Cotangent Sheaf

In algebraic geometry, given a morphism ''f'': ''X'' → ''S'' of schemes, the cotangent sheaf on ''X'' is the sheaf of $\mathcal_X$-modules $\Omega_$ that represents (or classifies) ''S''- derivations in the sense: for any $\mathcal_X$-modules ''F'', there is an isomorphism
:$\operatorname_(\Omega_, F) = \operatorname_S(\mathcal_X, F)$
that depends naturally on ''F''. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d: \mathcal_X \to \Omega_$ such that any ''S''-derivation $D: \mathcal_X \to F$ factors as $D = \alpha \circ d$ with some $\alpha: \Omega_ \to F$.

In the case ''X'' and ''S'' are affine schemes, the above definition means that $\Omega_$ 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 $\Theta_X$.

There are two important exact sequences:
#If ''S'' →''T'' is a morphism of schemes, then
#:$f^* \Omega_ \to \Omega_ \to \Omega_ \to 0.$
#If ''Z'' is a closed subscheme of ''X'' with ideal sheaf ''I'', then
#:$I/I^2 \to \Omega_ \otimes_ \mathcal_Z \to \Omega_ \to 0.$

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:
:$\Omega_ = \Delta^* (I/I^2)$
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 $\mathbf^n_R$ for the projective space over a ring ''R'',
:$0 \to \Omega_ \to \mathcal_(-1)^ \to \mathcal_ \to 0.$

(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 eigensheave

see also: § 3 of http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf
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, $\mathbf(\operatorname(\check))$ is the algebraic vector bundle corresponding to ''E''.)

See also: Hitchin fibration (the cotangent stack of $\operatorname_G(X)$ is the total space of the Hitchin fibration.)

Notes

See also

*cotangent complex

References

*
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External links

*{{cite web , title=Questions about tangent and cotangent bundle on schemes , date=November 2, 2014 , work=Stack Exchange , url=https://math.stackexchange.com/q/1001941

Algebraic geometry