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In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings,
- φ:R → S,
that provides an analogue of the construction of differential forms (1-forms). The idea is that there should be an S-module homomorphism
- d:S → Ω1S/R
that is a derivation over R, that is best possible; and that this therefore should be a purely algebraic analogue of the exterior derivative. This approach was introduced by Erich Kähler, originally in the 1930s. It was adopted as standard, in commutative algebra and algebraic geometry, somewhat later, in particular given the need to adapt methods from geometry over the complex numbers, and their free use of calculus methods, to contexts where those are not available.
The actual construction of Ω1S/R can proceed by introducing formal generators ds for s in S, and imposing relations
- dr = 0 for r in R,
- d(s + t) = ds + dt,
and
- d(st) = sdt + tds.
Another equivalent construction, that makes sense in terms of affine schemes, is to take the ideal defining the diagonal I in the fiber product of Spec(S) with itself over Spec(R); and set
- Ω1S/R = I/I2.
This is more geometric, 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 details).
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 LHS 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 as R, for example.