nLab spectral affine line

Idea

The concept of spectral affine line is the generalization of the concept of affine line to spectral geometry, in the sense of E-∞ geometry.

Given a (connective) E-∞ ring $R$, then the spectral affine line over $R$ is the affine spectral scheme given by the spectral symmetric algebra on a single generator:

$$\mathbb{A}^1_R = Spec( Sym_R(R) ) \,.$$

(e.g. Lurie Schemes, below prop. 2.20)

For $R = \mathbb{S}$ the sphere spectrum, then this may be called the absolute spectral affine line. This is discussed in Strickland-Turner 97.

References

Some general comments are in

• Jacob Lurie, below prop. 2.20 Spectral Schemes.

The absolute spectral affine line over $R = \mathbb{S}$ the sphere spectrum is discussed in

• Neil Strickland, Paul Turner, Rational Morava $E$-theory and $D S^0$, Topology Volume 36, Issue 1, January 1997, Pages 137-151 (pdf)