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Let $X$ be a hypersurface in $\mathbb P^{n}$ defined by the vanishing set of a homogeneous degree $k$ polynomial.

Why is the sequence

$0 \rightarrow \mathcal O(-k) \rightarrow \mathcal O_{\mathbb P^{n}} \rightarrow i_{*} \mathcal O_{X} \rightarrow 0$

(where $i_{*}$ is the inclusion of $X$ into $\mathbb P^{n}$)

exact?

I have seen this or a very similar statement (slightly more/less general) referenced in several sources including Hartshorne and it is always stated as fact.

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The ideal sheaf of a degree $k$ hypersurface is isomorphic to $\mathcal O(-k)$. More precisely, the ideal sheaf $\mathcal I_X$ is a subsheaf of $\mathcal O_{\mathbb P^n}$, and the isomorphism $\mathcal O(-k) \cong \mathcal I_X \subset \mathcal O_{\mathbb P^n}$ is given by multiplication by $f$. (Note that $f$, being a degree $k$ polynomial, is a global section of $\mathcal O(k)$, and so multiplication by $f$ embeds $\mathcal O(-k)$ into $\mathcal O_{\mathbb P^n}$.)

Now the short exact sequence $0 \to \mathcal I_X \to \mathcal O_{\mathbb P^n} \to i_* \mathcal O_X \to 0$ is the standard short exact sequence that relates the ideal sheaf of the subvariety $X$ to its structure sheaf.

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Note that if $f$ is a polynomial of degree $e$, then we have the following exact sequence:

$0 \longrightarrow S(-e) \longrightarrow S \longrightarrow S/(f) \longrightarrow 0$,

where $S$ is the polynomial ring $k[x_0,\ldots ,x_n]$.

[small edit:This comes from considering free graded resolutions of $S/(f)$. If you can't see this, a great source for this sort of things is "Using algebraic geometry" by Cox, Little and O'Shea.]

If we then apply the exact functor $\tilde{}$ described in Hartshorne on Chapter 2,section 5, which associates a $\mathcal{O}_{\mathbb{P}^{n}}$-module to each graded $S$-module(Here $\tilde{S} = \mathcal{O}_{\mathbb{P}^{n}}$), we get the desired exact sequence.

This short exact sequence is interesting since it allows one to compute sheaf cohomology of hypersurfaces.

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