Mathematical Research Letters

Volume 1 (1994)

Number 2

Commutative algebra in stable homotopy theory and a completion theorem

Pages: 225 – 239



A. D. Elmendorf

P. C. Greenlees

I. Kriz

J. P. May


We construct a category of spectra that has all limits and colimits and also has a strictly associative and commutative smash product. This provides the ground category for a new theory of structured ring and module spectra that allows the wholesale importation of techniques of commutative algebra into stable homotopy theory. Applications include new constructions of basic spectra, new generalized universal coefficient and Künneth spectral sequences, and a new construction of topological Hochschild homology. The theory works equivariantly, where it allows the construction of equivariant versions of Brown-Peterson, Morava $K$-theory, and other module spectra over $MU$. Via a topological realization of “local homology and cohomology groups”, the general theory leads to a completion theorem for the computation of $M_{*}(BG)$ and $M^{*}(BG)$ in terms of equivariant cobordism groups, where $M$ is $MU$, $BP$, $k(n)$, $K(n)$, or any other module spectrum over $MU$. (The reader most interested in the equivariant applications may wish to read the last section first.)

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