Kähler differentials

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

Let R and S be commutative rings and φ : R → S be a ring homomorphism. An important example is for R a field and S a unital algebra over R (such as the coordinate ring of an affine variety). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms. This observation can be turned into a definition of the module

of differentials in different, but equivalent ways.

An R-linear derivation on S is an R-module homomorphism ${\displaystyle d\colon S\to M}$ to an S-module M with the image of R in its kernel, satisfying the Leibniz rule ${\displaystyle d(fg)=f\,dg+g\,df}$. The module of Kähler differentials is defined as the S-module ${\displaystyle \Omega _{S/R}}$ for which there is a universal derivation ${\displaystyle d\colon S\to \Omega _{S/R}}$. As with other universal properties, this means that d is the best possible derivation in the sense that any other derivation may be obtained from it by composition with an S-module homomorphism. In other words, the composition with d provides, for every S-module M, an S-module isomorphism

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