On the \(K\)-theory of nilpotent endomorphisms (Q2752872)

This paper gives a calculation of the relative \(K\)-theory of a truncated polynomial algebra \(\Lambda = A[x]/(x^{n})\) where \(A\) is a smooth algebra over a perfect field \(k\) of positive characteristic. This work extends previous work of the authors, which considered the case \(A=k\) [\textit{L. Hesselholt} and \textit{I. Madsen}, Invent. Math. 130, No. 1, 73-97 (1997; Zbl 0884.19004)]. The main result of this paper says that the relative \(K\)-groups \(K_{\ast}(A[x]/(x^{n}),(x))\) sit in a long exact sequence involving direct sums of instances of the Verschiebung homomorphism \(V_{n}: \mathbf{W}_{m}\Omega_{A}^{\ast} \to \mathbf{W}_{mn}\Omega_{A}^{\ast}\) on the big de Rham-Witt complex. This result specializes to a characterization of the nil groups \(N_{\ast}(A[x]/(x^{n}))\). It is well known that the nil groups of a ring \(\Lambda\) are the obstructions to the \(K\)-theory of \(\Lambda\) having the homotopy property.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].