Let \(\characteristic k = p\) with \(k = \kbar\) and let \(F : \PP^n_k \to \PP^n_k\) be the Frobenius, given on coordinates by \(x \mapsto x^p\). Then \(F\) is finite, flat, surjective, and a homeomorphism, and it is nowhere smooth. Since \(d(t^p) = 0\), the map \(F^* \Omega_{\PP^n/k} \to \Omega_{\PP^n/k}\) is zero, so \(\Omega_{\PP^n/\PP^n} = \Omega_{\PP^n/k}\) is locally free of rank \(n\) while the relative dimension of \(F\) is \(0\).
This example separates smoothness from flatness and from the topology of the underlying map. \(F\) is finite flat of degree \(p^n\) and bijective on points, so no condition on fibres as sets or on flatness detects the failure; only the differentials do. It is the sharpest illustration that smoothness is not a topological or even a flat-family condition.
It also shows that “étale equals flat plus finite fibres” is false: \(F\) is flat with one point in every fibre and is not étale, because the fibres are the non-reduced \(\Spec k[x]/(x^p)\) rather than reduced points. The separability clause in the definition of unramified is precisely what rules this out.
Dependencies
D-MORSMSmooth morphisms and relative dimensionuses
Backlinks
D-IV2FROBTWISTThe $k$-linear Frobenius and the Frobenius twistrelated-toD-CRVHASSEThe Hasse invariant, and ordinary versus supersingularrelated-toT-MORGENGeneric smoothnessuses