# Problems

In this page, I raise some problems in the representation theory of algebras, which are interesting to me, but which seem so difficult that I can’t answer by myself. I hope someone will solve (or at least consider) them.

I DO NOT claim the originality of the problems below, hence you’re free to solve and write papers about them.

## Self-orthogonal modules and Wakamatsu tilting modules

I list some conjectures and questions raised in my new paper, below. For additional details, please refer to the paper.

A $\Lambda$-module $T$ is self-orthogonal if $\operatorname{Ext}^{>0}(T, T) = 0$. $T$ is maximal self-orthogonal if $T$ is self-orthogonal, and if $T \oplus M$ is self-orthogonal, then $M \in \mathsf{add} \, T$. As usual, $|T|$ denotes the number of non-isomorphic indecomposable direct summands of $T$.

• Boundedness conjecture (BC): If $T$ is self-orthogonal, then $|T| \leq |\Lambda|$.
• Proj=Inj conjecture (PIC): Let $\mathcal{C}$ is a subcategory of $\mathsf{mod}\,\Lambda$ closed under extensions and summands which has an Ext-progenerator $P$ and an Ext-injective cogenerator $I$. Then $|P| = |I|$ holds. Equivalently, if $T$ is Wakamatsu tilting, then $|T| = |\Lambda|$ holds.
• Maximal self-orthogonal conjecture (MSOC): If $T$ is Wakamatsu tilting, then $T$ is maximal self-orthogonal.

It is shown that (BC) implies (PIC) and (MSOC), which in turn imply the famous Auslander-Reiten Conjcture and the Generalized Nakayama conjecture (so they should be very difficult…)

I also interested in the following variant of (PIC) (which is essentially same as the one raised by Auslander and Smalo’s paper):

• Let $\mathcal{C}$ is a subcategory of $\mathsf{mod}\,\Lambda$ closed under extensions and summands. Suppose that $\mathcal{C}$ has an Ext-progenerator $P$ and has enough Ext-injectives. Is it true that $\mathcal{C}$ has an Ext-injective cogenerator $I$ (that is, there are only fintely many Ext-injectives), at least under some assumptions like functorial finiteness? If so, does $|P| = |I|$ hold?

The following question on Iwanaga-Gorenstein algebras should be much more tractable. $\Lambda$ is Iwanaga-Gorenstein if the injective dimension of $\Lambda$ as $\Lambda$-module (both as left and right) is finite.

• If $\Lambda$ is Iwanaga-Gorenstein, then does every self-orthogonal $\Lambda$-module have finite projective dimension?
• If $\Lambda$ is Iwanaga-Gorenstein, then is every Wakamatsu tilting module tilting?

One more technical question:

• Suppose that $T$ is an Ext-progenerator of $T^\perp$. Then is $T^\perp$ covariantly finite? Here $T^\perp$ consists of $X$ such that $\operatorname{Ext}^{>0}(T, X) = 0$.

It is shown that the set of Wakamatsu tilting modules with the binary relation $T_1 \geq T_2$ by $\operatorname{Ext}^{>0}(T_1, T_2)$ is not a poset in general (not transitive, even for the rep-fin case). This is in contrast to tilting modules with finite projective dimension. So the following questions.

• When does the set of Wakamatsu tilting modules become a poset? (some combinatorial criterion for Nakayama algebras may be interesting)

• Is there a theory of mutation of Wakamatsu tilting modules, which enables us to compute all Wakamatsu tilting modules by starting from a ring itself? I observed that two summands may differ at the Hasse arrow.

## Combinatorics of Auslander-Reiten quivers

Suppose that the Auslander-Reiten quiver of the module category is given. Then can we compute the following only from the AR quiver?

• Extension-closed subcategories closed under direct summands
• Syzygy for a given indecomposable module
• Projective dimension and injective dimension, global dimension
• Fac M and Sub M for a given M
• Kernel of add M-approximation of an indecomposable X

## Torsion classes

• Characterize the lattice of torsion classes of a tau-tilting finite artin algebra. For example, it is known that such a lattice satisfies the following two conditions. Conversely, if $L$ satisfies the above two, then is there exists some artin alg $A$ such that tors $A$ is isomorphic to $L$?
• L is semidistributive (or more strongly, congruence-uniform).
• L is Hasse-regular.

NO. Consider the following lattice, generated by congruence-uniform lattice generator in SageMath.

Then this is congruence-uniform (since this can be constructed by iteratind interval doublings) and Hasse-regular, but cannot be isomorphic to the lattice of torsion classes in an abelian length category, because (for example) its core-label order is not a lattice, which should be isomorphic to the lattice of wide subcategories of $A$ (see this paper).

• In the lattice of torsion classes of artin algebras, completely join-irreducibles are in bijection with finitely generated bricks. Is there a similar characterization of not necessarily completely join-irreducibles? Maybe one can use infinitely generated bricks, see F. Sentieri, A brick version of a theorem of Auslander. It will be very nice if there’s bijection between not necessarily finitely generated (good? e.g. pure-injective?) bricks and join-irreducible elements.

• Is there an analogue of $\tau$-tilting theory for exact categories? More precisely, can we classify torsion classes of exact categories using projective objects?

• Is there a purely poset-theoretical characterization of torsion classes in the poset of torsion classes which are

• functorially finite?
• faithful?
• functorially finite and faithful (thus corresponds to a tilting module)? In particular, can we construct the simplicial complex of tilting modules only using the poset structure?
• Is there a categorical characterization of the heart of intervals in the poset of torsion classes (Tattar’s twin torsion heart)?

• If we consider $\tau$-tilting finite algebras, then the Hasse quiver of the poset of torsion classes is a regular graph by the theory of mutation (Adachi-Iyama-Reiten). On the other hand, if we consider an arbitrary abelian length category, some properties of torsion classes of artin algebras are also satisfied (mainly those about bricks), e.g. DIRRT’s brick labeling and bijection between join-irreducible torsion classes and bricks, Asai-Pfeifer’s characterization of wide intervals, and a bijection between torsion classes and wide subcategories provided that there are only finitely many torsion classes (see my paper). Thus,

• Is the Hasse quiver of torsion classes of an abelian length category a regular graph if there are only finitely many torsion classes?
• To what extent can we generalize many papers on the poset of torsion classes over artin algebras?
• It is interesting to investigate torsion classes in the category of modules with finite length over non-artinian ring (e.g. commutative Cohen-Macaulay ring, etc.).

## Wide subcategories

• Is there any wide subcategory $\mathcal{W}$ in the module category of an artin algebra such that $\mathcal{W}$ has enough projectives and there are infinitely many indecomposable projective objects in $\mathcal{W}$?

• Suppose that there is a common upper bound on the cardinality of semibricks. Then is this algebra necessarily $\tau$-tilting finite? (The converse of course holds by the result of e.g. Asai.)

• By computer computation, I conjecture that for $\tau$-tilting finite case, the lattice of wide subcategory
• is strongly Sperner (so becomes a Peck poset known in combinatorics), or more strongly,
• has a symmetric Boolean decomposition. Since the non-crossing partition lattice and the shard intersection lattice are realized as lattices of wide subcategories, this will unify several known results by combinatorialists.
• Or are there more properties such that the lattice of wide subcategory has? Especially, what can be said if we consider $\tau$-tilting infinite case, or we only consider left finite wide subcategories?
• For the hereditary case, there is a self-duality on the lattice of wide subcategories. For a general algebra $\Lambda$, is there any algebra $\Gamma$ such that the lattices of wide subcategories over $\Lambda$ and $\Gamma$ are dual to each other? This will make things easier to deal with.
• Is there a combinatorial (or surface) model for the lattice of wide subcategories over gentle algebras?

## ICE-closed subcategories

• Characterize the number of Hasse arrows starting at an ICE-closed subcategory in the poset of ICE-closed subcategories, or characterize a Hasse arrow (minimal inclusion of ICE-closed subcategories). In the poset of torsion classes, the number of arrows starting at a torsion class is equal to the number of indecomposable split projective objects (for $\tau$-tilting finite case). I and Sakai conjectured that the number of Hasse arrows is equal to the number of indecomposable projective objects in the case of ICE-closed subcategories, which are true for Dynkin path algebras (by our paper) and Nakayama algebras (in preparation), but the computer experiment shows that this conjecture is not true in general. Thus

• Is there any class of algebras such that this conjecture is true?
• Is there a combinatorial (or surface) model for the lattice of wide subcategories over gentle algebras?
• How many ICE-closed subcategories the preprojective algebras of Dynkin type have? Is there any Lie-theoretic interpretation of ICE-closed subcategories?

## Extension-closed subcategories

• Is there any extension-closed subcategory (closed under summands) $\mathcal{E}$ in the module category of an artin algebra such that $\mathcal{E}$ has enough projectives and there are infinitely many indecomposable projective objects in $\mathcal{E}$?

• YES in general. Consider the subcategory of n-preprojective modules in any n-representation infinite algebra with n>1. Then this category is extension-closed and every object is both projective and injective. In particular, it trivially has enough projectives and there are infinitely many indecomposable projectives. See Question 1.2 and Corollary 4.11 on Herschend, Iyama, Oppermann, n-representation infinite algebras. I thank Sondre Kvamme for letting me know this example.
• Let $\mathcal{E}$ be a functorially finite extension-closed subcategory of the module category of an artin algebra. Then it’s known that $\mathcal{E}$ has both enough projectives and enough injectives. Are the numbers of indecomposable projectives and injectives finite in this case? And do the numbers coincide?

• Are there any classes of subcategories which we can classify? How about subcategories closed under images and extensions (see our paper) for the non-hereditary case?

## Higher homological algebra

Can we classify all the possible structure of $n$-exact categories on a given additive category like my paper on exact structures? In particular, does the unique maximum $n$-exact structure always exists?

## Grothendieck monoid

• Compute the Grothendieck monoids of a special class of exact categories, for example, torsion(-free) classes over path algebras of Dynkin type, or even type A. In particular, are these monoids reduced?

• We can define the Grothendieck monoids of extriangulated categories. It seems that triangulated categories behave in contrast to exact categories with respect to the Grothendieck monoid, e.g. the Grothendieck monoid of exact categories is always reduced (invertible element is only 0), but every element is invertible in the Grothendieck monoid of triangulated category. Therefore,

• Is there any way to compute the Grothendieck monoids of extriangulated categories (at least those embedded in some triangulated categories)?
• Is there any characterization of objects whose images in the Grothendieck monoid are invertible?
• Since the Grothendieck monoid is a commutative monoid, there is a unique abelian subgroup s.t. the quotient monoid is reduced. It would be very nice if such a factorization can be realized categorically (e.g. describe an extriangulated category as a some kind of semi-direct product of a triangulated category and an exact category).

Some of the above questions are addressed in our recent paper with S. Saito, but not fully answered (see Section 6: Questions).

## Lean Theorem Prover

Teaching Auslander-Reiten theory to computer using Lean, see also Lean community, and a very good introductory material Formalising Mathematics

## SageMath

• Create a system which can deal with the Auslander-Reiten quivers as translation quivers. In particular, compute the dimension of Hom spaces using AR quiver.
• Create or modify programs which enables us to deal with string and band modules over special biserial (or string) algebras.
• More concretely, develop a SageMath version of String Applet!

I know that QPA are useful, but I think there’s no special functions which can applied to special biserial, string, gentle algebras. Since SageMath can deal with other materials including posets, polytopes, lattices, root system, and so on (and since it’s written in Python), I hope that special computational experiments on e.g. string algebras can be done inside SageMath.