The plus-construction as a localization, by A J Berrick

To appear in Algebraic K-Theory and its Applications, Proc Symp ICTP, Trieste, World Scientific (Singapore). An initial survey contrasts two points of view in the historical development of the theory of localization. The first, starting with inversion of elements in a ring, leads to quotient categories and indirectly to the Q-construction. The second considers idempotent functors. This leads to the Berrick-Casacuberta description of the plus-construction on X as the idempotent functor that is nullification of X with respect to an acyclic space W. Focus on the case X = BGLR produces new results, including the classification of perfect normal subgroups of GLR. When R is a group ring AG, links are obtained between these perfect normal subgroups and the A-representability of the group G. A final section studies the relationship between the plus-construction on BGLR and acyclicity of the space W. This prompts some general questions on the K-theory of rings.

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