Generalized non-commutative degeneration conjecture
Alexander I. Efimov
TL;DR
This paper extends the Kontsevich--Soibelman degeneration phenomenon from smooth, proper DG categories to arbitrary small DG categories by proposing a generalized degeneration conjecture governed by the boundary map $\delta$ in the long exact sequence of negative cyclic homology and the induced maps $\varphi_n=(\mathrm{id}\otimes\delta)\circ ch$ on $K$-theory. The authors formalize the setting via the mixed Hochschild complex, define the relevant bi-additive invariants, and establish Morita-invariance and gluing additivity to relate different DG categories. They prove that the generalized degeneration conjecture follows from the original KS conjecture together with a smooth categorical compactification conjecture, using a stepwise program that passes from homotopically finite categories to general ones through localization, gluing, and colimits, and then to higher $K$-groups via inductive arguments. The work situates generalized degeneration within noncommutative Hodge theory and offers a pathway to extend degeneration phenomena to broader classes of DG categories with potential implications for noncommutative geometry and representation theory.
Abstract
In this paper we propose a generalization of the Kontsevich--Soibelman conjecture on the degeneration of Hochschild-to-cyclic spectral sequence for smooth and compact DG category. Our conjecture states identical vanishing of a certain map between bi-additive invariants of arbitrary small DG categories over a field of characteristic zero. We show that this generalized conjecture follows from the Kontsevich--Soibelman conjecture and the so--called conjecture on smooth categorical compactification.