TITLE: Lie algebra of Derivations as a functor?
QUESTION [7 upvotes]: To an associative algebra $A$ one can associate a Lie algebra $\operatorname{Der} A$ of all derivations $D:A\to A$.
To any morphism of associative algebras $\alpha:A\to B$, how can one associate a morphism of Lie algebras $\alpha_\ast:\operatorname{Der} A\to \operatorname{Der} B$ or $\alpha^\ast:\operatorname{Der} B\to \operatorname{Der} A$ in a functorial way?

REPLY [3 votes]: You can't. You should think of $\text{Der}(A)$ as infinitesimal automorphisms of $A$, or equivalently as vector fields on $\text{Spec } A$, and so it isn't functorial with respect to arbitrary maps in the same way that taking automorphism groups or vector fields of manifolds isn't functorial. (However, like taking automorphism groups or vector fields, it is functorial with respect to isomorphisms.)