On the algebraic K-theory of truncated polynomial algebras in several variables

Abstract

We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K(k[x1......xn]/(x1a1…….xnan). This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕ n . If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ. To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write TC(k[x1…….xn]/(x1a1……..xnan))as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.