John Baez and James Dolan, 2023-09-14

Kähler differentials generalize 1-forms to an arbitrary commutative algebra A over a general commutative ring k:
en.wikipedia.o...
But there are even more general differentials that work for an arbitrary algebra A over k, used in noncommutative geometry. They are a universal object for derivations from A to (A,A)-bimodules. This universal derivation is obtained from D: A → A⊗A where D(a) = a⊗1 - 1⊗a, but the range of this D is really just I = ker(m) where m: A⊗A → A, and the universal derivation is D: A → I.
When A is commutative I is an ideal in A, and the Kähler differentials are d: I → Ω¹(A) where Ω¹(A) = I/I² and d comes from D. This is the universal derivation to a symmetric bimodule of A, i.e. one where the left and right actions of A agree.
The following are the same when k is a field:
1) finite separable extensions K of k,
2) finite extensions K of k with Ω¹(K) = 0,
3) finite-dimensional commutative algebras K over k that admit these structure of a special commutative Frobenius algebra.
Algebras that admit the structure of a special Frobenius algebra are called 'separable algebras'. This is a different meaning of 'separable' than in 1), but related.
All this is explained here more detail with some proofs here:
golem.ph.utexa...
golem.ph.utexa...
golem.ph.utexa...
Classification of ℤ/3 torsors over ℚ, as an example of Artin reciprocity.
For more on this whole series of conversations, go here:
math.ucr.edu/h...