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.
An R-linear derivation on S is a morphism of R-modules with R in its kernel, and satisfying Leibniz rule . The module of Kähler differentials is defined as the R-linear derivation that factors all others.
- d:S → Ω1S/R
- 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 Ω1S/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 , defined as the kernel of the multiplication map: , given by . Then the module of Kähler differentials of "S" can be equivalently defined by Ω1S/R = I/I2, together with the morphism
To see that this construction is equivalent to the previous one, note that I is the kernel of the projection , given by . Thus we have:
Then may be identified with I, by the map induced by the complimentary projection which is given by .
Thus this map identifies I with the S module generated by the formal generators ds for s in S, subject to the first two relations given above (with the second relation strengthened to demanding that d is R-linear). The elements set to zero by the final relation map to precisely I2 in I.
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 Ω1S/R leads to a natural isomorphism
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 ΩpS/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.
Use in algebraic number theory
In algebraic number theory, the Kähler differentials may be used to study the ramification in an extension of algebraic number fields. If L/K is a finite extension with rings of integers O and o respectively then the different ideal δL/K, which encodes the ramification data, is the annihilator of the O-module Ω1O/o:
- Neukirch (1999) p.200
- Neukirch (1999) p.201
- Johnson, J. (1969). "Kähler differentials and differential algebra". Annals of Mathematics 89: 92–98. doi:10.2307/1970810. Zbl 0179.34302.
- Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
- Rosenlicht, M. (1976). "On Liouville's theory of elementary functions". Pacific J. Math. 65: 485–492. doi:10.2140/pjm.1976.65.485. Zbl 0318.12107.
- Fu et al., G., et al (2011). "Some remarks on Kähler differentials and and ordinary differentials in nonlinear control systems". Systems and Control Letters 60: 699–703. doi:10.1016/j.sysconle.2011.05.006.