Positselski duality in $\infty$-categories
Torgeir Aambø
TL;DR
This work extends Positselski duality from abelian settings to presentably symmetric monoidal $\infty$-categories by defining comodules and contramodules over cocommutative coalgebras. It proves a monoidal version of the comodule–contramodule correspondence for coidempotent coalgebras, yielding a natural symmetric monoidal equivalence between these two viewpoints and providing a new proof of local duality in stable, compactly generated contexts. The authors then develop an analogue of contramodules over topological (pro-dualizable) algebras, showing an equivalence between contramodules over a coalgebra and its linear dual, and apply this to chromatic homotopy theory and derived completion. The framework unifies local duality, topological algebra contramodules, and concrete examples such as $K(n)$-local and $T(n)$-local categories, offering a flexible, higher-categorical lens for dualities in homotopy theory and algebra.
Abstract
We introduce the notion of a contramodule over a cocommutative coalgebra in a presentably symmetric monoidal $\infty$-category $\mathcal{C}$, and prove a symmetric monoidal $\infty$-categorical version of Positselski's comodule-contramodule correspondence when the coalgebra is coidempotent. This gives a new perspective on, and a new proof of local duality -- in the sense of Hovey--Palmieri--Strickland and Dwyer--Greenlees -- whenever $\mathcal{C}$ is stable and compactly generated. We further consider an analog of Positselski's definition of contramodules over topological rings in the $\infty$-categorical setting, and show that the two perspectives on contramodules are equivalent. As examples we describe the categories of $K(n)$-local spectra, $T(n)$-local spectra and the derived complete category of a ring $R$, as categories of contramodules.