# ag.algebraic geometry – Are common geometric equivalences of DM stacks affine? Answer

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ag.algebraic geometry – Are common geometric equivalences of DM stacks affine?

Let $$f:Xto Y$$ breathe a map of Deligne-Mumford stacks. Let’s say that the map $$f$$ is a geometric equivalence if the induced map on diminutive étale topoi is a geometrical equivalence. Moreover, for instance that the map $$f$$ is a common geometric equivalence if for each morphism $$Ato Y$$ with $$A$$ an affine strategy, the projection map $$Xtimes_Y A to A$$ is a geometrical equivalence (so with out lack of generality, then, we will occupy that $$Y$$ is affine). So then the query:

Is it undoubted that given a common geometric equivalence $$f:Xto Y$$, the map is an affine morphism?

It would in actual fact breathe sufficient to display that the map is schematic (utilizing the classification of common homeomorphisms between schemes along with the discount to $$Y$$ affine), however I’m not positive if that is so limpid both.

Bonus query: If it’s undoubted for Deligne-Mumford stacks, does it stay undoubted for algebraic stacks?

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