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Let $\pi: Y \to X$ be an etale morphism of finite-type schemes over a field $k$. Let $Z$ be a $k$-scheme. Suppose we have a morphism $f:Y \to Z$. When does $f$ descend to a morphism $g: X \to Z$? Is the answer simpler if any of $X,Y, Z$ are affine?

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For a faithfully flat morphism $\pi\colon Y\to X$ of finite type, the morphism $f$ will descend if and only if the composites of it with the two projection morphisms $Y\times_X Y \to Y$ coincide. Etale maps are flat, and faithfully flat if surjective. If your etale morphism $\pi$ is Galois, then the second condition just says that $f$ is invariant under the action of the Galois group.

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$\pi$ only has to be faithfully flat and quasi-compact, i.e. fpqc (in fact a weaker version of fpqc also works, see Angelo Vistoli's notes on descent theory). – Martin Brandenburg Nov 11 '13 at 16:31

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