Canonical bundle of projective space
= Canonical bundle of projective space
{title2=$\omega_{\mathbf P^n}\cong\mathcal O(-n-1)$}
Over any field, the <canonical bundle> of <projective space> is $\mathcal O(-n-1)$. Indeed, dualizing the <Euler sequence> gives $0\to\Omega^1_{\mathbf P^n}\to\mathcal O(-1)^{\oplus(n+1)}\to\mathcal O\to0$; taking determinants gives the result. Hence <Serre duality> identifies $H^i(\mathbf P^n,\mathcal O(m))$ with the dual of $H^{n-i}(\mathbf P^n,\mathcal O(-m-n-1))$.