Cotangent sheaf


In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents 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 is through a diagonal morphism 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. :
  3. If Z is a closed subscheme of X with ideal sheaf I, then
  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 ΩX is a locally free sheaf of rank n.

Construction through a diagonal morphism

Let be a morphism of schemes as in the introduction and Δ: X → X ×S X the diagonal morphism. Then the image of Δ is locally closed; i.e., closed in some open subset W of X ×S X. Let I be the ideal sheaf of Δ in W. One then puts:
and checks this sheaf of modules satisfies the required universal property of a cotangent sheaf. The construction shows in particular that the cotangent sheaf is quasi-coherent. It is coherent if S is Noetherian and f is of finite type.
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.
See also: bundle of principal parts.

Relation to a tautological line bundle

The cotangent sheaf on a projective space is related to the tautological line bundle O by the following exact sequence: writing for the projective space over a ring R,

Cotangent stack

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.
See also: Hitchin fibration