# Cotangent sheaf

In algebraic geometry, given a morphism *f*: *X* → *S* 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:

- If
*S*→*T*is a morphism of schemes, then - 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 Ω_{X} 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: *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:

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.)

## Notes

- https://stacks.math.columbia.edu/tag/08RL
- In concise terms, this means:
- Hartshorne, Ch. II, Proposition 8.12.
- https://mathoverflow.net/q/79956 as well as (Hartshorne, Ch. II, Theorem 8.17.)
- Hartshorne, Ch. II, Theorem 8.15.
- see also: § 3 of http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf

## See also

## References

- "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

## External links

- "Questions about tangent and cotangent bundle on schemes".
*Stack Exchange*. November 2, 2014.