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.

Note: This problem will be also be explained on my research talk on Wednesday. An algebra is said to be g-tame if the union of all the cones spanned by a τ-tilting pairs is dense in R^n. The statement of the second τ-Brauer-Thrall conjecture states that if an algebra is τ-tilting finite, then there exists an infinite family of bricks (i.e. modules whose endomorphism algebra is a division ring) {B_j : j \in J} such that dim_k B_j = d for all j \in J. We want to show the second τ-Brauer-Thrall conjecture for g-tame algebras. In order to do this, we will first see that every τ-tilting infinite algebra has an infinite chain of torsion classes in their lattice of torsion classes. Then we will realise this infinite chain of torsion classes as an infinite set of distinct points in the wall-and-chamber of the algebra in such a way that this sequence of points has a limit. We then will study the category of stable modules in this limit point. If we are lucky, we can find an invite family of stable objects at this point. Since every stable module is necessarily a brick, then we would have proven the second τ-Brauer-Thrall conjecture for g-tame algebras.