Lemma 37.20.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

1. $f$ is regular,

2. $f$ is flat and its fibres are geometrically regular schemes,

3. 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,

4. 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

5. 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. $\square$

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