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.



Ramification divisor and degenerate locus of jacobian
I don't know it is exactly what you want or not. But a reference not in the context of complex manifolds but smooth schemes is for example "Neron models" by Bosch-Lütkebomert, setion 2.2: If $f:X\rightarrow Y$ is a morphism of smooth schemes, then $f$ is etale at $x\in X$ if and only if the canonical homomorphism $(f^{\*}\Omega\_{Y}^{1})\_{x}\rightarrow (\Omega\_{X}^{1})\_{x}$ is isomorphism. Now this last condition is equivalent to smoothness of $f$ at $x$ (proposition 8) or the fact that $Jac(f)\neq0$ at $x$. This shows that in fact $J\_{f}$ and $R\_{f}$ are the same at least as sets. But it is also proved there that the map $(f^{\*}\Omega\_{Y}^{1})\_{x}\rightarrow (\Omega\_{X}^{1})\_{x}$ is isomorphism if and only if the canonical homomorphism $(f^{\*}\Omega\_{Y}^{1})\otimes k(x)\rightarrow (\Omega\_{X}^{1})\otimes k(x)$ is injective. This in turn shows that $R\_{f}$ and $J\_{f}$ can both be thought of as the kernel of this restricted homomorphism at each point. So $R\_{f}$ and $J\_{f}$ are the same as schemes. In that book, they also mention (p.40) that this is the counterpart of Jacobi criterion in differential geometry.


