Let $f : X \to Y$ be a finite morphism between compact complex manifolds of the same dimension $n$. We denote by $R_f \subset X$ the ramification divisor of $f$ and by $J_f \subset X$ the set of points where $\operatorname{Jac} f = 0$.

It is easy to see that $R_f = J_f$ as *sets* ($J_f \subset R_f$ is trivial; the other inclusion follows from the inverse function theorem). Do the two also have the same scheme structure?

This is true in the case of curves, pretty much by local inspection. I have a vague feeling one can prove this holds in the general case too by first reducing to affine spaces and then inducting on the dimension: given a point of interest we'd first pick a hyperplane $X_1$ in $X$ that intersects the sets nicely, then project away from a point in $Y$ to an affine space $Y_1$ and the composition $f_1: X_1 \to X \to Y \to Y_1$ would be finite of the right order at the corresponding point of interest. Eventually we'd be dealing with curves where things are true and try to bootstrap our way up again. This feels messy (and maybe doesn't work) so if there's a simpler argument I'd love to see it.