Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.

The category of finitely affine schemes admits such an adjunction into the category of profinite spaces given by taking the Pierce spectrum (connected components). In the book Galois Theories, the authors (Borceux and Janelidze) write at the top of page 309 that the covering morphisms w.r.t this adjunction are the componentially locally strongly separable algebras.

Definitions. A (commutative) ring homomorphism $R\to S$ is:

  • an extension if $S$ is a faithful $R$-module.
  • separable if $S$ is a projective $S/R$ bimodule.
  • strongly separable if it's separable and $S$ is a finitely generated projective $R$-module.
  • locally strongly separable extension if it's a direct limit of strongly separable subextensions.
  • componentially locally strongly separable extension if it's each localization at an element of the Pierce spectrum is a locally strongly separable extension.

Since "locally" seems analogoue to ind-étale I suppose the covering morphisms for the analogous adjunctions from the opposite of finitely presented commutative rings to sets given by taking idempotents are precisely the componentially strongly separable extensions.

What is the geometric intuition for such ring homomorphisms? What's the relationship with finite étale morphisms? What are some crucial geometric differences?


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