Cotangent sheaf

In algebraic geometry, given a morphism f: XS of schemes, the cotangent sheaf on X 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 sheaf on X and is sometimes denoted by .[2]

There are two important exact sequences:

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

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]

Construction through a diagonal morphism

Let be a morphism of schemes as in the introduction and Δ: XX ×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:

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.

See also: bundle of principal parts.

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 for the projective space over a ring R,

(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 [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, is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of is the total space of the Hitchin fibration.)


  2. In concise terms, this means:
  3. Hartshorne, Ch. II, Proposition 8.12.
  4. as well as (Hartshorne, Ch. II, Theorem 8.17.)
  5. Hartshorne, Ch. II, Theorem 8.15.
  6. see also: § 3 of

See also


  • "Sheaf of differentials of a morphism".
  • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
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