Let (I, ≤) be a totally ordered set. A stratifying system in an abelian (exact) category is a set of indecomposable objects Θ={θ_i : i ε Θ} such that Hom(θ_i, θ_j)=0 if i > j and Ext^1(θ_i, θ_j)=0 if i ≥ j.
In a joint work with Octavio Mendoza we have shown that every τ-rigid module T induces a stratifying system Θ_T in mod A, where A is an Artin algebra (see https://arxiv.org/pdf/1904.11903.pdf). The main idea behind the proof of this result is the existence of a partial order on the indecomposable direct summands of T, which is a consequence of the good behaviour of support τ-tilting modules under mutation.
In a recent work, Angler-Hügel, Laking, Marks and Vitoria have shown the existence of a notion of mutation for cosilting modules in Mod R, where R is a unitary associative ring (see https://arxiv.org/pdf/2201.02147.pdf). In this problem session we will try to show that this notion of mutation induces a partial order on the indecomposable direct summands of a silting module S in Mod R. If this is possible we would like to conclude that some (hopefully every) silting object S induces a stratifying system Θ_S.