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,

$\mathrm{Hom}(S,\mathrm{Spec}(A))\simeq \mathrm{Hom}(A,\mathrm{\Gamma }(S,{𝒪}_S))$,

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 $(S,{𝒪}_S)\mapsto \mathrm{\Gamma }(S,{𝒪}_S)$. 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.Template:SfnpTemplate:Sfnp

Over a field of characteristic zero, Behrend constructs the étale spectrum of a graded algebra called a perfect resolving algebra.Template:Sfnp 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

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