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Completed power operations for Morava \(E\)-theory. (English) Zbl 1326.55018
The authors consider a completed version of the algebraic approximation functors which is due to C. Rezk [Homology Homotopy Appl. 11, No. 2, 327–379 (2009; Zbl 1193.55010)] by taking completions into account. By Morava \(E\)-theory \(E=E_h\) is meant a commutative ring spectrum with coefficients \[ E_*=W\mathbb{F}_{p^h}[[u_1, \cdots, u_{h-1}]][u^\pm] \] for \(u\) of degree 2 where \(W\mathbb{F}_{p^h}\) is the ring of Witt vectors on \(\mathbb{F}_{p^h}\). Let \(\text{Mod}_E\) denote the category of \(E\)-modules. The functor \(\mathbb{P} : \text{Mod}_E \to \text{Mod}_E\) is given by \(\mathbb{P}(M)=\bigvee_{n\geq 0}\mathbb{P}_n(M)=\bigvee_{n\geq 0} (E\Sigma_n)_+\wedge_{\Sigma_n} M^{\wedge n}\), which defines a monad on its homotopy category \(h\,\text{Mod}_E\). Let \(L_K\) be the Bousfield localization with respect to Morava \(K\)-theory \(K(h)\) and \(j : \text{id} \to L_K\) be the localization functor. Let \(\pi_*\) denote the natural functor \(h\, \text{Mod}_E \to \text{Mod}_{E_*}\). Then \(\pi_*j\) can be uniquely factorized into the composite \[ \pi_{*} @>{\eta}>> L_0\pi_* \to \pi_*L_K, \] \(L_0\) denoting the 0th derived functor. The authors construct a new version of the algebraic approximation functors \(\mathbb{T}_n : \text{Mod}_{E_*} \to \text{Mod}_{E_*}\), which is defined as the Kan extension of \(\pi_*i\) along \(\pi_*L_K\mathbb{P}_ni\) where \(i\) denotes the natural functor \(\text{Mod}_E \to h\, \text{Mod}_E\). Using these functors the main theorem of this paper can be stated as follows: The natural map \(L_0\mathbb{T}\eta\) gives an isomorphism \[ L_0\mathbb{T}(M) \cong L_0\mathbb{T}L_0(M) \] for all \(E_*\)-modules \(M\) where \(\mathbb{T}=\bigoplus_{n \geq 0}\mathbb{T}_n\). Write \(\widehat{\text{Mod}}_{E_*}\) for the subcategory of \(\text{Mod}_{E_*}\) consisting of \(L\)-complete \(E_*\)-modules and denote by \(\iota\) its inclusion, wherein \(M\) is called \(L\)-complete if \(\eta : M \to L_0M\) is an isomorphism. Then this result tells us that the completed algebraic approximation functor \(\widehat{\mathbb{T}} : \widehat{\text{Mod}}_{E_*} \to \widehat{\text{Mod}}_{E_*}\), which can be written as \(\widehat{\mathbb{T}}=L_0\mathbb{T}\iota\), admits a natural monad structure being compatible with that of \(L_K\mathbb{P}\). The proof of this uses the fact that if \(M\) is a flat \(E\)-module then there are natural isomorphisms \(\widehat{\mathbb{T}}_n\pi_*M \cong \pi_*L_K\mathbb{P}_nM\), which exhibits an advantage of using \(\widehat{\mathbb{T}}\). The authors say that “the completed algebraic approximation functor \(\widehat{\mathbb{T}}_n\) resembles the structure on the homotopy groups of \(K(h)\)-local commutative \(E\)-algebras more closely than \(\mathbb{T}_n\)”. The second part of this paper discusses the case of \(E_1=K^\wedge_p\) from the point of view of making the usefulness of \(\widehat{\mathbb{T}}\) explicit and consequently obtains the following result: In the case \(h=1\), \(\mathbb{T} : \text{Mod}_{E_*} \to \text{Mod}_{E_*}\) becomes the free \(\mathbb{Z}/2\)-graded \(\theta\)-ring over the ground \(\theta\)-ring \(\mathbb{Z}_p\).

55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
55S12 Dyer-Lashof operations
13B35 Completion of commutative rings
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