I have two automorphisms $f$ and $g$ of a scheme $X$ over a scheme $Y$. Both schemes are $1$-dimensional, smooth and integral, and the map is flat and finite. (In fact, $Y=X/G$ where $G$ is a finite group of automorphisms to which $f$ and $g$ belong.) I know that the two automorphisms coincide on an open (dense) subset of $X$.
Now, I know that the "dense" hypothesis is not sufficient to conclude that $f$ and $g$ coincide. But what if the set is open? Does this assure me that $f=g$?
Thank you in advance.