Commutative ring spectrum

In algebraic topology, a commutative ring spectrum, roughly equivalent to a $E_{\infty }$-ring spectrum, is a commutative monoid in a good category of spectra. For example, the sphere spectrum is a commutative ring spectrum.

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

References

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