Kähler differential

The concept of the Kähler differential (after E. Kähler ) is an algebraic abstraction of the Leibniz rule from the mathematical sub-area of differential calculus .

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 .

definition

Let it be a ring and an algebra . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$

For a module a is - linear Derivation of having values in defined as a linear map , that is, for which the rule applies Leibniz ${\displaystyle B}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle D\colon B\to M}$

${\displaystyle D(b_{1}b_{2})=b_{1}D(b_{2})+b_{2}D(b_{1}).}$

The set of all such derivatives forms a module, which with ${\displaystyle B}$

${\displaystyle \mathrm {Der} _{A}(B,M)}$

referred to as.

Be further

${\displaystyle I:=\ker(B\otimes _{A}B\to B)}$

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 ${\displaystyle B}$

${\displaystyle \Omega _{B/A}:=I/I^{2}.}$

The universal derivation is the picture

${\displaystyle \mathrm {d} \colon B\to \Omega _{B/A},\quad b\mapsto \mathrm {d} b:=b\otimes 1-1\otimes b.}$

It is a linear derivation. ${\displaystyle A}$

Universal property

The following applies:

${\displaystyle \mathrm {Hom} _{B}(\Omega _{B/A},M)\to \mathrm {Der} _{A}(B,M),\quad f\mapsto f\circ \mathrm {d} ,}$

is an isomorphism. This can also be formulated as follows: The functor by the pair shown . In particular, this property is essentially uniquely determined. ${\displaystyle \mathrm {Der} _{A}(B,{-})}$ ${\displaystyle (\Omega _{B/A},\mathrm {d} )}$ ${\displaystyle \Omega _{B/A}}$

The exact sequences

• If a ring, an algebra, an algebra and a module, the following sequence is exact : ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle M}$ ${\displaystyle C}$

${\displaystyle 0\longrightarrow \mathrm {Der} _{B}(C,M)\longrightarrow \mathrm {Der} _{A}(C,M)\longrightarrow \mathrm {Der} _{A}(B,M).}$

As a result, the corresponding sequence of relative differentials is exact:

${\displaystyle \Omega _{B/A}\otimes _{B}C\longrightarrow \Omega _{C/A}\longrightarrow \Omega _{C/B}\longrightarrow 0.}$

• Is special for an ideal in , so is , but one can give one more term in the exact sequence: ${\displaystyle C=B/I}$ ${\displaystyle I}$ ${\displaystyle B}$ ${\displaystyle \mathrm {Der} _{B}(C,M)=0}$

${\displaystyle 0\longrightarrow \mathrm {Der} _{A}(B/I,M)\longrightarrow \mathrm {Der} _{A}(B,M)\longrightarrow \mathrm {Hom} _{B/I}(I/I^{2},M)}$

As a result, the following sequence of the modules of the Kähler differentials is exact:

${\displaystyle I/I^{2}\longrightarrow \Omega _{B/A}\otimes _{B}B/I\longrightarrow \Omega _{(B/I)/A}\longrightarrow 0.}$

Differentials and body extensions

It is an extension of the body . ${\displaystyle L/K}$

• If characteristic has 0, then is equal to the degree of transcendence of . ${\displaystyle K}$ ${\displaystyle \dim _{L}\Omega _{L/K}}$ ${\displaystyle L/K}$
• 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. ${\displaystyle K}$ ${\displaystyle p>0}$ ${\displaystyle L/K}$ ${\displaystyle \Omega _{L/K}=0}$ ${\displaystyle L/K}$ ${\displaystyle L=K({\sqrt[{p}]{a}})}$ ${\displaystyle \Omega _{L/K}}$ ${\displaystyle L}$

Examples

• Is , then is a free module with producers . ${\displaystyle B=A[X_{1},\ldots ,X_{n}]}$ ${\displaystyle \Omega _{B/A}}$ ${\displaystyle B}$ ${\displaystyle \mathrm {d} X_{1},\ldots ,\mathrm {d} X_{n}}$