Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities

We introduce the categories $\text{CM}(X)$ and $\text{SCM}(X)$ of reflexive sheaves on a minimal partial resolution $f\text{\hspace{0.17em}⁣}:X\to \text{Spec}R$ of a rational surface singularity $\text{Spec}R$. The main result of this paper establishes that $\text{SCM}(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen–Macaulay $R$-modules, introduced by Iyama–Wemyss and Iyama–Kalck–Wemyss–Yang. Utilizing this geometric framework, we establish an exact equivalence between $\text{SCM}(X)$ and the category of special Cohen–Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama–Kalck–Wemyss–Yang equivalence and yields a Buchweitz-type equivalence $\underset{‾}{\text{SCM}}(X)\simeq D_{\text{sg}}(X)$.