Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring. Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the dual sheaf.

How can I see that the determinant map is a map from $A^*$ to $O_X((1-\frac{1}{e})C)$? And how to understand the invertible sheaf $O_X((1-\frac{1}{e})C)$? How to handle the rational coefficients?

Since $A$ only ramifies in C, we have that $A$ is Azumaya on $U:=X\backslash C$. So on $U$ we have $A^*\cong A$. There the determinant map induces a map $A^{\*} \rightarrow O_U$ since $A$ is just a matrix algbera etale locally. So I see that we have a map $A^* \rightarrow O_X(rC)$. But how to find $r=1-\frac{1}{e}$? How can i determine $A^*$ on $C$?

The question arose reading Theorem 7.1.4. on page 157 of this script: http://www.math.lsa.umich.edu/courses/711/ordersms-num.pdf