---
tag: ZKSPYN2CRJ
kind: definition
name: Sheaf of differentials
msc: [14F10, 14F06]
defines: 6VC6HQTFSA
uses: [JXRKH6P1Z4, 1DZMT4S9FB, KPKND0K7W6, 7HCFARHG7D]
sources:
  - url: https://en.wikipedia.org/wiki/Cotangent_sheaf
---

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
Let $\Delta \colon X \to X \times_Y X$ be the diagonal morphism into the [[fiber product|1DZMT4S9FB]].
The morphism $\Delta$ is an immersion, so there is an open subscheme $W \subseteq X \times_Y X$ such that $\Delta$ is a closed immersion of $X$ into $W$.
Let $\mathcal{I} \subseteq \mathcal{O}_W$ be the ideal sheaf of $\Delta(X)$ in $W$.
The *sheaf of relative differentials* of $X$ over $Y$ is
$$\Omega_{X/Y} = \Delta^*(\mathcal{I}/\mathcal{I}^2),$$
a [[quasi-coherent|KPKND0K7W6]] $\mathcal{O}_X$-module.

## Remarks

If $U = \operatorname{Spec} B \subseteq X$ maps into $V = \operatorname{Spec} A \subseteq Y$, then $\Omega_{X/Y}|_U$ is the quasi-coherent sheaf associated to the module of [[Kähler differentials|7HCFARHG7D]] $\Omega_{B/A}$.
There is a universal derivation $d \colon \mathcal{O}_X \to \Omega_{X/Y}$, and $\Omega_{X/Y}$ is generated by the sections $dg$.
If $X$ is a variety over a field $k$, one writes $\Omega_X = \Omega_{X/k}$, the *cotangent sheaf*.

Example: for $X = \mathbb{A}^n_k = \operatorname{Spec} k[x_1, \dots, x_n]$, $\Omega_{X/k}$ is free of rank $n$ with basis $dx_1, \dots, dx_n$.
This matches $\Omega_{B/k}$ for the polynomial ring $B$.

Non-example of local freeness: for the cuspidal curve $X = \operatorname{Spec} k[x, y]/(y^2 - x^3)$ with $\operatorname{char} k \ne 2, 3$, $\Omega_{X/k}$ is generated by $dx, dy$ with the relation $2y \, dy = 3x^2 \, dx$.
Its fiber at the origin has dimension $2$, while the generic rank is $1$, so $\Omega_{X/k}$ is not locally free there.
An irreducible variety $X$ over a perfect field $k$ is smooth over $k$ if and only if $\Omega_{X/k}$ is locally free of rank $\dim X$.