Definitions and examples of derivatives and derivations

Derivatives. Let A be any ring. There is an A-linear map from A[X] to A[X], called the derivative with respect to X, taking Xn to nXn-1 for each nonnegative integer n. We use the familiar notation f' for the derivative of f. Besides A-linearity, the derivative satisfies the product rule (fg)'=fg'+f'g. This can be proved either directly (and somewhat laboriously), or by giving an alternative definition of the derivative of f that naturally yields the product rule and agrees with the nXn-1 formula.

This alternative definition is an algebraic interpretation of the familiar limit definition from the differential calculus. Adjoin a further indeterminate h, and consider the polynomial f(X+h) in A[X,h]. Expand it in powers of h: f(X+h)=f0(X)+hf1(X)+h2f2(X)+...+hnfn(X) (where n is the degree of f). Clearly f0(X)=f(X+0)=f(X). By definition, f' is the polynomial f1(X). That is, f' is the unique element of A[X] for which f(X+h) is congruent to f(X)+hf'(X) mod h2 in A[X,h]. It is readily checked that f' is an A-linear function from A[X] to A[X] that takes A to 0 and X to 1 and satisfies the product rule. The formula for the derivative of Xn then follows by induction on n.

Derivations. More generally, let B be any ring containing A, and consider the notion of an A-linear map D on B that satisfies the product rule D(fg)=f*Dg+g*Df. This makes sense even if D takes values not in B but in some B-module M. Such a function D is called a derivation (or an A-derivation) from B to M. An example from differential geometry is the map d from the ring B of smooth functions on a manifold to the B-module of smooth 1-forms on the same manifold (here A=R, the ring of constant functions). The set of all A-derivations from B to M is itself a B-module, which we may call DerA(B,M). For example, if B=A[X] and M=B then DerA(B,M) consists of all maps of the form D(f)=af' for some a in B. More generally, if B is the polynomial ring in k variables over A then DerA(B,B) consists of all B-linear combinations of the k ``partial derivatives'' with respect to those variables.

Let D be any A-derivation on B, and let C be its kernel. By taking f=g=1 in the product rule we see that D(1)=0. Thus by linearity D(a)=0 for all a in A. Hence C contains A. Also by the product rule, C is closed under multiplication, and is thus a subring of B, called the ``ring of constants'' for the derivation D; and D is C-linear. For instance, if A is a field, B=A[X], and D is the usual derivative, then the ring of constants is A if the field has characteristic zero, and A[Xp] if it has characteristic p.

What about the derivative on the field F(X) of rational functions in one variable? If that is to be an F-derivation from F(X) to F(X) extending the derivative on F[X], we can determine (f/g)' for any f,g in F[X] by the product rule for f=(f/g)g. Differentiating, we find f'=(f/g)g'+(f/g)'g; solving for (f/g)' yields the ``quotient rule'' (f/g)'=(f'g-fg')/g^2. We can then check that (f/g)' does not depend on the choice of representation of a given rational function as f/g, and does yield an F-derivation on F(X). More generally, if A is a subring of B which in turn is contained in somce field F, and D is an A-derivation from B to an F-vector space V, we can extend D to the quotient field of B by defining D(f/g)=(f'g-fg')/g^2, and check that this is well-defined and yields an A-derivation on that quotient field.

What happens in algebraic field extensions? Now suppose K is an algebraic extension of the field F, and D is an F0-derivation from F to some K-vector space V (where F0 is some subfield of F; for instance, F might be the transcendental extension F0(X), and we could have V=K and D = the derivative from F0(X) to F0(X) in V). When can we extend D to an F0-derivation on K?