Definition 37.20.1. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.

Let $x \in X$, and $y = f(x)$. We say that $f$ is

*regular at $x$*if $f$ is flat at $x$, and the scheme $X_ y$ is geometrically regular at $x$ over $\kappa (y)$ (see Varieties, Definition 33.12.1).We say $f$ is a

*regular morphism*if $f$ is regular at every point of $X$.

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