Derivations

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

Last updates: 25 November 2011

Derivations

Let $𝔽$ be a field. A vector space over $𝔽$ is an abelian group $V$ with a function $$\begin{array}{ccc}𝔽\times V& ⟶& V\\ (c,v)& ⟼& cv\end{array}$$ such that

1. if $c_1,c_2\in 𝔽$ and $v\in V$ then $(c_1+c_2)v=c_{1}v+c_{2}v$,
2. if $c\in 𝔽$ and $v_1,v_2\in V$ then $c(v_1+v_2)=c{v}_1+c{v}_2$,
3. if $c_1,c_2\in 𝔽$ and $v\in V$ then $c_1\left(c_{2}v\right)=\left(c_1{c}_2\right)v$, and
4. if $v\in V$ then $1\cdot v=v$.

Let $𝔽$ be a field. Let $V,W$ be vector spaces over $𝔽$. An $𝔽$-linear map from $V$ to $W$ is a function $\phi :V\to W$ such that

1. $\phi$ is a group homomorphism,
2. if $c\in 𝔽$ and $v\in V$ then $\phi \left(cv\right)=c\phi \left(v\right)$,

Let $𝔽$ be a field. An algebra is a vector space $A$ over $𝔽$ with an function $$\begin{array}{ccc}A\times A& ⟶& A\\ (a_1,a_2)& ⟼& a_1{a}_2\end{array}$$ such that $A$ is a ring and scalar multiplication is the composition of the map $$\begin{array}{ccc}𝔽& ⟶& A\\ \xi & ⟼& \xi \cdot 1\end{array}$$ and the multiplication in $A$.

Let $𝔽$ be a field. Let $A$ be an $𝔽$-algebra. A derivation of $A$ is an $𝔽$-linear map $d:A\to A$ such that $$\text{if}{a}_1,a_2\in A\text{then}d\left(a_1{a}_2\right)=a_{1}d\left(a_2\right)+d\left(a_1\right)a_2.$$

1. There is a unique derivation ${\displaystyle \frac{d}{dx}}$ if $𝔽\left[x\right]$ such that ${\displaystyle \frac{dx}{dx}}$ =1.
2. If $p\in 𝔽\left[x\right]$ then $$\frac{dp}{dx}=\left(\text{coefficient of}y\text{in}p\right(x+y\left)\right).$$
3. If $p\in 𝔽\left[x\right]$ then $$p=\sum _{k\in {\mathbb{Z}}_{\ge 0}}{\left({\left(\frac{d}{dx}\right)}^{k}p\right)|}_{x=0}{x}^k$$
4. There is a unique extension of ${\displaystyle \frac{d}{dx}}$ to a derivation of $𝔽\left(x\right)$.
5. There is a unique extension of ${\displaystyle \frac{d}{dx}}$ to a derivation of $𝔽\left[\right[x\left]\right]$.
6. There is a unique extension of ${\displaystyle \frac{d}{dx}}$ to a derivation of $𝔽\left(\right(x\left)\right)$.
7. If $p\in 𝔽\left[\right[x\left]\right]$ then $$\frac{dp}{dx}=\left(\text{coefficient of}y\text{in}p\right(x+y\left)\right).$$
8. If $p\in 𝔽\left[\right[x\left]\right]$ then $$p=\sum _{k\in {\mathbb{Z}}_{\ge 0}}{\left({\left(\frac{d}{dx}\right)}^{k}p\right)|}_{x=0}{x}^k$$

Notes and References

These notes provide a bridge between an introductory calculus course and the use of derivations in the definitions of tangent spaces in algebraic geometry.

References

[BouTop] N. Bourbaki, General Topology, Chapter VI, Springer-Verlag, Berlin 1989. MR?????

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