Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am currently studying varieties over $\mathbb{C}$, i know some scheme theory.

  • Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition only at the closed points of $Y$? If yes, could you give an argument, if not, could you give an example where $f$ is flat over all closed points of $Y$, but not over some non-closed point?

  • Same question with $f$ smooth of relative dimension $n$, using the following characterization of smooth (Hartshorne III.10.5):

$f$ is called smooth if it is flat and for all $y \in Y$ the "algebraic closure of the fibre" $X_{\overline{y}} := X_y \times_{k(y)} \overline{k(y)}$ is regular (and equidimensional of dimension $n$). Now it must be clear what is meant by smooth "at a point" $y \in Y$.

The question originally continued after this, but i decided to split it since it became too long. Second part: (Continued:) finiteness of étale morphisms

Thanks a lot!

PS tag "complex-geometry" is included since i'm happy to assume $k=\mathbb{C}$.

share|cite|improve this question
up vote 2 down vote accepted

The answer is yes.

  1. For the flatness: a module $M$ over a ring $A$ is flat if and only if for all maximal ideal $\mathfrak m$ of $A$, $M_\mathfrak m$ is flat over $A_\mathfrak m$.

  2. The smooth locus is open in $X$. If it contains all closed points of $X$, then it is equal to $X$.

share|cite|improve this answer
Happy to see you back, QiL: I missed you! – Georges Elencwajg Aug 15 '12 at 13:54
Remarkably simple, easy to check, thanks a lot! – Joachim Aug 15 '12 at 14:10
Thanks @GeorgesElencwajg ! Je vois que tu es toujours aussi dévoué :). – user18119 Aug 16 '12 at 14:58

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.