Hochschild–Serre spectral sequence

Definition

The Hochschild–Serre spectral sequence is a powerful tool in algebraic topology and homological algebra that provides a way to compute the Hochschild cohomology of a module or an algebraic structure through a filtration process. It connects the cohomology of groups with that of their subgroups, revealing deep relationships between their structures and allowing for the transfer of information across different levels.

5 Must Know Facts For Your Next Test

1. The Hochschild–Serre spectral sequence is derived from a fibration or a short exact sequence of groups, connecting the cohomologies of these groups.
2. It is particularly useful in studying group extensions and can provide information about the derived functors of the Hom functor.
3. The first page of the spectral sequence consists of the derived functors of the cohomology groups of the quotient and subgroup.
4. The spectral sequence converges to the Hochschild cohomology of the original group, providing an effective means to compute it in complex cases.
5. One important application of this spectral sequence is in computing invariants in algebraic geometry and representation theory.

Review Questions

• How does the Hochschild–Serre spectral sequence relate to group extensions and what role does it play in understanding their properties?
  • The Hochschild–Serre spectral sequence is crucial for studying group extensions as it helps compute the Hochschild cohomology related to a group through its subgroups. When analyzing a short exact sequence of groups, this spectral sequence connects the cohomology of the whole group with that of its normal subgroup and quotient. By establishing this connection, it reveals how properties like extensions can be tracked from smaller subgroups to larger groups, offering insights into their structure.
• Discuss how the construction of the first page of the Hochschild–Serre spectral sequence aids in computing cohomology groups.
  • The first page of the Hochschild–Serre spectral sequence consists of derived functors that relate to the cohomology groups associated with a normal subgroup and its quotient. By explicitly computing these derived functors, one can gather information that serves as input for subsequent pages. This iterative process simplifies complex computations by breaking them down into manageable parts, ultimately leading to the desired Hochschild cohomology groups through convergence.
• Evaluate how the application of the Hochschild–Serre spectral sequence enhances our understanding of invariants in algebraic geometry and representation theory.
  • Applying the Hochschild–Serre spectral sequence provides deep insights into invariants within algebraic geometry and representation theory by linking cohomological properties across different structures. This connection enables mathematicians to extract information from smaller, simpler components and apply it to more complex scenarios, facilitating computations that might otherwise be infeasible. As such, it acts as a bridge between various areas of mathematics, enriching our understanding of underlying principles and enabling more robust theoretical developments.

Related terms

Hochschild cohomology: A type of cohomology theory used in algebra, which captures information about the extensions and deformations of algebras.

Spectral sequence: A computational tool in homological algebra that allows for the calculation of homology or cohomology groups through a sequence of pages, converging to the desired results.

Filtration: A way of breaking down a mathematical object into simpler parts, allowing for the analysis and computation of its properties step by step.