Cotangent sheaf

rdf:langString Cotangent sheaf

rdf:langString In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents (or classifies) S-derivations in the sense: for any -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 such that any S-derivation factors as with some . In the case X and S are affine schemes, the above definition means that 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 . There are two important exact sequences: 1. * If S →T is a morphism of schemes, then 2. * If Z is a closed subscheme of X with ideal sheaf I, then 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 ΩX is a locally free sheaf of rank n.