ScalingStacks

By [BS21, Theorems 3.11, 3.56] the family of tilting modules $\mathcal{T}(\mathcal{R})$ is tilting in $\operatorname{D}^{b}(\mathcal{R})$ in the sense of Definition 2.3. Therefore Corollary 2.16 implies that the weight complex functor induces an equivalence of categories

(2.6)

$\displaystyle t:\langle\mathcal{T}(\mathcal{R})\rangle_{\cong,\Delta}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D_{perf}}(E).$

Now assume that $\mathcal{T}(\mathcal{R})$ generates $\operatorname{D}^{b}(\mathcal{R})$ as a triangulated category and $E$ has finite cohomological dimension. For example, this is the case if $\mathcal{R}$ is a highest weight category. Then (2.6) yields a derived equivalence, called Ringel duality, between $\mathcal{R}$ and its Ringel dual $\mathcal{R}^{\prime}$

$$\operatorname{D}^{b}(\mathcal{R})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D}^{b}(\mathcal{R}^{\prime}).$$

This interpretation of Ringel duality in terms of weight complex functors has interesting applications. For example, one can use Proposition 2.10 to show that Ringel duality commutes with functors preserving tilting modules (under the correct technical assumptions).