Commutative ring spectrum

In the mathematical field of 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. 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

• P. Goerss, Topological Modular Forms [after Hopkins, Miller, and Lurie]
• J.P. May, What precisely are ${\displaystyle E_{\infty }}$ ring spaces and ${\displaystyle E_{\infty }}$ ring spectra? arXiv:0903.2813