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.
Definition 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
Ω S / R {\displaystyle \Omega _{S/R}}
of differentials in different, but equivalent ways.
Definition using derivations An R-linear derivation on S is an R-module homomorphism d : S → M {\displaystyle d:S\to M} to an S-module M satisfying the Leibniz rule d ( f g ) = f d g + g d f {\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 Ω S / R {\displaystyle \Omega _{S/R}} for which there is a universal derivation d : S → Ω S / R {\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
Hom S ( Ω S / R , M ) → ≅ Der R ( S , M ) . {\displaystyle \operatorname {Hom} _{S}(\Omega _{S/R},M){\xrightarrow {\cong }}\operatorname {Der} _{R}(S,M).}
One construction of ΩS/R and d proceeds by constructing a free S-module with one formal generator ds for each s in S, and imposing the relations
dr = 0, d(s + t) = ds + dt, d(st) = s dt + t ds, 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.
Definition using the augmentation ideal Another construction proceeds by letting I be the ideal in the tensor product S ⊗ R S {\displaystyle S\otimes _{R}S} defined as the kernel of the multiplication map
{ S ⊗ R S → S ∑ s i ⊗ t i ↦ ∑ s i ⋅ t i {\displaystyle {\begin{cases}S\otimes _{R}S\to S\\\sum s_{i}\otimes t_{i}\mapsto \sum s_{i}\cdot t_{i}\end{cases}}}
Then the module of Kähler differentials of S can be equivalently defined by
Ω S / R = I / I 2 , {\displaystyle \Omega _{S/R}=I/I^{2},}
and the universal derivation is the homomorphism d defined by
d s = 1 ⊗ s − s ⊗ 1. {\displaystyle ds=1\otimes s-s\otimes 1.}
This construction is equivalent to the previous one because I is the kernel of the projection
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