$R$-linear triangulated categories and stability conditions

TL;DR

This paper introduces the concept of support in $R$-linear triangulated categories and investigates the conditions under which Bridgeland stability conditions do not exist in these categories.

Contribution

It defines support for objects in $R$-linear triangulated categories and applies this to analyze the non-existence of Bridgeland stability conditions.

Findings

Support concept helps understand stability conditions.

Bridgeland stability conditions do not exist in certain $R$-linear categories.

Provides criteria for non-existence of stability conditions.

Abstract

Let $R$ be a commutative ring. We introduce the notion of support of objects in an $R$-linear triangulated category. As an application, we study the non-existence of Bridgeland stability conditions on $R$-linear triangulated categories.