dérivations

Dérivations are linear operators that generalize the familiar derivative from calculus to algebraic settings. Let R be a commutative ring and A an R-algebra. An R-derivation with values in an A-module M is an R-linear map D: A → M that satisfies the Leibniz rule D(ab) = a D(b) + D(a) b for all a, b in A. When M = A, D is simply called a derivation of A.

Examples illustrate the concept. For the polynomial ring k[x1, ..., xn] over a field k, the partial derivatives ∂/∂xi are derivations. In differential geometry, on a smooth manifold M, the C∞-algebra of functions C∞(M) has derivations that correspond to vector fields: each vector field X defines a derivation X(f) = df(X) and satisfies X(fg) = X(f)g + f X(g).

Structure and basic properties. For an R-algebra A and an A-module M, the set Der_R(A, M) of R-derivations is an A-module with (a·D)(b) = a D(b). If M = A and A is commutative, Der_R(A) forms a Lie algebra under the commutator [D1, D2] = D1∘D2 − D2∘D1. In noncommutative algebras, inner derivations ad_x(y) = [x, y] = xy − yx are always derivations, and inner derivations need not exhaust all derivations; in the commutative case all inner derivations vanish.

Relation to differentials. There is a universal object, the module of Kähler differentials Ω_{A/R}, together with a universal derivation d: A → Ω_{A/R}. For any A-module M, Der_R(A, M) ≅ Hom_A(Ω_{A/R}, M). This universal property underpins many constructions in algebraic geometry and deformation theory.

Applications. Dérivations play a central role in algebraic geometry, differential geometry, deformation theory, and the study of dynamical systems, providing a formal language for infinitesimal changes and symmetries of algebras and spaces.