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is the spectrum, in the sense of spectrum of a commutative ring in E-∞ geometry, of the sphere spectrum , 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 any other E-∞ ring, then the essentially unique -ring homomorphism (the unit, exhibiting as an E-∞ algebra over ) corresponds to an essentially unique morphism
The 1-image of such a morphism is known as (the formal dual of) the -nilpotent completion of .
For instance for H, then the 1-image of is , the formal dual of the p-completion of , hence the infinitesimal neighbourhood of in . The tool that computes (hence the -primary stable homotopy groups of spheres) by regarding it this way is the original -Adams spectral sequence.
Or for instance for MU, then the 1-image of is already all of , hence is covering. The tool that computes , 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 09:48:09. See the history of this page for a list of all contributions to it.