# 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 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