Derivations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 25 June 2012

Let $𝕂$ be a commutative ring, let $A$ be a $𝕂 -$ algebra and let $E$ be an $(A, A) -$ bimodule.

A $𝕂 -$ derivation from $A$ to $E$ is a $𝕂 -$ linear map $d : A \to E$ such that $if x, y \in A then d (x y) = x (d y) + (d x) y .$

A $𝕂 -$ derivation of $A$ is a $𝕂 -$ linear map $d : A \to E$ such that $if x, y \in A then d (x y) = x (d y) + (d x) y .$

Let $A$ be a $𝕂 -$ algebra with multiplication $m : A \otimes_{𝕂} A \to A and let I = \ker m .$

$\begin{array}{rrcl} δ_{A} : & A & \to & I \\ x & \mapsto & x \otimes 1 - 1 \otimes x \end{array}$ is a derivation and $I = A \cdot im δ_{A} .$
If $E$ is an $(A, A) -$ bimodule and $d : A \to E$ is a $𝕂 -$ derivation then there exists a unique $(A, A) -$ bimodule homomorphism $f : I \to E$ such that $d = f \circ δ_{A} .$ $\begin{array}{rcl} {Hom}_{(A, A)} (I, E) & \tilde{\to} & {Der}_{𝕂} (A, E) \\ f & \mapsto & f \circ δ_{A} \end{array} \begin{matrix} \end{matrix}$
If $A$ is commutative then $A \otimes_{𝕂} A$ is a $𝕂 -$ algebra with product $(a_{1} \otimes a_{2}) (b_{1} \otimes b_{2}) = a_{1} b_{1} \otimes a_{2} b_{2} .$

c1. $m : A \otimes A \to A$ provides a $𝕂 -$ algebra isomorphism $\frac{A \otimes A}{I} ≃ A .$
c2. $\begin{array}{rrcl} d_{A / 𝕂} : & A & \to & I / I^{2} \\ x & \mapsto & x \otimes 1 - 1 \otimes x \end{array}$ is a $𝕂 -$ derivation.
c3. If $E$ is an $A -$ module and $D : A \to E$ is a $𝕂 -$ derivation then there exists a unique $A -$ module homomorphism $g : I / I^{2} \to E$ such that $D = g \circ d_{A / 𝕂} .$ ${Hom}_{(A, A)} (I, E) ≃ {Hom}_{A \otimes_{𝕂} A} (I, E) ≃ {Hom}_{A} (I / I^{2}, E)$ and $\begin{matrix} \end{matrix}$

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