# Cotangent sheaf

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of ${\mathcal {O}}_{X}$-modules $\Omega _{X/S}$ that represents (or classifies) S-derivations in the sense: for any ${\mathcal {O}}_{X}$-modules F, there is an isomorphism
    $\operatorname {Hom} _{{\mathcal {O}}_{X}}(\Omega _{X/S},F)=\operatorname {Der} _{S}({\mathcal {O}}_{X},F)$
that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d:{\mathcal {O}}_{X}\to \Omega _{X/S}$ such that any S-derivation $D:{\mathcal {O}}_{X}\to F$ factors as $D=\alpha \circ d$ with some $\alpha :\Omega _{X/S}\to F$.
In the case X and S are affine schemes, the above definition means that $\Omega _{X/S}$ 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 $\Theta _{X}$.
There are two important exact sequences:
* If S →T is a morphism of schemes, then
    $f^{*}\Omega _{S/T}\to \Omega _{X/T}\to \Omega _{X/S}\to 0.$
* If Z is a closed subscheme of X with ideal sheaf I, then
    $I/I^{2}\to \Omega _{X/S}\otimes _{O_{X}}{\mathcal {O}}_{Z}\to \Omega _{Z/S}\to 0.$
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.

## Construction through a diagonal morphism

Let $f:X\to S$ 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 (the image is closed if and only if f is separated). Let I be the ideal sheaf of Δ(X) in W. One then puts:
    $\Omega _{X/S}=\Delta ^{*}(I/I^{2})$
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 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.

## Relation to a tautological line bundle

The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing $\mathbf {P} _{R}^{n}$ for the projective space over a ring R,
    $0\to \Omega _{\mathbf {P} _{R}^{n}/R}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}(-1)^{\oplus (n+1)}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}\to 0.$
(See also Chern class#Complex projective space.)

## Cotangent stack

For this notion, see § 1 of
    A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves Archived 2015-01-05 at the Wayback Machine
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, $\mathbf {Spec} (\operatorname {Sym} ({\check {E}}))$ is the algebraic vector bundle corresponding to E.)
See also: Hitchin fibration (the cotangent stack of $\operatorname {Bun} _{G}(X)$ is the total space of the Hitchin fibration.)