Unitary SK1 of semiramified graded and valued division algebras

“…Remark 4.12. In a sequel to this paper [28], the following will be shown: With the hypotheses of Theorem 4.7, suppose that E is semiramified with a graded maximal subfield L totally ramified over T such that τ (L) = L, and suppose that Gal(E 0 /T 0 ) is bicyclic, say E 0 = N ⊗ T0 N with N and N each cyclic Galois over T 0 . Then [31,Theorem 5.6].)…”

Section: )mentioning

confidence: 99%

“…In the sequel to this paper [28], the very interesting special case will be treated where E/T is semiramified (and T/R is unramified) and Gal(E 0 /T 0 ) is bicyclic. This case was the setting of essentially all of Platonov's specifically computed examples with non-trivial SK 1 (D) (see [18,19]), and likewise Yanchevskiȋ's unitary examples in [32] where the non-trivial SK 1 (D, τ ) was fully computed.…”

Section: Introductionmentioning

confidence: 99%

Unitary SK₁ of graded and valued division algebras

Hazrat

Wadsworth

2011

Proceedings of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields.

show abstract