Comparing Invariants of SK1

Abstract

In this text, we compare several invariants of the reduced Whitehead group SK 1 of a central simple algebra.For biquaternion algebras, we compare a generalised invariant of Suslin as constructed by the author in [Wou] to an invariant introduced by Knus-Merkurjev-Rost-Tignol [KMRT]. Using explicit computations, we prove these invariants are essentially the same.We also prove the non-triviality of an invariant introduced by Kahn [Kah2]. To obtain this result, we compare Kahn's invariant to an invariant introduced by Suslin in 1991 [Sus1] which is non-trivial for Platonov's examples of non-trivial SK 1 [Pla]. We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras [Mer1].

Unitary SK1 of semiramified graded and valued division algebras

“…See, e.g., the paper [RTY] where Suslin's conjecture is reduced to the case of cyclic algebras. See also [W,Th. 4.11], where the proof of nontriviality of a cohomological invariant of Kahn uses a careful analysis of SK 1 (D) for the D in Platonov's original example.…”