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.



Canonical Sheaf of Projective Space
det of the middle term of a short exact sequence is the tensor product of the dets of the left and right terms (det = top wedge). 


The canonical bundle is det of \Omega, det of O is O. 


