zbMATH — the first resource for mathematics

The \(E_{2}\)-term of the \(K(n)\)-local \(E_{n}\)-Adams spectral sequence. (English) Zbl 1348.55008
In chromatic homotopy theory, the \(K(n)\)-local category is for a fixed prime \(p\) an important approximation of the stable homotopy category. Thus, it is an important problem how to compute homotopy groups of \(K(n)\)-local spectra or more generally maps between them.
Denote by \(E\) Morava \(E\)-theory at height \(n\). This has by the Hopkins-Miller theorem an action by the (profinite) Morava stabilizer group \(\mathbb{G}_n\). E. S. Devinatz and M. J. Hopkins [Topology 43, No. 1, 1–47 (2004; Zbl 1047.55004)] have constructed for any spectrum \(X\) a strongly convergent spectral sequence \[ H^*_c(\mathbb{G}_n, E^*X)\Rightarrow \pi_*L_{K(n)}X, \] where \(H^*_c\) denotes continuous cohomology.
The aim of the paper under review is to formulate a homological version of this spectral sequence. We use the notation \(E^{\vee}_*X\) for \(\pi_* L_{K(n)}E \wedge X\), which can be seen as a completed version of \(E\)-homology adapted to the \(K(n)\)-local context. This functor takes values in so-called \(L\)-complete \(E^{\vee}E\)-comodules, where \(L\)-completness means that a module coincides with the \(0\)-th left derived functor of its completion. It is shown that for spectra \(X\) and \(Y\) the \(E^2\)-term of the \(K(n)\)-local \(E\)-based Adams spectral sequence computing \(\pi_*F(X, L_{K(n)}Y)\) can be computed via Ext in the category of \(L\)-complete \(E^{\vee}_*E\)-comodules if \(E_*^{\vee}X\) is pro-free and \(E^{\vee}_* Y\) satisfies some more general conditions. Note that \(E_*^{\vee}X\) is, for example, pro-free if \(K(n)_*X\) is concentrated in even degrees.
In the next step, the authors identify this \(E^2\)-term in the case \(X = S^0\) and under the same conditions on \(Y\) with \(H_c^*(\mathbb{G}_n; E^{\vee}_*Y)\). The corresponding spectral sequence \[ H^s_c(\mathbb{G}_n; E^{\vee}_tY) \Rightarrow \pi_{t-s}L_{K(n)}Y \] generalizes most of the known homological spectral sequences for computing \(\pi_*L_{K(n)}Y\). Note that in [D. G. Davis and T. Lawson, Glasg. Math. J. 56, No. 2, 369–380 (2014; Zbl 1295.55007)] there is a spectral sequence converging to \(\pi_*L_{K(n)}Y\) for any \(Y\) with \(E^2\)-term computed as continuous \(\mathbb{G}_n\)-cohomology of a Morava module, which is rather complicated though.
In the last section, the present paper provides in the case \(n=1\) a spectral sequence computing Ext between \(E^{\vee}_*X\) and \(E^{\vee}_*Y\) in the category of \(L\)-complete \(E^{\vee}E\)-comodules starting with the uncompleted \(Ext^*_{E_*E}(E_*X, E_*Y)\) in the case where \(E_*X\) is projective and \(E_*Y\) is flat. This is illustrated in the example \(X=Y =S^0\) and \(p=2\).
We note that most of the work in this article is actually about homological algebra of \(L\)-complete modules and comodules and the comparison of the latter to Morava modules. It summarizes and extends many known results in a nice way.

55P60 Localization and completion in homotopy theory
55Q10 Stable homotopy groups
13J10 Complete rings, completion
Full Text: DOI arXiv
[1] Baker, Andrew, L-complete Hopf algebroids and their comodules, alpine perspectives on algebraic topology, Contemp. Math., vol. 504, 1-22, (2009), Amer. Math. Soc. Providence, RI · Zbl 1192.55004
[2] Behrens, Mark; Davis, Daniel G., The homotopy fixed point spectra of profinite Galois extensions, Trans. Am. Math. Soc., 362, 9, 4983-5042, (2010) · Zbl 1204.55007
[3] Barthel, Tobias; Frankland, Martin, Completed power operations for Morava E-theory, Algebraic Geom. Topol., 15, 4, 2065-2131, (2015) · Zbl 1326.55018
[4] Baker, Andrew; Jeanneret, Alain, Brave new Hopf algebroids and extensions of MU-algebras, Homol. Homotopy Appl., 4, 1, 163-173, (2002) · Zbl 1380.55009
[5] Bousfield, A. K., The localisation of spectra with respect to homology, Topology, 18, 4, 257-281, (1979) · Zbl 0417.55007
[6] Daniel, G. Davis, Homotopy fixed points for \(L_{K(n)}(E_n \wedge X)\) using the continuous action, J. Pure Appl. Algebra, 206, 3, 322-354, (2006) · Zbl 1103.55005
[7] Devinatz, Ethan S., Morava’s change of rings theorem, (The Čech Centennial, Boston, MA, 1993, Contemp. Math., vol. 181, (1995), Amer. Math. Soc. Providence, RI), 83-118 · Zbl 0829.55004
[8] Devinatz, Ethan S.; Hopkins, Michael J., Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology, 43, 1, 1-47, (2004) · Zbl 1047.55004
[9] Davis, Daniel G.; Lawson, Tyler, A descent spectral sequence for arbitrary \(K(n)\)-local spectra with explicit \(E_2\)-term, Glasg. Math. J., 56, 2, 369-380, (2014) · Zbl 1295.55007
[10] Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P., Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, (1997), American Mathematical Society Providence, RI, with an appendix by M. Cole · Zbl 0894.55001
[11] Eilenberg, Samuel; Moore, J. C., Foundations of relative homological algebra, Mem. Am. Math. Soc., 55, 39, (1965) · Zbl 0129.01101
[12] Goerss, P. G.; Hopkins, M. J., Moduli spaces of commutative ring spectra, (Structured Ring Spectra, London Math. Soc. Lecture Note Ser., vol. 315, (2004), Cambridge Univ. Press Cambridge), 151-200 · Zbl 1086.55006
[13] Goerss, P.; Henn, H.-W.; Mahowald, M.; Rezk, C., A resolution of the \(K(2)\)-local sphere at the prime 3, Ann. of Math. (2), 162, 2, 777-822, (2005) · Zbl 1108.55009
[14] Husemoller, Dale; Moore, John C., Differential graded homological algebra of several variables, (Symposia Mathematica, Vol. IV, INDAM, Rome, 1968/69, (1970), Academic Press London), 397-429
[15] Hopkins, Michael J.; Mahowald, Mark; Sadofsky, Hal, Constructions of elements in Picard groups, (Topology and Representation Theory, Evanston, IL, 1992, Contemp. Math., vol. 158, (1994), Amer. Math. Soc. Providence, RI), 89-126 · Zbl 0799.55005
[16] Hovey, Mark, Bousfield localisation functors and hopkins’ chromatic splitting conjecture, (The Čech Centennial, Boston, MA, 1993, Contemp. Math., vol. 181, (1995), Amer. Math. Soc. Providence, RI), 225-250 · Zbl 0830.55004
[17] Hovey, Mark, Operations and co-operations in Morava E-theory, Homol. Homotopy Appl., 6, 1, 201-236, (2004) · Zbl 1063.55003
[18] Hovey, Mark, Some spectral sequences in Morava E-theory, available at: · Zbl 1063.55003
[19] Hovey, Mark, Morava E-theory of filtered colimits, Trans. Am. Math. Soc., 360, 1, 369-382, (2008), (electronic) · Zbl 1128.55005
[20] Hovey, Mark; Sadofsky, Hal, Invertible spectra in the \(E(n)\)-local stable homotopy category, J. Lond. Math. Soc. (2), 60, 1, 284-302, (1999) · Zbl 0947.55013
[21] Hovey, Mark; Strickland, Neil P., Morava K-theories and localisation, Mem. Am. Math. Soc., 139, (1999), viii+100 · Zbl 0929.55010
[22] Hovey, Mark; Strickland, Neil, Comodules and Landweber exact homology theories, Adv. Math., 192, 2, 427-456, (2005) · Zbl 1075.55001
[23] Lam, T. Y., Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, (1999), Springer-Verlag New York · Zbl 0911.16001
[24] Lawson, Tyler; Naumann, Niko, Commutativity conditions for truncated Brown-Peterson spectra of height 2, J. Topol., 5, 1, 137-168, (2012) · Zbl 1280.55007
[25] Miller, Haynes R., On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra, 20, 3, 287-312, (1981) · Zbl 0459.55012
[26] Morava, Jack, Noetherian localisations of categories of cobordism comodules, Ann. of Math. (2), 121, 1, 1-39, (1985) · Zbl 0572.55005
[27] Miller, Haynes R.; Ravenel, Douglas C., Morava stabilizer algebras and the localisation of Novikov’s \(E_2\)-term, Duke Math. J., 44, 2, 433-447, (1977) · Zbl 0358.55019
[28] Palmieri, John H., Stable homotopy over the Steenrod algebra, Mem. Am. Math. Soc., 151, (2001), xiv+172 · Zbl 0966.55013
[29] Quick, Gereon, Continuous homotopy fixed points for Lubin-Tate spectra, Homol. Homotopy Appl., 15, 1, 191-222, (2013) · Zbl 1278.55018
[30] Ravenel, Douglas C., Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, (1986), Academic Press Inc. Orlando, FL · Zbl 0608.55001
[31] Rezk, Charles, Notes on the hopkins-Miller theorem, (Homotopy Theory via Algebraic Geometry and Group Representations, Evanston, IL, 1997, Contemp. Math., vol. 220, (1998), Amer. Math. Soc. Providence, RI), 313-366 · Zbl 0910.55004
[32] Charles Rezk, Analytic completion, draft available at: http://www.math.uiuc.edu/ rezk/papers.html.
[33] Shimomura, Katsumi; Yabe, Atsuko, The homotopy groups \(\pi_\ast(L_2 S^0)\), Topology, 34, 2, 261-289, (1995) · Zbl 0832.55011
[34] Tate, John, Relations between \(K_2\) and Galois cohomology, Invent. Math., 36, 257-274, (1976) · Zbl 0359.12011
[35] Valenzuela, Gabriel, Homological algebra of complete and torsion modules, (2015), Wesleyan University, PhD thesis
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.