TITLE: Canonical Sheaf of Projective Space
QUESTION [15 upvotes]: I am stuck on one step that occurs without explanation in several Algebraic geometry books. 
Starting from the exact sequence
$$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus n+1}\rightarrow \mathcal{O}_{\mathbb{P}^n}\rightarrow 0$$
it is concluded that $$\omega_{\mathbb{P}^n}=\wedge^n \Omega_{\mathbb{P}^n}\cong \mathcal{O}_{\mathbb{P}^n}(-n-1)$$
How does this follow and in particular how does $\Omega_{\mathbb{P}^1}\cong \mathcal{O}_{\mathbb{P}^1}(-2)$ follow ?
Thanks in advance.

REPLY [4 votes]: One could see this in the following way. We have
$$\omega_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(c_1)$$
where $c_1 = c_1(\omega_{\mathbb{P}^n}) = c_{1}(\bigwedge^n\Omega_{\mathbb{P}^n}) = c_1(\Omega_{\mathbb{P}^n})$ is the first Chern class.
Now, by the Euler's exact sequence 
$$0\mapsto\Omega_{\mathbb{P}^n}\rightarrow\mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (n+1)}\rightarrow \mathcal{O}_{\mathbb{P}^n}\mapsto 0$$
we get
$$c_1(\omega_{\mathbb{P}^n}) = c_1(\mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (n+1)})-c_1(\mathcal{O}_{\mathbb{P}^n}) = -n-1.$$
Therefore 
$$\omega_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(-n-1).$$