cotangent sheaf について

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

.〔In concise terms, this means:
:

\Theta_X \oversetom_(\Omega_X, \mathcal_X) = \mathcaler(\mathcal_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.

〔http://mathoverflow.net/questions/79956/jacobian-criterion-for-smoothness-of-schemes as well as 〕
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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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