## Derivations

Last update: 25 June 2012

## Derivations

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

A $𝕂-$derivation from $A$ to $E$ is a $𝕂-$linear map $d:A\to E$ such that

A $𝕂-$derivation of $A$ is a $𝕂-$linear map $d:A\to E$ such that

Let $A$ be a $𝕂-$algebra with multiplication $m:A{\otimes }_{𝕂}A\to A\phantom{\rule{2em}{0ex}}and let\phantom{\rule{2em}{0ex}}I=\mathrm{ker}m.$

1. $\begin{array}{rrcl}{\delta }_{A}:& A& \to & I\\ & x& ↦& x\otimes 1-1\otimes x\end{array}$ is a derivation and $I=A\cdot \mathrm{im}{\delta }_{A}.$
2. If $E$ is an $\left(A,A\right)-$bimodule and $d:A\to E$ is a $𝕂-$derivation then there exists a unique $\left(A,A\right)-$bimodule homomorphism $f:I\to E$ such that $d=f\circ {\delta }_{A}.$ $Hom(A,A)(I,E) →∼ Der𝕂(A,E) f ↦ f∘δA A E I d δA f$
3. If $A$ is commutative then $A{\otimes }_{𝕂}A$ is a $𝕂-$algebra with product $(a1⊗a2) (b1⊗b2) = a1b1⊗a2b2.$
1. c1. $m:A\otimes A\to A$ provides a $𝕂-$algebra isomorphism $A⊗A I ≃ A.$
2. c2. $\begin{array}{rrcl}{d}_{A/𝕂}:& A& \to & I/{I}^{2}\\ & x& ↦& x\otimes 1-1\otimes x\end{array}$ is a $𝕂-$derivation.
3. 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 $D=g\circ {d}_{A/𝕂}.$ $Hom(A,A)(I,E) ≃ HomA⊗𝕂A(I,E) ≃ HomA( I/I2,E )$ and $A E I/I2 D dA/𝕂 g$

References?