On the Adams \(E_2\)-term for elliptic cohomology (Q2752865)

Let \(\text{Ell}^*( \enskip)\) denote level 1 elliptic cohomology with coefficient ring \(\text{Ell}_* = {\mathbb Z}[\frac{1}{6}] [Q,R, \Delta^{-1}]\). This paper gives a chromatic description of the \(E_2\) term of the Adams spectral sequence based on \(\text{Ell}\), i.e. a description modeled on that for Brown-Peterson theory given by \textit{H. R. Miller, D. C. Ravenel} and \textit{W. S. Wilson} in [Ann. Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)]. The basic tool for constructing this description is a general change of rings theorem which, for \((A, \Gamma)\) a Hopf algebroid, over \(k\), \(f: A \to B\) a \(k\)-algebra homomorphism and \(\Sigma_f = B \otimes_f \Gamma \otimes_f B\), gives conditions insuring that \(\text{Ext}_{\Sigma_f} (B, f^*N) \simeq \text{Ext}_{\Gamma}(A, N)\). A similar result is proved by \textit{M. Hovey} and \textit{H. Sadofsky} in [J. Lond. Math. Soc., II. Ser. 60, No. 1, 284-302 (1999; Zbl 0947.55013)] where they credit it to M. Hopkins. This result, together with results of N. Katz on the ring of divided congruences suffice to show that the \(E_2\) term is of chromatic length \(2\) and to compute the groups. A notable feature of this paper for topologists is an appendix giving an easily accessible proof of the algebraic part of Katz's results need for these computations. The calculation of the groups in the \(E_2\) term is left unfinished (the \(2\)-line is barely mentioned) and the reader should consult the paper of \textit{G. Laures} [Topology 38, No. 2, 387-425 (1999; Zbl 0924.55004)] for more complete results.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].