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(1) "For affine variety $V$ of $\mathbb{A}^{n}$ such that its coordinate ring is UFD, closed subvariety of $V$ which has codimension 1 is cut out by a single equation."

I looked at the proof of this statement, Where I have been using UFD in I do not know...

(2) I want to see proof of follwing statement....

"Any closed subvariety of affine normal variety with codimension 1 is cut out by a single equation."

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2 Answers 2

up vote 2 down vote accepted

As for (1): a closed subvariety of $V$ is defined by a prime ideal $p$ of the coordinate ring $k[V]$ of height $1$. If $k[V]$ is a UFD such a prime ideal is principal.

As for (2): the statement is false. The closed subvariety is cut out by a single equation locally but not globally as in the case of a UFD.

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Thanks, Hagen.. Where I can see it's proof? –  Sang Cheol Lee Aug 18 '11 at 13:53
    
Just use that a noetherian, normal, local ring of dimension $1$ is a UFD - see for example Matsumura's Commutative ring theory Chapter 4 for a proof of this result. –  Hagen Aug 18 '11 at 14:34

Here is a counterexample to 2) :

Let $E$ be an elliptic curve over $\mathbb C$ and $P\in E$ a non-torsion point, i.e. such that the abelian group inequality $nP\neq O $ holds for all $n\geq 1$ in the abelian group $E$.
The variety $X=E\setminus \{O\}$ is affine (like all non-projective curves) and normal (since it is smooth), but there is no function $f\in \Gamma(X,\mathcal O)$ which has as zero set exactly the one-codimensional subvariety $\{P\}$.
Indeed such an $f$ can be seen as a rational function on $E$ (with only pole the origin $O$) and should have a divisor whose image in the group $E$ is zero, like all rational functions on an elliptic curve.
But the divisor of $f$ is of the form $nP-nO$, so that the choice of $P$ implies that its image in the group $E$ is not zero.
So $\{P\}$ cannot be cut out by a single function on $X$.

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