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.
Goodwillie, Thomas G. (1986), "Relative algebraic K-theory and cyclic homology", Annals of Mathematics, Second Series, 124 (2): 347–402,
doi:
10.2307/1971283,
JSTOR1971283,
MR0855300
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,
Bibcode:
1971IzMat...5..859V,
doi:
10.1070/IM1971v005n04ABEH001121,
MR0296140, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859–887)