Let $f: V \to W$ be a finite surjective morphism of varieties, with $W$ proper and normal. Does there exist a trace map $$ \operatorname{trace}\colon f_* \mathcal O_V \to \mathcal O_W? $$
I know that such a thing exists if $f$ is etalé, or (more generally) finite and flat. Does normality of $W$ help here?
Lipman seems to assume the existence in his exposition Dualizing sheaves, differentials, and residues on algebraic varieties, but doesn't explain it further.
In general, if $A \subset B$ is a finite ring extension, there is a finite fileld extension $K \subset L$ of fraction fields, and a trace map $\operatorname{trace}: L \to K$. I guess one wants to restrict this to $B$ and hopes it ends up in $A$?
I find it hard to find examples where $f$ is not finite and flat, but according to Wikipedia,
$$\mathbb C[x^2, xy, y^2] \subset \mathbb C[x,y]$$
gives an example where $X$ is regular, $Y$ is normal, but $f$ is not flat. However, considering the extension of quotient fields $K \subset L$ one sees that $L = K(x)$, and $x$ has degree $2$. The trace mapping is then given by \begin{align*} \operatorname{trace}: L & \to K \\ 1 & \mapsto \operatorname{trace} \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} = 2 \\ x & \mapsto \operatorname{trace}\begin{pmatrix} 0 & x^2 \\ 1 & 0 \end{pmatrix} = 0 \\ y & \mapsto \operatorname{trace}\begin{pmatrix} 0 & xy \\ \frac{xy}{x^2} & 0 \end{pmatrix} = 0. \end{align*} Since $\mathbb C[x,y]$ is generated by $x,y$ as a $\mathbb C[x^2, xy, y^2]$-module, we see indeed that the trace morphism descends to a morphism $$ \mathbb C[x,y] \to \mathbb C[x^2, xy, y^2].$$
Are there examples where the trace map does not descend?