Tensor stable moduli stacks and refined representations of quivers
Abstract
In this paper, we look at the problem of modular realisations of derived equivalences, and more generally, the problem of recovering a DeligneMumford 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 Serrestable 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.07070
 Bibcode:
 2019arXiv190707070A
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory
 EPrint:
 27 pages, comments are welcome