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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.

##### MSC:
 55P60 Localization and completion in homotopy theory 55Q10 Stable homotopy groups 13J10 Complete rings, completion
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