Geometric Phantom Categories

TL;DR

This paper constructs special subcategories called phantom categories within derived categories of smooth projective varieties, showing they have trivial Hochschild homology, Grothendieck group, and K-motives, revealing new insights into their structure.

Contribution

It introduces a method to construct phantom categories with trivial Hochschild homology, Grothendieck group, and K-motives, advancing understanding of their properties.

Findings

Phantom categories have trivial Hochschild homology.

They also have trivial Grothendieck group.

All higher K-groups of these categories are trivial.

Abstract

In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense, namely, they have trivial K-motives and, hence, all their higher K-groups are trivial too.