Derivations, module of

module of Kähler derivations

An algebraic analogue of the concept of the differential of a function. Let $ A $ be a commutative ring regarded as an algebra over a subring $ B $ of it. The module of derivations of the $ B $- algebra $ A $ is defined as the quotient module $ \Omega _ {A/B} ^ {1} $ of the free $ A $- module with basis $ ( dx ) _ {x \in A } $ by the submodule generated by the elements of the type

$$ d ( x + y ) - dx - dy ,\ d ( xy ) - x dy - y dx,\ db , $$

where $ x, y \in A $, $ b \in B $. The canonical homomorphism of $ A $- modules $ d: A \rightarrow \Omega _ {A/B} ^ {1} $ is a $ B $- derivation in the ring $ A $( cf. Derivation in a ring) with values in the $ A $- module $ \Omega _ {A/B} ^ {1} $ having the following universality property: For any $ B $- derivation $ \partial : A \rightarrow M $ with values in an $ A $- module $ M $ there exists a uniquely defined homomorphism of $ A $- modules $ \overline \partial \; : \Omega _ {A/B} ^ {1} \rightarrow M $ such that $ \overline \partial \; \circ d = \partial $. The correspondence $ \partial \rightarrow \overline \partial \; $ defines an isomorphism of $ A $- modules:

$$ \mathop{\rm Der} _ {B} ( A , M) \simeq \mathop{\rm Hom} _ {A} ( \Omega _ {A/B} ^ {1} , M). $$

In particular, the module of derivations of a ring $ A $ into itself is isomorphic to the dual $ A $- module to the module $ \Omega _ {A/B} ^ {1} $.

If $ A \otimes _ {B} A $ is regarded as an $ A $- algebra with respect to the homomorphism

$$ A \rightarrow A \otimes _ {B} A \ ( a \rightarrow a \otimes 1 ) $$

and $ I $ is the ideal generated by the elements of the type

$$ a \otimes 1 - 1 \otimes a , $$

then the $ A $- module $ \Omega _ {A/B} ^ {1} $ is isomorphic to the $ A $- module $ I / I ^ {2} $.

The module $ \Omega ^ {1} $ of derivations has the following properties:

1) If $ S $ is a multiplicatively closed set in $ A $ and $ T = S \cap B $, then there is a canonical localization isomorphism:

$$ ( \Omega _ {A/B} ^ {1} ) _ {S} \simeq \Omega _ {A _ {S} / B _ {T} } ^ {1} . $$

2) If $ \phi : A \rightarrow A _ {1} $ is a homomorphism of $ B $- algebras, then there is a canonical exact sequence of $ A _ {1} $- modules:

$$ \Omega _ {A/B} ^ {1} \otimes _ { A } A _ {1} \mathop \rightarrow \limits ^ \alpha \Omega _ {A _ {1} / B } ^ {1} \rightarrow \Omega _ {A _ {1} / A } \rightarrow 0 . $$

3) If $ I $ is an ideal of the ring $ A $ and $ A _ {1} = A/I $, then there is an exact canonical sequence of $ A _ {1} $- modules:

$$ I / I ^ {2} \rightarrow ^ { {d _ 1} } \Omega _ {A/B} ^ {1} \otimes _ { A } A _ {1} \rightarrow \ \Omega _ {A _ {1} / B } ^ {1} \rightarrow 0 , $$

where the homomorphism $ d _ {1} $ is induced by the derivation $ d: A \rightarrow \Omega _ {A/B} ^ {1} $.

4) A field $ K $ is a separable extension of a field $ k $ of finite transcendence degree $ n $ if and only if there is a $ K $- space isomorphism $ \Omega _ {K/k} ^ {1} \simeq K ^ {n} $.

5) If $ A = B [ T _ {1} \dots T _ {n} ] $ is an algebra of polynomials, then $ \Omega _ {A/B} ^ {1} $ is a free $ A $- module with as basis $ dT _ {1} \dots dT _ {n} $.

6) An algebra $ A $ of finite type over a perfect field $ k $ is a regular ring if and only if the $ A $- module $ \Omega _ {A/k} ^ {1} $ is projective.

7) Concerning 2) above, the $ A $- algebra $ A _ {1} $ of finite type is smooth over $ A $ if and only if the homomorphism $ \alpha $ is injective while the module $ \Omega _ {A _ {1} / A } ^ {1} $ of derivations is projective and its rank is equal to the relative dimension of $ A _ {1} $ over $ A $.

The $ i $- th exterior power $ \wedge ^ {i} \Omega _ {A/B} ^ {1} $ of the module $ \Omega _ {A/B} ^ {1} $ of derivations is said to be the module of (differential) $ i $- forms of the $ B $- algebra $ A $ and is denoted by $ \Omega _ {A/B} ^ {i} $.

By virtue of 1) it is possible to define, for any morphism of schemes $ X \rightarrow Y $, the sheaf of relative (or Kähler) derivations $ \Omega _ {X/Y} ^ {1} $ and its exterior powers $ \Omega _ {X/Y} ^ {i} $.

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 schémas et des morphismes de schémas" Publ. Math. IHES , 20 (1964) MR0173675
[4]	E. Kähler, "Algebra und Differentialrechnung" , Deutsch. Verlag Wissenschaft. (1958) MR0094593 Zbl 0079.05701
[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001