Brauer Group

Definition

The Brauer group is a fundamental concept in algebraic geometry and number theory that classifies central simple algebras over a field, up to isomorphism. It provides important insights into the structure of algebras and relates to various cohomological properties, especially in contexts like the study of Galois cohomology and results like the Merkurjev-Suslin theorem, showing how algebraic structures can reflect the properties of underlying fields.

5 Must Know Facts For Your Next Test

1. The Brauer group consists of equivalence classes of central simple algebras over a given field, where two algebras are equivalent if they become isomorphic after tensoring with some separable extension of the field.
2. Elements of the Brauer group can be represented by division algebras, which are non-commutative algebras where every non-zero element has a multiplicative inverse.
3. The Brauer group plays a critical role in understanding the obstruction to splitting of vector bundles over algebraic varieties.
4. The relationship between the Brauer group and Galois cohomology highlights how algebraic structures capture symmetries inherent in field extensions.
5. In the context of the Merkurjev-Suslin theorem, one can interpret elements of the Brauer group as obstructions to finding rational points on algebraic varieties.

Review Questions

• How does the Brauer group connect to the concepts of central simple algebras and their classification over a field?
  • The Brauer group is fundamentally about classifying central simple algebras over a field, which means it groups these algebras based on their properties under extension. Each element in the Brauer group represents an equivalence class of central simple algebras, illustrating how these structures reflect underlying field characteristics. This classification provides insights into both algebraic geometry and number theory, particularly in understanding extensions and their behaviors.
• Discuss the significance of Galois cohomology in relation to the Brauer group and its applications.
  • Galois cohomology is significant for the Brauer group because it provides tools for understanding how algebraic structures behave under field extensions. The Brauer group can be interpreted through Galois cohomological techniques, revealing how certain obstructions arise when attempting to lift properties from base fields to larger fields. This relationship not only enhances our understanding of both areas but also illustrates how they interplay in solving problems related to algebraic varieties.
• Evaluate the implications of the Merkurjev-Suslin theorem for the structure of the Brauer group and its role in K-theory.
  • The Merkurjev-Suslin theorem has profound implications for the structure of the Brauer group by establishing links between this group and K-theory. It asserts that certain elements in the Brauer group correspond to matrix algebras over fields, thus giving a concrete representation for abstract objects. This connection allows mathematicians to utilize K-theoretical methods to study problems related to central simple algebras and vector bundles, highlighting how deep results in K-theory can illuminate aspects of algebraic structures encapsulated by the Brauer group.

Related terms

Central Simple Algebra: A central simple algebra is an algebra that is simple (has no two-sided ideals other than zero and itself) and has its center being a field.

Galois Cohomology: Galois cohomology studies the cohomological properties of field extensions, particularly how they interact with Galois groups.

Merkurjev-Suslin Theorem: The Merkurjev-Suslin theorem establishes a connection between the Brauer group of a field and the K-theory of that field, asserting that certain elements of the Brauer group can be represented by matrices.