×

Noetherian localisations of categories of cobordism comodules. (English) Zbl 0572.55005

The author presents, in polished form, his results on \(MU_*MU\)- comodules. This work began in the early 1970’s and has already enjoyed a wide influence, most notably in the work by H. Miller, D. C. Ravenel and W. Wilson on periodic phenomena in the Adams-Novikov spectral sequence [Ann. Math. 106, 469-516 (1977; Zbl 0374.55022)] and by D. C. Ravenel on localization with respect to periodic homology theories obtained from MU and BP [Am. J. Math. 106, 351-414 (1984)]. In turn, the formulation offered here benifits from the studies that have been made by D. Johnson, Yosimura and the reviewer, in addition to the works just mentioned.
It is no easy task to describe the main results of this paper. The fine structure of the category \({\mathcal C}\) of \(MU_*MU\)-comodules is the main object of study. One can consult the reviewer’s papers [ibid. 98, 591-610 (1976; Zbl 0355.55007); Lect. Notes Math. 741, 449-460 (1979; Zbl 0482.55010)] for the main algebraic properties of this category, but the author is aiming for deeper results. Letting \((v_ n\)-torsion) denote the full subcategory of \({\mathcal C}\) generated by \(v_ n\)-torsion comodules, one obtains a decreasing filtration of \({\mathcal C}\) and so also quotient categories \((v_{n-1}\)-torsion) denoted by \(\hat {\mathcal C}(n)\). In turn, \(\hat {\mathcal C}(n)\) is extended to a wider category \({\mathcal C}(n)\), and the main results concern the concise description of the category of torsion objects in \({\mathcal C}(n).\)
The purpose of all this is to reduce the homological algebra one meets in dealing with the Adams-Novikov spectral sequence to group cohomology, the latter being the topic in section 2. Formal groups play an essential role in carrying out this plan; what is needed is presented in section 1. One won’t find any topology here; the tools are algebraic. One also won’t find mention of the Morava K-theories \(K(n)_*(X)\), which derive from this study and play a central role in current work on the fine structure of the stable homotopy category initiated by Ravenel in the paper mentioned above.
Reviewer: P.Landweber

MSC:

55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R77 Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism)
14L05 Formal groups, \(p\)-divisible groups
55P42 Stable homotopy theory, spectra
PDFBibTeX XMLCite
Full Text: DOI