Lemma 33.12.5. Let k be a field. Let f : X \to Y be a morphism of locally Noetherian schemes over k. Let x \in X be a point and set y = f(x). If X is geometrically regular at x and f is flat at x then Y is geometrically regular at y. In particular, if X is geometrically regular over k and f is flat and surjective, then Y is geometrically regular over k.
Proof. Let k' be finite purely inseparable extension of k. Let f' : X_{k'} \to Y_{k'} be the base change of f. Let x' \in X_{k'} be the unique point lying over x. If we show that Y_{k'} is regular at y' = f'(x'), then Y is geometrically regular over k at y', see Lemma 33.12.3. By Morphisms, Lemma 29.25.7 the morphism X_{k'} \to Y_{k'} is flat at x'. Hence the ring map
\mathcal{O}_{Y_{k'}, y'} \longrightarrow \mathcal{O}_{X_{k'}, x'}
is a flat local homomorphism of local Noetherian rings with right hand side regular by assumption. Hence the left hand side is a regular local ring by Algebra, Lemma 10.110.9. \square
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