nLab E-∞ geometry

Idea

What may be called $E_\infty$-geometry, or spectral algebraic geometry is the “full” version of derived algebraic geometry where the spaces are locally modelled on E-infinity rings (as opposed to simplicial commutative rings or dg-algebras).

(In view of existing terms “arithmetic geometry” and “higher geometry” one might also tend call this higher arithmetic geometry, but notice that the term higher arithmetic geometry already has a different traditional usage. Maybe $E_\infty$-arithmetic geometry works well.)

That is to say, it is the higher geometry in the (∞,1)-topos over the (∞,1)-site of formal duals of E-∞ rings, equipped with the etale topology.

An E-∞ scheme ( spectral scheme ) in this sense is a structured (∞,1)-topos in $E_\infty$-geometry.

Terminology

The term spectral algebraic geometry is used in the literature. On the nLab we use the term $E_\infty$-geometry due to a possible terminology clash discussed at spectral geometry.

Properties

Higher analogs of Fermat derivations and Frobenius maps

The refinement to $E_\infty$-arithmetic geometry of the power map $(-)^p$ which gives the Fermat quotient p-derivation and the Frobenius homomorphism in plain arithmetic geometry is the power operations on multiplicative cohomology theory (Lurie “Adic”, remark 2.2.9).

Cohesion

see at differential cohesion and idelic structure

cohesion in E-∞ arithmetic geometry:

cohesion modality	symbol	interpretation
flat modality	$\flat$	formal completion at
shape modality	$ʃ$	torsion approximation
dR-shape modality	$ʃ_{dR}$	localization away
dR-flat modality	$\flat_{dR}$	adic residual

the differential cohomology hexagon/arithmetic fracture squares:

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

Examples

• Spec(S)

References

The basic definitions are in

• Jacob Lurie, Spectral Schemes

Fundamental properties of $E_\infty$-geometry are discussed in

• Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems

• Jacob Lurie, Closed Immersions

• Jacob Lurie, Descent Theorems

• Jacob Lurie, Proper Morphisms, Completions, and the Grothendieck Existence Theorem

• Jacob Lurie, Representability Theorems

• Jacob Lurie, Rational and p-adic Homotopy Theory

• Jacob Lurie, Spectral Algebraic Geometry (pdf)

See also the overview

• Clark Barwick, Applications of derived algebraic geometry to homotopy theory, lecture notes, mini-course in Salamanca, 2009, pdf.