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 $k(P)=\mathcal{O}_P/\mathcal{m}_P$.
Best Answer
Ok, so first of all I'm just learning this stuff so I'm kind of shaky. If anyone has any comments for me or notices any errors please point them out. As far as I understand there are many different choices for ideal sheaf on $Y$ and they correspond to different subscheme structures. Since Hartshorne does not specify a choice of subscheme structure, I think we can make our own choice (since he only uses the existence of such a SES to derive something unrelated to the subscheme structure) and think of the ideal sheaf of $Y$ as the collection of sections of $\mathcal O_X$ that are not invertible at the stalks of elements of $Y$. We also should make the same choice for the ideal sheaf of $Y\cup\{p\}$.
With this choice in mind it is clear that on the complement of $p$, which is open since $p$ is a closed point, the ideal sheaves agree. Since the skyscraper sheaf is zero here, we have exactness. Thus we only need exactness at the stalk at $p$. Looking at the complement of $Y$ we reduce to the case where we have an affine scheme $U$ and our ideal sheaf on a closed point $p$ injecting into the sheaf of regular functions:
$$0\to I_p\to \mathcal O_U\to k(p)\to 0$$
But by our choice of ideal sheaf, $I_p$ is just the maximal ideal corresponding to $p$. This shows exactness on stalks at $p$.