Geometric phantom categories do not admit Noetherian t-structures
Yeqin Liu
TL;DR
The paper addresses whether geometric phantom or quasi-phantom categories can admit bounded $t$-structures whose hearts are Noetherian or Artinian. It develops a generator-based framework in which the heart $\mathcal{A}$ of a bounded $t$-structure on such a category must have a generator-derived object $G' = \bigoplus_{i\in\mathbb{Z}} \mathcal{H}^{i}_{\mathcal{A}}(G)$ that generates $\mathcal{A}$, and then uses Krull-Gabriel dimension $\mathrm{KGdim}(\mathcal{A})$ to force a contradiction with $K_{0}(\mathcal{A})=0$ or torsion. The main result shows that geometric phantom or quasi-phantom categories do not admit Noetherian or Artinian bounded $t$-structures, a conclusion that extends to any small triangulated category with $K_{0}$ vanishing (or torsion) and a classical generator. This clarifies a fundamental obstruction to stability conditions and related moduli problems in these categories, impacting the understanding of semi-orthogonal decompositions and related homological invariants.
Abstract
There are no Noetherian or Artinian bounded t-structures on geometric phantom or quasi-phantom categories.