Let $X=\mathbb{P}^n_k$ be the projective space over a field $k$ of dimension $n$ and denote by $\Omega$ the sheaf of differentials. I want to show that, $$\operatorname{dim}_k H^i(X,\Lambda^j\Omega)\begin{cases}1 \text{ if } i=j\leq n\\ 0 \text{ otherwise}\end{cases}$$
I can easily derive the case $j=1$ with the help of Euler's exact sequence but for the case of $j>1$ if I apply $\Lambda^j$ to the Euler's exact sequence I get this description Eisenbud's proof of right-exactness of the exterior algebra which is not very useful.
How should I proceed?
If $F=\mathcal{O}_{\mathbb{P}^n}(-1)^{n+1}$, then you have an exact sequence, $$0\to \Omega^j\to\wedge^j F\to\Omega^{j-1}\to 0,$$ and then you should be able to calculate everything using induction on $j$.