TOPICS
Search

Kähler Differential

A Kähler differential is an element of the module Omega_(S/R) representing derivations of a commutative R-algebra S. The module is generated by symbols ds, with additive and Leibniz relations d(a+b)=da+db, d(ab)=adb+bda, and dr=0 for elements from R. Every R-derivation D:S->M factors uniquely as D=alpha degreesd for an S-linear map alpha:Omega_(S/R)->M (Stacks Project).

For S=R[x_1,...,x_n], this module is free with basis dx_1, ..., dx_n. On an algebraic variety these modules glue to the sheaf of differentials. Its dual is the tangent sheaf, which coincides with tangent-bundle sections on a smooth variety.

See also

Derivation, Module, Polynomial Ring, Sheaf, Tangent Sheaf

Explore with Wolfram|Alpha

References

Stacks Project. "Differentials." Tag 00RM. https://stacks.math.columbia.edu/tag/00RM.

Cite this as:

Weisstein, Eric W. "Kähler Differential." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KaehlerDifferential.html

Subject classifications