= Algebraic cotangent bundle
{title2=$\Omega^1_{X/k}$}
= Cotangent bundle of a smooth algebraic variety
{synonym}
For a smooth <algebraic variety> of dimension $r$ over a field, the <sheaf of Kähler differentials over a field> is locally free of rank $r$ and defines its algebraic cotangent bundle. Its dual is the <tangent sheaf of a scheme>. For a smooth hypersurface of degree $d$ in $\mathbb P^n$, the <Conormal exact sequence for Kähler differentials> is the short exact sequence $0\to\mathcal O_X(-d)\to\Omega^1_{\mathbb P^n}|_X\to\Omega^1_X\to0$.
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