Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, $y^n=v$ and $xy=\xi_n yx$, where $\xi_n$ is an n-root of unity. (But i think this should be true in more general situations.)

Then we can look at the sheaf $\mathcal{A}$ associated to $A$ on $U=Spec(\mathcal{O})\backslash \lbrace \mathfrak{m} \rbrace$.

Why does $H^1_{zar}(U,\mathcal{A}^{\times})$ classify fractional left $A$-ideals $L$, which are reflexive as $\mathcal{O}$-modules?

Shouldn't this group classify modules, which look locally like $\mathcal{A}$, as some kind of Picard group for $\mathcal{A}$? But then i think this group should correspond to principal left ideals. But this seems wrong, because then this group would be zero, but in the text it is proven seperately that every such ideal is principal. And why are these ideals reflexive?

This is stated in M.Artin's article "Local structure of maximal orders on surfaces".It says on page 26: "And OF COURSE, $H^1_{zar}(U,\mathcal{A}^{\times})$ classifies reflexive fractional left ideals in $A$." Or on page 31 it says: "locally principal $\mathcal{A}$-module (or left ideal)".

Is this so obvious? Am I missing something?