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It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this ...

Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$? If $n=1$, I think this should be a finite etale ...