nLab
E-∞ geometry

Context

Higher geometry

Higher algebra

Arithmetic geometry

Contents

Idea

What may be called E 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 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 E_\infty-geometry.

Terminology

The term spectral algebraic geometry is used in the literature. On the nLab we use the term E 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 E_\infty-arithmetic geometry of the power map () p(-)^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 modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃʃtorsion approximation
dR-shape modalityʃ dRʃ_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \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

References

The basic definitions are in

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

See also the overview

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

Revised on February 5, 2016 06:54:11 by Urs Schreiber (86.187.77.97)