Formality and strongly unique enhancements
Antonio Lorenzin
TL;DR
The paper develops a framework connecting triangulated formality and formal standardness of DG-categories to strong uniqueness of DG-enhancements, generalizing intrinsic formality to a DG-context and to linearity over any commutative ring. The main result characterizes when a graded category has a strongly unique enhancement in terms of triangulated formality and formal standardness, and then derives broad consequences for derived categories, including new examples and a unifying view of D-standardness and K-standardness. Key contributions include a general criterion for strong uniqueness, links between enhancement uniqueness and autoequivalences via lifts, and extensive applications to exact and hereditary categories, as well as to free generator constructions and their periodizations. The results provide a robust toolkit for understanding when triangulated categories arising from algebraic sources admit a unique DG-enhancement, with implications for both algebraic and geometric contexts and for deriving categories over arbitrary commutative rings.
Abstract
Inspired by the intrinsic formality of graded algebras, we prove a necessary and sufficient condition for strongly uniqueness of DG-enhancements. This approach offers a generalization to linearity over any commutative ring. In particular, we obtain several new examples of triangulated categories with a strongly unique DG-enhancement. Moreover, we also show that the bounded derived category of an exact category has a unique enhancement.