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date: 2022-04-04 00:24
modification date: Monday 4th April 2022 00:24:41
title: "examples of computations of etale cohomology"
aliases: [examples of computations of etale cohomology]

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# examples of computations of etale cohomology

- For a [smooth](smooth) [connected](connected) [complete](complete%20scheme) curve $X$ over an algebraically closed field of characteristic coprime to $n$:
$$\cocomplex{H}(X; C_n) \cong C_n \oplus C_n\cartpower{2g}[1] \oplus C_n[2]$$
and
$$\cocomplex{H}(X; \GG_m{}\slice k) \cong k\units \oplus \Pic(X)[1]
$$

- For any sheaf $\mcf$ and $x= \spec k$ a geometric point, 
$$\tau_{\geq 1}\cocomplex{H}_\et(x; \mcf) = 0.$$

- $H^i_{\et, c}(\AA^1\slice k \tensor_k k^\sep;  \QQladic) = \QQladic(-1)[2]$.

# Etale fundamental groups

![](attachments/Pasted%20image%2020220404010008.png)
![](attachments/Pasted%20image%2020220404010029.png)

# Elliptic curves

See <https://people.math.ethz.ch/~kowalski/deligne.pdf#page=12>


![](attachments/Pasted%20image%2020220424125342.png)
![](attachments/Pasted%20image%2020220424125513.png)
![](attachments/Pasted%20image%2020220424125542.png)
![](attachments/Pasted%20image%2020220424125559.png)