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Spec(S)

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Idea

Spec(𝕊)Spec(\mathbb{S}) is the spectrum, in the sense of spectrum of a commutative ring in E-∞ geometry, of the sphere spectrum 𝕊\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 EE any other E-∞ ring, then the essentially unique E E_\infty-ring homomorphism 𝕊E\mathbb{S} \longrightarrow E (the unit, exhibiting EE as an E-∞ algebra over 𝕊\mathbb{S}) corresponds to an essentially unique morphism

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

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

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

Or for instance for E=E = MU, then the 1-image of Spec(MU)Spec(𝕊)Spec(MU) \to Spec(\mathbb{S}) is already all of Spec(S)Spec(S), hence Spec(MU)Spec(MU) is covering. The tool that computes 𝕊\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.

Last revised on September 11, 2018 at 05:48:09. See the history of this page for a list of all contributions to it.