Kderivations

K-derivations, or derivations over a base ring, are a fundamental concept in commutative algebra and algebraic geometry. Let k be a commutative ring, and let R be a k-algebra and M an R-module. A k-derivation is a map d: R → M that is k-linear and satisfies the Leibniz rule d(ab) = a d(b) + b d(a) for all a, b in R. When M = R, a k-derivation is a derivation of R into itself.

Der_k(R, M) denotes the set of all k-derivations from R to M. This set is naturally an R-module, with (r · d)(a) = r d(a) for r in R and a in R. The concept generalizes the ordinary notion of differentiation to algebraic structures and remains meaningful in singular settings.

A central construction is the module of Kähler differentials, Ω_{R/k}, together with the universal derivation d_{R/k}: R → Ω_{R/k}. This pair has the property that for every R-module M, there is a natural isomorphism Der_k(R, M) ≅ Hom_R(Ω_{R/k}, M). In other words, Ω_{R/k} represents the functor of k-derivations.

Concrete example: if R = k[x_1, ..., x_n] is a polynomial ring over k, then Ω_{R/k} is a free R-module with basis {dx_1, ..., dx_n}, and any derivation d is determined by its values on the generators x_i, with d(f) = ∑ (∂f/∂x_i) dx_i. Consequently Der_k(R, R) is a free R-module of rank n, generated by the formal partial derivatives ∂/∂x_i.

In geometry, Der_k(R, κ(p)) describes the tangent space at a point p of the affine scheme Spec R over k, linking algebraic and geometric notions of differentiation.