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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be a scheme that is [[smooth|EXKGG9X6J0]] over $k$ and of pure dimension $n$.\nThen the [[sheaf of differentials|6VC6HQTFSA]] $\\Omega_{X/k}$ is locally free of rank $n$.\nThe *canonical bundle* of $X$ is the [[invertible sheaf|7JCF2TE3V5]]\n$$\\omega_X = \\bigwedge^n \\Omega_{X/k},$$\nthe top [[exterior power|0MDMZ965VE]] of $\\Omega_{X/k}$.\nA *canonical divisor* is a divisor $K_X$ with $\\mathcal{O}_X(K_X) \\cong \\omega_X$.\n\n## Remarks\n\nLocally, if $x_1, \\dots, x_n$ are local coordinates (an étale chart), then $\\omega_X$ is free with basis $dx_1 \\wedge \\dots \\wedge dx_n$.\n\nExample: 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, $\\omega_X \\cong \\mathcal{O}(-n-1)$.\nFor $n = 1$ this is the degree $-2$ bundle on $\\mathbb{P}^1_k$.\n\nNon-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.\nFor singular varieties one uses instead the dualizing sheaf, which agrees with $\\omega_X$ on smooth projective varieties.\n",
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