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Commutative ring spectrum

From Wikipedia, the free encyclopedia

In algebraic topology, a commutative ring spectrum, roughly equivalent to a -ring spectrum, is a commutative monoid in a good[1] category of spectra.

The category of commutative ring spectra over the field of rational numbers is Quillen equivalent to the category of differential graded algebras over .

Example: The Witten genus may be realized as a morphism of commutative ring spectra MStringtmf.

See also: simplicial commutative ring, highly structured ring spectrum and derived scheme.

Terminology

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Almost all reasonable categories of commutative ring spectra can be shown to be Quillen equivalent to each other.[citation needed] Thus, from the point view of the stable homotopy theory, the term "commutative ring spectrum" may be used as a synonymous to an -ring spectrum.

Notes

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  1. ^ symmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of symmetric spectra

References

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  • Goerss, P. (2010). "1005 Topological Modular Forms [after Hopkins, Miller, and Lurie]" (PDF). Séminaire Bourbaki : volume 2008/2009, exposés 997–1011. Société mathématique de France.
  • May, J.P. (2009). "What precisely are ring spaces and ring spectra?". arXiv:0903.2813.