Let $X$ be a noetherian integral separated scheme which is regular of codimension one. Let $K$ be the function field of $X$.
Now let $f \in K^*$, (I am interpreting $K^*$ to be the set of field automorphisms from $K \to K$) be any nonzero rational function on $X$.
I am very confused by the statement after the parenthesis. I thought a rational function on $X$ was an element of the function field $K$.
How can $f \in K$ and $f \in K^{*}$ ?
It is likely that I am not interpreting $K^*$ correctly.
The intended interpretation is that $K^*=K\setminus\{0\}$ (the group of multiplicative units of $K$), so an element of $K^*$ is just a nonzero element of $K$. Another common notation for this set is $K^\times$.