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Kähler differentials

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Definition

Let A\to B be an algebra. An A differential of B into an A-module M is a map D:B\to M such that

  1. D(a) = 0 for all a\in A
  2. D(b + b') = D(b) + D(b') for b,b'\in B
  3. D(bb') = b'D(b) + bD(b')

Observe that the set of all such maps DerA(B,M) is a B-module. Moreover, DerA(B, − ) is a representable functor; we call the representative ΩB / A the module of Kähler differentials. That is, ΩB / A satisfies the following universal property:

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