96j:55011 55P60

Greenlees, J. P. C. ^{(}4-SHEF^{)}; May, J. P. ^{(}1-CHI^{)} Completions in algebra and topology. Handbook of algebraic topology, 255–276, North-Holland, Amsterdam, 1995. Localization and completion play a fundamental role in algebraic topology. A common general framework starts from ideals in the coefficient ring of a generalized cohomology theory. The authors explain some old and new algebra that arises in this context and they show how this algebra can be mimicked topologically. The topological constructions require the foundations described in the previous chapter of this book, which deals with the algebraically familiar theory of localization at multiplicatively closed subsets. The authors here explain the deeper and less familiar theory of completion, together with an ideal-theoretic variant of localization. The algebraic fact that completion is not exact in general forces topologists to work with the derived functors of completion. The authors explain how topological completions of spectra mimic algebraic descriptions of these derived functors in terms of local homology groups. These constructs are designed for the study of cohomology theories. They describe dual constructs that are designed for the study of homology theories and involve Grothendieck’s local cohomology groups. There are concomitant notions of Cech homology and cohomology groups which fit into algebraic fibre sequences that they mimic by interesting fibre sequences of spectra. These lead to a theory of localizations of spectra away from ideals. When specialized to MU-module spectra, these new localizations shed considerable light on the chromatic filtration that is at the heart of the study of periodic phenomena in stable homotopy theory, which is the subject of the next chapter of this book.

{For the entire collection see MR 96g:55002}.

Jean-Claude Thomas (F-LILL)

\begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 1996, 1997