Homology of twisted quiver bundles with relations

TL;DR

This paper investigates the Ext modules in the category of modules over twisted quiver algebras with relations on a ringed space, introducing a spectral sequence connecting these Ext modules with those of sheaves, and analyzing their properties.

Contribution

It introduces a spectral sequence relating Ext modules over twisted quiver algebras with sheaf Ext modules, accounting for relations and non-degeneration phenomena.

Findings

Spectral sequence relates Ext modules in twisted quiver categories to sheaf Ext.

In presence of relations, the spectral sequence may not degenerate early.

Under certain conditions, Ext modules are represented as hypercohomology groups.

Abstract

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space $(X,\mathcal O_X)$, allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that category with the Ext modules in the category of $\mathcal O_X$-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups