Cotangent sheaf

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of ${\mathcal {O}}_{X}$ -modules that represents (or classifies) S-derivations [1] in the sense: for any ${\mathcal {O}}_{X}$ -modules F, there is an isomorphism

\operatorname {Hom} _{{\mathcal {O}}_{X}}(\Omega _{X/S},F)=\operatorname {Der} _{S}({\mathcal {O}}_{X},F)

that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d:{\mathcal {O}}_{X}\to \Omega _{X/S}$ such that any S-derivation $D:{\mathcal {O}}_{X}\to F$ factors as $D=\alpha \circ d$ with some ${\displaystyle \alpha$ .

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:

If S →T is a morphism of schemes, then
$f^{*}\Omega _{S/T}\to \Omega _{X/T}\to \Omega _{X/S}\to 0.$
If Z is a closed subscheme of X with ideal sheaf I, then
$I/I^{2}\to \Omega _{X/S}\otimes _{O_{X}}{\mathcal {O}}_{Z}\to \Omega _{Z/S}\to 0.$ [3][4]

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 $f:X\to S$ 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.

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 $\mathbf {P} _{R}^{n}$ for the projective space over a ring R,

0\to \Omega _{\mathbf {P} _{R}^{n}/R}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}(-1)^{\oplus (n+1)}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}\to 0.

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, $\mathbf {Spec} (\operatorname {Sym} ({\check {E}}))$ 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.)

Notes

https://stacks.math.columbia.edu/tag/08RL
In concise terms, this means:
$\Theta _{X}{\overset {\mathrm {def} }{=}}{\mathcal {H}}om_{{\mathcal {O}}_{X}}(\Omega _{X},{\mathcal {O}}_{X})={\mathcal {D}}er({\mathcal {O}}_{X}).$
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

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.

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[1] ttps://stacks.math.columbia.edu/tag/08RL

[2] In concise terms, this means:
$\Theta _{X}{\overset {\mathrm {def} }{=}}{\mathcal {H}}om_{{\mathcal {O}}_{X}}(\Omega _{X},{\mathcal {O}}_{X})={\mathcal {D}}er({\mathcal {O}}_{X}).$

[3] Hartshorne, Ch. II, Proposition 8.12.

[4] ttps://mathoverflow.net/q/79956 as well as (Hartshorne, Ch. II, Theorem 8.17.)

[5] Hartshorne, Ch. II, Theorem 8.15.

[6] see also: § 3 of http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf