Kähler differential

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 ${\displaystyle d:S\to M}$ to an S-module M satisfying the Leibniz rule ${\displaystyle d(fg)=f\,dg+g\,df}$ (it automatically follows from this definition that the image of R is in the kernel of d ). The module of Kähler differentials is defined as the S-module ${\displaystyle \Omega _{S/R}}$ for which there is a universal derivation ${\displaystyle d:S\to \Omega _{S/R}}$. 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

One construction of Ω_S/R and d proceeds by constructing a free S-module with one formal generator ds for each s in S, and imposing the relations

for all r in R and all s and t in S. The universal derivation sends s to ds. The relations imply that the universal derivation is a homomorphism of R-modules.

Another construction proceeds by letting I be the ideal in the tensor product ${\displaystyle S\otimes _{R}S}$ defined as the kernel of the multiplication map

Then the module of Kähler differentials of S can be equivalently defined by