A Cartan-Eilenberg spectral sequence for non-normal extensions (Q2332159)

The paper under review is devoted to spectral sequences associated with extensions \(\Phi \to \Gamma\to \Sigma\) of Hopf algebras over a commutative ring \(k\). If the extension is conormal, then for an arbitrary \(\Gamma\)-comodule \(M\) there is the Cartan-Eilenberg spectral sequence \(E_2= \mathrm{Ext}_{\Phi}(k, \mathrm{Ext}_{\Sigma}(k,M))\Rightarrow \mathrm{Ext}_{\Gamma}(k, M)\). This spectral sequence is a standard tool for computing the Hopf algebra cohomology of \(\Gamma\) with coefficients in \(M\) in terms of the cohomology of \(\Phi\) and \(\Sigma\). Recall that if \(\Sigma\) is a quotient coalgebra of \(\Gamma\) and if \(M\) is a \(\Sigma\)-module, then the cotensor product \(\Gamma\square_{\Sigma}M\) is defined as \(ker(1_{\Gamma}\otimes\psi_M-\psi_{\Gamma}\otimes 1_M)\), where \(\psi_{\Gamma}\) is the right \(\Sigma\)-comodule structure map on \(\Gamma\), and \(\psi_{\Sigma}\) is the left \(\Sigma\)-comodule structure map on \(M\). In this paper, a generalization of the Cartan-Eilenberg sequence is constructed that can be defined when \(\Phi=\Gamma{\square}_{\Sigma}k\) is compatibly an algebra and a \(\Gamma\)-comodule. This is related to a construction independently developed by \textit{R. Bruner} and \textit{J. Rognes} [``The Adams spectral sequence for topological modular forms'' (in preparation)], whose paper is being prepared for publication. For a surjection \(\Gamma\to \Sigma\) of Hopf algebras, we consider the map \(\Phi:=\Gamma\square_{\Sigma}k \to \Gamma\) as a map of \(\Gamma\)-comodule algebras. In the paper, for a \(\Gamma\)-comodule \(M\), three spectral sequences convering to \(\mathrm{Ext}^*_{\Gamma}(k, M)\) are considered: \begin{itemize} \item[(1)] A generalization of the classical Cartan-Eilenberg spectral sequence [\textit{H. Cartan} and \textit{S. Eilenberg}, Homological algebra. Princeton, NJ: Princeton University Press (1999; Zbl 0933.18001), XVI 5.2(4)] converging to \(\mathrm{Ext}_{\Gamma}(M,N)\) (Section 3). \item[(2)] \textit{H. R. Margolis} [Spectra and the Steenrod algebra. Modules over the Steenrod algebra and the stable homotopy category. Amsterdam - New York - Oxford: North-Holland (1983; Zbl 0552.55002)] and \textit{J. H. Palmieri} [Stable homotopy over the Steenrod algebra. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0966.55013)] have studied the generalized Adams spectral sequence constructed in the category of stable \(\Gamma\)-modules, a close cousin of the derived category of \(\Gamma\)-comodules. If \(\Phi\) is a \(\Gamma\)-comodule algebra and \(M\) is a \(\Gamma\)-comodule, then the \(\Phi\)-based Adams spectral sequence in \(Stable(\Gamma)\) for \(M\) converges to \(\mathrm{Ext}_{\Gamma}(k, M)\). \item[(3)] The third construction is a filtration spectral sequence on the cobar complex on \(\Gamma\) due to \textit{J. F. Adams} [Ann. Math. (2) 72, 20--104 (1960; Zbl 0096.17404), \S2.3]. Though originally studied in the case where \(\Phi\to \Gamma\to \Sigma\) is an extension of Hopf algebras, the filtration spectral sequence itself may be defined in the setting here without modification. \end{itemize} The main result of this paper can be summarizied as follows. Theorem 1.1. The spectral sequences (1), (2), and (3) coincide at the \(E_1\) page, and have the form \[ E^{s,t}_1= \mathrm{Ext}^t_{\Sigma}(k, \overline{\Phi}^{\otimes{s}}\otimes M)\Rightarrow \mathrm{Ext}^{s+t}_{\Gamma}(k, M) \] where \(\overline{\Phi}=coker(k\to \Phi)\) is coaugmentation coideal. Section 2 begins by discussing the motivating application for this work, a localization of the Adams \(E_2\) page for the sphere. Section 3 contains a review of the classical construction of the Cartan-Eilenberg spectral sequence for an extension of Hopf algebras \(\Phi\to \Gamma\to \Sigma\), and a definition of a variation (the spectral sequence mentioned in (1)) that makes sense when \(\Phi\) is only a \(\Gamma\)-comodule algebra. The main step is to replace the \(\Phi\)-cobar resolution of a \(\Phi\)-comodule \(M\) (which does not make sense when \(\Phi\) does not have a coalgebra structure) with a \(\Gamma\)-comodule resolution. In Section 4, the Margolis and Palmieri's Adams spectral sequence is reviewed, and it is proved that the spectral sequence (1) coincides with this one at \(E_1\). This extends a remark of \textit{J. H. Palmieri} [Stable homotopy over the Steenrod algebra. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0966.55013), \S1.4], who notes that the spectral sequence he studies coincides with the Cartan-Eilenberg spectral sequence in the case that the extension is conormal (the coalgebra analogue of the normality condition mentioned above). Section 5 is devoted to comparing the spectral sequences (1) and (3). Adams mentions (without proof) that (3) coincides with the classical Cartan-Eilenberg spectral sequence in the case when the latter is defined. A proof of this fact is given in [\textit{D. C. Ravenel}, Complex cobordism and stable homotopy groups of spheres. Orlando etc.: Academic Press, Inc. (1986; Zbl 0608.55001), A1.3.16], attributed to Ossa. The proposed comparison proof generalizes Ossa's argument to the considered setting. This involves the use of explicit formulas for the iterated shear isomorphism and its inverse, which are established in the appendix.