Cotangent sheaf explained

\Omega _{X/S} that represents (or classifies) S-derivations^[1] in the sense: for any l{O} _{X} -modules F, there is an isomorphism\operatorname{Hom} _{l{O} _{X}(\Omega} _{X/S} , F) = \operatorname{Der} _{S(l{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: l{O} _{X} \to \Omega _{X/S} such that any S-derivation D: l{O} _{X} \to F factors as D = \alpha \circ d with some \alpha: \Omega _{X/S} \to F .

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

\Omega _{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 \Theta _{X} .^[2]

There are two important exact sequences:

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

f ^{*} \Omega _{S/T} \to \Omega _{X/T} \to \Omega _{X/S} \to 0 .

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

I/I ^{2} \to \Omega _{X/S} ⊗ _{O _{X}} l{O} _{Z} \to \Omega _{Z/S} \to 0 . ^[3]

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

f: X \to 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:\Omega _{X/S} = \Delta ^{*} (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.

See also: bundle of principal parts.

Relation to a tautological line bundle

See main article: Euler sequence.

The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing

P ^{n} _{R} for the projective space over a ring R,0 \to \Omega _{P ^{n} _{R/R}} \to l{O} _{P ^{n} _{R}} ( -1 ) ^{⊕ (n+1)} \to l{O} _{P ^{n} _{R}} \to 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 http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf ^[4] 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,

Spec (\operatorname{Sym}(\check{E})) is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of

\operatorname{Bun} _{G(X)} is the total space of the Hitchin fibration.)

See also

• cotangent complex

References

• Web site: Sheaf of differentials of a morphism .

External links

• Web site: Questions about tangent and cotangent bundle on schemes . November 2, 2014 . Stack Exchange .

Notes and References

1. Web site: Section 17.27 (08RL): Modules of differentials—The Stacks project.
2. In concise terms, this means:\Theta _{X} \overset{ def } = \mathcalom_(\Omega_X, \mathcal_X) = \mathcaler(\mathcal_X).
3. https://mathoverflow.net/q/79956 as well as
4. see also: § 3 of http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf