Cohomological Invariants of Simply Connected Groups of Rank 3

✍ Scribed by Alexander Merkurjev

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 152 KB
• Volume: 227
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8241

✦ Synopsis

Let G be a linear algebraic group defined over a field F. One can define an equivalence relation (called R-equivalence) on the group G F of points over F as follows (cf. [4, 9, 14]). Two points g 0 g 1 ∈ G F are R-equivalent, if there is a rational morphism f 1 F → G of algebraic varieties over F defined at points 0 and 1 such that f 0 = g 0 and f 1 = g 1 . The group of R-equivalence classes is denoted by G F /R. For example, if G = SL 1 A is the special linear group of a central simple F-algebra A, then the group of R-equivalence classes G F /R is equal to the reduced Whitehead group in algebraic K-theory (cf. [27])

We say that the group G is R-trivial if G E /R = 1 for any field extension L/F (cf. [14]). The group G F /R measures complexity of G F . One of the major properties of G F /R is that this group is rigid, i.e., G F /R = G F t /R. In other words, any rational family of elements in G F /R is constant. On the other hand, any R-trivial element (i.e., an element in the kernel of G F → G F /R) can be connected to the identity of the group by a rational family of elements in G.

It is known that an algebraic group G of rank (dimension of a maximal torus) at most 2 is rational (cf. [2, Theorem 7.9, 27, Theorem 4.74]) and hence G is R-trivial by [4, 14, Proposition 1]. In the present paper we 1 Partially supported by the N.S.F. 614