nLab E-∞ geometry



Higher geometry

Higher algebra

Arithmetic geometry



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.


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.


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).


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 } \,,



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.

Last revised on July 26, 2023 at 13:11:29. See the history of this page for a list of all contributions to it.