Derived Ringel duality through finite subquotients of modular category $\mathcal O$
We study derived Ringel duality for modular category $\mathcal O$ on downward-closed, quasi-bounded supports. We reconstruct global tilting objects and their morphisms from compatible images in finite highest-weight Serre subquotients, and realize the complete Ringel algebra as an inverse limit of finite-dimensional algebras. The duality identifies the bounded homotopy category of finitely generated projectives over the associated locally unital algebra with the bounded derived category of the Ringel dual category. We characterize tiltings recovered, together with their outgoing morphisms, by a single finite subquotient, and give a sharp criterion for equivalence under Frobenius restriction. For $SL_2$, we compute tilting endomorphism algebras and their quotient maps, and obtain sharp recovery bounds for finite tilting families.