-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
In the case X and S are affine schemes, the above definition means that
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.)
[2] There are two important exact sequences: 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.[5] Let
Let I be the ideal sheaf of Δ(X) in W. One then puts: and checks this sheaf of modules satisfies the required universal property of a cotangent sheaf (Hartshorne, Ch II.
The construction shows in particular that the cotangent sheaf is quasi-coherent.
The above definition means that the cotangent sheaf on X is the restriction to X of the conormal sheaf to the diagonal embedding of X over S. The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing
For this notion, see § 1 of There, the cotangent stack on an algebraic stack X is defined as the relative Spec of the symmetric algebra of the tangent sheaf on X.
(Note: in general, if E is a locally free sheaf of finite rank,
is the algebraic vector bundle corresponding to E.[citation needed]) See also: Hitchin fibration (the cotangent stack of