Simplicial groups that are models for algebraic K-theory (Q1882754)

Let \(X\) be a simplicial set and \(\Omega(X)\) it loop space as defined by \textit{C. Berger} [Bul. Soc. Mat. Fr. 123, 1-32 (1995; Zbl 0820.18005)]. Let \(R\) be a ring with \(1\), the author studies a simplicial group \(GR\) functorially associated to \(R\) defined as a certain subgroup of the loop space of the simplicial set \(NQPR\) (the nerve of Quillen's \(Q\)-construction on the category of finitely projective \(R\)-modules). One of the main results in this work is the following theorem: Let \(R\) be a ring with \(1\), and \(\Sigma R\) its suspension ring. Then, there is a homomorphism of simplicial groups \[ \alpha_{R}:GR\to \Omega(G\Sigma R) \] which is a natural weak equivalence as well. Moreover, the author constructs explicit group isomorphisms \(\theta:K_{0}(R) \to \pi_{0}(GR)\) and \(\xi :K_{1}(R)\to \pi_{1}(GR)\)