---
date: 2022-04-08 12:43
modification date: Friday 8th April 2022 12:43:27
title: "Galois cohomology"
aliases: [Galois cohomology]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- <https://wstein.org/edu/2010/582e/refs/tate-galois_cohomology.pdf>
- Links:
	- [Hilbert 90](Unsorted/Hilbert%2090.md)
	- [Brauer group](Unsorted/Brauer%20group.md)
	- [Tate-Shafarevich group](Unsorted/Tate-Shafarevich%20group.md)
	- [Unsorted/herbrand quotient](Unsorted/herbrand%20quotient.md)

---

# Galois cohomology

# Notation

![](attachments/Pasted%20image%2020220408124437.png)
![](attachments/Pasted%20image%2020220408132445.png)

# Definition

![](attachments/Pasted%20image%2020220408124330.png)

## H_2

![](attachments/Pasted%20image%2020220408124408.png)

![](attachments/Pasted%20image%2020220408124357.png)

# Profinite groups

![](attachments/Pasted%20image%2020220408132433.png)
![](attachments/Pasted%20image%2020220408125522.png)

# Exact sequences

![](attachments/Pasted%20image%2020220408125549.png)
![](attachments/Pasted%20image%2020220408125620.png)


# Applications

Proving [weak Mordell-Weil](weak%20Mordell-Weil)
![](attachments/Pasted%20image%2020220408125731.png)
![](attachments/Pasted%20image%2020220408125741.png)

# Inflation and Restriction

See also [transfer](Unsorted/transfers.md), [norm](Unsorted/field%20norm.md).
![](attachments/Pasted%20image%2020220408125833.png)

![](attachments/Pasted%20image%2020220408130132.png)

In terms of the [transgression](Unsorted/Serre%20spectral%20sequence.md):
![](attachments/Pasted%20image%2020220408130159.png)

![](attachments/Pasted%20image%2020220408130217.png)
![](attachments/Pasted%20image%2020220408130227.png)

# Cup product

![](attachments/Pasted%20image%2020220408130356.png)
![](attachments/Pasted%20image%2020220408130415.png)

# Duals

See [Pontrayagin dual](Pontrayagin%20dual), [Cartier dual](Cartier%20dual), [Weil Pairing](Weil%20Pairing).
![](attachments/Pasted%20image%2020220408130515.png)

# Local fields

For a [local field](Unsorted/global%20field.md):
![](attachments/Pasted%20image%2020220408130629.png)
See [Hasse invariant](Hasse%20invariant)

## Pairing

See also [tate duality](Unsorted/tate%20duality.md).

![](attachments/Pasted%20image%2020220408130811.png)

Relation to [Artin reciprocity](Unsorted/Artin%20symbol.md):
![](attachments/Pasted%20image%2020220408130832.png

# Examples

A use of [dimension shifting](Unsorted/dimension%20shifting.md)

![](attachments/Pasted%20image%2020220408132540.png)
![](attachments/Pasted%20image%2020220408132546.png)
![](attachments/Pasted%20image%2020220408132557.png)

## Main theorem of CFT

Proving the main theorem of [class field theory](Unsorted/class%20field%20theory.md):
![](attachments/Pasted%20image%2020220408132850.png)Defining the [Artin map](Unsorted/Artin%20symbol.md):
![](attachments/Pasted%20image%2020220408132925.png)

The [local reciprocity](local%20reciprocity) map and the [norm-residue symbol](norm-residue%20symbol):
![](attachments/Pasted%20image%2020220408133006.png)
![](attachments/Pasted%20image%2020220408133059.png)
![](attachments/Pasted%20image%2020220408133108.png)

See [quadratic reciprocity](Unsorted/quadratic%20reciprocity.md)

# How to compute dimensions

![](attachments/Pasted%20image%2020220410154747.png)
![](attachments/Pasted%20image%2020220410154756.png)

# H_1 unramified

![](attachments/Pasted%20image%2020220410154848.png)
![](attachments/Pasted%20image%2020220410154912.png)
![](attachments/Pasted%20image%2020220410154922.png)