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On the one hand, we may define $\mathcal{O}_{\mathbb{P}^{n}}(l)$ as the invertible sheaf with trivializing cover $\{D(X_i): i\in \{0,...,n\}\}$ and transition functions $\left(\frac{X_i}{X_j}\right)^l$.

On the other hand, we may define it as the sheaf of modules induced by the graded module $K[X_0,...,X_n](l)$.

I would like to proof that these definitions are equivalent. One way could be finding transition functions for the sheaf of the second definition. If they are $\left(\frac{X_i}{X_j}\right)^l$ we are done, but I am finding hard to show this.

Howard
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  • let me add that good references (more understandable than Hartshorne) on this subject are notes by Gathman, available on internet. Also, Mile Reid wrote fantastic notes on algebraic surfaces (on arxiv). –  May 10 '17 at 17:54

2 Answers2

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For any sheaf of $\mathcal{O}_{\mathbb{P}^n}$ modules $\mathcal{F}$ on $\mathbb{P}^n$, let $\Gamma_*(\mathcal{F})= \bigoplus_{n\in\mathbb{Z}} \Gamma(\mathcal{F}\otimes\mathcal{O}(n))$. Now, let $\mathcal{F}$ be a quasicoherent sheaf of $\mathcal{O}_{\mathbb{P}^n}$ modules. Then there is a natural isomorphism $\beta:\widetilde{\Gamma_*(\mathcal{F})}\to\mathcal{F}$ (this is Hartshorne II.5.15). After identifying $\Gamma_*(\mathcal{O}(l))=K[X_0,\cdots,X_n](l)$, we'll apply this statement to show what you want.

Identifying $\Gamma_*(\mathcal{O}(l))$: recall that $\Gamma(\mathcal{O}(a))$ is the set of homogeneous polynomials of degree $a$ in $X_0,\cdots,X_n$. This means that $\Gamma_*(\mathcal{O}(l))=\bigoplus_{n\in\mathbb{Z}} (K[X_0,\cdots,X_n])_{(n+l)}$, which exactly means that $\Gamma_*(\mathcal{O}(l))=K[X_0,\cdots,X_n](l)$.

Honestly calculating the transition functions should be possible using the isomorphism $\Gamma(\widetilde{M(l)}|_{D(f)},D(f))\cong (M_f)_l$ where the first subscript is localization and the second is "take the $l^{th}$ graded piece", but I get confused by this too.

KReiser
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I have to say I find it pretty confusing me too, especially when I was reading Hartshorne.

Let's compute global section of the sheaf $\mathcal O_{P^n}(l)$. Let $s$ be such a section, i.e a family of regular map $s_i : U_i \to \mathbb C$ such that $\phi_{ij} s_i = s_j$, where $\phi_{ij} = (\frac{x_i}{x_j})^l$. (Usually people wrote $\phi_{ji}$ what you wrote $\phi_{ij}$ but this is not so important) .

Let's look at $U_0$ and $U_1$, we have $s_0 = f/x_0^a$ and $s_1 = g/x_1^b$. Since $(\frac{x_0}{x_1})^l s_0 = s_1$ we have $x_0^{l-a}f = x_1^{l-b} g$. In particular, $0 \leq a, b \leq l$ as e.g the right hand side has non poles at $x_0$ so the LHS too. If $a < l$ we can multiply both numerator and denominator by $x_0^{l-a}$, i.e we can assume $a = b = l$. So we have the equality $f/x_0^l = (\frac{x_1}{x_0})^l g/x_1^l$, this exactly means that in fact $f = g$, and similar argument shows that for any $i$, $s_i = f/x_i^l$. It follows that a section of $\mathcal O_{P^n}(l)$ is exactly an homogeneous $f \in k[x_0, \dots, x_n]_l$.