Severi-Brauer varieties

Definition

Severi-Brauer varieties are algebraic varieties that represent the projective space associated with a division algebra over a field. They serve as a crucial link between algebraic geometry and the theory of central simple algebras, particularly in understanding the structure of Brauer groups and their relation to K-theory.

5 Must Know Facts For Your Next Test

1. Severi-Brauer varieties can be seen as geometric representations of division algebras, capturing their essential features in a visual form.
2. The relationship between Severi-Brauer varieties and the Brauer group is significant because it allows for a geometric interpretation of cohomological data related to division algebras.
3. Over a field that is algebraically closed, every Severi-Brauer variety is isomorphic to projective space, reflecting the nature of division algebras in that context.
4. Severi-Brauer varieties play an important role in the Merkurjev-Suslin theorem, which deals with the relationship between K-theory and the Brauer group.
5. The study of Severi-Brauer varieties helps in understanding the behavior of rational points on algebraic varieties, especially over fields that are not algebraically closed.

Review Questions

• How do Severi-Brauer varieties connect division algebras and projective spaces?
  • Severi-Brauer varieties provide a geometric framework for understanding division algebras by associating them with projective spaces. Each division algebra corresponds to a Severi-Brauer variety that can be interpreted as a projective space over the field, capturing the essence of the algebra's structure. This connection highlights how these varieties represent classes in the Brauer group while facilitating an exploration of their properties within the realm of algebraic geometry.
• Discuss the role of Severi-Brauer varieties in the context of the Merkurjev-Suslin theorem.
  • The Merkurjev-Suslin theorem states that for a field with a certain property, every element of its Brauer group can be represented by a severi-brauer variety. This implies that the K-theory and Brauer group are tightly intertwined through these varieties. By establishing this connection, one can utilize Severi-Brauer varieties to study rational points and their properties within K-theory, effectively linking algebraic structures with geometric interpretations.
• Evaluate how Severi-Brauer varieties influence our understanding of rational points on algebraic varieties over various fields.
  • Severi-Brauer varieties provide insight into how rational points behave on algebraic varieties, especially when dealing with fields that are not algebraically closed. By examining these varieties, one can analyze conditions under which rational points exist or fail to exist, particularly in relation to division algebras. This evaluation not only contributes to a deeper understanding of K-theory but also highlights the complexities arising from different field characteristics, shaping our comprehension of arithmetic geometry.

Related terms

Division Algebra: A type of algebra where division is possible, except by zero, and is a central simple algebra over a field.

Brauer Group: A group consisting of equivalence classes of division algebras over a given field, which provides important information about the field's structure.

Central Simple Algebra: An algebra that is simple (having no two-sided ideals other than zero and itself) and has its center as a field.