Nonsmooth Calabi-Yau structures for algebras and coalgebras
Matt Booth, Joseph Chuang, Andrey Lazarev
TL;DR
The paper develops a comprehensive framework for generalized Calabi–Yau structures on dg (co)algebras, showing that nonsmooth CY algebras are Koszul dual to symmetric coalgebras (and vice versa), while Gorenstein and Frobenius properties are likewise dual. It systematically builds the necessary Koszul duality machinery, including one- and two-sided dualities, derived Picard groups, and twists, and connects these to smoothness, properness, and locality notions across the algebra/coalgebra divide. A central achievement is a new characterisation of Poincaré duality spaces via Poincaré duality for chain coalgebras and Hopf-algebra methods, extending Félix–Halperin–Thomas to non-simply connected settings. The results unify CY, Gorenstein, and Frobenius phenomena in both algebraic and coalgebraic contexts, with broad topology applications spanning string topology, rational homotopy theory, and Lie theory. Overall, the work provides a robust, non-smooth–non-proper generalization of noncommutative CY dualities and yields new topological characterisations grounded in derived (co)algebra structures.
Abstract
We show that generalised Calabi-Yau dg (co)algebras are Koszul dual to generalised symmetric dg (co)algebras, without needing to assume any smoothness or properness hypotheses. Similarly, we show that Gorenstein and Frobenius are Koszul dual properties. As an application, we give a new characterisation of Poincaré duality spaces, which extends a theorem of Félix- Halperin-Thomas to the non-simply connected setting.