Kähler differential

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

Presentation

The idea was introduced by Erich Kähler in the 1930s. It was adopted as standard, in commutative algebra and algebraic geometry, somewhat later, following the need to adapt methods from geometry over the complex numbers, and the free use of calculus methods, to contexts where such methods are not available.

Let R and S be commutative rings and φ:R → S a ring homomorphism. An important example is for R a field and S a unital algebra over R (such as the coordinate ring of an affine variety).

An R-linear derivation on S is a morphism of R-modules $\mathrm d \colon S \to M$ with R in its kernel, and satisfying Leibniz rule $\mathrm d (fg) = f \mathrm \, \mathrm dg + g \, \mathrm df$. The module of Kähler differentials is defined as the R-linear derivation $\mathrm d \colon S \to \Omega_{S/R}$ that factors all others.

Construction

The idea is now to give a universal construction of a derivation

d:S → Ω¹_S/R

over R, where Ω¹_S/R is an S-module, which is a purely algebraic analogue of the exterior derivative. This means that d is a homomorphism of R-modules such that

d(st) = s dt + t ds

for all s and t in S, and d is the best possible such derivation in the sense that any other derivation may be obtained from it by composition with an S-module homomorphism.

The actual construction of Ω¹_S/R and d can proceed by introducing formal generators ds for s in S, and imposing the relations

• dr = 0 for r in R,
• d(s + t) = ds + dt,
• d(st) = s dt + t ds

for all s and t in S.

Another construction proceeds by letting I be the ideal in the tensor product $S \otimes_R S$, defined as the kernel of the multiplication map: $S \otimes_R S\to S$, given by $\Sigma s_i \otimes t_i \mapsto \Sigma s_i.t_i$. Then the module of Kähler differentials of "S" can be equivalently defined by :Ω¹_S/R = I/I², together with the morphism

$\mathrm ds = 1 \otimes s - s \otimes 1. \,$

Use in algebraic geometry

Geometrically, in terms of affine schemes, I represents the ideal defining the diagonal in the fiber product of Spec(S) with itself over Spec(S) → Spec(R). This construction therefore has a more geometric flavor, in the sense that the notion of first infinitesimal neighbourhood of the diagonal is thereby captured, via functions vanishing modulo functions vanishing at least to second order (see cotangent space for related notions).

For any S-module M, the universal property of Ω¹_S/R leads to a natural isomorphism

$\operatorname{Der}_R(S,M)\cong \operatorname{Hom}_S(\Omega^1_{S/R},M), \,$

where the left hand side is the S-module of all R-linear derivations from S to M. As in the case of adjoint functors (though this isn't an adjunction), this is more than just an isomorphism of modules; it commutes with S-module homomorphisms M → M' and hence is an isomorphism of functors.

To get Ω^p_S/R, the Kähler p-forms for p > 1, one takes the R-module exterior power of degree p. The behaviour of the construction under localization of a ring (applied to R and S) ensures that there is a geometric notion of sheaf of (relative) Kähler p-forms available for use in algebraic geometry, over any field R.

References

• J. Johnson (1969): Kähler differentials and differential algebra. Annals of Mathematics, 89:92–98.
• M. Rosenlicht (1976): On Liouville's theory of elementary functions. Pacific J. Math., 65:485–492.
• A thread devoted to the question on MathOverflow