nLab Spec(S)

Idea

$Spec(\mathbb{S})$ is the spectrum, in the sense of spectrum of a commutative ring in E-∞ geometry, of the sphere spectrum $\mathbb{S}$, in the sense of stable homotopy theory, regarded canonically as the initial E-∞ ring.

This is the refinement to E-∞ arithmetic geometry of what Spec(Z) is in arithmetic geometry.

It is the absolute base space, in that it is the terminal object among E-∞ schemes. Notably for $E$ any other E-∞ ring, then the essentially unique $E_\infty$-ring homomorphism $\mathbb{S} \longrightarrow E$ (the unit, exhibiting $E$ as an E-∞ algebra over $\mathbb{S}$) corresponds to an essentially unique morphism

$$Spec(E) \longrightarrow Spec(\mathbb{S}) \,.$$

The 1-image of such a morphism is known as (the formal dual of) the $E$-nilpotent completion of $\mathbb{S}$.

For instance for $E =$ H$\mathbb{F}_p$, then the 1-image of $Spec(H\mathbb{F}_p) \to Spec(\mathbb{S})$ is $Spec(\mathbb{S}_{(p)})$, the formal dual of the p-completion of $\mathbb{S}$, hence the infinitesimal neighbourhood of $(p)$ in $Spec(S)$. The tool that computes $\mathbb{S}_{(p)}$ (hence the $p$-primary stable homotopy groups of spheres) by regarding it this way is the original $\mathbb{F}_p$-Adams spectral sequence.

Or for instance for $E =$ MU, then the 1-image of $Spec(MU) \to Spec(\mathbb{S})$ is already all of $Spec(S)$, hence $Spec(MU)$ is covering. The tool that computes $\mathbb{S}$, hence the full stable homotopy groups of spheres, using this covering is the MU-Adams spectral sequence, hence the Adams-Novikov spectral sequence.

See at Adams spectral sequence – As derived descent for more on this.

Related concepts

• spectrum of the category of spectra

• Spec(Z), field with one element

• moduli stack of formal groups

• chromatic homotopy theory

• The Music of the Spheres