Kähler differential

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In mathematics, Kähler differentials provide a generalization of differential forms to arbitrary commutative rings (or schemes). 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. For example R could be a field and S could be an algebra over R (such as the coordinate ring of an affine variety). The idea is now to give a universal construction of a derivation

d:S → Ω1S/R

over R, where Ω1S/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 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 of S with itself over R defined by the kernel of the multiplication map to S, and defining

Ω1S/R = I/I2,

and

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

Geometrically, in terms of affine schemes, I represents the ideal defining the diagonal in the fiber product of Spec(S) with itself over 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).

The universal property leads to a defining relation

DerR(S,M) = HomS(Ω1S/R,M)

for any S-module M. As in the case of adjoint functors (though this isn't precisely an adjunction) the equality sign here means only that there is a (canonical) identification of the two sets. The left hand side is the set of derivations over R (i.e., treating R as constants) of S into M.

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, over any field R.

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