In the mathematical field of
algebraic topology, a commutative ring spectrum, roughly equivalent to a
-ring spectrum, is a
commutative monoid in a good
[symmetric 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
of rational numbers is
Quillen equivalent to the category of
differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
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Definition
A differential graded alg ...
s over
.
Example: The
Witten genus may be realized as a
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
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
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
, the term "commutative ring spectrum" may be used as a synonymous to an
-ring spectrum.
Notes
References
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{{topology-stub
Algebraic topology