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Cotangent sheaf

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In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X -modules that represents (or classifies) S-derivations in the sense: for any O X -modules F, there is an isomorphism

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Hom O X ⁡ ( Ω X / S , F ) = Der S ⁡ ( 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 : O X → Ω X / S such that any S-derivation D : O X → F factors as D = α ∘ d with some α : Ω X / S → F .

In the case X and S are affine schemes, the above definition means that Ω 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 Θ X .

There are two important exact sequences:

  1. If S →T is a morphism of schemes, then f ∗ Ω S / T → Ω X / T → Ω X / S → 0.
  2. If Z is a closed subscheme of X with ideal sheaf I, then I / I 2 → Ω X / S ⊗ O Z → Ω Z / S → 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 ΩX is a locally free sheaf of rank n.

Construction through a diagonal morphism

Let f : X → 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:

Ω X / S = Δ ∗ ( 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 P R n for the projective space over a ring R,

0 → Ω P R n / R → O P R n ( − 1 ) ⊕ ( n + 1 ) → O P R n → 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]

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, S p e c ( Sym ⁡ ( E ˇ ) ) is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of Bun G ⁡ ( X ) is the total space of the Hitchin fibration.)

References

Cotangent sheaf Wikipedia
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