nLab Bousfield equivalence

Definition

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

Examples

Global arithmetic fracture square

There is a Bousfield equivalence

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

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 $n$, the $n$th Morava E-theory $E(n)$ is Bousfield equivalence to $E(n-1) \times K(n)$, where the last factor is $n$th 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

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010; Lecture 23 The Bousfield Classes of $E(n)$ and $K(n)$ (pdf)