This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .
For a module a is - linear Derivation of having values in defined as a linear map , that is, for which the rule applies Leibniz
The set of all such derivatives forms a module, which with
referred to as.
Be further
the core of the multiplication, which is understood as a module via the left factor . The module of the Kähler differentials or the relative differentials is then
The universal derivation is the picture
It is a linear derivation.
Universal property
The following applies:
is an isomorphism. This can also be formulated as follows: The functor by the pair shown . In particular, this property is essentially uniquely determined.
The exact sequences
If a ring, an algebra, an algebra and a module, the following sequence is exact :
As a result, the corresponding sequence of relative differentials is exact:
Is special for an ideal in , so is , but one can give one more term in the exact sequence:
As a result, the following sequence of the modules of the Kähler differentials is exact:
Has characteristic , and is finitely generated, then if and only if is algebraic and separable. For example, if a nontrivial inseparable extension is, then is a one-dimensional vector space.