For a better format of the mathematical expressions, see the attached Pdf. file!
This problem was suggested by Francesco Sentieri, based on a question posed by
Haruhisa Enomoto on https://haruhisa-enomoto.github.io/problems/
Let A be a finite dimensional algebra. In the lattice of torsion classes tors(A), the completely meet-irreducible classes are in bijection with finitely generated bricks. More precisely, for a torsion pair (u,v) in mod-A, the class u is completely meet-irreducible if and only if there is a unique brick B in mod-A such that u = {M in mod-A | Hom_A(M;B) = 0} (see [2, Theorem 3.4]).
Is there a similar characterization of meet irreducible torsion classes, maybe involving infinite dimensional bricks? Let us fix a torsion pair (u,v) in mod-A. We will see in my lecture series that (u,v) is associated to a cosilting torsion pair (U,V) = (direct limit of u, direct limit of v) in Mod-A, and to a non-empty semibrick consisting of the modules which are torsion-free, almost torsion with respect to (U,V). The modules in this semibrick may be infinite dimensional, the finite-dimensional ones correspond to torsion classes covering u. For details we refer to [2, 1, 3].
Question: Is it true that u is meet-irreducible if and only if there is precisely one torsion-free, almost torsion module with respect to (U,V)?
As a first step, one should observe that if the class u is meet-irreducible but not completely meet-irreducible, then all modules which are torsion-free, almost torsion with respect to (U,V) are in finite dimensional. Then the question becomes:
Question: Is it true that u is meet-irreducible but not completely meet-irreducible if and only if there is precisely one torsion-free, almost torsion module with respect to (U,V) and this module is infinite dimensional?
Here are some more specific problems.
Questions:
(1): Is there a torsion pair (u,v) in mod-A such that u is not meet-irreducible and all torsion-free, almost torsion module with respect to (U,V) are infinite dimensional?
(2) Is there a torsion pair (u,v) in mod-A such that u is not meet-irreducible and there are only a finite number of infinite dimensional torsion-free, almost torsion module with respect to (U,V)?
References:
[1] E. Barnard, A. Carroll and S. Zhu, Minimal inclusions of torsion classes, J. Algebraic
Combin. 2 (2019) 879901.
[2] L. Demonet, O. Iyama, N. Reading, I. Reiten and H. Thomas, Lattice theory of torsion
classes, preprint (2017), arXiv:1711.01785.
[3] F. Sentieri, A brick version of a theorem of Auslander, preprint (2020), arxiv:2011.09253.