An exact sequence related to Adams-Novikov \(E_ 2\)-terms of a cofibering
type: Item
instance of: scholarly article
title: An exact sequence related to Adams-Novikov \(E_ 2\)-terms of a cofibering
zbMATH Open document ID: 0817.55004
author: Mizuho Hikida
author: Katsumi Shimomura
published in: Journal of the Mathematical Society of Japan
publication date: 1995-07-31T00:00:00Z
review text: Let \(L_ n X\) denote the Bousfield localization of a spectrum \(X\) with respect to the Johnson-Wilson spectrum \(E(n)\), and \(\eta_ X: X\to L_ n X\), the localization. Then \textit{D. C. Ravenel} gives the cofibration \(\Sigma^{-n-1} N_{n+1} X\to X@> \eta X>> L_ n X\) in [Am. J. Math. 106, 351-414 (1984; Zbl 0586.55003), Th. 5.10]. Take \(X\) to be the Toda- Smith spectrum \(V(j-1)\). Then the Brown-Peterson homology \(BP_ * (N_{n+1} X)\) of \(N_{n+1} X\) is \(N_ j^{n+1}\) of \textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson}'s sense [Ann Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)]. The \(E_ 2\)-terms of the Adams-Novikov spectral sequences converging to \(\pi_ * (N_{n+1} X)\), \(\pi_ * (X)\) and \(\pi_ * (L_ n X)\) are \(\text{Ext}_ \Gamma (A, N_ j^{n+1-j})\), \(\text{Ext}_ \Gamma (A, A/I_ j)\) and \(\text{Ext}_ \Sigma (B, B/I_ j)\), respectively, for the Hopf algebroids \((A, \Gamma)= (BP_ *, BP_ * (BP))\) and \((B, \Sigma)= (E(n)_ *, E(n)_ * (E(n)))\), and for the ideal \(I_ j=(p, v_ 1,\dots, v_{j-1})\). One of the results is the long exact sequence \[ \begin{multlined} \cdots\to\text{Ext}_ \Gamma^ t (A,N_ j^{n+1-j}) \overset \eta{\rightarrow} \text{Ext}_ \Gamma^{t+ n+1} (A,A/I_ j) \overset(\eta_ X)_ {*} {\longrightarrow} \text{Ext}_{\Sigma'}^{t+n +1} (B, B/I_ j)\\ \text{Ext}_ \Gamma^{t+1} (A, N_ j^{n+ 1-j}{)} \cdots ,\end{multlined} \] where \(\eta\) denotes the universal Greek letter map and \((\eta_ X )_ *\), the induced map from the localization. Here the universal Greek letter map is introduced by Miller, Ravenel and Wilson [loc. cit.]. This follows from the main result which claims the long exact sequence involving the universal Greek letter map and is proved algebraically. The result on \(V(j)\) is generalized to spectra \(W_ k(i)\) characterized by \(BP_ * (W_ k (i))= BP_ */ I_{i+1} [t_ 1, \dots, t_ k]\).
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55T25
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 758150
zbMATH Keywords: universal Greek letter map
zbMATH Keywords: Bousfield localization
zbMATH Keywords: Johnson-Wilson spectrum
zbMATH Keywords: Toda-Smith spectrum
zbMATH Keywords: Brown-Peterson homology
zbMATH Keywords: Adams-Novikov spectral sequences
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04640645
OpenAlex ID: W2029981053
DOI: 10.2969/JMSJ/04640645
description: scientific article; zbMATH DE number 758150