Kähler differential

The notion of Kähler differential ( after E. Kähler ) is an algebraic abstraction of the Leibniz rule from the mathematical subfield of differential calculus.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Definition

There was a ring and a - algebra.

For a module A - linear derivation of with values ​​in is defined as a linear map for which the Leibniz rule, ie

The set of all such derivations forms a module, with the

Is called.

Next was

The core of multiplication, would be perceived on the left factor as a module. The module of Kähler differentials or relative differentials is then

The universal derivation is the mapping

It is an - linear derivation.

Universal property

The following applies:

Is an isomorphism. One can also express this as: The functor is represented by the pair. In particular, is uniquely determined by this property essentially.

The exact sequences

• Is a ring, algebra, an algebra and a module, the following sequence is precisely:

• Is designed for an ideal in, then, but you can still specify another term in the exact sequence:

Differentials and field extensions

It should be a field extension.

• Has characteristic 0, it is equal to the transcendence degree of.
• Has characteristics, and is finitely generated, as accurately applies if is algebraic and separable. For example, if a non-trivial inseparable extension, so is a one-dimensional vector space.

Examples

• If, as is a free module with generators.

• Commutative Algebra