Definition of $ \ \mathcal{O}_{\mathbb{P}_k^n}(1)$ [duplicate]

Question: "I've found references to the object $\ \mathcal{O}_{\mathbb{P}_k^n}(1), \ $ where $\mathbb{P}_k^n \ $ is the usual scheme, but I'm not able to find its proper definition. Is it just a line bundle over $\ \mathbb{P}_k^n $?"

Answer: Let $S:=k[x_0,..,x_n]:=\oplus_{d \geq 0} k[x_0,..,x_n]_d:=\oplus_{d \geq 0}S_d$ and let $S(l)$ be the twist of $S$ by an integer $l$. By definition

$$S(l)_d:=S_{l+d}:=k[x_0,..,x_n]_{l+d}.$$

When you sheafify $S(l)$ you get an invertible sheaf $\mathcal{O}(l)$ on $\mathbb{P}^n$. It has the property that

$$I1.\text{ }\mathcal{O}(l)(D(x_i)) \cong k[\frac{x_0}{x_i},. ,\overline{\frac{x_i}{x_i}}.,\frac{x_n}{x_i}]x_i^l$$

is a free $\mathcal{O}(D(x_i))$ module of rank one on the element $x_i^l$. Here $D(x_i) \subseteq \mathbb{P}^n$ is the open subscheme where "$x_i \neq 0$". There is an isomorphism $D(x_i) \cong \mathbb{A}^n$.

A proof of $I1$ is found in Hartshorne, Prop.II.5.11b.