Galois Action On Cohomology

Introduction

In the realm of algebraic geometry and number theory, Galois cohomology plays a pivotal role in understanding the properties of algebraic extensions and their Galois groups. The concept of Galois action on cohomology is a fundamental aspect of this field, providing a powerful tool for studying the behavior of cohomology groups under the action of the Galois group. In this article, we will delve into the world of Galois cohomology and explore the Galois action on cohomology, its significance, and its applications.

Galois Extensions and Cohomology

Let $F/K$ be a Galois extension with Galois group $G$. The Galois group $G$ is a group of automorphisms of the field $F$ that fix the base field $K$. The cohomology groups $H^i(F,A)$, where $A$ is an $F$-module, are a fundamental object of study in algebraic geometry and number theory. These groups encode important information about the algebraic structure of the field $F$ and its extensions.

The Galois Action

The Galois action on cohomology is induced by the conjugation action of the Galois group $G$ on the cohomology groups $H^i(F,A)$. Specifically, for any $\sigma \in G$, we have a conjugation action $\sigma: H^i(F,A) \to H^i(F,A)$, defined by $\sigma(a) = \sigma(a)$ for any $a \in H^i(F,A)$. This action is a group homomorphism, meaning that it preserves the group operation.

Properties of the Galois Action

The Galois action on cohomology has several important properties that make it a powerful tool for studying the behavior of cohomology groups. Some of the key properties include:

• Transitivity: The Galois action is transitive, meaning that for any two elements $a, b \in H^i(F,A)$, there exists an element $\sigma \in G$ such that $\sigma(a) = b$.
• Invariance: The Galois action is invariant under the action of the Galois group, meaning that for any $\sigma, \tau \in G$, we have $\sigma(\tau(a)) = \sigma(a)$.
• Functoriality: The Galois action is functorial, meaning that it preserves the functorial structure of the cohomology groups.

Applications of the Galois Action

The Galois action on cohomology has numerous applications in algebraic geometry and number theory. Some of the key applications include:

• Galois cohomology: The Galois action on cohomology is a fundamental tool for studying the properties of Galois cohomology groups. It provides a powerful way to compute the cohomology groups and to study their properties.
• Class field theory: The Galois action on cohomology plays a crucial role in the study of class field theory, which is a fundamental area of study in number theory.
• Algebraic K-theory: The Galois action on cohomology is also used in the study of algebraic K-theory, which is a branch of algebraic geometry that studies the properties of algebraic structures.

Examples of the Galois Action

To illustrate the Galois action on cohomology, let us consider a few examples.

• The Brauer group: Let $F/K$ be a Galois extension with Galois group $G$. The Brauer group $Br(F)^G$ is a cohomology group that encodes important information about the algebraic structure of the field $F$. The Galois action on the Brauer group is given by the conjugation action of the Galois group on the Brauer group.
• The cohomology of the multiplicative group: Let $F/K$ be a Galois extension with Galois group $G$. The cohomology group $H^2(F, \mathbb{G}_m)^G$ is a fundamental object of study in algebraic geometry and number theory. The Galois action on this cohomology group is given by the conjugation action of the Galois group on the cohomology group.

Conclusion

In conclusion, the Galois action on cohomology is a fundamental concept in algebraic geometry and number theory. It provides a powerful tool for studying the properties of cohomology groups and their behavior under the action of the Galois group. The Galois action has numerous applications in various areas of mathematics, including Galois cohomology, class field theory, and algebraic K-theory. We hope that this article has provided a useful introduction to the Galois action on cohomology and its significance in mathematics.

References

• Milne, J. S. (1980). Etale cohomology. Princeton University Press.
• Serre, J.-P. (1979). Cohomologie galoisienne. Springer-Verlag.
• Tate, J. (1962). Duality in Galois cohomology. In Proceedings of the International Congress of Mathematicians (pp. 288-295).

Further Reading

For further reading on the Galois action on cohomology, we recommend the following resources:

• The book "Galois cohomology" by Jean-Pierre Serre provides a comprehensive introduction to the subject and its applications.
• The article "Galois cohomology and class field theory" by John Tate provides a detailed study of the Galois action on cohomology and its role in class field theory.
• The book "Algebraic K-theory" by Max Karoubi provides a comprehensive introduction to the subject and its applications, including the Galois action on cohomology.
  Galois Action on Cohomology: Q&A =====================================

Introduction

In our previous article, we explored the concept of Galois action on cohomology and its significance in algebraic geometry and number theory. In this article, we will address some of the most frequently asked questions about Galois action on cohomology, providing a deeper understanding of this fundamental concept.

Q: What is the Galois action on cohomology?

A: The Galois action on cohomology is a group homomorphism that induces a conjugation action of the Galois group on the cohomology groups. Specifically, for any $\sigma \in G$, we have a conjugation action $\sigma: H^i(F,A) \to H^i(F,A)$, defined by $\sigma(a) = \sigma(a)$ for any $a \in H^i(F,A)$.

Q: What are the properties of the Galois action on cohomology?

A: The Galois action on cohomology has several important properties, including:

• Transitivity: The Galois action is transitive, meaning that for any two elements $a, b \in H^i(F,A)$, there exists an element $\sigma \in G$ such that $\sigma(a) = b$.
• Invariance: The Galois action is invariant under the action of the Galois group, meaning that for any $\sigma, \tau \in G$, we have $\sigma(\tau(a)) = \sigma(a)$.
• Functoriality: The Galois action is functorial, meaning that it preserves the functorial structure of the cohomology groups.

Q: What are the applications of the Galois action on cohomology?

A: The Galois action on cohomology has numerous applications in algebraic geometry and number theory, including:

• Galois cohomology: The Galois action on cohomology is a fundamental tool for studying the properties of Galois cohomology groups.
• Class field theory: The Galois action on cohomology plays a crucial role in the study of class field theory.
• Algebraic K-theory: The Galois action on cohomology is also used in the study of algebraic K-theory.

Q: How does the Galois action on cohomology relate to the Brauer group?

A: The Galois action on cohomology is closely related to the Brauer group. Specifically, the Brauer group $Br(F)^G$ is a cohomology group that encodes important information about the algebraic structure of the field $F$. The Galois action on the Brauer group is given by the conjugation action of the Galois group on the Brauer group.

Q: What are some examples of the Galois action on cohomology?

A: Some examples of the Galois action on cohomology include:

• The cohomology of the multiplicative group: Let $F/K$ be a Galois extension with Galois group $G$. The cohomology group $H^2(F, \mathbb{G}_m)^G$ is a fundamental object of study in algebraic geometry and number theory. The Galois action on this coology group is given by the conjugation action of the Galois group on the cohomology group.
• The cohomology of the additive group: Let $F/K$ be a Galois extension with Galois group $G$. The cohomology group $H^2(F, \mathbb{G}_a)^G$ is a fundamental object of study in algebraic geometry and number theory. The Galois action on this cohomology group is given by the conjugation action of the Galois group on the cohomology group.

Q: What are some open problems in the study of Galois action on cohomology?

A: Some open problems in the study of Galois action on cohomology include:

• The computation of the Galois action on cohomology: The computation of the Galois action on cohomology is a fundamental problem in the study of Galois cohomology.
• The study of the Galois action on cohomology in positive characteristic: The study of the Galois action on cohomology in positive characteristic is an open problem in the study of Galois cohomology.
• The study of the Galois action on cohomology in the context of algebraic K-theory: The study of the Galois action on cohomology in the context of algebraic K-theory is an open problem in the study of algebraic K-theory.

Conclusion

In conclusion, the Galois action on cohomology is a fundamental concept in algebraic geometry and number theory. It provides a powerful tool for studying the properties of cohomology groups and their behavior under the action of the Galois group. We hope that this Q&A article has provided a useful introduction to the Galois action on cohomology and its significance in mathematics.

References

• Milne, J. S. (1980). Etale cohomology. Princeton University Press.
• Serre, J.-P. (1979). Cohomologie galoisienne. Springer-Verlag.
• Tate, J. (1962). Duality in Galois cohomology. In Proceedings of the International Congress of Mathematicians (pp. 288-295).

Further Reading

For further reading on the Galois action on cohomology, we recommend the following resources:

• The book "Galois cohomology" by Jean-Pierre Serre provides a comprehensive introduction to the subject and its applications.
• The article "Galois cohomology and class field theory" by John Tate provides a detailed study of the Galois action on cohomology and its role in class field theory.
• The book "Algebraic K-theory" by Max Karoubi provides a comprehensive introduction to the subject and its applications, including the Galois action on cohomology.