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    Algebraic K-theory - Wikipedia

    • Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to com… See more

    History

    The history of K-theory was detailed by Charles Weibel.
    In the 19th century, Bernhard Riemann and … See more

    Lower K-groups

    The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring.
    The functor K0 takes a ring A to the Grothendie… See more

    Milnor K-theory

    The above expression for K2 of a field k led Milnor to the following definition of "higher" K-groups by
    thus as graded parts of a quotient of the tensor algebra of the multiplicative group k by the … See more

    Higher K-theory

    The accepted definitions of higher K-groups were given by Quillen (1973), after a few years during which several incompatible definitions were suggested. The object of the program was to find definitions of K(R) and K(R,I) i… See more

    Examples

    While the Quillen algebraic K-theory has provided deep insight into various aspects of algebraic geometry and topology, the K-groups have proved particularly difficult to compute except in a few isolated but interesting … See more

    Applications and open questions

    Algebraic K-groups are used in conjectures on special values of L-functions and the formulation of a non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.
    Parshin's c… See more

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    Table of Contents
    History
    Lower K-groups
    Milnor K-theory
    Higher K-theory
    Examples
    Applications and open questions
    See also
     
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