Kähler differential Guide, Meaning , Facts, Information and Description
In
mathematics, the
Kähler differentials are a
universal construction Ω
1S/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
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.
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.
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