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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 8th 2025

    Created:

    Idea

    The abstract notion of a derivation corresponding to that of a Beck module.

    Definition

    Given a category C with finite limits, a Beck module in C over an object AC is an abelian group object in the slice category C/A.

    The forgetful functor from modules to rings is modeled by the forgetful functor

    UA:Ab(C/A)C/A.

    Given MAb(C/A), a Beck derivation AM is a a morphism idAUA(M) in C/A.

    If UA has a left adjoint ΩA, then ΩA is known as the Beck module of differentials over A. Thus, Beck derivations AM are in bijection with morphisms of Beck modules

    ΩAM,

    generalizing the universal property of Kähler differentials.

    Examples

    For ordinary commutative algebras, Beck derivations coincide with ordinary derivations.

    For C^∞-rings, Beck derivations coincide with C^∞-derivations.

    References

    The original definition is due to Jon Beck. An exposition can be found in Section 6.1 of

    v1, current