In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheafonX is the sheaf of -modules
that represents (or classifies) S-derivations[1] 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 sheafonX and is sometimes denoted by
.[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 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 WofX ×S X (the image is closed if and only if fisseparated). 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. Remark 8.9.2). The construction shows in particular that the cotangent sheaf is quasi-coherent. It is coherent if SisNoetherian 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.
The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing for the projective space over a ring R,
(See also Chern class#Complex projective space.)
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 is the total space of the Hitchin fibration.)