Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 2

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.

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.