Derivations

Derivations are algebraic analogues of differentiation that appear in many areas of mathematics. Let k be a commutative ring and A an associative k-algebra. A derivation of A is a k-linear map D: A → M, where M is an A-bimodule, satisfying the Leibniz rule D(ab) = D(a)b + a D(b) for all a, b in A. When M = A with its standard bimodule structure, D is simply a derivation of A.

A familiar example is the usual derivative on the polynomial ring k[x]: D(f) = f′, which is k-linear and satisfies D(fg) = f′g + fg′. In any associative algebra, a common source of derivations is inner derivations: for a fixed a ∈ A, the map ad(a)(b) = ab − ba is a derivation because ad(a)(bc) = [a,b]c + b[a,c]. In a commutative algebra, all inner derivations vanish.

The collection Der_k(A, M) of k-derivations from A to an A-bimodule M carries natural algebraic structure. In the commutative case, Der_k(A) ⊆ End_k(A) forms an A-module and, with the commutator bracket [D1, D2] = D1D2 − D2D1, a Lie algebra. The set Der_k(A) can be studied via cohomology (as part of Hochschild cohomology) and through exact sequences relating inner and outer derivations.

Geometrically, derivations encode tangent directions. If A is the coordinate ring of a variety or the structure sheaf of a scheme X over k, Der_k(A) corresponds to vector fields, and the sheaf Der_k(O_X) is the tangent sheaf. In differential geometry, the derivations of smooth functions on a manifold yield the usual notion of tangent vectors.

Variants include graded or superderivations, and derivations play key roles in deformation theory, algebraic geometry, and the theory of differential operators.