Explicit computation of exterior power of vector bundle

Question

I am following the excellent book by Ellinsgrud and Ottem, Introduction to Schemes.

When proving that the $n$-th exterior power of the cotangent bundle of $\mathbb{P}^n$ is $\mathcal{O}_{\mathbb{P}^n}$, they simply say that:

$ \wedge^n \Omega_{\mathbb{P}^n} = \wedge^n \mathcal{O}_{\mathbb{P}^n}(-1)^{n+1},$

and that the later is equals to $\mathcal{O}_{\mathbb{P}^n}(-n-1)$. Can someone provide a reference for this equality, or gave me some hint on how to do it?

Thank you in advance

Answer 1

Use the Euler exact sequence $$ 0 \to \Omega \to \mathcal{O}(-1)^{\oplus (n+1)} \to \mathcal{O} \to 0 $$ and multiplicativity of determinant with respect to exact sequences.