The \(E_{2}\)-term of the \(K(n)\)-local \(E_{n}\)-Adams spectral sequence.

*(English)*Zbl 1348.55008In 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.

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.

Reviewer: Lennart Meier (Bonn)

##### MSC:

55P60 | Localization and completion in homotopy theory |

55Q10 | Stable homotopy groups |

13J10 | Complete rings, completion |

##### Keywords:

\(K(n)\)-local homotopy theory; Morava \(E\)-theory; Adams spectral sequence; \(L\)-complete comodules##### References:

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