Étale spectrum

In mathematics, specifically algebraic geometry, the étale spectrum of a commutative ring or of an E_∞-ring, denoted by Spec^ét or Spét, is an analog of the prime spectrum Spec of a commutative ring that is obtained by replacing the Zariski topology with the étale topology.

The usual prime spectrum Spec enjoys the following relation: for a scheme (S, O_S) and a commutative ring A,

$\operatorname {Hom} (S,\operatorname {Spec} (A))\simeq \operatorname {Hom} (A,\Gamma (S,{\mathcal {O}}_{S}))$,

where Hom on the left is for morphisms of schemes and Hom on the right ring homomorphisms. That is, Spec is the right adjoint to the global section functor $(S,{\mathcal {O}}_{S})\mapsto \Gamma (S,{\mathcal {O}}_{S})$. So, roughly, one can (and typically does) simply define the étale spectrum Spét to be the right adjoint to the global section functor on the category of "spaces" with étale topology.

Over a field of characteristic zero, Behrend constructs the étale spectrum of a graded algebra called a perfect resolving algebra. He then defines a differential graded scheme (a type of a derived scheme) as one that is étale-locally such an étale spectrum.

The notion makes sense in the usual algebraic geometry but appears more frequently in the context of derived algebraic geometry.

References

• Behrend, Kai (2002). "Differential Graded Schemes II: The 2-category of Differential Graded Schemes". arXiv:math/0212226.

• Lurie, Jacob (2018). "Spectral Algebraic Geometry (under construction)" (PDF).