On the Cooperation Algebra of the Connective Adams Summand

The aim of this paper is to gain explicit information about the multiplicative structure of l∗l, where l is the connective Adams summand. Our approach differs from Kane’s or Lellmann’s because our main technical tool is the MU -based Künneth spectral sequence. We prove that the algebra structure on l∗l is inherited from the multiplication on a Koszul resolution of l∗BP . Introduction Our goal in these notes is to shed light on the structure, in particular on the multiplicative structure, of l∗l, where we work at an odd prime p and l is the Adams summand of the plocalization of the connective K-theory spectrum ku. This was investigated by Kane [5] and Lellmann [9] using Brown-Gitler spectra. Our approach is different and exploits the fact that MU is a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell and May [4] and l is a MU -ring spectrum (in fact it is even an MU -algebra). As a calculational tool, we make use of a Künneth spectral sequence (2.2) converging to l∗l where we work with a concrete Koszul resolution. Our approach bears some similarities to old work of Landweber [8], who worked without the benefit of the modern development of structured ring spectra. The multiplicative structure on the Koszul resolution gives us control over the convergence of the spectral sequence and the multiplicative structure of l∗l. In particular, it sheds light on the torsion. The outline of the paper is as follows. We recall some basic facts about complex cobordism, MU , in Section 1 and describe the Künneth spectral sequence in Section 2. Some background on the Bockstein spectral sequence is given in Section 3. The multiplicative structure on the E2-term of this spectral sequence is made precise in section 4 where we introduce the Koszul resolution we will use later in terms of its multiplicative generators. We study the torsion part in l∗l and the torsion-free part separately. The investigation of ordinary and L-homology of l in Section 5 leads to the identification of the p-torsion in l∗l with the u-torsion where l∗ = Z(p)[u] with u being in degree 2p− 2. In Section 6 we show how to exploit the cofibre sequence l p −→ l −→ l/p to analyse the Künneth spectral sequence and relate the simpler spectral sequence for l/p to that for l. To that end we prove an auxiliary result on connecting homomorphisms in the Künneth spectral sequence, which is analogous to the well-known geometric boundary theorem (see for instance [14, chapter 2, §3]). We summarize our calculation of l∗l at the end of that section. In Section 7 we use classical tools from the Adams spectral sequence in order to study torsion phenomena in l∗l. We use the fact that the p and u-torsion is all simple to show that the Künneth spectral sequence for l∗l collapses at the E 2-term and that there are no extension issues. We can describe the torsion in l∗l in terms of familiar elements which are certain coaction-primitives in the Fp-homology of l. 2000 Mathematics Subject Classification. Primary 55P43, 55N15; Secondary 55N20, 18G15.