Lemma 33.25.7. Let X be a scheme over a field k. If X is locally of finite type and geometrically reduced over k then X contains a dense open which is smooth over k.
Proof. The problem is local on X, hence we may assume X is quasi-compact. Let X = X_1 \cup \ldots \cup X_ n be the irreducible components of X. Then Z = \bigcup _{i \not= j} X_ i \cap X_ j is nowhere dense in X. Hence we may replace X by X \setminus Z. As X \setminus Z is a disjoint union of irreducible schemes, this reduces us to the case where X is irreducible. As X is irreducible and reduced, it is integral, see Properties, Lemma 28.3.4. Let \eta \in X be its generic point. Then the function field K = k(X) = \kappa (\eta ) is geometrically reduced over k, hence separable over k, see Algebra, Lemma 10.44.2. Let U = \mathop{\mathrm{Spec}}(A) \subset X be any nonempty affine open so that K = A_{(0)} is the fraction field of A. Apply Algebra, Lemma 10.140.5 to conclude that A is smooth at (0) over k. By definition this means that some principal localization of A is smooth over k and we win. \square
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Comment #1931 by Keenan Kidwell on