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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
-3
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Isomorphism in fibers and flatness
This should be false without additional assumptions. Take a field $k$. Let $A=k[[x^2]]$, $Y_A=\mathrm{Spec}\:A$, $X_A=\mathrm{Spec}\:k[[x]]$. It is not hard to see that $X_A$ is flat over $A$. The map …