On the topological Hochschild homology of bu. I

The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(`). Here p is an odd prime, ` is the Adams summand of p-local connective K-theory (see for example [25]) and THH is the topological Hochschild homology construction introduced by Bökstedt in [3]. In the present paper we will determine the mod p homotopy groups of THH(`) and also the integral homotopy groups of THH(L) (where L denotes the periodic Adams summand). In the sequel we will investigate the integral homotopy groups of THH(`) using our present results as a starting point. The THH construction appears to be of basic importance in algebraic K-theory because it combines two useful properties: it can be used to construct good approximations to the algebraic K-theory functor, and it is very accessible to calculation. We shall review what is known about the first property in a moment; the second property was demonstrated by Bökstedt’s calculation, in his paper [4], of the homotopy groups of THH(HZ/p) and THH(HZ) (here HZ/p and HZ denote the evident Eilenberg-Mac Lane spectra). It is natural to ask about THH(R) for other popular ring spectra R, and our work is a first step in this direction. We pay special attention to the connective case because this is the case which is likely to be relevant in applications (see Subsection 1.4 below). The calculation which we present in this paper is a homotopy-theoretic one which uses the Adams spectral sequence (hereafter abbreviated ASS). This calculation has several interesting features; in particular it is a pleasing example of an ASS calculation in which, although there are infinitely many differentials, it is still possible to get the complete answer. Here is a summary of the contents of the paper. In Section 2 we review the facts we need to know about ordinary Hochschild homology. In Section 3 we do the same