The microstable Adams-Novikov spectral sequence (Q2707515)

Recent results in stable homotopy theory, such as the proof of the nilpotence theorem, have involved a sequences of spectra \(T(m)\), originally defined in [the author, Complex cobordism and the stable homotopy groups of spheres (1986; Zbl 0608.55001)], whose Brown-Peterson homology is given by \(BP_* (T(m)) = BP_* [t_i, \ldots, t_m]\). Some properties of the stable homotopy groups, \(\pi_k(T(m))\), of these spectra are dependent only on \(k-m\), and the paper under review constructs an algebraic framework with which to study this phenomenon. If \((A, \Gamma)\) denotes the Hopf algebroid \((BP_*, BP_*BP)\) and \(\Gamma(m+1) = \Gamma/(t_1, \ldots t_m)\) then, by a change of rings formula, the \(E_2\) term of the Adams-Novikov spectral sequence of \(T(m)\) is \(\text{Ext}_{\Gamma(m+1)}(A,A)\). The author constructs a bigraded Hopf algebroid \((\widehat{A}, \widehat{\Gamma})\) with a chain of sub Hopf algebroids \(\{\widehat{G}(1,m)\}\), of which it is the union, and with \(\widehat{G}(1,m)\) mapping to \(\Gamma(m+1)\). It is conjectured that there is a spectral sequence, called the microstable Adams-Novikov spectral sequence, whose \(E_2\) term is \(\text{Ext}_{\widehat{\Gamma}}(\widehat{A}, \widehat{A})\) and which is compatible, in a range of dimensions depending on \(m\), with the Adams-Novikov spectral sequence of \(T(m)\) . There is an analog of the chromatic spectral sequence for this situation which allows the author to compute \(\text{Ext}_{\widehat{\Gamma}}^s (\widehat{A}, \widehat{A})\) for \(s = 0\) and \(1\). The paper concludes with some results about the image of \(\text{Ext}_{\widehat{\Gamma}}(\widehat{A}, \widehat{A})\) in \(\text{Ext}_{\widehat{\Gamma}}(\widehat{A}, \widehat{A}/I)\) where \(I = (p, v_1, v_2, \ldots)\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].