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

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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 Nⁿ. 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. Updated: This is a substantial revision. We corrected several errors in the description of the Witt vectors on a truncation set on Nⁿ and modified the key proofs accordingly. We also replaces several topological statement with purely algebraic ones. Most arguments have been reworked and streamlined.