Cotangent sheaf

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of $\mathcal{O}_X$ -modules that represents (or classifies) S-derivations in the sense: for any $\mathcal{O}_X$ -modules F, there is an isomorphism

that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d: \mathcal{O}_X \to \Omega_{X/S}$ such that any S-derivation $D: \mathcal{O}_X \to F$ factors as $D = \alpha \circ d$ with some $\alpha: \Omega_{X/S} \to F$ .

单词	Tangent sheaf
释义	Tangent sheaf 英语百科 Cotangent sheaf In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of $\mathcal{O}_X$ -modules that represents (or classifies) S-derivations in the sense: for any $\mathcal{O}_X$ -modules F, there is an isomorphism that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d: \mathcal{O}_X \to \Omega_{X/S}$ such that any S-derivation $D: \mathcal{O}_X \to F$ factors as $D = \alpha \circ d$ with some $\alpha: \Omega_{X/S} \to F$ .
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Tangent sheaf

Cotangent sheaf