Three Discussion Problems (from Eric Hanson)

Below are three problems (all of different flavors) related to mutations and torsion classes.

Semibrick pairs and subsets of 2-term simple-minded collections

Let A be a finite-dimensional algebra, and let D and U be sets of bricks in mod(A). We say that (D,U) is a canonical pair if there exists a torsion class T in tors(A) such that every brick in D labels a downward cover (or left/green mutation) of T and every brick in U labels an upward cover (or right/red mutation) of T. We likewise say that (D,U) is a semibrick pair if (i) Hom(X,Y) = 0 for all X \neq Y in D, (ii) Hom(X,Y) = 0 for all X \neq Y in U, and (iii) Hom(X,Y) = 0 = Ext^1(X,Y) for all X in D and U in Y. It is known that every canonical pair is a semibrick pair, but not vice versa [5, Counterexample 1.9]. This leads to the following problem.

Problem: determine which semibrick pairs are also canonical pairs.

Some cases of this problem have already been solved. For example, every semibrick pair of the form (D,\emptyset) or (\emptyset,U) is a canonical pair [2, Theorem 1.8]. For tau-tilting finite algebras, the canonical pairs are precisely the subsets of the so-called 2-term simple-minded collections [1, Theorem 3.3], and there is an algorithm (based on concepts from mutation) for determining whether an arbitrary semibrick pair is such a subset [4, Section 3]. Moreover, a very recent result shows that for silting discrete algebras (which are in particular tau-tilting finite), every semibrick pair which satisfies |D| + |U| = |A| is a canonical pair [6, Corollary 6.13(3)]. Thus as starting points, one could consider either (or both) of the following.

Problem: extend the algorithm for determining whether a semibrick pair is a 2-term simple-minded collection to tau-tilting infinite algebras.

Problem: given a semibrick pair (D,U), determine if (D,U) is a subset of a 2-term simple-minded collection without computing other elements of its mutation class.

Let me also mention that the semibrick pairs and 2-term simple-minded collections of both Nakayama algebras [5, Section 3] and preprojective algebras of type A [3, Section 7] [7] have combinatorial descriptions that make them a good option for computing examples.

References:

[1] S. Asai. Semibricks, IMRN (2020), no. 16, 4993-5054. https://doi.org/10.1093/imrn/rny150.

[2] E. Barnard, A. Carroll, and S. Zhu. Minimal inclusions of torsion classes, Algebra. Comb. 2 (2019), no. 5, 879-901. https://doi.org/10.5802/alco.72.

[3] E. Barnard and E. J. Hanson. Pairwise compatibility for 2-simple minded collections II: preprojective algebras and semibrick pairs of full rank, Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00585-4.

[4] E. J. Hanson and K. Igusa. Pairwise compatibility for 2-simple minded collections, J. Pure Appl. Alg. 225 (2021). https://doi.org/10.1016/j.jpaa.2020.106598.

[5] E. J. Hanson and K. Igusa. tau-cluster morphism categories and picture groups, Comm. Alg. 49 (2021), no. 10, 4376-4415. https://doi.org/10.1080/00927872.2021.1921184.

[6] W. Hara and M. Wemyss. Spherical objects in dimensions two and three, https://arxiv.org/abs/2205.11552.

[7] Y. Mizuno, Arc diagrams and 2-term simple-minded collections of preprojective algebras of type A, J. Algebra 495 (2022), 444-478. https://doi.org/10.1016/j.jalgebra.2021.12.029.

Big picture groups

Let A be a finite-dimensional algebra. If A is tau-tilting finite, the picture group of A is the group G(A) with a generator x(T) for every torsion class T in tors(A), a generator y(B) for every brick B in mod(A), a relation that x(0) is the identity of the group, and a relation x(T) = y(B) x(U) whenever there is an arrow T --> U in the Hasse quiver of tors(A) (or equivalently in the exchange quiver) which is labeled by B. (This presentation is given in [1, Proposition 4.4], but the original definition in [3] actually comes from wall-and-chamber structures!) There are also many other presentations of the picture group, for example using maximal green sequences and (tau-)cluster morphism categories.

There are many choices for generalizing the definition of the picture group to the tau-tilting infinite case. For example, one can restrict to only the functorially finite torsion classes and bricks labeling arrows in the exchange quiver to obtain what we will call the (small) picture group. Alternatively, one can consider all torsion classes and bricks to obtain what we will call the big picture group. One interesting problem is to verify or find a counterexample for the following.

Conjecture: Let A be a finite-dimensional algebra. Then the following are equivalent.

1. A is tau-tilting finite.

2. The (small) picture group of A is finitely presented.

3. The (small) picture group of A is finitely generated.

4. The big picture group of A is finitely presented.

5. The big picture group of A is finitely generated.

6. The (small) picture group of A and the big picture group of A are isomorphic.

In fact, each presentation of the picture group in the tau-tilting finite case can be generalized to define many "infinite-type picture groups" which may or may not be isomorphic, see e.g. [2, Section 3.7]. Thus a less well precise, but nevertheless interesting, problem is the following.

Problem: Determine which presentations of the picture group in the tau-tilting finite case generalize to isomorphic groups in the infinite case.

References:

[1] E. J. Hanson and K. Igusa. tau-cluster morphism categories and picture groups, Comm. Alg. 49 (2021), no. 10, 4376-4415. https://doi.org/10.1080/00927872.2021.1921184.

[2] K. Igusa, M. Kim, G. Todorov, and J. Weyman. Periodic trees and propictures, unpublished preprint available at: https://people.brandeis.edu/~igusa/Papers/PeriodicProPics.pdf.

[3] K. Igusa, G. Todorov, and J. Weyman. Picture groups of finite type and cohomology in type A_n, https://arxiv.org/abs/1609.02636.

tau-perpendicular wide subcategories

Let A be a finite-dimensional algebra. Following [3] and [4, Section 4.2], to each tau-rigid pair (M,P) associate the subcategory J(M,P) consisting of those modules X for which:

1. Hom(M,X) = 0

2. Hom(X, tau M) = 0

3. Hom(P, X) = 0.

That is, J(M,P) is the intersection of the torsion-free class corresponding to the torsion class Fac(M) with the torsion class whose Ext-projectives form the Bongartz completion of (M,P). We refer to those categories which can be expressed as J(M,P) as the tau-perpendicular subcategories.

Every tau-perpendicular subcategory is equivalent to the module category of some finite-dimensional algebra, and thus has its own notions of torsion classes and tau-rigid pairs. Moreover, the tau-tilting theory of J(M,P) is closely related to that of mod(A). In particular, suppose (M,P) is a tau-rigid pair in mod(A) and (N,Q) is a tau-rigid pair in W = J(M,P). (Caution: (N,Q) will in generally not be a tau-rigid pair in mod(A)!) Then there exists a tau-rigid pair (N',Q') in mod(A) such that J(N',Q') = J_W(N,Q), where J_W denotes that the tau-pependicular subcategory is being computed in W [1, Corollary 1.2]. It is an open problem whether the converse is true. More precisely, we have the following conjecture, which is verified in the tau-tilting finite case in [3, Theorem 4.18].

Conjecture: Let (M,P) and (N,Q) be tau-rigid pairs in mod(A), and suppose that J(N,Q) is contained in W = J(M,P). Then there exists a tau-rigid pair (N',Q') in W such that J(N',Q') = J_W(N,Q).

For some additional motivation, tau-perpendicular subcategories are in particular (exact embedded) abelian subcategories of mod(A); that is, they are wide. As with the torsion classes, the wide subcategories of mod(A) form a lattice under containment. Validating this conjecture would then say that the relation V \leq W if V is a tau-perpendicular subcategory of W is a subposet of this lattice.

In cases where this conjecture is verified, one can then compare the subposet of tau-perpendicular subcategories with other subposets of wide subcategories. For example, it is known that every tau-perpendicular wide subcategory is functorially finite, but not vice versa. On the other hand, functorially finite wide subcategories are precisely those which are equivalent to module categories of finite-dimensional algebras, meaning they each have their own tau-tilting theory. This leads to the following problem.

Problem: Let W be a functorially finite wide subcategory of mod(A). Is there any relationship between the tau-perpendicular wide-subcategories of W and those of mod(A)?

References:

[1] A. Buan and E. J. Hanson. tau-perpendicular wide subcategories. https://arxiv.org/abs/2107.01141.

[2] G. Jasso. Reduction of tau-tilting modules and torsion pairs, IMRN (2015), no. 16, 7190-7237. https://doi.org/10.1093/imrn/rnu163.

[3] L. Demonet, O. Iyama, N. Reading, I. Reiten, and H. Thomas. Lattice theory of torsion classes, https://arxiv.org/abs/1711.01785.