Manuscripta Mathematica

, Volume 139, Issue 3–4, pp 343–389 | Cite as

Unitary SK1 of semiramified graded and valued division algebras

  • A. R. WadsworthEmail author


We prove formulas for SK1(E, τ), which is the unitary SK1 for a graded division algebra E finite-dimensional and semiramified over its center T with respect to a unitary involution τ on E. Every such formula yields a corresponding formula for SK1(D, ρ) where D is a division algebra tame and semiramified over a Henselian valued field and ρ is a unitary involution on D. For example, it is shown that if \({\sf{E} \sim \sf{I}_0 \otimes_{\sf{T}_0}\sf{N}}\) where I 0 is a central simple T 0-algebra split by N 0 and N is decomposably semiramified with \({\sf{N}_0 \cong L_1\otimes_{\sf{T}_0} L_2}\) with L 1, L 2 fields each cyclic Galois over T 0, then
$${\rm SK}_1(\sf{E}, \tau) \,\cong\ {\rm Br}(({L_1}\otimes_{\sf{T}_0} {L_2})/\sf{T}_0;\sf{T}_0^\tau)\big/ \left[{\rm Br}({L_1}/\sf{T}_0;\sf{T}_0^\tau)\cdot {\rm Br}({L_2}/\sf{T}_0;\sf{T}_0^\tau) \cdot \langle[\sf{I}_0]\rangle\right].$$

Mathematics Subject Classification (2000)



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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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