Geometric models for the K-theory of fields
Quillen's space \(BGL^+(k)\) is difficult to study even when k is a field. This paper contains the following model for \(BGL^+(k)\). Denote by \(F_ N(n)\) the Grassmann scheme of subspaces of dimension n in \(k^ N\). It makes sense to go to the limit \(F_{\infty}(n)\). Now consider the simplicial algebra \(k_*\) with n-simplices \(k_ n=k[x_ 0,...,x_ n]/(\sum x_ i-1)\) and operators \(d_ i(x_ j)=x_ j (j<i)\), \(=0 (j=i)\), \(=x_{j-1} (j>i)\), and \(s_ i(x_ j)=x_ j (j<i)\), \(=x_ i+x_{i+1} (j=1)\), \(=x_{j+1} (j>i)\), respectively. Applying \(F_{\infty}(n)\) to \(k_*\) and using a stabilization map \(F_{\infty}(n)(k_*)\to F_{\infty}(n+1)(k_*)\) we get some Kan complex which is homotopically equivalent to \(BGL(k_*)\) and consequently to \(BGL^+(k_*)\) [cf. \textit{S. M. Gersten}, Lect. Notes Math. 341, 3-42 (1973; Zbl 0285.18010)]. This is the main result. As the author remarks it was developed in an attempt to calculate K-theory of an algebraically closed field with finite coefficients. Meanwhile this problem (the Quillen-Lichtenbaum conjecture) was solved by \textit{A. A. Suslin} [Invent. Math. 73, 241-245 (1983; Zbl 0514.18008)].
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