition7.3.5.

Let $p$ be prime and $G$ profinite then

\begin{equation*} cd_p (G) = \sup \lb n : \exists\text{discrete finite }G\text{-module }A\text{ killed by }p \text{ s.t. }H^n(G, A) \ne 0\}\text{.} \end{equation*}

We have an isomorphism

\begin{equation*} H^1_\et(\aff^1_k , \FF_p) \xrightarrow \sim H^1_\et(\Spec K, \FF_p) \end{equation*}

which descends to

\begin{equation*} H^1(\pi _1^\et(\aff^1_k)(p) , \FF_p) \xrightarrow \sim H^1(G_ K(p), \FF_p) \end{equation*}

so use above theorem.