Commutative ring spectrum

In algebraic topology, a commutative ring spectrum, roughly equivalent to a ${\displaystyle E_{\infty }}$-ring spectrum, is a commutative monoid in a good^[1] category of spectra.

The category of commutative ring spectra over the field ${\displaystyle \mathbb {Q} }$ of rational numbers is Quillen equivalent to the category of differential graded algebras over ${\displaystyle \mathbb {Q} }$.

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

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

Terminology

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 ${\displaystyle E_{\infty }}$-ring spectrum.

Notes

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

References

• 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 ${\displaystyle E_{\infty }}$ ring spaces and ${\displaystyle E_{\infty }}$ ring spectra?". arXiv:0903.2813.