Limits and colimits in silting theory with applications to the wall and chamber structure of an algebra

TL;DR

This paper explores the construction of silting objects via homotopy limits and colimits, applying these to understand the wall and chamber structure of finite-dimensional algebras, especially in tame cases.

Contribution

It introduces a method to construct silting objects from nested t-structures and applies this to describe the wall and chamber structure of an algebra's Grothendieck group.

Findings

Constructed silting objects as homotopy colimits and limits.

Applied constructions to describe walls and chambers in algebra's structure.

In tame cases, characterized all numerical torsion pairs using these methods.

Abstract

In this paper we consider a family of nested t-structures given by silting objects and construct a silting object corresponding to the intersection of aisles of these t-structures as a homotopy colimit. The dual construction for the cosilting case is given as a homotopy limit. The results are applied to construct two-term large silting objects corresponding to the numerical torsion pairs and the limiting walls in the wall and chamber structure of the real Grothendieck group of a finite dimensional algebra. In particular, in case the algebra is tame we can describe any numerical torsion pair in this way by combining our results with results of Plamondon and Yurikusa.