The Stacks project

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

  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. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 37.21: Regular morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07R8. Beware of the difference between the letter 'O' and the digit '0'.