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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    Extended the entry Cohn localization now starting with the ring viewpoint. Urs: I hope you will now agree that it is justified to call it a localization of a ring RΣ 1RR\to \Sigma^{-1} R.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    One should also point out (see Andrew Ranicki’s slides in the references at Cohn localization and his papers,specially the series with Amnon Neeman) that universal localization is much used in algebraic K-theory. algebraic L-theory and surgery theory.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012

    One should

    So please do! :-) (in the entry!)

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012

    Oops. Error. Invert all matrices not entries, of course.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    If we have just a set SS of elements in RR and want to invert them, we take as Σ S\Sigma_S the smallest set of matrices contining unit matrices with elements in SS such that if A,BΣ SA,B\in \Sigma_S and CC another matrix so that we can make a triangular block matrix with AA and BB as diagonal block matrix, then this triangular matrix is also there. Now inverting all such matrices is a little stronger requirement than to invert just elements in SS. Thus Σ S\Sigma_S-inverting is stronger property than SS-inverting, hence the category is smaller so there is a different chance for having an initial object, and indeed, it always exists, by the result of Cohn. Making the nondiagonal triangular block matrices amounts to be using Gauss elimination procedure to be solving systems of linear equations over a noncommutative ring with matrix operations from one side only. One side only is in a spirit of Ore, but it has the Gauss elimination procedure advantage over the scalar case.

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