Commutative ring spectrum

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

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

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.Template:Fact Thus, from the point view of the stable homotopy theory, the term "commutative ring spectrum" may be used as a synonymous to an $E_{\mathrm{\infty }}$-ring spectrum.

References

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