@YCor I am interested in general case, to simplify the notion here I only consider affine spaces. Thank you, now it's fixed.

But we know maximal orders are hereditary rings, so I am very skeptical about the theorem used in that paper.

@inkspot，thank you. I checked that book but it defines $Cl$ as stable isomorphism classes of ideals and always input the Eichler condition. However, here $R$ is the maximal order in the quaternion algebra over $\mathbb Q$ ramified at $p$ at $\infty$, and it's class number is not $1$ when $p$ grows. The theorem only states for higher rank, which is quite wild.

Thank you, is there a general bound such as exponential bound for any noetherian local ring?