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Let $X$ be a scheme and $Y$ a closed subscheme. Let $i : Y \hookrightarrow X$ be the inclusion.

Then, we define the ideal sheaf of $Y$, denoted $\mathcal {I}_Y$to be the kernel of the morphism $i^{\sharp} : \mathcal{O}_X \rightarrow i_{*} \mathcal{O}_Y$.

Let $P$ be a closed point of $X$, let $U$ be an affine open subet of $X$ containing $P$,and let $Y=X-U$.

Then is the following sequence exact??

$$0 \rightarrow \mathcal{I}_{Y\cup \{P\} } \rightarrow \mathcal{I}_Y \rightarrow k(P) \rightarrow 0$$ where $k(P)=\mathcal{O}_P/\mathcal{m}_P$.

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For $Y=\{0\}, X=\mathbb{A}^1_k, P=\{1\},$ and $U=X-Y,$ is $0\to (x,x-1) \to (x) \to k[x]_{(x-1)}/(x-1) \to 0$ exact? –  Ehsan M. Kermani Apr 8 '13 at 3:43
    
Then, not eacxt?? –  Sang Cheol Lee Apr 9 '13 at 1:42
    
What is $(x,x-1)$ the ideal generated by $x,x-1$ in $k[x]?$ then can $(x,x-1) \to (x)$ be injective? –  Ehsan M. Kermani Apr 9 '13 at 3:10
    
$(x,x-1)=k[x]$.... –  Sang Cheol Lee Apr 9 '13 at 15:55
    
Yes, so the $k[x]$ doesn't inject into $(x)$ right? –  Ehsan M. Kermani Apr 10 '13 at 1:04
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