Derivations, module of

module of Kähler derivations

An algebraic analogue of the concept of the differential of a function. Let be a commutative ring regarded as an algebra over a subring of it. The module of derivations of the -algebra is defined as the quotient module of the free -module with basis by the submodule generated by the elements of the type

where , . The canonical homomorphism of -modules is a -derivation in the ring (cf. Derivation in a ring) with values in the -module having the following universality property: For any -derivation with values in an -module there exists a uniquely defined homomorphism of -modules such that . The correspondence defines an isomorphism of -modules:

In particular, the module of derivations of a ring into itself is isomorphic to the dual -module to the module .

If is regarded as an -algebra with respect to the homomorphism

and is the ideal generated by the elements of the type

then the -module is isomorphic to the -module .

The module of derivations has the following properties:

1) If is a multiplicatively closed set in and , then there is a canonical localization isomorphism:

2) If is a homomorphism of -algebras, then there is a canonical exact sequence of -modules:

3) If is an ideal of the ring and , then there is an exact canonical sequence of -modules:

where the homomorphism is induced by the derivation .

4) A field is a separable extension of a field of finite transcendence degree if and only if there is a -space isomorphism .

5) If is an algebra of polynomials, then is a free -module with as basis .

6) An algebra of finite type over a perfect field is a regular ring if and only if the -module is projective.

7) Concerning 2) above, the -algebra of finite type is smooth over if and only if the homomorphism is injective while the module of derivations is projective and its rank is equal to the relative dimension of over .

The -th exterior power of the module of derivations is said to be the module of (differential) -forms of the -algebra and is denoted by .

By virtue of 1) it is possible to define, for any morphism of schemes , the sheaf of relative (or Kähler) derivations and its exterior powers .

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	A. Grothendieck (ed.) et al. (ed.) , Revêtements étales et groupe fondamental. SGA 1 , Lect. notes in math. , 224 , Springer (1971) MR0354651 Zbl 1039.14001
[3]	A. Grothendieck, "Eléments de géométrie algébrique IV. Etude locale des schémes et des morphismes de schémes" Publ. Math. IHES , 20 (1964) MR0173675
[4]	E. Kähler, "Algebra und Differentialrechnung" , Deutsch. Verlag Wissenschaft. (1958) MR0094593 Zbl 0079.05701

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001