Tangent sheaf

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of ${\displaystyle {\mathcal {O}}_{X}}$-modules that represents (or classifies) S-derivations in the sense: for any ${\displaystyle {\mathcal {O}}_{X}}$-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 ${\displaystyle d:{\mathcal {O}}_{X}\to \Omega _{X/S}}$ such that any S-derivation ${\displaystyle D:{\mathcal {O}}_{X}\to F}$ factors as ${\displaystyle D=\alpha \circ d}$ with some ${\displaystyle \alpha :\Omega _{X/S}\to F}$.

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