2012/06/01 by Vigleik Angeltveit, Angeltveit, Vigleik, Teena Gerhardt +5
Mathematics · Physics and Astronomy · #19D55 #55Q91 #Advanced Differential Equations and Dynamical Systems #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1206.0247
openalex publication_date 2012/06/01 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider the algebraic K-theory of a truncated polynomial algebra in\nseveral commuting variables, K(k[x1, ..., xn]/(x1a1, ..., xnan)). This\nnaturally leads to a new generalization of the big Witt vectors. If k is a\nperfect field of positive characteristic we describe the K-theory computation\nin terms of a cube of these Witt vectors on Nn. If the characteristic of k\ndoes not divide any of the ai we compute the K-groups explicitly. We also\ncompute the K-groups modulo torsion for k=Z. To understand this K-theory\nspectrum we use the cyclotomic trace map to topological cyclic homology, and\nwrite TC(k[x1, ..., xn]/(x1a1, ..., xnan)) as the iterated homotopy\ncofiber of an n-cube of spectra, each of which is easier to understand.\n Updated: This is a substantial revision. We corrected several errors in the\ndescription of the Witt vectors on a truncation set on Nn and modified the key\nproofs accordingly. We also replaces several topological statement with purely\nalgebraic ones. Most arguments have been reworked and streamlined.\n