In mathematics, more specifically in topology, the Volodin space of a ring R is a subspace of the classifying space given by

where is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and a permutation matrix thought of as an element in and acting (superscript) by conjugation.[1] The space is acyclic and the fundamental group is the Steinberg group of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction in algebraic K-theory.

Application

An analogue of Volodin's space where GL(R) is replaced by the Lie algebra was used by Goodwillie (1986) to prove that, after tensoring with Q, relative K-theory K(A, I), for a nilpotent ideal I, is isomorphic to relative cyclic homology HC(A, I). This theorem was a pioneering result in the area of trace methods.

Notes

  1. ^ Weibel 2013, Ch. IV. Example 1.3.2.

References


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