Bloch-Lichtenbaum Spectral Sequence

Definition

The Bloch-Lichtenbaum spectral sequence is a computational tool in algebraic topology and algebraic geometry that helps in understanding the cohomology of schemes, particularly in the context of étale cohomology. It is a spectral sequence that arises from a filtered complex, providing a way to compute the associated graded objects and linking various cohomological dimensions through its convergence properties.

5 Must Know Facts For Your Next Test

1. The Bloch-Lichtenbaum spectral sequence provides tools to study the relationship between étale cohomology and other types of cohomology theories, bridging gaps between different fields of mathematics.
2. It is particularly useful in the computation of motivic cohomology, allowing for connections between algebraic cycles and cohomological operations.
3. This spectral sequence typically arises from the study of smooth projective varieties over a field, making it significant in both algebraic geometry and number theory.
4. Convergence of the Bloch-Lichtenbaum spectral sequence provides important information about the dimensions of various cohomology groups associated with schemes.
5. The construction involves considering both the Frobenius and the Galois actions on the relevant cohomological groups, leading to rich interactions between arithmetic and geometry.

Review Questions

• How does the Bloch-Lichtenbaum spectral sequence help in computing cohomological dimensions for schemes?
  • The Bloch-Lichtenbaum spectral sequence helps compute cohomological dimensions by providing a framework that organizes complex relationships between various cohomology theories. By analyzing its terms, we can derive significant information about the dimensions of associated graded objects, leading to an understanding of how different types of cohomology interact. This framework simplifies computations and allows mathematicians to extract key properties of schemes that might not be immediately obvious.
• Discuss the significance of the convergence properties of the Bloch-Lichtenbaum spectral sequence in understanding algebraic cycles.
  • The convergence properties of the Bloch-Lichtenbaum spectral sequence are crucial for understanding algebraic cycles because they reveal how these cycles relate to various cohomological dimensions. The spectral sequence converges to groups that reflect important invariants associated with cycles, providing insight into their geometric and arithmetic properties. This understanding allows mathematicians to study deeper connections between geometry, topology, and number theory, leading to advances in various areas of research.
• Evaluate the role of Frobenius and Galois actions within the Bloch-Lichtenbaum spectral sequence and their implications for arithmetic geometry.
  • The role of Frobenius and Galois actions within the Bloch-Lichtenbaum spectral sequence is pivotal in linking arithmetic geometry with geometric properties. These actions allow for a deeper analysis of how cohomological groups change under field extensions, which has profound implications for studying rational points on varieties and other arithmetic questions. By examining these actions within the context of the spectral sequence, researchers can gain insights into how algebraic structures behave under various symmetries, fostering further developments in arithmetic geometry.

Related terms

Spectral Sequence: A spectral sequence is a sequence of complex of abelian groups or modules that converges to a certain object, often used in homological algebra to compute homology or cohomology groups.

Cohomology:

Cohomology is a mathematical concept that provides a way to associate algebraic invariants to topological spaces, capturing their global properties through a sequence of abelian groups.

Étale Cohomology: Étale cohomology is a type of cohomology used in algebraic geometry that generalizes the notion of topological cohomology to schemes over fields, allowing the study of their geometric properties.