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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 $k=\mathbb{C}$.

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1 Answer 1

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 $X$.

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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

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