Weak equivalence class (awaiting review)

Two fractional $S$-ideals $I$ and $J$ are in the same weak equivalence class if $I_\mathfrak{l}=I\otimes_S S_\mathfrak{l}$ and $J_\mathfrak{l}=J\otimes_S S_\mathfrak{l}$ are isomorphic as $S_\mathfrak{l}$-modules for every maximal ideal $\mathfrak{l}$ of $S$. In the literature, weak equivalence classes are also called local isomorphism classes of genera.