ScalingStacks

Similarly to the last paragraph, weight structures and weight complex functors can be applied to the theory of tilting objects and Ringel duality for highest weight categories and their semi-infinite and stratified generalisation. For the general setup we refer to Brundan–Stroppel [BS21] and freely use the terminology from there. Let $\mathcal{R}$ be a lower finite or essentially finite $\epsilon$-stratified category over an algebraically closed field (in particular it could be a highest weight category).

Then by [BS21, Theorems 4.2, 4.13, 4.18] tilting modules in the sense of highest weight categories exist in $\mathcal{R}$ and form an additive subcategory generated by the family $\mathcal{T}(\mathcal{R})$ of representatives of isomorphism classes of indecomposable tilting modules. For $E=\operatorname{End}_{\mathcal{R}}(\mathcal{T}(\mathcal{R})),$ see (2.2), the category $\mathcal{R}^{\prime}=\operatorname{mod_{lfd}-}E$ of locally finite dimensional right modules is called the Ringel dual of $\mathcal{R}.$

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).