Hello !

If $X$ is a scheme, we can consider the etale topos of $X$ whose object are etale scheme above $X$ with the etale topology.

My question is : is there a know way to express this topos as the classifying topos of some geometric theory ? Of course it is possible, just because it's a grothendieck topos, but I'm looking for an explicit theory at least on some particular case (like when $X$ is affine, or when $X$ is the spectrum of the ring of integer of a number field, or when $X$ is a projective curve over a finite field... )

For example, if $A$ is a ring, then the Zariski topos of $Spec A$ (topos of finite presentation scheme above $Spec A$ with the Zariski topology) is the classifying topos of the theory of local $A$ algebra. (the universal local $A$ algebra being the structural sheaf).


I cannot give the details, but my guess is that the etale topos should be the classifying topos of the theory of strictly local $A$ algebras. By a strictly local $A$ algebra I mean a henselian local algebra with separably closed residue field. I don't know if this is a honest algebraic theory.

Bonus: in this vein the Nisnevich topos should be the classifying topos of the theory of henselian local $A$ algebras. The proof should follow similar lines to the previous one.

| cite | improve this answer | |
  • 1
    $\begingroup$ The link by Achilleas K supports your guess. $\endgroup$ – Martin Brandenburg Mar 23 '12 at 17:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.