The group \(SK_1\) for simple algebras.
type: Item
instance of: scholarly article
title: The group \(SK_1\) for simple algebras.
zbMATH Open document ID: 1117.16006
published in: \(K\)-Theory
publication date: 2007-02-13T00:00:00Z
review text: An algebra \(A\) over a field \(F\) is a `central simple algebra' if, over the algebraic closure of \(F\), \(A\) is isomorphic to a matrix algebra. The square root of the dimension of \(A\) is called its `degree'. It is known that \(A\simeq M_n(D)\) for a central division algebra \(D\) over \(F\). We call the degree of \(D\) the `index' of \(A\). Let \(A\) be a central simple algebra over \(F\) and let \(\text{SL}_1(A)=\{a\in A^*\mid\text{Nrd}_A(a)=1\}\). This subgroup of \(A^*\) contains the commutator subgroup \([A^*,A^*]\) of \(A^*\). The quotient \(\text{SK}_1(A)=\text{SL}_1(A)/[A^*,A^*]\) is known as the `reduced Whitehead group' of \(A\). It is known that \(\text{SK}_1(A)=0\) if \(A\) is `split' (i.e., is isomorphic to a matrix algebra over \(F\)) and \(A\neq M_2(\mathbb{F}_2)\), or if the index of \(A\) is squarefree (this is due to \textit{S. Wang}, [Am. J. Math. 72, 323-334 (1950; Zbl 0040.30302)]). In the case when the index of \(A\) is not squarefree, \(\text{SK}_1(A)\) may be trivial for some classes of fields (e.g., global and local fields). \textit{A. A. Suslin} [in Algebraic K-theory, Adv. Sov. Math. 4, 75-99 (1991; Zbl 0758.11047)] conjectured that if the index of \(A\) is not squarefree, then \(\text{SK}_1(A)\) is nontrivial `generically', i.e. there is an extension \(L/F\) such that \(\text{SK}_1(A_L)\neq 0\). In the paper under review, the author proves a special case of this conjecture: Theorem: Let \(A\) be a central simple algebra over a field \(F\). If the index of \(A\) is divisible by \(4\), then \(\text{SK}_1(A_L)\neq 0\) for some field extension \(L/F\). -- This theorem was proved [in \textit{A. S. Merkurjev}, K-Theory 7, No. 1, 1-3 (1993; Zbl 0811.16011)] in the case when \(\text{char}(F)\neq 2\), using algebraic theory of quadratic forms. The present proof does not depend on quadratic forms and is a step towards the proof of the general conjecture.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 19B14
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 14C15
zbMATH DE Number: 5125382
zbMATH Keywords: reduced Whitehead groups
zbMATH Keywords: Severi-Brauer varieties
zbMATH Keywords: cycle modules
zbMATH Keywords: Chow groups
author: Alexander Merkurjev
reviewed by: Maneesh Thakur
MaRDI profile type: Publication
full work available at URL: s10977-006-0021-4
OpenAlex ID: W2052985249
cites work: Function fields of generalized brauer-severi varieties
cites work: Q3963168
cites work: Q4416651
cites work: Q4210545
cites work: Invariants of algebraic groups
cites work: Higher algebraic K-theory: I
cites work: Q3222951
cites work: Q3974026
cites work: Q3974027
DOI: 10.1007/S10977-006-0021-4
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description: scientific article; zbMATH DE number 5125382