From: instanton@mail.ru
Subject: Re: Commutators in GL_n(R) [Was: Commutators in GL_n(K)]
Date: Sun, 11 Feb 2001 12:01:09 GMT
Newsgroups: sci.math
Summary: Algebraic K-theory of rings (commutators of unions of matrix rings)

In article <96428k$kar$1@news1.math.niu.edu>,
  rusin@vesuvius.math.niu.edu (Dave Rusin) wrote:
> In article <963p42$7na$1@wisteria.csv.warwick.ac.uk>,
>  <mareg@mimosa.csv.warwick.ac.uk> wrote:
> >In article <3A8416A9.5F25C3D0@jps.net>,
> >	alperin@jps.net writes:
> >>Questions of this kind are considered in a recent preprint by U. Rehmann
> >>and N. Gordeeva, available from the preprint server at Bielefeld
> >>University, Germany.
> >>
> >>Roger  Alperin
> >
> >I just looked this up.
>
> What's this? Actual scholarly activity in sci.math? Begone, impostor!
>
> >In fact the answer to the original question, of
> >whether every element of SL_n(K) is a commutator in GL_n(K) is yes, except
> >when n=|K|=2. This was proved by R.C.Thomson in 1961, in
> >"Commutators in the special and general linear groups", Trans. Amer. Math.
> >Soc. 101 (1961), 16-23. There are some cases however when certain
> >scalar matrices are not commutators within the subgroup SL_n(K).
>
> Thank you for the summary; I had been under the misapprehension this
> result was due to Serre.
>
> What is the situation for commutative rings  R  (with unit element)?
> I was under the impression that when  K-theory  was translated from
> topology to ring theory, the construct for the translation was the set
> of groups  SL_n(R), or more precisely the direct limit  SL(R)  of the
> ordered family of them. One of the K-theoretic invariants of a ring
> amounts to the difference between  SL(R)  and the subgroup generated by
> the elementary matrices, no? (transvections, permutations, and
> stretchings along one axis). Is there a K-theoretic measure of the
> difference between one of these and  [GL(R),GL(R)] ? Or between
> the latter and the actual set of commutators?





  let us denote GL(R) - infinite matrices

  Whitehead lemma:
  1) [GL(R),GL(R)]=the subgroup generated by
   the elementary matrices ( denoted by E(R))
  2) [E(R),E(R)]=E(R)


  Definition:
  K_1(R)=GL(R)/[GL(R),GL(R)]


  R- commutative

  SL(R)/E(R)
  sometimes called reduced K_1 - denoted by SK_1

  Theorem:
  K_1(R)=R^* + SK_1(R)


  ***
  Section 2 of text below  is nice, short list of such results

  http://www.math.uiuc.edu/K-theory/

  420: 2000 Jun 5, Algebraic K-theory of rings from a topological
  viewpoint, by Dominique
       Arlettaz.


>
> dave
>


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