higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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.
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.
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).
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:
The basic definitions are in
Fundamental properties of $E_\infty$-geometry are discussed in
Jacob Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems
Jacob Lurie, Proper Morphisms, Completions, and the Grothendieck Existence Theorem
See also the overview