In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of ${\displaystyle {\mathcal {O))_{X))$-modules ${\displaystyle \Omega _{X/S))$ that represents (or classifies) S-derivations^[1] in the sense: for any ${\displaystyle {\mathcal {O))_{X))$-modules F, there is an isomorphism

${\displaystyle \operatorname {Hom} _((\mathcal {O))_{X))(\Omega _{X/S},F)=\operatorname {Der} _{S}({\mathcal {O))_{X},F)}$

that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential ${\displaystyle d:{\mathcal {O))_{X}\to \Omega _{X/S))$ such that any S-derivation ${\displaystyle D:{\mathcal {O))_{X}\to F}$ factors as ${\displaystyle D=\alpha \circ d}$ with some ${\displaystyle \alpha :\Omega _{X/S}\to F}$.

In the case X and S are affine schemes, the above definition means that ${\displaystyle \Omega _{X/S))$ 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 ${\displaystyle \Theta _{X))$.^[2]

There are two important exact sequences:

1. If S →T is a morphism of schemes, then

   ${\displaystyle f^{*}\Omega _{S/T}\to \Omega _{X/T}\to \Omega _{X/S}\to 0.}$

2. If Z is a closed subscheme of X with ideal sheaf I, then

   ${\displaystyle I/I^{2}\to \Omega _{X/S}\otimes _{O_{X)){\mathcal {O))_{Z}\to \Omega _{Z/S}\to 0.}$^[3][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 Ω_X is a locally free sheaf of rank n.^[5]

Construction through a diagonal morphism

Let ${\displaystyle f:X\to S}$ 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:

${\displaystyle \Omega _{X/S}=\Delta ^{*}(I/I^{2})}$

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 ${\displaystyle \mathbf {P} _{R}^{n))$ for the projective space over a ring R,

${\displaystyle 0\to \Omega _{\mathbf {P} _{R}^{n}/R}\to {\mathcal {O))_{\mathbf {P} _{R}^{n))(-1)^{\oplus (n+1)}\to {\mathcal {O))_{\mathbf {P} _{R}^{n))\to 0.}$

(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 [1] Archived 2015-01-05 at the Wayback Machine^[6]

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, ${\displaystyle \mathbf {Spec} (\operatorname {Sym} ({\check {E))))}$ is the algebraic vector bundle corresponding to E.^{[citation needed]})

See also: Hitchin fibration (the cotangent stack of ${\displaystyle \operatorname {Bun} _{G}(X)}$ is the total space of the Hitchin fibration.)