The trace discriminant detects étaleness of a finite free algebra

Trace and its determinant commute with every scalar extension because they are the trace and determinant of matrices of multiplication on the specified free module. Over an algebraically closed field, each local Artinian factor $C$ has residue field that field. If its nilradical is nonzero, each nonzero nilpotent $z$ is orthogonal under trace to every $w\in C$, since multiplication by $zw$ is nilpotent and has trace zero; hence the pairing is degenerate. If the nilradical is zero, the algebra is a product of copies of the field, with trace pairing the ordinary diagonal nondegenerate pairing. Therefore the determinant is nonzero exactly when the geometric fibre is reduced, or equivalently regular of dimension zero. By [F1] this is exactly étaleness.

Under the local hypotheses, generic étaleness says $\mathrm{\Delta }\mathrm{\ne }0$. If $\mathrm{\Delta }$ were not a unit, a prime minimal over $(\mathrm{\Delta })$ would have height at most one by [F2]; because $A$ is a domain and $\mathrm{\Delta }\mathrm{\ne }0$, its height is one. It is therefore in the punctured spectrum of the local ring of dimension at least two, contradicting step 1.1 and the assumed étaleness there. Thus $\mathrm{\Delta }$ is a unit and $D$ is étale everywhere.