For derivations in logic, see deduction.


Derivations on an algebra

For AA an algebra (over some ring kk), a derivation on AA is a kk-linear morphism

d:AA d : A \to A

such that for all a,bAa,b \in A we have

d(ab)=d(a)b+ad(b), d(a b) = d(a) b + a d (b) \,,

This identity is called the Leibniz rule; compare it to the product rule in ordinary calculus (first written down by Gottfried Leibniz).

Derivations with values in a bimodule

For AA an algebra (over some ring kk) and NN a bimodule over AA, a derivation of AA with values in NN is a kk-linear morphism

d:AN d : A \to N

such that for all a,bAa,b \in A we have

d(ab)=d(a)b+ad(b), d(a b) = d(a) \cdot b + a \cdot d (b) \,,

where on the dot on the right-hand side denotes the right (first term) and left (second term) action of AA on the bimodule NN.

The previous definition is a special case of this one, where the bimodule is N=AN = A, the algebra itself with its canonical left and right action on itself.

A special case is where AA is a group algebra kGk G of a group GG, and NN is a left GG-module, regarded as a bimodule where the right action is trivial. Here a derivation is also called a 1-cocycle, as used in group cohomology.

Graded derivations

A graded derivation of degree pp on a graded algebra AA is a degree-pp graded-module homomorphism d:AAd: A \to A such that

d(ab)=d(a)b+(1) pqad(b) d(a b) = d(a) b + (-1)^{p q} a d(b)

whenever aa is homogeneous of degree qq. (By default, the grade is usually 11, or sometimes 1-1.)

Augmented derivations

An augmented derivation on an algebra AA, augmented by an algebra homomorphism ϵ:AB\epsilon: A \to B, is a module homomorphism d:ABd: A \to B such that

d(ab)=d(a)ϵ(b)+ϵ(a)d(b). d(a b) = d(a) \epsilon(b) + \epsilon(a) d(b) .

If you think about it, you should be able to figure out the definition of an augmented graded derivation.


Let kAlgk Alg denote the category of commutative kk-algebras, and kAlg/kk Alg/k the slice, i.e., the category of commutative kk-algebras AA equipped with an augmentation ϵ:Ak\epsilon: A \to k. Then the functor

Der:(kAlg/k) opkModDer: (k Alg/k)^{op} \to k Mod

which takes an augmented algebra to the module of augmented derivations d:Akd: A \to k is represented by the augmented algebra eval 0:k[x]/(x 2)keval_0: k[x]/(x^2) \to k.


Indeed, an algebra map δ:Ak[x 2]/(x 2)\delta: A \to k[x^2]/(x^2) which renders the triangle

A δ k[x]/(x 2) ϵ eval 0 k\array{ A & \stackrel{\delta}{\to} & k[x]/(x^2) \\ & \mathllap{\epsilon} \searrow & \downarrow \mathrlap{eval_0} \\ & & k }

commutative may be uniquely written in the form δ(a)=ϵ(a)+d(a)x\delta(a) = \epsilon(a) + d(a)x for some kk-module map d:Akd: A \to k, and then we have

ϵ(ab)+d(ab)x = δ(ab) = δ(a)δ(b) = (ϵ(a)+d(a)x)(ϵ(b)+d(b)x) = ϵ(a)ϵ(b)+(ϵ(a)d(b)+d(a)ϵ(b))x+d(a)d(b)x 2 = ϵ(ab)+(ϵ(a)d(b)+d(a)ϵ(b))x\array{ \epsilon(a b) + d(a b)x & = & \delta(a b) \\ & = & \delta(a)\delta(b) \\ & = & (\epsilon(a) + d(a)x)(\epsilon(b) + d(b)x) \\ & = & \epsilon(a)\epsilon(b) + (\epsilon(a)d(b) + d(a)\epsilon(b))x + d(a)d(b) x^2 \\ & = & \epsilon(a b) + (\epsilon(a)d(b) + d(a)\epsilon(b))x }

and we conclude d(ab)=ϵ(a)d(b)+d(a)ϵ(b)d(a b) = \epsilon(a)d(b) + d(a)\epsilon(b), so that dd is an augmented derivation. Of course the calculation may be run in reverse, so that δ\delta is an augmented algebra homomorphism precisely when dd is a derivation.

Further variations

There are many further extensions, for examples derivations with values in an AA-bimodule MM forming Der k(A,M)Hom k(A,M)Der_k(A,M) \subset Hom_k(A,M) (see also double derivation), skew-derivations in ring theory (with a twist in the Leibniz rule given by an endomorphism of a ring) and the dual notion of a coderivation of a coalgebra. The latter plays role in Koszul-dual definitions of A A_\infty-algebras and L L_\infty-algebras. See also derivation on a group, which uses a modified Leibniz rule: d(ab)=d(a)+ad(b)d(a b) = d(a) + a d(b).

Derivations on algebras over a dg-operad

More generally, there is a notion of derivation for every kind of algebra over an operad over a dg-operad (at least).


Let 𝒪\mathcal{O} be a dg-operad (a chain complex-enriched operad). For AA an 𝒪\mathcal{O}-algebra over an operad and NN a module over that algebra a derivation on AA with values in NN is a morphism

v:AN v : A \to N

in the underlying category of graded vector spaces, such that for each n>0n \gt 0 we have a commuting diagram

𝒪(n)A n A a+b=n1idid avid b v a+b=n1𝒪(n)A aNA b N, \array{ \mathcal{O}(n) \otimes A^{\otimes n} &\stackrel{}{\to}& A \\ {}^{\mathllap{\sum_{a+b=n-1} id \otimes id^{\otimes a} \otimes v \otimes id^{\otimes b}}}\downarrow && \downarrow^{\mathrlap{v}} \\ \oplus_{a+ b = n-1} \mathcal{O}(n) \otimes A^{\otimes a} \otimes N \otimes A^{\otimes b} &\to& N } \,,

where the top horizontal morphism is that given by the 𝒪\mathcal{O}-algebra structure of AA and the bottom that given by the AA-module structure of NN.

This appears as (Hinich, def. 7.2.1).

The theory of tangent complexes, Kähler differentials, etc. exists in this generality for derivations on algebras over an operad.

Generalization to arbitrary (,1)(\infty,1)-categories

Another equivalent reformulation of the notion of derivations turns out to be useful for the vertical categorification of the concept:

for NN an RR-module, there is the nilpotent extension ring G(N):=NRG(N) := N \oplus R, equipped with the product operation

(r 1,n 1)(n 2,r 2):=(r 1,r 2,n 1r 2+n 2r 1). (r_1, n_1) \cdot (n_2, r_2) := (r_1, r_2, n_1 r_2 + n_2 r_1) \,.

This comes with a natural morphism of rings

G(N)R G(N) \to R

given by sending the elements of NN to 0. One sees that a derivation on RR with values in NN is precisely a ring homomorphism RG(N)R \to G(N) that is a section of this morphism.

In terms of the bifibration p:ModRingp : Mod \to Ring of modules over rings, this is the same as a morphism from the module of Kähler differentials Ω K(R)\Omega_K(R) to NN in the fiber of pp over RR.

While this is a trivial restatement of the universal property of Kähler differentials, it is this perspective that vastly generalizes:

we may replace ModRingMod \to Ring by the tangent (∞,1)-category projection p:T CCp : T_C \to C of any (∞,1)-category CC. The functor that assigns Kähler differentials is then replaced by a left adjoint section of this projection

Ω:CT C. \Omega : C \to T_C \,.

An (,1)(\infty,1)-derivation on an object RR with coefficients in an object NN in the fiber of T CT_C over RR is then defined to be morphism Ω(R)N\Omega(R) \to N in that fiber.

More discussion of this is at deformation theory.


Derivations on an algebra

  • Let AA consist of the smooth real-valued functions on an interval in the real line. Then differentiation is a derivation; this is the motivating example.
  • Let AA consist of the holomorphic functions on a region in the complex plane. Then differentiation is a derivation again.
  • Let AA consist of the meromorphic functions on a region in the complex plane. Then differentiation is still a derivation.
  • Let AA consist of the smooth functions on a manifold (or generalized smooth space) XX. Then any tangent vector field on XX defines a derivation on AA; indeed, this serves as one definition of tangent vector field.
  • Let AA consist of the germs of differentiable functions near a point pp in a smooth space XX. Then any tangent vector at aa on XX defines a derivation on AA augmented by evaluation at aa; again, this serves to define tangent vectors.
  • Let AA consist of the smooth differential forms on a smooth space XX. Then exterior differentiation is a graded derivation (of degree 11).
  • In any of the above examples containing the adjective ‘smooth’, replace it with C kC^k and augment AA by the inclusion of C kC^k into C k1C^{k-1}. Then we have an augmented derivation.

There should be some more clearly algebraic examples (other than obvious things like restricting the above to polynomials), but I don't know how to state them.

Derivations with values in a bimodule

The standard example of a derivation not on an algebra, but with values in a bimodule is a restriction of the above case of the exterior differential acting on the deRham algebra of differential forms. Restricting this to 0-fomrs yields a morphism

d:C (X)Ω 1(X) d : C^\infty(X) \to \Omega^1(X)

where Ω 1(X)\Omega^1(X) is the space of 1-forms on XX, regarded as a bimodule over the algebra of functions in the obvious way.

A variation of this example is given by the Kähler differentials. These provide a universal derivation in some sense.

Derivations of smooth functions


Let XX be a smooth manifold and C (X)C^\infty(X) its algebra of smooth functions. Then the morphism

Vect(X)Der(C (X)) Vect(X) \to Der(C^\infty(X))

that sends a vector field vv to the derivation v():C (X)C (X)v(-) : C^\infty(X) \to C^\infty(X) is a bijection.

See also at derivations of smooth functions are vector fields.


This is true because C (X)C^\infty(X) satisfies the Hadamard lemma.

Since every smooth manifold is locally isomorphic to n\mathbb{R}^n, it suffices to consider this case. By the Hadamard lemma every function fC ( n)f \in C^\infty(\mathbb{R}^n) may be written as

f(x)=f(0)+ ix ig i(x) f(x) = f(0) + \sum_i x_i g_i(x)

for smooth {g iC (X)}\{g_i \in C^\infty(X)\} with g i(0)=fx i(0)g_i(0) = \frac{\partial f}{\partial x_i}(0). Since any derivation δ:C (X)C (X)\delta : C^\infty(X) \to C^\infty(X) satisfies the the Leibniz rule, it follows that

δ(f)(0)= iδ(x i)fx i(0). \delta(f)(0) = \sum_i \delta(x_i) \frac{\partial f}{\partial x_i}(0) \,.

Similarly, by translation, at all other points. Therefore δ\delta is already fixed by its action of the coordinate functions {x iC (X)}\{x_i \in C^\infty(X)\}. Let v δT nv_\delta \in T \mathbb{R}^n be the vector field

v δ iδ(x i)x i v_\delta \coloneqq \sum_i \delta(x_i) \frac{\partial}{\partial x_i}

then it follows that δ\delta is the derivation coming from v δv_\delta under Vect(X)Der(C (X))Vect(X) \to Der(C^\infty(X)).

Derivations of continuous functions

Let now XX be a topological manifold and C(X)C(X) the algebra of continuous real-valued functions on XX.


The derivations δ:C(X)C(X)\delta : C(X) \to C(X) are all trivial.


Observe that generally every derivation vanishes on the function 1 that is constant on 11 \in \mathbb{R}. Therefore it is sufficient to show that if fC(X)f \in C(X) vanishes at x 0Xx_0 \in X also δ(f)\delta(f) vanishes at x 0x_0, because we may write every function gg as (gg(x 0))+g(x 0)(g - g(x_0)) + g(x_0).

So let fC(X)f \in C(X) with f(x 0)=0f(x_0) = 0. Then we may write ff as a product

f=g 1g 2 f = g_1 g_2


g 1=|f| g_1 = \sqrt{|f|}


g 2:x{f(x)/|f(x)| |f(x)0 0 |f(x)=0. g_2 : x \mapsto \left\{ \array{ f(x)/\sqrt{|f(x)|} & | f(x) \neq 0 \\ 0 & | f(x) = 0 } \right. \,.

Notice that indeed both functions are continuous. (But even if XX is a smooth manifold and ff a smooth function, g 1g_1 will in general not be smooth.)

But also both functions vanish at x 0x_0. This implies that

δ(f)(x 0)=δ(g 1)(x 0)g 2(x 0)+g 1(x 0)δ(g 2(x 0))=0. \delta(f)(x_0) = \delta(g_1)(x_0) g_2(x_0) + g_1(x_0) \delta(g_2(x_0)) = 0 \,.


Derivations on algebras over a dg-operad are discussed in section 7 of

Revised on August 10, 2016 14:06:27 by Todd Trimble (