nLab Volodin model

Volodin model for K-theory

This is the simplicial set / complex constructed by Volodin, using a construction similar to that of the Vietoris complex. It is the Volodin space of the family of subgroups of the stable general linear group described as follows:

We let T n σ(R)T_n^\sigma(R) be the subgroup of G n(R)G\ell_n(R) formed by the σ\sigma-triangular matrices, (discussed at higher generation by subgroups), and then look at all such subgroups for all nn, considering the stable general linear group G(R)G\ell(R) as the colimit of the nested sequence of all the G n(R)G\ell_n(R), take G=G(R)G = G\ell(R). Considering the family, \mathcal{H}, of all the T n σ(R)T_n^\sigma(R), form the corresponding Volodin space.


  • A. A. Suslin and M. Wodzicki, Excision in algebraic K-theory, The Annals of Mathematics, 136, (1992), 51 – 122.

  • I. Volodin, Algebraic K-theory as extraordinary homology theory on the category of associative

    rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859-887)

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