In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules Ω X / S {\displaystyle \Omega _{X/S}} that represents S-derivations in the sense: for any O X {\displaystyle {\mathcal {O}}_{X}} -modules F, there is an isomorphism Hom O X ⁡ = Der S ⁡ {\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{X}}=\operatorname {Der} _{S}} that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential d : O X → Ω X / S {\displaystyle d:{\mathcal {O}}_{X}\to \Omega _{X/S}} such that any S-derivation D : O X → F {\displaystyle D:{\mathcal {O}}_{X}\to F} factors as D = α ∘ d {\displaystyle D=\alpha \circ d} with some α : Ω X / S → F {\displaystyle \alpha :\Omega _{X/S}\to F}. Wikipedia