Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy groups. Friedlander has shown how to associate an etale topological type (a pro-simplicial set) to reasonable schemes, thereby refining Artin--Mazur theory.

Presumably, one could consider invariants of etale topological type which are not homotopy invariant (roughly speaking something like Reidemeister torsion). Were such invariants considered in the literature? Do some of them admit an interpretation independent of Friedlander construction (like etale fundamental group can be defined without referring to etale homotopy type and it is also connected to Galois theory in the case of fields)?

  • $\begingroup$ The Krull dimension of a scheme is invariant under passing to 'etale covers, but it is not a homotopy invariant. $\endgroup$ – Jason Starr Dec 26 '18 at 12:39

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