Topological Hochschild homology of extensions of \(\mathbb{Z}/p\mathbb{Z}\) by polynomials, and of \(\mathbb{Z} [x]/(x^ n)\) and \(\mathbb{Z} [x]/(x^ n-1)\)
The paper explicitly calculates the topological Hochschild homology spectrum \(THH(R)\) in terms of Eilenberg-MacLane spectra for the rings \(R=\mathbb{Z}[x]/(x^n)\), \(\mathbb{Z}[x]/(x^n-1)\), \(\mathbb{F}_p[x]/(f)\), where \(f\) is an arbitrary monic polynomial. The calculation follows the line of Bökstedt's calculation of \(THH(\mathbb{Z})\): A. Lindenstrauss first computes the spectrum homology of \(THH(R)\) with coefficients in \(\mathbb{Z}/p\), \(p\) prime, using Bökstedt's spectral sequence. For the rings \(R\) considered the computations give a splitting. \[ H^S_*(THH(R);\mathbb{Z}/p)\cong HH_*(R;\mathbb{Z}/p)\otimes H^S_*(THH(\mathbb{Z});\mathbb{Z}/p)\tag{*} \] where \(HH_*\) is ordinary Hochschild homology. Bökstedt's result and the known calculations of Hochschild homology determine the homology of \(THH(R)\). Since \(THH(R)\) is an \(H\mathbb{Z}\)-module spectrum (\(H\mathbb{Z}\) is the Eilenberg-MacLane spectrum of \(\mathbb{Z}\)) it is a retract of \(H\mathbb{Z}\wedge THH(R)\). The calculations of the homology of \(THH(R)\) determine this retract as a weak product of Eilenberg-MacLane spectra. Independently, \textit{T. Pirashvili} [J. Algebra 170, 422-428 (1994; Zbl 0811.18009)], \textit{L. Hesselholt} and \textit{I. Madsen} [Topology 36, 29-101 (1997; Zbl 0866.55002), see the review above], and \textit{R. Schwänzl}, the reviewer and \textit{F. Waldhausen} [Preprint 1996] discovered splittings similar to \((*)\) on the homotopy level from which the results above can be deduced.
scientific article; zbMATH DE number 895836
scientific article