Tensor stable moduli stacks and refined representations of quivers

TL;DR

This paper introduces tensor stable moduli stacks and refined representations to improve the modular realization of derived equivalences and stack recovery, incorporating monoidal structures into moduli problems.

Contribution

It develops a new tensor stable moduli stack framework and explores its application to derived equivalences and stack reconstruction, especially for tilting bundles composed of line bundles.

Findings

Tensor stable moduli stacks generalize Serre-stable moduli stacks.

The new approach aids in stack recovery and derived equivalence realization.

Relations between tensor stable and refined representation moduli are established.

Abstract

In this paper, we look at the problem of modular realisations of derived equivalences, and more generally, the problem of recovering a Deligne-Mumford stack $\mathbb{X}$ and a bundle $\mathcal{T}$ on it, via some moduli problem (on $\mathbb{X}$ or $A = \operatorname{End}_{\mathbb{X}} \mathcal{T}$). The key issue is, how does one incorporate some of the monoidal structure of $\operatorname{Coh}(\mathbb{X})$ into the moduli problem. To this end, we introduce a new moduli stack, the tensor stable moduli stack which generalises the notion of the Serre-stable moduli stack. We then show how it can be used both for stack recovery and the modular realisation problem for derived equivalences. We also study the moduli of refined representations and how it addresses these problems. Finally, we relate the two approaches when $\mathcal{T}$ is a tilting bundle which is a direct sum of line bundles.