[Paper Review] Topologies on a triangulated category

So Okada|ArXiv.org|Jan 18, 2007

Homotopy and Cohomology in Algebraic Topology19 references3 citations

TL;DR

This paper introduces a topology on the objects of a triangulated category equipped with a stability condition, using the central charge to define a topological structure. It shows that under faithful or numerically faithful stability conditions, this topology is compatible with the Grothendieck or numerical Grothendieck group, and proves that the space of objects is connected under this topology.

ABSTRACT

On objects of a triangulated category with a stability condition, we construct a topology.

Motivation & Objective

• To define a topology on the objects of a triangulated category using the central charge of a stability condition.
• To establish compatibility of this topology with the Grothendieck group or numerical Grothendieck group under faithful or numerically faithful stability conditions.
• To demonstrate that the space of objects in a triangulated category with a stability condition is connected under the induced topology.
• To analyze the structure of preimages of central charges, particularly for semistable objects and their S-equivalence classes.

Proposed method

• Define a topology on the set of objects in a triangulated category using the central charge map, treating phases as topological coordinates.
• Use the Harder-Narasimhan filtration and Jordan-Hölder decompositions to analyze the structure of semistable objects and their cycles.
• Introduce the notion of faithful and numerically faithful stability conditions to ensure distinct phases for linearly independent objects in $K({ m T})_{{f Q}}$ or $N({ m T})_{{f Q}}$.
• Analyze the preimage $\tilde{Z}^{-1}(\tilde{Z}(E))$ for objects $E$, showing that under faithfulness, such preimages are isomorphic to direct sums of point objects in $\mathbb{P}^1$.
• Apply autoequivalences and twist functors (e.g., $T_{{\mathcal{O}}_{{\mathbb{P}}^{1}}(w-1)}$) to relate objects across different phases and preserve topological structure.
• Use the Euler pairing and vanishing of Hom spaces in negative and high degrees to constrain possible extensions and prove triviality of certain morphism spaces.

Experimental results

Research questions

• RQ1 How can a topology be naturally defined on the objects of a triangulated category equipped with a stability condition?
• RQ2 Under what conditions is this topology compatible with the Grothendieck group or numerical Grothendieck group?
• RQ3 Is the space of objects in a triangulated category with a stability condition connected under the induced topology?
• RQ4 What is the structure of the preimage of the central charge map for semistable objects?
• RQ5 How do autoequivalences and twist functors affect the topological structure of objects?

Key findings

• The induced topology on the objects of a triangulated category via the central charge makes the space of objects connected.
• For a faithful stability condition, the preimage $\tilde{Z}^{-1}(\tilde{Z}(E))$ consists of objects isomorphic to direct sums of point objects in $\mathbb{P}^1$, up to S-equivalence and autoequivalences.
• When $\sigma$ is numerically faithful, the topology is compatible with the numerical Grothendieck group $N({\mathcal{T}})$.
• The Euler pairing $\chi(E, E)$ vanishes for stable objects $E$ and $\mathcal{O}_x$ of the same phase, implying $\dim\operatorname{Hom}^1(E, \mathcal{O}_x) = 0$.
• The Harder-Narasimhan filtration ensures that any object $E_x$ with $[E_x] = [{\mathcal{O}}_{{\mathbb{P}}^{1}}(w-1)[1]] + [{\mathcal{O}}_{{\mathbb{P}}^{1}}(w)]$ must be semistable, as no nontrivial morphisms exist between the factors.
• For $|n| > 1$, a semistable object $E$ with $[E] = n[\mathcal{O}_x]$ in $K({\mathcal{T}})$ has a Jordan-Hölder decomposition with composition factors isomorphic to $\mathcal{O}_x$, so $E$ is S-equivalent to an $n$-fold direct sum of points.