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.