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This is an exercise in Hartshorne's book.

For a quasi projective variety $Y$ with dimension $\geq 2$ and $p \in Y$ a normal point, if $f$ is regular on $Y-\{p\}$ then $f$ can be extended to a regular function on $Y$.

I want to get some hint to prove this problem....

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2 Answers 2

Since the problem is local around $p$, you can assume that $Y=Spec(A)$ where $A$ is a noetherian domain (quasi-projectiveness is irrelevant).

Clearly, every point $\mathfrak q \in Spec(A)$ of height one is distinct from $p$ (since $\mathfrak q$ it corresponds to a subvariety of codimension $1$). So every function $f$ defined on $Spec(A)\setminus \lbrace p\rbrace $ is defined at $\mathfrak q $.
You can then conclude that $f\in A$, that is $f$ extends regularly through $p$, thanks to the formula valid for a noetherian normal domain (Matsumura, Commutative ring theory, Theorem 11.5, page 81)

$$ A=\bigcap_{ht(\mathfrak q)=1} A_ \mathfrak q $$

A general result in this vein is that if $X$ is a locally noetherian normal integral scheme and $Y\subset X$ a closed subset of codimension $\geq 2$, the restriction morphism $\mathcal O_X(X)\to O_X(X\setminus F)$ is bijective.

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Thank Georges, But, This problem is one of exercise of Hartshorne's book "Algebraic geometry" chapterI.1.3.... Maybe there is s proof with out scheme theory...?? –  Sang Cheol Lee Mar 8 '12 at 15:36
@Georges Elencwajg : Could you please show me where to find the proof of the above general result ? –  Arsenaler Jan 27 '13 at 18:46

This is a simple corollary of so called, algebraic Hartog's lemma. And it requires fair amount of commutative algebra......You can find more informations and geometric intuition in the Vakil's lecture note 12.3.10

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