Canonical bundle

Remarks

Locally, if $x_1, \dots, x_n$ are local coordinates (an étale chart), then $\omega_X$ is free with basis $dx_1 \wedge \dots \wedge dx_n$.

Example: for $X = \mathbb{P}^n_k$, the Euler sequence $0 \to \Omega_X \to \mathcal{O}(-1)^{\oplus (n+1)} \to \mathcal{O}_X \to 0$ gives, by taking determinants, $\omega_X \cong \mathcal{O}(-n-1)$. For $n = 1$ this is the degree $-2$ bundle on $\mathbb{P}^1_k$.

Non-example: for the cuspidal curve $y^2 = x^3$, $\Omega_{X/k}$ is not locally free at the cusp, so the definition does not apply. For singular varieties one uses instead the dualizing sheaf, which agrees with $\omega_X$ on smooth projective varieties.