Compare with Section 37.20. The algebraic version of this notion is discussed in More on Algebra, Section 15.41.
Definition 37.21.1. 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 $X$.
The condition that the morphism $X \to Y$ is regular is stronger than just requiring all the fibres to be regular locally Noetherian schemes.
Lemma 37.21.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent
$f$ is regular,
$f$ is flat and its fibres are geometrically regular schemes,
for every pair of affine opens $U \subset X$, $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}(V) \to \mathcal{O}(U)$ is regular,
there exists an open covering $Y = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$ is regular, and
there exists an affine open covering $Y = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring maps $\mathcal{O}(V_ j) \to \mathcal{O}(U_ i)$ are regular.
Proof. The equivalence of (1) and (2) is immediate from the definitions. Let $x \in X$ with $y = f(x)$. By definition $f$ is flat at $x$ if and only if $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is a flat ring map, and $X_ y$ is geometrically regular at $x$ over $\kappa (y)$ if and only if $\mathcal{O}_{X_ y, x} = \mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ is a geometrically regular algebra over $\kappa (y)$. Hence Whether or not $f$ is regular at $x$ depends only on the local homomorphism of local rings $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$. Thus the equivalence of (1) and (4) is clear.
Recall (More on Algebra, Definition 15.41.1) that a ring map $A \to B$ is regular if and only if it is flat and the fibre rings $B \otimes _ A \kappa (\mathfrak p)$ are Noetherian and geometrically regular for all primes $\mathfrak p \subset A$. By Varieties, Lemma 33.12.3 this is equivalent to $\mathop{\mathrm{Spec}}(B \otimes _ A \kappa (\mathfrak p))$ being a geometrically regular scheme over $\kappa (\mathfrak p)$. Thus we see that (2) implies (3). It is clear that (3) implies (5). Finally, assume (5). This implies that $f$ is flat (see Morphisms, Lemma 29.25.3). Moreover, if $y \in Y$, then $y \in V_ j$ for some $j$ and we see that $X_ y = \bigcup _{i \in I_ j} U_{i, y}$ with each $U_{i, y}$ geometrically regular over $\kappa (y)$ by Varieties, Lemma 33.12.3. Another application of Varieties, Lemma 33.12.3 shows that $X_ y$ is geometrically regular. Hence (2) holds and the proof of the lemma is finished. $\square$
Lemma 37.21.3. A smooth morphism is regular.
Proof. Let $f : X \to Y$ be a smooth morphism. As $f$ is locally of finite presentation, see Morphisms, Lemma 29.34.8 the fibres $X_ y$ are locally of finite type over a field, hence locally Noetherian. Moreover, $f$ is flat, see Morphisms, Lemma 29.34.9. Finally, the fibres $X_ y$ are smooth over a field (by Morphisms, Lemma 29.34.5) and hence geometrically regular by Varieties, Lemma 33.25.4. Thus $f$ is regular by Lemma 37.21.2. $\square$
Lemma 37.21.4. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is regular” is local in the fppf topology on the target and local in the smooth topology on the source.
Proof. We have $\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f) \wedge \mathcal{P}_3(f)$ where $\mathcal{P}_1(f)=$“the fibres of $f$ are locally Noetherian”, $\mathcal{P}_2(f)=$“$f$ is flat”, and $\mathcal{P}_3(f)=$“the fibres of $f$ are geometrically regular”. We have already seen that $\mathcal{P}_1$ and $\mathcal{P}_2$ are local in the fppf topology on the source and the target, see Lemma 37.20.4, and Descent, Lemmas 35.23.15 and 35.27.1. Thus we have to deal with $\mathcal{P}_3$.
Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fpqc covering of $Y$. Denote $f_ i : X_ i \to Y_ i$ the base change of $f$ by $\varphi _ i$. Let $i \in I$ and let $y_ i \in Y_ i$ be a point. Set $y = \varphi _ i(y_ i)$. Note that
Hence in this situation we have that $X_ y$ is geometrically regular if and only if $X_{i, y_ i}$ is geometrically regular, see Varieties, Lemma 33.12.4. This fact implies $\mathcal{P}_3$ is fpqc local on the target.
Let $\{ X_ i \to X\} $ be a smooth covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is a smooth covering of the fibre. Hence the locality of $\mathcal{P}_3$ for the smooth topology on the source follows from Descent, Lemma 35.18.4. Combining the above the lemma follows. $\square$
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