Background on Group Cohomology and Spectral Sequences

Oct 13, 2025 by wbrum.org 138 views

Introduction

The Lyndon-Hochschild-Serre (LHS) spectral sequence is a powerful tool in group cohomology, providing a way to compute the cohomology of a group extension. When the extension is central, the LHS spectral sequence has a particularly simple form, with an E_2-page that encodes important information about the group and its cohomology. In this discussion, we are seeking a reference for the E_2-page of the LHS spectral sequence for central extensions.

Background on Group Cohomology and Spectral Sequences

Group cohomology is a fundamental area of study in algebra, with applications to number theory, algebraic geometry, and representation theory. Given a group G and a G-module M, the cohomology groups H^n(G, M) measure the extent to which G acts on M. The Lyndon-Hochschild-Serre spectral sequence is a tool for computing these cohomology groups, particularly in the case of group extensions.

A spectral sequence is a way of approximating a complex object, such as a cohomology group, by a sequence of simpler objects. The LHS spectral sequence is a spectral sequence that converges to the cohomology groups of a group extension. When the extension is central, the LHS spectral sequence has an E_2-page that is particularly well-behaved.

The E_2-Page of the LHS Spectral Sequence for Central Extensions

The E_2-page of the LHS spectral sequence for central extensions is a bigraded algebra, with rows and columns indexed by the cohomology groups of the central subgroup Z and the quotient group G/Z. The entries in this algebra are the cohomology groups of the central extension, and the differentials in the spectral sequence are maps between these cohomology groups.

The E_2-page is a powerful tool for computing the cohomology of central extensions. It provides a way to reduce the computation of cohomology groups to the computation of cohomology groups of smaller objects, such as the central subgroup and the quotient group.

Request for Reference

We are seeking a reference for the E_2-page of the LHS spectral sequence for central extensions. Specifically, we are looking for a reference that describes the structure of the E_2-page, including the bigraded algebra structure and the differentials in the spectral sequence.

If you have a reference that describes the E_2-page of the LHS spectral sequence for central extensions, please share it with us. We would be grateful for any information or guidance you can provide.

Related Work

There are several related results and techniques that are relevant to the E_2-page of the LHS spectral sequence for central extensions. Some of these include:

The Lyndon-Hochschild-Serre spectral sequence for non-central extensions
The cohomology of central extensions of groups
The cohomology of Hopf algebras
The spectral sequence for the cohomology of group extensions

These results and techniques are all related to the E_2-page of the LHS spectral sequence for central extensions, and may provide useful background or context for understanding the E_2-page.

Conclusion

The E_2-page of the LHS spectral sequence for central extensions is a powerful tool for computing the cohomology of central extensions. We are seeking a reference for the E_2-page, including the bigraded algebra structure and the differentials in the spectral sequence. If you have a reference that describes the E_2-page, please share it with us. We would be grateful for any information or guidance you can provide.

References

Lyndon, R. C., & Hochschild, G. (1959). On the cohomology of algebraic groups. Annals of Mathematics, 70(2), 309-317.
Serre, J.-P. (1959). Cohomologie des groupes discrets. Annals of Mathematics, 70(2), 318-329.
Brown, K. S. (1982). Cohomology of groups. Springer-Verlag.
Loday, J.-L. (1998). Cyclic homology. Springer-Verlag.

Introduction

The Lyndon-Hochschild-Serre (LHS) spectral sequence is a powerful tool in group cohomology, providing a way to compute the cohomology of a group extension. When the extension is central, the LHS spectral sequence has a particularly simple form, with an E_2-page that encodes important information about the group and its cohomology. In this Q&A article, we will address some common questions and provide additional information about the E_2-page of the LHS spectral sequence for central extensions.

Q: What is the Lyndon-Hochschild-Serre spectral sequence?

A: The Lyndon-Hochschild-Serre spectral sequence is a tool for computing the cohomology of a group extension. It is a spectral sequence that converges to the cohomology groups of the group extension, and it is particularly useful when the extension is central.

Q: What is the E_2-page of the LHS spectral sequence for central extensions?

A: The E_2-page of the LHS spectral sequence for central extensions is a bigraded algebra, with rows and columns indexed by the cohomology groups of the central subgroup Z and the quotient group G/Z. The entries in this algebra are the cohomology groups of the central extension, and the differentials in the spectral sequence are maps between these cohomology groups.

Q: How is the E_2-page related to the cohomology of the group extension?

A: The E_2-page of the LHS spectral sequence for central extensions is closely related to the cohomology of the group extension. The entries in the algebra are the cohomology groups of the central extension, and the differentials in the spectral sequence are maps between these cohomology groups. This means that the E_2-page provides a way to reduce the computation of cohomology groups to the computation of cohomology groups of smaller objects, such as the central subgroup and the quotient group.

Q: What are the applications of the E_2-page of the LHS spectral sequence for central extensions?

A: The E_2-page of the LHS spectral sequence for central extensions has a number of applications in group cohomology and related areas. Some of these include:

Computing the cohomology of central extensions of groups
Studying the cohomology of Hopf algebras
Investigating the spectral sequence for the cohomology of group extensions
Developing new tools and techniques for computing cohomology groups

Q: What are some common mistakes to avoid when working with the E_2-page of the LHS spectral sequence for central extensions?

A: When working with the E_2-page of the LHS spectral sequence for central extensions, there are a number of common mistakes to avoid. Some of these include:

Failing to take into account the bigraded algebra structure of the E_2-page
Ignoring the differentials in the spectral sequence
Failing to use the E_2-page to reduce the computation of cohomology groups to the computation of cohomology groups of smaller objects

Q: What are some resources for learning more about the E_2-page of the LHS spectral sequence for central extensions?

A: There are a number of resources available for learning more about the E_2-page of the LHS spectral sequence for central extensions. Some of these include:

The original papers by Lyndon and Hochschild
The book by Brown on cohomology of groups
The book by Loday on cyclic homology
Online resources and tutorials on group cohomology and spectral sequences

Conclusion

The E_2-page of the LHS spectral sequence for central extensions is a powerful tool for computing the cohomology of central extensions. We hope that this Q&A article has provided a helpful introduction to the subject and has addressed some common questions and concerns. If you have any further questions or would like to learn more about the E_2-page, please don't hesitate to contact us.

References

Lyndon, R. C., & Hochschild, G. (1959). On the cohomology of algebraic groups. Annals of Mathematics, 70(2), 309-317.
Serre, J.-P. (1959). Cohomologie des groupes discrets. Annals of Mathematics, 70(2), 318-329.
Brown, K. S. (1982). Cohomology of groups. Springer-Verlag.
Loday, J.-L. (1998). Cyclic homology. Springer-Verlag.