commutative ring spectrum

In algebraic topology, a commutative ring spectrum, roughly equivalent to a $E_\infty$ -ring spectrum, is a commutative monoid in a goodsymmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of symmetric spectra 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.

Terminology

Almost all reasonable categories of commutative ring spectra can be shown to be Quillen equivalent to each other.{{fact|date=July 2024}} Thus, from the point view of the stable homotopy theory, the term "commutative ring spectrum" may be used as a synonymous to an $E_\infty$ -ring spectrum.

Notes

References

{{cite book |first=P. |last=Goerss |chapter-url=http://www.math.northwestern.edu/~pgoerss/papers/Exp.1005.P.Goerss.pdf |chapter=1005 Topological Modular Forms [after Hopkins, Miller, and Lurie] |title=Séminaire Bourbaki : volume 2008/2009, exposés 997–1011 |publisher=Société mathématique de France |year=2010}}
{{cite arXiv |first=J.P. |last=May |title=What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra? |year=2009 |arxiv=0903.2813}}

Category:Algebraic topology