nLab
Bousfield equivalence

Contents

Context

Equality and Equivalence

Stable Homotopy theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

Two generalized homology theories E E_\bullet, F F_\bullet, hence spectra EE, FF are called Bousfield equivalent if the homology groups of both always vanish simultaneously, hence if for every homotopy type/spectrum XX we have E (X)0E_\bullet(X) \simeq 0 precisely if F (X)0F_\bullet(X) \simeq 0.

Examples

Global arithmetic fracture square

There is a Bousfield equivalence

S(/) Bousf pS𝔽 p S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p

between the Moore spectrum of the quotient /\mathbb{Q}/\mathbb{Z} and the coproduct of the Moore spectra of all cyclic groups/finite fields of prime order (e.g. Strickland 12, MO comment).

This goversn the global arithmetic fracture theorem in stable homotopy theory.

Morava E-theory and Morava K-theory

For all nn, the nnth Morava E-theory E(n)E(n) is Bousfield equivalence to E(n1)×K(n)E(n-1) \times K(n), where the last factor is nnth Morava K-theory.

(Lurie 10, lect. 23, prop. 1)

References

The concept of Bousfield classes is due to (see at Bousfield localization of spectra)

  • Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

and was named such in

  • Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (pdf)

Discussion in the context of higher algebra is in

Last revised on February 2, 2016 at 03:54:45. See the history of this page for a list of all contributions to it.