Gluing silting objects along recollements of well generated triangulated categories

TL;DR

This paper introduces a method to glue silting objects across recollements in well generated triangulated categories, generalizing previous results and applying to classifications of modules and sheaves.

Contribution

It provides an explicit, compatible procedure for gluing silting objects in well generated triangulated categories, extending prior work and enabling new classifications.

Findings

Gluing procedure compatible with co-t-structures

Conditions for restricting to tilting objects

Classification of silting modules and tilting sheaves

Abstract

We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vit\'{o}ria and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a weighted noncommutative regular projective curve of genus 0.