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A Criterion for Phantomness of dg-categories

Keiho Matsumoto

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.

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 be a smooth proper variety over a field , and let be a -linear admissible full dg subcategory. We construct a non-compact motive and show that its -adic realization recovers the -local algebraic -theory of . Analogous statements are obtained for Betti and de Rham realizations, which recover topological -theory and periodic cyclic homology, respectively. As a consequence, assuming that the Chow motive of is Kimura-finite, we prove a criterion for phantomness: the vanishing of , of Hochschild homology in characteristic zero, or of rational topological -theory over implies that the rational noncommutative motive of 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.

Paper Structure

This paper contains 11 sections, 14 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Dg-categories and motives
  4. commutative motive of a dg-category
  5. Realization of $\mathcal{M}(\mathcal{T})$
  6. Construction of $F_1$
  7. Construction of $F_2$
  8. Equivalence of $F_1$ and $F_2$
  9. Realizations of $\mathcal{M}(\mathcal{T})$
  10. Phantom and motives
  11. deformation invariance of phantomness

Key Result

Theorem 1.2

Let $k$ be a field, and let $\mathcal{T}$ be a non-zero $k$-linear smooth proper dg category admitting a geometric realization $\mathcal{T} \hookrightarrow \operatorname{perf}_{\operatorname{dg}}(X)$. Assume that $X$ has Kimura-finite Chow motive. Then:

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 16 more