A Criterion for Phantomness of dg-categories

TL;DR

This paper establishes a criterion for phantomness of smooth proper dg categories based on the vanishing of various additive invariants, linking motives, algebraic K-theory, and realizations.

Contribution

It introduces a new criterion for phantomness using non-compact motives and their realizations, connecting algebraic and topological invariants.

Findings

Vanishing of certain K-theories implies the noncommutative motive vanishes.

Constructs a motive whose realizations recover key invariants like K-theory and cyclic homology.

Provides a deformation-invariance result for phantomness in families.

Abstract

We study the question of whether the vanishing of additive invariants characterizes phantomness for smooth proper dg categories admitting geometric realizations. More precisely, let $X$ be a smooth proper variety over a field $k$, and let $\sT\subset \perfdg(X)$ be a $k$-linear admissible full dg subcategory. We construct a non-compact motive $\sM(\sT)\in \DM(k,\Q)$ and show that its $l$-adic realization recovers the $K(1,l)$-local algebraic $K$-theory of $\sT$. Analogous statements are obtained for Betti and de Rham realizations, which recover topological $K$-theory and periodic cyclic homology, respectively. As a consequence, assuming that the Chow motive of $X$ is Kimura-finite, we prove a criterion for phantomness: the vanishing of $L_{K(1,l)}K(\sT_{\overline{k}})_\Q$, of Hochschild homology in characteristic zero, or of rational topological $K$-theory over $\mathbb{C}$ implies that the rational noncommutative motive of $\sT$ vanishes. In this way, our results provide a partial answer to a question raised by Sosna. We also establish a deformation-invariance result for phantomness in smooth proper families.