Cotangent 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 ${\displaystyle \Omega _{X/S}}$ that represents (or classifies) S-derivations[1] in the sense: for any ${\displaystyle {\mathcal {O}}_{X}}$-modules F, there is an isomorphism

${\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{X}}(\Omega _{X/S},F)=\operatorname {Der} _{S}({\mathcal {O}}_{X},F)}$

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} .

In the case X and S are affine schemes, the above definition means that ${\displaystyle \Omega _{X/S}}$ is the module of Kähler differentials. The standard way to construct a cotangent sheaf (e.g., Hartshorne, Ch II. § 8) is through a diagonal morphism (which amounts to gluing modules of Kähler differentials on affine charts to get the globally-defined cotangent sheaf.) The dual module of the cotangent sheaf on a scheme X is called the tangent sheaf on X and is sometimes denoted by ${\displaystyle \Theta _{X}}$.[2]

There are two important exact sequences:

1. If S →T is a morphism of schemes, then

   ${\displaystyle f^{*}\Omega _{S/T}\to \Omega _{X/T}\to \Omega _{X/S}\to 0.}$

2. If Z is a closed subscheme of X with ideal sheaf I, then

   ${\displaystyle I/I^{2}\to \Omega _{X/S}\otimes _{O_{X}}{\mathcal {O}}_{Z}\to \Omega _{Z/S}\to 0.}$[3][4]

The cotangent sheaf is closely related to smoothness of a variety or scheme. For example, an algebraic variety is smooth of dimension n if and only if Ω_X is a locally free sheaf of rank n.[5]