 
Summary: Algebraic Ktheory of the first Morava Ktheory
Christian Ausoni John Rognes
8. February 2011
Abstract
For a prime p 5, we compute the algebraic Ktheory modulo p and v1 of
the mod p Adams summand, using topological cyclic homology. On the way, we
evaluate its modulo p and v1 topological Hochschild homology. Using a localization
sequence, we also compute the Ktheory modulo p and v1 of the first Morava K
theory.
Keywords. Algebraic Ktheory, Morava Ktheory, topological cyclic homology, to
pological Hochschild homology
1 Introduction
In this paper we continue the investigation from [AR02] and [Aus10] of the algebraic K
theory of topological Ktheory and related Salgebras. Let p be the pcomplete Adams
summand of connective complex Ktheory, and let /p = k(1) be the first connective
Morava Ktheory. It has a unique Salgebra structure [Ang, Th. A], and we show in
Section 2 that /p is an palgebra (in uncountably many ways), so that K( /p) is a K( p)
module spectrum.
Let V (1) = S/(p, v1) be the type 2 SmithToda complex (see Section 4 below for
a definition). It is a homotopy commutative ring spectrum for p 5, with a preferred
