A Criterion for Phantomness of dg-categories

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.