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 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 .
There are two important exact sequences:
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.)
For this notion, see § 1 of
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.)
- 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
- "Questions about tangent and cotangent bundle on schemes". Stack Exchange. November 2, 2014.