nLab
connective cover

Contents

Contents

Idea

For plain spectra

Connective spectra form a coreflective sub-(∞,1)-category of the (∞,1)-category of spectra. The right adjoint (∞,1)-functor from spectra to connective spectra EE0E \mapsto E \langle 0 \rangle is called the connective cover construction. This comes with the coreflection morphism of spectra

(1)E0ϵ EE E\langle 0 \rangle \overset{ \epsilon_E }{\longrightarrow} E

which induces an isomorphism on homotopy groups of spectra in non-negative degrees:

π 0(E0)π 0(ϵ E)π 0(E). \pi_{\bullet \geq 0}\big( E\langle 0\rangle\big) \underoverset {\simeq} { \pi_{\bullet \geq 0} (\epsilon_E) }{\longrightarrow} \pi_{\bullet \geq 0}(E) \,.

For ring spectra

The connective cover functor extends from plain spectra to E-∞ ring spectra (May 77, Prop. VII 4.3, Lurie, Prop. 7.1.3.13), such that the coreflection E0EE \langle0\rangle \longrightarrow E (1) is a homomorphism of E-∞ rings.

Besides a canonically inherited ring structure, the connective cover may sometimes carry further ring structures, but in many examples of interest it is unique (Baker-Richter 05).

Examples

  • ku is the connective cover of KU

References

For plain spectra:

For ring spectra:

Last revised on January 20, 2021 at 04:55:25. See the history of this page for a list of all contributions to it.