Higher polyhedral \(K\)-groups (Q1410963)

This is a continuation of an earlier paper of the authors [\textit{W. Bruns} and \textit{G. Gubeladze}, Manuscr. Math. 109, 367-404 (2002; Zbl 1025.19002)], where the polyhedral \(K_2\)-groups were defined by using elementary graded automorphisms of polytopal algebras. In the present paper, the authors define higher polyhedral \(K\)-groups for commutative rings, starting from the stable groups of elementary automorphisms of polytopal algebras. It can be viewed as a generalization of the well-known Volodin's theory and Quillen's plus construction. In the special case of algebras with unit simplices one recovers the usual algebraic \(K\)-groups, while the general case of lattice polytopes reveals many new aspects, governed by polyhedral geometry. This explores the polyhedral geometry behind Suslin's proof of the coincidence of the classical Volodin's and Quillen's theories. The \(K\)-groups coming from two-dimensional polytopes are also determined.