ISSN:
1573-0514
Keywords:
Nonabelian K 1
;
noncommutative homotopy
;
general linear group
;
superspecial linear groups
;
descending central series
;
stability
;
relative normal subgroups
;
nilpotent sandwich classifications
;
quasi-finite algebras
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract A functorial filtration GL n =S−1L n $$ \supseteq$$ S0L n $$ \supseteq$$ ⋯ $$ \supseteq$$ S i L n $$ \supseteq$$ ⋯ $$ \supseteq$$ E n of the general linear group GL n, n ⩾ 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S−1 L n (A)/S0L n (A) is abelian, that S0L n (A) $$ \supseteq$$ S1L n (A) $$ \supseteq$$ ⋯ is a descending central series, and that S i L n (A) = E n(A) whenever i ⩾ the Bass-Serre dimension of A. In particular, the K-functors k 1 S i L n =S i L n /E n are nilpotent for all i ⩾ 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S i L n (A)/S i+1 L n (A)→S i L n+ 1(A)/S i+1 L n + 1 (A) is injective whenever n ⩾ i + 3, so that one has stability results without stability conditions, and if A is commutative then S0L n (A) agrees with the special linear group SL n (A), so that the functor S0L n generalizes the functor SL n to noncommutative rings. Applying the above to subgroups H of GL n (A), which are normalized by E n(A), one obtains that each is contained in a sandwich GL′ n (A, ρ) $$ \supseteq$$ H $$ \supseteq$$ E n(A, ρ) for a unique two-sided ideal ρ of A and there is a descending S0L n (A)-central series GL′ n (A, ρ) $$ \supseteq$$ S0L n (A, ρ) $$ \supseteq$$ S1L n (A, ρ) $$ \supseteq$$ ⋯ $$ \supseteq$$ S i L n (A, ρ) $$ \supseteq$$ ⋯ $$ \supseteq$$ E n(A, ρ) such that S i L n (A, ρ)=E n(A, ρ) whenever i ⩾ Bass-Serre dimension of A.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00533991
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