Cotangent sheaf

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

[math]\displaystyle{ \operatorname{Hom}_{\mathcal{O}_X}(\Omega_{X/S}, F) = \operatorname{Der}_S(\mathcal{O}_X, F) }[/math]

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

In the case X and S are affine schemes, the above definition means that [math]\displaystyle{ \Omega_{X/S} }[/math] 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 [math]\displaystyle{ \Theta_X }[/math].

There are two important exact sequences:

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

   [math]\displaystyle{ f^* \Omega_{S/T} \to \Omega_{X/T} \to \Omega_{X/S} \to 0. }[/math]

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

   [math]\displaystyle{ I/I^2 \to \Omega_{X/S} \otimes_{O_X} \mathcal{O}_Z \to \Omega_{Z/S} \to 0. }[/math]

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 [math]\displaystyle{ f: X \to S }[/math] 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:

[math]\displaystyle{ \Omega_{X/S} = \Delta^* (I/I^2) }[/math]

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 [math]\displaystyle{ \mathbf{P}^n_R }[/math] for the projective space over a ring R,

[math]\displaystyle{ 0 \to \Omega_{\mathbf{P}^n_R/R} \to \mathcal{O}_{\mathbf{P}^n_R}(-1)^{\oplus(n+1)} \to \mathcal{O}_{\mathbf{P}^n_R} \to 0. }[/math]

(See also Chern class.)

Cotangent stack

For this notion, see § 1 of

A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves [1]

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, [math]\displaystyle{ \mathbf{Spec}(\operatorname{Sym}(\check{E})) }[/math] is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math] is the total space of the Hitchin fibration.)

See also

• cotangent complex

References

• "Sheaf of differentials of a morphism". https://stacks.math.columbia.edu/tag/01UM. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 

External links

• "Questions about tangent and cotangent bundle on schemes". Stack Exchange. November 2, 2014. https://math.stackexchange.com/q/1001941.