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The Kähler differentials commute with extension of the base ring. A derivation of the base-changed algebra that kills $k^{\prime }$ is determined by its restriction to $A\otimes 1$, and the Leibniz rule provides its unique extension. No flatness hypothesis is required for this isomorphism.

A finite étale morphism is both a finite morphism and an étale morphism. Affine-locally it is represented by a finite locally free algebra with zero module of Kähler differentials. Its fibres are finite products of finite separable field extensions. For an elliptic curve with good reduction at a prime not dividing $m$, multiplication by $m$ on its smooth proper group model is finite étale of degree $m^2$: its differential is multiplication by the unit $m$. A division torsor of a section consequently gives an unramified local extension, the good-prime input to the Weak Mordell-Weil theorem.

The Kähler differentials of a polynomial ring are free on the differentials of its variables. Assigning arbitrary values to these generators uniquely defines a derivation. Applying the Conormal exact sequence for Kähler differentials to a quotient gives its presentation by the formal derivatives of its defining equations.

Kähler differentials commute with localization of a ring. The quotient rule $d(a/s)=(sda-ads)/s^2$ uniquely extends a derivation after inverting $s$; its universal property proves the displayed module isomorphism.

Let $K=k(X)$. The space of Kähler differentials ${\Omega }_{K/k}^1$ is the $K$-vector space generated by symbols $df$, $f\in K$, with relationsIts elements are rational differentials. A local parameter $t$ is a nonconstant element of $K$, so $k(t)$ is a rational function field and $K/k(t)$ is finite. In characteristic zero this extension is separable.

For any $f\in K$ let $P(t,f)=0$ be its minimal polynomial over $k(t)$. Differentiating givesSeparability ensures $P_f(t,f)\ne 0$, so $df$ is a multiple of $dt$, and hence ${\Omega }_{K/k}^1=Kdt$. To show $dt\ne 0$, the derivation $d/dt$ on $k(t)$ extends uniquely to the separable extension $K$, by the same differentiated minimal-polynomial formula; it sends $t$ to $1$, so the universal differential $dt$ cannot vanish. ThusFor a nonzero differential $\omega =fdt$, define its valuation of a rational differential at $p$ byConcretely $\omega =t^{m}udt$ with $u$ a unit in the local ring, and this valuation is $m$. For $\omega =0$ one conventionally assigns $+\infty$, not an integer.

The relative tangent sheaf is the dual of a sheaf of Kähler differentials. Its sections are derivations of the structure sheaf that kill base scalars. For a smooth morphism it is locally free and describes first-order deformations of sections into the scheme.