# nLab E-∞ geometry

## Theorems

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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

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

## References

The basic definitions are in

Fundamental properties of $E_\infty$-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 February 5, 2016 at 06:54:11. See the history of this page for a list of all contributions to it.