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 chec