---
tag: 7HCFARHG7D
kind: concept
name: Kähler differentials
msc: [13N05, 13N15]
sources:
  - url: https://en.wikipedia.org/wiki/Kähler_differential
same_as:
  wikidata: Q1609490
  wikipedia: Kähler differential
---

## Summary

The module of Kähler differentials $\Omega_{A/R}$ of an algebra $A$ over a commutative ring $R$ is the algebraic version of the module of differential $1$-forms.
It is generated by symbols $da$ for $a \in A$, subject only to the rules of differentiation.
It is the universal target of [[derivations|3J89REV9WQ]] of $A$ over $R$.
It is the basic tool to define [[smooth|286G0P1B69]], [[unramified|GY3QA6NH0E]] and [[étale|FTRW0679E2]] maps of rings without any analysis.

## Explanations

### L1 human

On a curve or surface given by polynomial equations, one can differentiate polynomial functions without limits, using only the product rule and linearity.
Kähler differentials collect all such formal differentials $df$ in one module.
For the polynomial ring $k[x, y]$, every differential is a combination $g \, dx + h \, dy$, and $d(x^2 y) = 2xy \, dx + x^2 \, dy$.
On the curve $y^2 = x^3$, one also has the relation $2y \, dy = 3x^2 \, dx$, which comes from differentiating the equation.
At the cusp $x = y = 0$ this relation becomes $0 = 0$, and this signals the singular point.