Local explicit Iwasawa main conjecture for Rubin–Stark elements

Assume condition $(*)$ and let $\Lambda$ be the Iwasawa algebra. For each height-one prime $\mathfrak{p}$, let $V_\mathfrak{p}$ and $r_\mathfrak{p}$ be defined from a character above $\mathfrak{p}$, and let $\epsilon^\mathfrak{p}_{K_\infty/k,S,T}$ be the corresponding localized Rubin–Stark element.

Local explicit Iwasawa main conjecture. For every height-one prime ideal $\mathfrak{p}$ of $\Lambda$,

$$\Lambda_\mathfrak{p}\cdot\epsilon^\mathfrak{p}_{K_\infty/k,S,T}={\rm Fitt}_\Lambda^0(A_S^T(K_\infty)){\rm Fitt}_\Lambda^0(\mathcal{X}_{K_\infty,S\setminus V_\mathfrak{p}})\cdot(\bigwedge_\Lambda^{r_\mathfrak{p}}U_{K_\infty,S,T})_\mathfrak{p}.$$

This is an explicit local form of the Iwasawa main conjecture, expressing localized Rubin–Stark elements through Fitting ideals of ray class and divisor modules.