Contravariant Koszul duality between non-positive and positive dg algebras
Riku Fushimi
TL;DR
The paper develops contravariant Koszul duality between locally finite non-positive dg algebras and locally finite positive dg algebras, introducing and exploiting the Koszul dual $A^!$ and the dual functor to connect derived categories. It characterizes when the dual of a positive dg algebra remains locally finite (the pvd-finite condition) and proves that the Koszul dual functor yields equivalences between $\mathsf{per}(A)$ and $\mathsf{pvd}((A^!)^{\mathrm{op}})^{\mathrm{op}}$ under suitable finiteness hypotheses. This duality underpins a robust ST-correspondence, linking silting objects, simple-minded objects, algebraic $t$-structures, and co-$t$-structures across the two derived frameworks. The results further provide a structural criterion for when perfectly valued derived categories arise from such algebras and establish that functorially finite hearts are length, with a triangulated symmetry that generalizes Smalø’s results. Altogether, the work extends Koszul duality to a graded-dg context and offers a cohesive view of finiteness, hearts, and t-structures in this setting, with implications for silting theory and ST-complete correspondences.
Abstract
The Koszul dual of locally finite non-positive dg algebra is locally finite positive dg algebra. However, the Koszul dual of locally finite positive dg algebra is not necessary locally finite. We characterize locally finite positive dg algebras whose Koszul dual is locally finite. Moreover, we show that the Koszul dual functor induces contravariant equivalences between the perfect derived category and the perfectly valued derived category. As an application of Koszul dualities, we establish an ST-correspondence. We also show that, under some assumption, every covariantly finite bounded heart is a length heart, and the triangulated analogy of Smalø's symmetry holds.