37.21 Regular morphisms
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}_ Y(V) \to \mathcal{O}_ X(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}_ Y(V_ j) \to \mathcal{O}_ X(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
X_{i, y_ i} = \mathop{\mathrm{Spec}}(\kappa (y_ i)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} X_ y.
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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