On the First Galois Cohomology Group of the Algebraic Group SL₁(D)

Abstract

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.

Key Words:

• Central simple algebras
• Reduced K-theory
• Reduced norm
• Valuation theory

2000 Mathematics Subject Classification:

• 16K20
• 16E20
• 19B99

Notes

Communicated by V. A. Artamonov.