Ring spectrum

From Wikipedia, the free encyclopedia
Jump to: navigation, search
For the concept of spectrum of a ring in algebraic geometry, see spectrum of a ring.

In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map


and a unit map


where S is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy much in the same way as the multiplication of a ring is associative and unital. That is,

μ (id ∧ μ) ∼ μ (μ ∧ id)


μ (id ∧ η) ∼ id ∼ μ(η ∧ id).

Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory.

See also[edit]


  • Adams, J. Frank (1974), Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-00523-2, MR 402720