Volodin space

In mathematics, more specifically in topology, the Volodin space ${\displaystyle X}$ of a ring R is a subspace of the classifying space ${\displaystyle BGL(R)}$ given by

${\displaystyle X=\bigcup _{n,\sigma }B(U_{n}(R)^{\sigma })}$

where ${\displaystyle U_{n}(R)\subset GL_{n}(R)}$ is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and ${\displaystyle \sigma }$ a permutation matrix thought of as an element in ${\displaystyle GL_{n}(R)}$ and acting (superscript) by conjugation.[1] The space is acyclic and the fundamental group ${\displaystyle \pi _{1}X}$ is the Steinberg group ${\displaystyle \operatorname {St} (R)}$ of R. In fact, Suslin (1981) showed that X yields a model for Quillen's plus-construction ${\displaystyle BGL(R)/X\simeq BGL^{+}(R)}$ in algebraic K-theory.

Application

An analogue of Volodin's space where GL(R) is replaced by the Lie algebra ${\displaystyle {\mathfrak {gl}}(R)}$ 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. harvnb error: no target: CITEREFWeibel2013 (help)

References

• Goodwillie, Thomas G. (1986), "Relative algebraic K-theory and cyclic homology", Annals of Mathematics, Second Series, 124 (2): 347–402, doi:10.2307/1971283, MR 0855300
• C. Weibel, The K-book: an introduction to algebraic K-theory
• Suslin, A. A. (1981), "On the equivalence of K-theories", Comm. Algebra, 9 (15): 1559–1566
• Volodin, I. (1971), "Algebraic K-theory as extraordinary homology theory on the category of associative rings with unity", Izv. Akad. Nauk. SSSR, 35 (4): 844–873, MR 0296140, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)