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An invariant differential on a group scheme is a differential one-form preserved by every left translation. For a smooth group scheme $G$ over a field $k$, translation identifiesThus a covector at the identity extends uniquely to a left-invariant form. On the additive algebraic group the invariant form is $dx$; on the multiplicative algebraic group it is $dx/x$, since $d(ax)/(ax)=dx/x$. An abelian variety of dimension $g$ therefore has $g$ linearly independent invariant one-forms, providing the local linear data behind its global group geometry.

For $x:S\to G$, defineThen $T_{x/S}=(T_x,{\mathrm{pr}}_2)$ is an $S$-morphism, and translation by $x^{-1}$ is its inverse.

Because an isomorphism preserves relative differentials and $T_{x/S}$ lies over $S$,Using base change, ${\Omega }_{G\times S/S}\cong {\mathrm{pr}}_1^{\ast }{\Omega }_{G/k}$, while the left side is $T_x^{\ast }{\Omega }_{G/k}$. Pulling this isomorphism back along the identity section $e\times 1_S$ givesFinally take $S=G$ and $x=1_G$. This yields the invariant differential on a group scheme trivializationso ${\Omega }_{G/k}$ is a free ${\mathcal{O}}_G$-module.