Kessar–Linckelmann's refined Broué conjecture

Let $\mathcal{O}$ be an arbitrary complete discrete valuation ring, let $G$ be a finite group, and let $b$ be a block idempotent of $G$ over $\mathcal{O}$ with an abelian defect group. Kessar–Linckelmann's refined Broué conjecture. There is a splendid Rickard equivalence between $\mathcal{O}Gb$ and its Brauer correspondent. This refines Broué's abelian defect group conjecture by allowing arbitrary complete discrete valuation rings rather than only rings large enough for $G$; the paper proves it for the RoCK blocks under consideration.