Cotangent sheaf

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

${\mathrm{Hom}}_{{{𝒪}}_X}({\mathrm{\Omega }}_{X/S},F)={\mathrm{Der}}_S({{𝒪}}_X,F)$

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

In the case X and S are affine schemes, the above definition means that ${\mathrm{\Omega }}_{X/S}$ 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 ${\mathrm{\Theta }}_X$.

There are two important exact sequences:

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

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

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

   $I/I^2\to {\mathrm{\Omega }}_{X/S}{\otimes }_{O_X}{{𝒪}}_Z\to {\mathrm{\Omega }}_{Z/S}\to 0.$

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:

${\mathrm{\Omega }}_{X/S}={\mathrm{\Delta }}^{*}(I/I^2)$

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,

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

(See also Chern class.)

Cotangent stack

For this notion, see § 1 of

A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves [1]

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, $\mathbf{𝐒}\mathbf{𝐩}\mathbf{𝐞}\mathbf{𝐜}(\mathrm{Sym}(\check{E}))$ is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of ${\mathrm{Bun}}_G(X)$ is the total space of the Hitchin fibration.)

See also

• cotangent complex

References

• "Sheaf of differentials of a morphism". https://stacks.math.columbia.edu/tag/01UM. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 

External links

• "Questions about tangent and cotangent bundle on schemes". Stack Exchange. November 2, 2014. https://math.stackexchange.com/q/1001941.