Volodin space
In mathematics, more specifically in topology, the Volodin space [math]\displaystyle{ X }[/math] of a ring R is a subspace of the classifying space [math]\displaystyle{ BGL(R) }[/math] given by
- [math]\displaystyle{ X = \bigcup_{n, \sigma} B(U_n(R)^\sigma) }[/math]
where [math]\displaystyle{ U_n(R) \subset GL_n(R) }[/math] is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and [math]\displaystyle{ \sigma }[/math] a permutation matrix thought of as an element in [math]\displaystyle{ GL n(R) }[/math] and acting (superscript) by conjugation.[1] The space is acyclic and the fundamental group [math]\displaystyle{ \pi_1 X }[/math] is the Steinberg group [math]\displaystyle{ \operatorname{St}(R) }[/math] of R. In fact, (Suslin 1981) showed that X yields a model for Quillen's plus-construction [math]\displaystyle{ BGL(R)/X \simeq BGL^+(R) }[/math] in algebraic K-theory.
Application
An analogue of Volodin's space where GL(R) is replaced by the Lie algebra [math]\displaystyle{ \mathfrak{gl}(R) }[/math] 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
- ↑ Weibel 2013, Ch. IV. Example 1.3.2.
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
- Weibel, Charles (2013). "The K-book: an introduction to algebraic K-theory". http://www.math.rutgers.edu/~weibel/Kbook.html.
- Suslin, A. A. (1981), "On the equivalence of K-theories", Comm. Algebra 9 (15): 1559–66, doi:10.1080/00927878108822666
- 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, doi:10.1070/IM1971v005n04ABEH001121, Bibcode: 1971IzMat...5..859V, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)
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