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Brown-Peterson cohomology from Morava \(E\)-theory. (English) Zbl 1373.55002
Let \(P(m)\) denote the \(BP\)-module spectrum with coefficients \(P(m)^* = BP^*/I_m\), where each \(I_m = (p, v_1, \ldots , v_{m-1})\) is an invariant prime ideal in \(BP^*\). The invariant prime ideals of \(P(m)^*\) are then \(I_{m,n} = (v_m, \ldots , v_{n-1})\) (for \(n \geq m\)). Denoting Morava \(K\)-theory by \(K(m)\) (for each \(m>0\)), we define \({\hat E} (m,n)\) as the \(K(n)\)-localization of the spectra \(E(m,n)\) that are constructed (for \(n \geq m\)) from the \(P(m)\) by modding out the ideal \((v_{n+1}, v_{n+2}, \ldots)\) and inverting \(v_n\).
The main result in this work appears as Corollary 3.4, stating that the diagonal \(P(m) \rightarrow \prod_{n \in I} {\hat E}(m,n)\) splits for any infinite set \(I\) of integers greater than \(m\). The most relevant particular case is \(m=0\): the \(p\)-completion of \(BP\) is a retract of the product over all Morava \(E_n\)-theories \(\prod_{n>0}E_n\). The corollary comes as a result of Theorem 3.2, stating that a morphism \(f : P(m) \rightarrow D\), for \(D\) a \(p\)-complete Landweber flat \(P(m)\)-module, must be a split inclusion of spectra whenever the induced maps \(P(m)_*/I_{m,n} \rightarrow D_*/I_{m,n}\) are injective for all \(n>m\). This theorem is proved by extending a result of Hovey using Brown-Comenetz duality, a lift of Pontryagin duality for abelian groups to the category of spectra. From Hovey’s result and further work with Strickland, one could get a splitting \(B_p \rightarrow \prod_{n>0} L_{K(n)} \Big( \bigvee_{r \in S(n)} \Sigma^r L_{K(n)}E(n) \Big)\) (the \(S(n)\) are some indexing sets of even integers), but the splitting from Corollary 3.4 here is much smaller (the authors claim it is as small as possible with the techniques at hand) and thus more convenient for the applied results that follow.
The splitting from Corollary 3.4 allows for the generalization of some structural theorems of Ravenel-Wilson-Yagita from spaces to spectra. This starts in Theorem 3.9: For \(X\) a spectrum, if \(K(n)^*(X)\) is even for infinitely many \(n\), then \(P(m)^*(X)\) is even and Landweber flat for all \(m\). Further results lifting information on the Morava \(K\)-theory of spectra and maps of spectra to their \(P(m)\)-theory are obtained, mainly in Theorem 3.15. The first part of the paper concludes with an application to the free commutative algebra spectrum \(\mathbb{P}X\) on a spectrum \(X\), giving conditions on \(X\) that guarantee that \(P(m)^*(\mathbb{P}X)\) is Landweber flat (Proposition 3.20); for a large class of spaces \(X\), Theorem 3.18 shows that \(BP_p^*(\mathbb{P}X)\) is functorially determined by \(BP_p^*(X)\).
The second part of the paper deals with the rationalization \(\mathbb{Q} \otimes BP^*_p(X) \rightarrow \mathbb{Q} \otimes \prod_{n>0} E_n^*(X)\) of the above retract for \(m=0\), whenever \(X\) is the classifying space \(BA\) of a finite abelian group \(A\). Whenever \(X=BA\) or \(X=B\Sigma_m\), the retract result is used to lift factorizations of \(\mathbb{Q} \otimes \prod_{n>0} E_n^*(X)\) in order to get factorizations of \(\mathbb{Q} \otimes BP_p^*(X)\) (Theorems 4.17 and 4.18). These results are shown to be of bounded torsion, independently of the height \(n\) that is used. A rational isomorphism \(\prod_n E_n^*(BA) \rightarrow \prod_n \prod_{H \subseteq A} E^*_n(BH)/I\) is achieved, where \(H\) runs through the subgroups of \(A\) and \(I\) is the transfer ideal (Corollary 4.20), which extends to a split inclusion \(BP^*(BG)/I \rightarrow \prod_{n>0} E^*_n(BG)/I\), for \(G\) a finite group, that is compatible with the one in Corollary 3.4 (Lemma 4.21).

MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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