On the triangulated structure of stable monomorphism categories

TL;DR

This paper explores the triangulated structure of stable monomorphism categories over Frobenius categories, revealing symmetries, decompositions, and auto-equivalences with applications to lifting representing objects.

Contribution

It uncovers the semiorthogonal decompositions and auto-equivalences in stable monomorphism categories, extending the understanding of their triangulated structure.

Findings

Identifies polygons of recollements with auto-equivalences

Shows a power of auto-equivalence equals the square of suspension functor

Constructs chains of adjoint pairs from these decompositions

Abstract

We investigate the triangulated structure of stable monomorphism categories (filtered chain categories) over a Frobenius category. The high degree of symmetry of linear quivers leads to a plethora of semiorthogonal decompositions into smaller categories of the same type. These form polygons of recollements, in which a full turn of mutations is a power of a particular auto-equivalence of the stable monomorphism category. A certain power of this auto-equivalence is the square of the suspension functor. We describe the infinite chains of adjoint pairs obtained from the polygons. As an application, we explicate the construction of Bondal and Kapranov for lifting representing objects of dualized hom-functors in our setup.