Cyclic polytopes and the $K$ -theory of truncated polynomial algebras

Abstract

This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K -theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate. The result is best expressed in terms of big Witt vectors. Let Wm (k ) denote the big Witt vectors in k of length m , i.e. the multiplicative group Wm (k ) = (1 + xk [[x ]])×/(1 + x k [[x ]])×,