= Invariant differential on a group scheme
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 identifies
$$
\Omega^1_{G/k}\cong\mathcal O_G\otimes_k e^*\Omega^1_{G/k}.
$$
Thus 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.
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