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derived smooth geometry
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symmetric monoidal (∞,1)-category of spectra
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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
in :
symbol | interpretation | |
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$\flat$ | at | |
$ʃ$ | ||
$ʃ_{dR}$ | away | |
$\flat_{dR}$ |
the /:
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
Last revised on February 5, 2016 at 06:54:11. See the history of this page for a list of all contributions to it.