Pairwise Compatibility for 2-Simple Minded Collections
Eric J. Hanson, Kiyoshi Igusa

TL;DR
This paper develops an algorithm to determine when a set of bricks forms a 2-simple minded collection in $ au$-tilting finite algebras, with applications to gentle algebras and classifying spaces.
Contribution
It extends mutation definitions to semibrick pairs and characterizes 2-simple minded collections via pairwise compatibility for certain gentle algebras.
Findings
Algorithm for checking semibrick pairs in 2-simple minded collections.
Characterization of collections in gentle algebras with quivers of degree at most 2.
Classifying space of the $ au$-cluster morphism category is an Eilenberg-MacLane space under certain conditions.
Abstract
In -tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a -tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a -tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the -cluster morphism category of aβ¦
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J. Pure Appl. Algebra 225 (2021), no. 6. DOI:10.1016/j.jpaa.2020.106598.
Pairwise Compatibility for 2-Simple Minded Collections
Eric J. Hanson
Brandeis University, Department of Mathematics, 415 South Street, Waltham MA 02453, USA
Β andΒ
Kiyoshi Igusa
Brandeis University, Department of Mathematics, 415 South Street, Waltham MA 02453, USA
(Date: October 14, 2020)
Abstract.
In -tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a -tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a -tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the -cluster morphism category of a -tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.
Key words and phrases:
simple minded collections, -tilting, gentle algebras, representations of quivers, torsion classes, picture groups
2010 Mathematics Subject Classification:
16G20, 05E15
© 2020 Elsevier. This work is licensed under CC-BY-NC-ND 2.0. See https://creativecommons.org/licenses/by-nc-nd/2.0/ for a copy of this license.
Contents
Introduction
This paper furthers the study of the connection between -tilting theory and semi-invariant pictures described in [HI]. In that paper, the authors study a finitely presented group, called the picture group, associated to an arbitrary -tilting finite algebra. Picture groups (and picture spaces) were first defined in a special case by Loday in [Lod00], then by the second author, Todorov, and Weyman in the general hereditary case in [ITW]. In [HI], this is extended to the non-hereditary case. The picture group of an algebra can be realized as the fundamental group of a topological space, constructed as the classifying space of the -cluster morphism category of the algebra, as defined by Buan and Marsh in [BM19] to generalize a construction of the second author and Todorov in [IT].
Crucial to relating the cohomology of the picture group of an algebra to the cohomology of the associated topological space is an understanding of the algebraβs 2-simple minded collections, as defined in [KY14, BY13]. These collections generalize the idea of a complete collection of non-isomorphic simple modules, and are known to be in bijection with many other objects in representation theory, for example 2-term silting complexes, support -tilting objects, and functorially finite torsion classes (see [BY13, Asa20] for a more detailed list). In [HI], the authors show that the topological space and picture group have isomorphic cohomology when the 2-simple minded collections of the algebra are given by pairwise compatibility conditions, generalizing results of [IT] and [Igu].
One of the main results of this paper is to show that for most -tilting finite gentle algebras, the 2-simple minded collections cannot be defined using pairwise compatibility conditions, disproving a conjecture from [HI]. This is in stark contrast to many of the other associated structures in representation theory. For example, both support -tilting objects (see [AIR14, Thm. 2.12]) and 2-term silting objects (see [Aih13, Prop. 2.16]) are given by pairwise compatibility conditions. In light of this, our results indicate that 2-simple minded collections are in many ways much more subtle than some of these other structures.
Gentle algebras form a natural class to study due to the long known combinatorial description of their indecomposable modules and the morphisms between them in terms of strings (see [BR87, CB89, Sch99]). More recently, a basis has been given for extensions between indecomposable modules in [ΓPS20, BDM*+*19] and the -tilting theory of gentle algebras has been described in terms of non-kissing complexes (see [PPP19, BDM*+*19]). This has influenced further study of the relationship between gentle algebras and various combinatorial objects, such as ribbon graphs (see [Sch15, OPS]), marked punctured surfaces (see [BS19, LP20, PPP19, APS]), and biclosed sets (see [GMM20, GM20]). One of the central results of [GM20] relates the 2-simple minded collections of certain gentle algebras to the data of a noncrossing tree partition and its Kreweras complement. We remark that our work does not rely on this interpretation.
Notation and Terminology
Let be a finite dimensional, basic algebra over an arbitrary field . When we write , we assume that is a quiver and that is an admissible ideal unless otherwise stated. Our convention is to multiply paths left to right. We denote by the category of finitely generated (right) -modules. Throughout this paper, all subcategories are assumed to be full and closed under isomorphisms. For , we denote by (resp. the subcategory of direct summands (resp. factors, submodules) of finite direct sums of . Moreover, refers to the subcategory of objects admitting a (finite) filtration by the direct summands of . Given a subcategory , we define , and analogously.
We denote by the bounded derived category of . The symbol will denote the shift functor in all triangulated categories. We identify with the subcategory of consisting of stalk complexes centered at zero.
For an object in a category , we define the left-perpendicular category of as . We define the right-perpendicular category, , dually. For a subcategory , we define and analogously. If we say the objects and are Hom orthogonal. Likewise if the category is triangulated, , and , we say and are Ext orthogonal. We say and are Hom-Ext orthogonal if they are both Hom and Ext orthogonal. We denote by the category of indecomposable objects of .
Organization and Main Results
The contents of this paper are as follows. In Section 1, we recall the definitions and preliminary results we will use regarding semibricks, 2-simple minded collections, and gentle algebras. We also give the definition of a semibrick pair, which drops from the definition of a 2-simple minded collection the assumption that it generates the bounded derived category.
In Section 2, we define a notion of mutation for certain well-behaved semibrick pairs that agrees with that for 2-simple minded collections. This allows us to prove our first main theorem.
Theorem A** (Theorem 2.9).**
Let be -tilting finite. Then a semibrick pair of is a subset of a 2-simple minded collection if and only if it is mutation compatible (see Definitions 2.2, 2.7).
We conclude Section 2 by showing that the 2-simple minded collections of representation finite hereditary algebras can be defined using pairwise compatibility conditions, giving a new proof of a result of [IT].
In Section 3, we discuss a class of algebras we refer to as Nakayama-like algebras (Definition 3.1). In particular, we use our results on mutation compatibility to prove our second main theorem.
Theorem B** (Theorem 3.3).**
Let be a Nakayama-like algebra. Then the 2-simple minded collections of can be defined using pairwise compatibility conditions.
This implies a known result, namely this holds from [HI] in the case that is Nakayama and from [IT], [Igu] in the case that .
In Section 4, we use our results on mutation compatibility to prove the central theorem of this paper, disproving a conjecture from [HI].
Theorem C** (Theorem 4.1).**
Let be a -tilting finite gentle algebra such that contains no loops or 2-cycles. Then the 2-simple minded collections of can be defined using pairwise compatibility conditions if and only if every vertex of has degree at most 2.
Amongst the simplest algebras for which the 2-simple minded collections cannot be defined using pairwise compatibility conditions are cluster tilted algebras of type for . This is in contrast to the cyclic cluster tilted algebras of type , which were shown to have this property in [HI].
In Section 5, we discuss picture groups and construct faithful group functors for Nakayama-like and gentle algebras. We do so by constructing a group homomorphism from the picture group to the group of units of the power series 0-Hall algebra of (Definition 5.7), a variant of the Hall algebra constructed by evaluating the Hall polynomials at . This, together with the results on pairwise compatibility, allows us to prove our final main theorem.
Theorem D** (Corollary 5.13).**
- (a)
Let be a Nakayama-like algebra. Then the classifying space of the -cluster morphism category of is a for the picture group of . 2. (b)
Let be a -tilting finite gentle algebra such that contains no loops or 2-cycles. Then the classifying space of the -cluster morphism category of is a for the picture group of if every vertex of has degree at most 2.
1. Background
Recall that a subcategory is called a torsion class if it is closed under extensions and factors. Likewise, a subcategory is called a wide subcategory if it is closed under extensions, kernels, and cokernels. We denote by (resp. ) the poset of torsion classes (resp. wide subcategories) ordered by inclusion. We assume throughout this paper that is a finite set, or equivalently (see [DIJ19, Thm 3.8]) that is -tilting finite.
1.1. Semibricks and 2-Simple Minded Collections
Recall that a (necessarily indecomposable) object (or more generally ) is called a brick if is a division algebra. A set of bricks is called a semibrick if it consists of pairwise Hom-orthogonal bricks. Depending on context, the term semibrick can refer to either the set or the object . We denote by (resp. ) the set of isoclasses of bricks (resp. semibricks) in . Central to this paper are the closely related 2-simple minded collections, defined as follows by [BY13, Rmk. 4.11].
Definition 1.1**.**
Let with . Then is called a 2-simple minded collection if
- (a)
For all , we have for . Equivalently, for . 2. (b)
, where is the smallest triangulated subcategory of containing which is closed under direct summands.
Remark 1.2**.**
2-simple minded collections are examples of the more general simple-minded collections (see [KY14]). In this general context, condition (a) is replaced with the requirement that for . In our context, either or is a module (and likewise for ), so we need only consider .
We denote by the set of isoclasses of 2-simple minded collections for . This leads us to the following definition.
Definition 1.3**.**
[HI, Def. 1.8] Let . We say that is a semibrick pair if for (or equivalently, for all and ). In particular, a semibrick pair is a 2-simple minded collection if and only if . We say the semibrick pair is completable if it is a subset of a 2-simple minded collection.
It is in general difficult to determine when a semibrick pair is completable; we do, however, have the following, deduced from [Asa20, Thm. 2.3]. We remark that as stated, this result depends on the fact that is -tilting finite.
Theorem 1.4**.**
Let be a semibrick pair. If or , then is completable.
We conclude this section with the following result describing the relationship between semibricks and torsion classes. As before, our version of this statement requires that be -tilting finite.
Proposition 1.5**.**
[BCZ19, Prop. 3.2.5, Cor. 3.2.7]** There is a bijection given by .
1.2. String and Gentle Algebras
Two of the central classes of algebras studied in this paper are string algebras and the subclass of gentle algebras. We begin with their definitions.
Definition 1.6**.**
- (a)
A finite dimensional algebra is called a string algebra if
- (i)
The ideal is generated by monomials. 2. (ii)
Every vertex of is the source of at most two arrows and the target of at most two arrows. 3. (iii)
For every arrow , there is at most one such that and at most one such that . 2. (b)
A string algebra is called a gentle algebra if
- (i)
The ideal is generated by paths of length two. 2. (ii)
For every arrow , there is at most one such that and at most one such that .
In this paper, we will be interested only in gentle algebras whose quivers contain no loops or oriented 2-cycles. As we are assuming our algebras are all -tilting finite, we have already excluded all algebras whose quivers contain multiple arrows with the same source and target. Under these restrictions, we see that an algebra is gentle if and only if for all , the local picture at containing all arrows incident to is a subquiver of the following, where the dotted lines represent relations and all five vertices are distinct.
1432
The following standard definition allows for a simple description of the indecomposable modules of a string algebra.
Definition 1.7**.**
Let be a string algebra. A sequence , where each and is called a string if
- (a)
There is no subsequence of the form or 2. (b)
Considering as the arrow and as its formal inverse, we have for . 3. (c)
No subsequence (or its inverse) belongs to the ideal .
We also allow for sequences of the form , the constant string at the vertex .
We will consider string algebras which are representation finite. In this case, it is a well-known result of [BR87] (see also [GP68]) that there is a bijection between strings of (up to taking inverses) and isoclasses of . This bijection allows for a nice combinatorial description of the morphisms between irreducible modules, given first more generally in [CB89] and later for string algebras in [Sch99]. In lieu of summarizing this construction here, we will cite results about the construction as necessary. We refer interested readers to [BDM*+*19, Sec. 2], which contains a well-written summary.
We end this section with the following result that will be crucial in what follows.
Theorem 1.8**.**
[Pla19, Thm. 1.1]** A gentle algebra is -tilting finite if and only if it is representation finite.
In particular, this means that in order for a gentle algebra to be -tilting finite, every (not necessarily oriented) cycle of must contain a relation.
2. Mutation Compatibility of Semibrick Pairs
The goal of this section is to determine when a semibrick pair is completable. To do so, we define a notion of mutation for semibrick pairs. This intentionally mirrors the corresponding notion for 2-simple minded collections defined by Koenig and Yang in [KY14, Sec. 7.2]. We begin with the following generalization of Lemma 7.8 in their paper. We remark that our proof depends on the fact that is -tilting finite.
To simplify notation, for we denote by the underlying module of ; that is,
[TABLE]
Lemma 2.1**.**
Let be a semibrick pair. Then we have the following.
- (a)
For and , there exists a left minimal -approximation . Moreover,
- (i)
For any , the induced map is a bijection. 2. (ii)
The induced map is injective. 2. (b)
For and , there exists a right minimal -approximation . Moreover,
- (i)
For any , the induced map is a bijection. 2. (ii)
The induced map is injective.
Proof.
We show only (a) as the proof of (b) is similar. Let and . If , then is completable by Theorem 1.4, and hence the result follows from [KY14, Lem. 7.8]. Thus, we can assume , so both and are modules. Moreover, since is -tilting finite, the wide subcategory is functorially finite by [MΕ 17, Cor. 3.11] and [DIJ19, Thm. 3.8]. Therefore, there exists a left minimal -approximation .
(i): Let and let such that . This means . Moreover, we see that . Thus since is a left minimal -approximation, it must be the case that and hence . This means the induced map is injective (and hence is a bijection by the definition of a left minimal -approximation).
(ii): Suppose there exists such that . We consider as a short exact sequence as shown below. Then means there exists so that . But by (i), so there is a unique morphism so that . But then . Since by (i), we conclude that must be the identity on . Thus, (i.e., the corresponding short exact sequence is split).
{S}$${E^{\prime}}$${T[-1]}$${S}$${E}$${S_{|T|}}$$\scriptstyle{g_{S,|T|}^{+}}$$\scriptstyle{h}$$\scriptstyle{q}$$\scriptstyle{h^{\prime}}
β
We are now ready to define mutation.
Definition 2.2**.**
- (a)
[HI, Def. 1.8] Let be a semibrick pair. We say that is singly left mutation compatible at if for all , a left minimal -approximation is either a monomorphism or an epimorphism. We say is singly left mutation compatible111In [HI] the term mutation compatible semibrick pair is used. We have reserved this terminology for a more subtle property. if it is singly left mutation compatible at every . 2. (b)
Let be singly left mutation compatible at . We define the left mutation of at , denoted , as follows.
- β’
.
- β’
For define , where is a left minimal -approximation. In particular, there is an exact sequence .
- β’
For define , where is a left minimal -approximation. In particular, if is mono, then and if is epi, then . 3. (c)
Let be a semibrick pair. We say that is singly right mutation compatible at if for all , a right minimal -approximation is either a monomorphism or an epimorphism. We say is singly right mutation compatible if it is singly right mutation compatible at every . 4. (d)
Let be a singly right mutation compatible at . We define the right mutation of at , denoted , as follows.
- β’
.
- β’
For define , where is a right minimal -approximation. In particular, there is an exact sequence .
- β’
For define , where is a right minimal -approximation. In particular, if is mono, then and if is epi, then .
Remark 2.3**.**
The assumption that, for , the map is mono or epi is equivalent to .
Remark 2.4**.**
If then is both singly left mutation compatible and singly right mutation compatible. In this case, the new definitions of mutation agree with those for 2-simple minded collections. Moreover, these definitions are pairwise in the following sense. Suppose and are semibrick pairs with and let . If (and hence ) is singly left mutation compatible at , then for , we have .
Proposition 2.5**.**
Let be a semibrick pair which is singly left mutation compatible at . Then
- (a)
* is a semibrick pair which is singly right mutation compatible at .* 2. (b)
.
Likewise, the dual result holds for any singly right mutation compatible semibrick pair.
The proof is similar to that of [KY14, Prop. 7.6] and [Dug14, Thm. 6.2], but we include it here for completeness and to emphasize that the result does not depend on starting with a 2-simple minded collection.
Proof.
Let be a semibrick pair which is singly left mutation compatible at . For , let be the defining triangle for .
Claim 1: for all and . Indeed, since and , we have that automatically. Moreover, applying to the defining triangle for , we have an exact sequence
[TABLE]
which proves the claim.
Claim 2: for all and . Indeed, applying to the defining triangle for , we have an exact sequence
[TABLE]
Moreover, by Lemma 2.1, we have an exact sequence
[TABLE]
This proves the claim since .
Claim 3: for and . Indeed, applying to the defining triangle of , we have an exact sequence
[TABLE]
for by Claim 2. Likewise, applying to the defining triangle of , we have an exact sequence
[TABLE]
for . This proves Claim 3. In particular, taking , we have that is a brick.
Remark 2.3, together with Claims 1, 2, and 3, implies that is a semibrick pair. To show that is singly right mutation compatible at and , we observe that in the triangle , the map is a left minimal (-approximation with cone if and only if the map is a right minimal -approximation with cocone . β
Before we propose our new definition of mutation compatibility for a semibrick pair, we need the following results.
Lemma 2.6**.**
- (a)
Let be singly left mutation compatible at . Write . Then . 2. (b)
Let be a semibrick pair. If is singly left mutation compatible and , choose and let . Repeat this process for . Then after finitely many mutations, we reach a semibrick pair for which either is not singly left mutation compatible or for some .
Proof.
(a) Let . By Definition 2.2(b), there are two possibilities. If with , then there exists and an exact sequence . Otherwise, with and there exists and an exact sequence (note that in this case, the map must be mono). In either case, we have and hence . Moreover, we observe that .
(b) If this process did not terminate, we would end up with an infinite descending chain of torsion classes by (a). This violates the assumption that is -tilting finite. β
We now propose the following definition.
Definition 2.7**.**
Let be a semibrick pair. We call mutation compatible if any of the following hold:
- (a)
, that is, is a shifted semibrick. 2. (b)
, that is, is a semibrick. 3. (c)
is singly left mutation compatible and there exists such that is a mutation compatible semibrick pair.
Remark 2.8**.**
- (a)
This notion is well-defined by the previous lemma. 2. (b)
We could have chosen to define left mutation compatible and right mutation compatible separately. The reason we chose not to do so is that these conditions turn out to be equivalent as a corollary of the following theorem.
We are now ready to prove our first main theorem (Theorem A in the introduction).
Theorem 2.9**.**
Let be a semibrick pair. Then is completable if and only if is mutation compatible.
Proof.
Let be a mutation compatible semibrick pair. We already know that is completable if or (Theorem 1.4). Otherwise, by Definition 2.7, there exists a sequence of left mutations
[TABLE]
such that is a shifted semibrick. It follows that is completable (say to ). Thus by the pairwise nature of mutation (see Remark 2.4), there exists a sequence of right mutations of 2-simple minded collections
[TABLE]
and therefore is completable.
Now let be a completable semibrick pair, so for some . If or is empty, then is mutation compatible, so assume both are nonempty. We now define a sequence of semibrick pairs and 2-simple minded collections as follows.
- β’
and .
- β’
If , then choose and define and .
It follows that each is contained in a 2-simple minded collection, and hence is singly left mutation compatible. Moreover, there exists some which is a shifted semibrick by Remark 2.8. Therefore by the pairwise nature of mutation (see Remark 2.4), is mutation compatible. β
Corollary 2.10**.**
Let be mutation compatible. Then for , the semibrick pair is mutation compatible.
Proof.
If is contained in the 2-simple minded collection , then is contained in the 2-simple minded collection . β
2.1. The Pairwise 2-Simple Minded Compatibility Property
An interesting problem is to determine when the 2-simple minded collections of an algebra are given by pairwise conditions. More precisely, we are interested in which algebras satisfy the following definition.
Definition 2.11**.**
Let be a semibrick pair. We say that has the pairwise 2-simple minded compatibility property if either is completable or there exists such that , considered as a semibrick pair, is not completable. We say the algebra has the pairwise 2-simple minded compatibility property if every semibrick pair for does.
Note that the negation of this property for is: There exists a semibrick pair which is not completable, but in which any pair of direct summands can be completed to a 2-simple minded collection.
In Section 4, we will show that not every -tilting finite algebra has the pairwise 2-simple minded compatibility property, disproving Conjecture 1.11 from [HI]. We end this subsection by using our previous results to rephrase Definition 2.11 in terms of mutation compatibility.
Proposition 2.12**.**
Let be a -tilting finite algebra. Then has the pairwise 2-simple minded compatibility property if and only if for every singly left mutation compatible semibrick pair , the following are equivalent.
- (i)
For all , the semibrick pair is mutation compatible. 2. (ii)
* is mutation compatible.*
Proof.
This follows directly from Theorem 2.9. β
Remark 2.13**.**
It may seem that the pairwise compatibility property should imply that every singly left mutation compatible semibrick pair is mutation compatible. However, a priori, the condition that be mutation compatible is necessary. Indeed, assume that is singly left mutation compatible and let be a left minimal -approximation. If is epi, then and is mutation compatible. However, if is mono, then , but it does not immediately follow that the left minimal -approximation is mono or epi. We do not, however, have an example where this is not the case.
We can further refine Proposition 2.12 for algebras with particularly well-behaved approximations, as shown in the following.
Proposition 2.14**.**
Let be a -tilting finite algebra such that for all , we have the following.
- (a)
If there exists a morphism which is mono or epi, then it is a left minimal -approximation. In particular, . 2. (b)
If there does not exist a morphism which is mono or epi, then there does not exist a nonzero left minimal -approximation which is mono or epi.
Then the following are equivalent.
- (i)
* has the pairwise 2-simple minded compatibility property.* 2. (ii)
Every singly left mutation compatible semibrick pair is mutation compatible. 3. (iii)
For every singly left mutation compatible semibrick pair and , the semibrick pair is singly left mutation compatible.
Proof.
Let be such an algebra and let be a singly left mutation compatible semibrick pair. Let and . We claim that is completable. Indeed, if , then and hence is mutation compatible. If , then by (a) and (b), we have that and there is a morphism which is mono or epi and is a left minimal -approximation. If is epi, then and hence is mutation compatible. If is mono, then and the quotient map is a left minimal -approximation by (a). This means is singly left mutation compatible and ; hence, is mutation compatible. This shows that every singly left mutation compatible semibrick pair of rank 2 is mutation compatible. Given this, the equivalence between (i) and (ii) is immediate from Proposition 2.12.
To see that (iii) implies (ii), let be singly left mutation compatible. If , we are done. Otherwise, choose and let . By (iii), is singly left mutation compatible, so we can repeat this process. Since this process must terminate by Lemma 2.6(b), we conclude that is mutation compatible.
Finally, it follows from Corollary 2.10 that (ii) implies (iii). Therefore, the three conditions are equivalent as claimed. β
The hypotheses of Proposition 2.14 include the simpler case where for all and bricks do not have nontrivial self-extensions. We will, however, need the weaker hypotheses when we discuss Nakayama-like algebras in Section 3.
We conclude this section with a new proof that every representation finite hereditary algebra has the pairwise 2-simple minded compatibility condition. This result has already been shown by the second author and Todorov in [IT] using the correspondence between bricks and -vectors. We give here a proof that does not rely on this machinery.
Proposition 2.15**.**
Let be hereditary. Then every semibrick pair is singly left mutation compatible and has the 2-simple minded compatibility property.
Proof.
Let be a semibrick pair. Let , and let be a left minimal -approximation. It follows from the proof of Proposition 2.5 that is a brick. Moreover, it is well known that since is hereditary, . Therefore, is either mono or epi. We conclude that is singly left mutation compatible. As in the proof of Proposition 2.14 above, this implies that has the pairwise 2-simple minded compatibility property. β
3. Pairwise 2-Simple Minded Compatibility for Nakayama-like Algebras
The goal of this section is to show that all Nakayma-like algebras have the pairwise 2-simple minded compatibility property.
Definition 3.1**.**
Consider the quivers
12n$$\gamma_{1}$$\gamma_{2}$$\cdots$$\gamma_{n-1}$$\gamma_{n}$$\Delta_{n}12n$$\gamma_{1}$$\gamma_{2}$$\cdots$$\gamma_{n-1}$$A_{n}
where each has arbitrary orientation. By a Nakayama-like algebra, we mean an algebra of the form or where is an admissible ideal generated by monomials and for some orientation of (the extended Dynkin diagram of type ).
Remark 3.2**.**
In the case that our algebra is of the form , we remark that the condition that be generated by monomials is superfluous. This is not the case for algebras of the form , where the resulting algebra is either Nakayama-like, extended Dynkin, or of the form , where is the quiver
{n}$${\cdots}$${i+2}$${1}$${i+1}$${2}$${\cdots}$${i}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{j-1}}$$\scriptstyle{\beta_{j}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\alpha_{i-1}}$$\scriptstyle{\alpha_{i}}
and , where the two signs are equivalent.
The name Nakayama-like comes from the fact that such an algebra is Nakayama if and only if each arrow is oriented the same direction (say ). We remark that we can consider Nakayama-like algebras with quiver to also be quotients of by relaxing the condition that be admissible.
The goal of this section is to prove the following theorem (Theorem B in the introduction).
Theorem 3.3**.**
Let be a Nakayama-like algebra. Then is -tilting finite and has the pairwise 2-simple minded compatibility property.
Nakayama algebras themselves have been shown to have the pairwise 2-simple minded compatibility property in [HI] and in [Igu], [IT] when . Algebras of the form are known to be -tilting infinite, which is why they are excluded from the definition of Nakayama-like algebras and the statement of Theorem 3.3. The fact that Nakayama-like algebras are -tilting finite is immediate, since they are representation-finite algebras.
For the remainder of this section, we fix a Nakayama-like algebra . We begin by giving a description of the indecomposable -modules and bricks, and the morphisms and extensions between them. This description is similar to the description for Nakayama algebras in [HI, Section 3.1] or [Ada16], and differs slightly from the standard description in terms of strings.
Recall that given a quiver with relations , there exists another quiver with relations called the universal cover of . Readers are referred to [MVdlP83] for details.
We begin by writing where . If , then is its own universal cover. Otherwise, the universal cover of is of the form , where is a quiver whose underlying graph has vertex set and contains an edge between and if and only if .
We now fix some notation that we will use to construct our model. We denote by the unique integer such that . We then define the ordered multisets
[TABLE]
and likewise for , etc. For example, and . Similarly, we define
[TABLE]
For example, . Lastly, given three marked points , we say if . Equivalently, either , and are distinct and in cyclic order or .
Alternatively, if we fix a section of the covering map we can label each with where and (that is, corresponds to the -th copy of the vertex ). Thus considering with this labeling, we can define, for example
[TABLE]
for all . We can likewise define intervals of the form in this way.
We are now ready to describe the indecomposable -modules. For , we denote the length of by . The following is immediate from the classification of (isomorphism classes of) indecomposable modules as (equivalence classes) of strings.
Proposition 3.4**.**
For every , there is an integer such that for there exists a unique string, either equal to or starting with , such that the corresponding module has length . Moreover, all (equivalence classes) of strings appear in this way. Thus there is a bijection
[TABLE]
We say a multiset of the form (resp. ) contains a relation if (resp. ).
We are now ready to construct our combinatorial model. Let be the punctured disk with marked points on its boundary, labeled counterclockwise . We label the marked point with a solid circle if is oriented from to and with a solid square if is oriented from to . If it is unknown or irrelevant which way is oriented, we label the marked point with a hollow circle222We also label with a hallow circle in the case that and the arrow does not exist.. The value of is as given in the above proposition and is called the length of the marked point labeled .
We now wish to relate to certain directed paths in . We first observe the following.
Lemma 3.5**.**
Let . Then is a brick if and only if . Moreover, in this case, .
Proof.
It is clear that is a brick if , thus suppose . Denote . If and are both circles or both squares, then as in [HI, Prop. 3.4], we have a chain or morphisms
[TABLE]
and hence is not a brick. Thus assume without loss of generality that is a circle and is a square. It follows that either or must contain a relation, contradicting that .
Now let be a brick. We wish to show that . We observe that since , the linear transformation corresponding to every arrow in the (nonempty) set is the zero map. Thus we can consider as a representation of the subquiver containing only the arrows , which is of type . The result then follows immediately. β
In light of Lemma 3.5, we propose the following definition.
Definition 3.6**.**
A directed path between two marked points of is called an arc if
- (a)
is homotopic to the counterclockwise boundary arc . 2. (b)
does not intersect itself unless , in which case the only intersection occurs at the endpoint. 3. (c)
.
We call the source of , denoted , and the target of , denoted . We call the length of , denoted , and the support of , denoted . Condition (3) can then be rephrased as . We denote the set of homotopy classes of arcs of by .
The following is automatic from the definition and Lemma 3.5.
Proposition 3.7**.**
There is a bijection given by sending an arc to the module .
Example 3.8**.**
Consider the algebra where is the quiver
{1}$${2}$${3}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha}
and . Then corresponds to , as partially shown below, where we have labeled each arc with the corresponding brick written in the form and as a string.
321321321
We now wish to give an overview of the morphisms and extensions between bricks in terms of how the corresponding arcs intersect. We remark that the pictures following the statements of the lemmas give examples of each type of intersection only. In each picture, we draw in solid blue and in dashed orange.
Lemma 3.9**.**
Let . Then there is an exact sequence with indecomposable if and only if one of the following holds.
- (i)
* is a circle and (resp. if ) does not contain a relation. In this case, we have if and if .* 2. (ii)
* is a square and (resp. if ) does not contain a relation. In this case, we have if and if .*
s(S)$$t(T)Case (i)t(S)$$s(T)Case (ii)
Proof.
In order for to be indecomposable, there must be an arrow from one of the ends of to one of the ends of (considering and as strings). This captures the two possibilities of this happening. We now consider three cases.
If , then we cannot have both and . Thus assume without loss of generality that . It is clear that any with an exact sequence must have (where we are identifying with its arc). This shows that there cannot be an indecomposable extension unless does not contain a relation. Thus assume does not contain a relation. It follows that, considered as representations, , and any extension between them are supported only on the arrows . Thus we can consider , and as representations of the subquiver containing only the arrows , which by assumption is of type . The result is then clear in this case.
Now suppose that , so we have both and . If the quiver of is , then the result follows as above. Thus assume the quiver of is . We now observe that neither nor is supported on the arrows and . Moreover, any with an exact sequence can only be supported on at most one of these arrows. The result then follows from an argument analogous to the first case.
Finally, suppose that , so without loss of generality we have and . Again, considered as an arc, any with an exact sequence clearly has . Thus, assume does not contain a relation. It follows that in the universal cover, we can consider , , and to be representations of the subquiver containing only the arrows , which by assumption is of type . The result is then immediate. β
Lemma 3.10**.**
Let . Then there is a monomorphism if and only if one of the following holds.
- (i)
* and is a square.* 2. (ii)
* and is a circle.* 3. (iii)
, is a circle, and is a square.
Moreover, in the third case, there is an exact sequence
[TABLE]
In the other cases, every exact sequence is split or has indecomposable.
t(T)$$t(S)Case (i)s(T)$$s(S)Case (ii)s(T)$$t(T)$$s(S)$$t(S)Case (iii)
Proof.
In all three cases, we observe that , and any extensions can be considered as representations of the subquiver consisting only of the arrows , which by assumption is of type . The result is then immediate. β
Lemma 3.11**.**
Let . Then there is an epimorphism if and only if one of the following holds.
- (i)
* and is a circle.* 2. (ii)
* and is a square.* 3. (iii)
, is a square, and is a circle.
Moreover, in the third case, there is an exact sequence
[TABLE]
In the other cases, every exact sequence is split or has indecomposable.
t(S)$$t(T)Case (i)s(S)$$s(T)Case (ii)s(S)$$t(S)$$s(T)$$t(T)Case (iii)
Proof.
The proof is nearly identical to that of Lemma 3.10. β
Remark 3.12**.**
It follows from the proofs of Lemma 3.10 and Lemma 3.11 that if (so that and do not intersect) and and are either both squares or both circles that and are Ext-orthogonal.
Lemma 3.13**.**
Let . Then there is a morphism which is neither mono nor epi if and only if one of the following holds.
- (i)
* and are both circles, , and .* 2. (ii)
* and are both squares, , and .*
Moreover, in either case, there is an exact sequence
[TABLE]
if and only if and neither nor contains a relation. Likewise, in case (i), there is an exact sequence
[TABLE]
if and only if and does not contain a relation. Finally, in case (ii), there is an exact sequence
[TABLE]
if and only if and does not contain a relation. Otherwise, every exact sequence is split or has indecomposable.
t(T)$$t(S)$$s(S)$$s(T)Case (i)s(T)$$t(S)Case (ii)t(T)$$s(T)$$s(S)$$t(S)Case (i) and Case (ii)
Proof.
Suppose there is a morphism which is neither mono nor epi. It follows that there exists and a chain of morphisms . As and are bricks, we observe that cannot be a submodule of nor a quotient of . It follows that and vice versa (where and are considered as arcs the supports are considered as ordered multisets). This shows that there can only be such a morphism if either both and or both and . It can then be verified directly that such a morphism exists only when and are both circles or and are both squares.
We now consider the results regarding extensions. Suppose first that the arcs corresponding to and intersect only once, so case (i) and (ii) cannot occur simultaneously. We consider only case (i), as the proof in case (ii) is nearly identical. Thus we have . We ignore for now that both and are circles. As before, we can then consider , and any extension as being representations of the subquiver containing only the arrows . Now clearly if there exists an exact sequence which is not split, is supported on every arrow in . Therefore must not contain a relation. The result then follows from the case. In particular, such an extension does not exist unless and are both circles.
Now consider the case that the arcs intersect twice, but one of the intersections occurs on the boundary of the disk. Again, case (i) and (ii) cannot occur simultaneously, so we consider only case (i). Thus we have . If the quiver of is , the result follows as above. Thus assume the quiver of is . There are then two generic ways to lift and to the universal cover. We denote by the lifts of and .
We can first consider the basepoint as . In this case, write , where we see that . We can then consider the lifting of as having , where again we see . It follows that , and any extension can be considered as representations of the subquiver of the universal cover containing only the arrows . By similar arguments as before, we observe that there is a non-split exact sequence if and only if does not contain a relation. In this case, we see that the projection of back to the quiver gives an exact sequence with .
To make sure we have accounted for all extensions, we also consider the other generic lifting, with basepoint . In this case, we see that and intersect only at their endpoint, so all additional extensions will be indecomposable as in Lemma 3.9.
The last case to consider is when the arcs corresponding to and intersect twice in the interior of the disk. This is nearly identical to the previous case, considering again the lifts with basepoints and . β
Remark 3.14**.**
It follows from the proof of Lemma 3.13 that if the ordering of and are given as in case (i) (resp. case (ii)) in the statement of the lemma, but and are not both circles (resp. and are not both squares), then every exact sequence is split or has indecomposable.
We now give two results that will be critical in showing that Nakayama-like algebras have the 2-simple minded pairwise compatibility property.
Proposition 3.15**.**
- (a)
Let . Then for any nonsplit exact sequence , is given in one of Lemmas 3.9, 3.10, 3.11, 3.13. 2. (b)
Let . Then .
Proof.
(1) Let . Lemma 3.9 has already characterized all possible indecomposable extensions between and . Thus all other extensions occur when . Thus either , in which case we are in the setting of Lemmas 3.10 and 3.11, or neither is contained in the other, in which case we are in the setting of Lemma 3.13.
(2) Suppose . If or , the result is clear. Thus in order for there to be at least two extensions, we must have that and are both circles, and are both squares, and neither (resp. ) nor (resp. ) contains a relation. However, since there exist both marked points which are circles and which are squares, one of these intervals will contains a relation, a contradiction. See the picture below for an illustration. β
s(S)$$t(S)$$s(T)$$t(T)
Corollary 3.16**.**
Let be a Nakayama-like algebra. If there exists such that has a nontrivial self-extension, then is a Nakayama algebra.
Proof.
Let and suppose there is a non-split exact sequence . It follows from Lemma 3.9 that and that does not contain a relation. In particular, the quiver of is . Therefore, since is Nakayama-like, . As for all , this implies that all arrows of the quiver of must be oriented in the same direction. We conclude that is a Nakayama algebra. β
Corollary 3.17**.**
Let be a Nakayama-like algebra. Then for all , we have the following.
- (a)
If there exists a morphism which is mono or epi, then it is a left minimal -approximation. 2. (b)
If there does not exist a morphism which is mono or epi, then there does not exist a left minimal -approximation which is mono or epi.
Proof.
(a) Such morphisms are completely characterized in Lemmas 3.10 and 3.11. We see that in all cases, . If does not have self extensions we are done. Otherwise, is Nakayama by Corollary 3.16. The result is shown explicitly in this case in [HI, Cor. 3.5].
(b) This follows immediately from Lemma 3.13 because in all three cases, and vice versa. β
We are now ready to prove that has the 2-simple minded pairwise compatibility property.
Theorem 3.18**.**
Let be a Nakayama-like algebra and let be a singly left mutation compatible semibrick pair. Then for every , the semibrick pair is singly left mutation compatible.
Proof.
Let be a singly left mutation compatible semibrick pair. If , then we are done. Otherwise, let and let . Assume there exists and such that there is a map which is not mono or epi. By Corollary 3.17, this is equivalent to assuming there exists a left minimal -approximation which is not mono or epi.
We first observe that is singly right mutation compatible at by Proposition 2.5(a), so . We can thus assume without loss of generality that . Moreover, by the above lemmas, the arcs corresponding to and (which we also refer to as and by abuse of notation) intersect (at least once) in the interior of the disk. By Lemma 3.13, we can thus assume without loss of generality that we have and , that and are circles, and that (resp. if ) contains a relation. Moreover, this relation cannot be contained in either interval or the interval , which implies that the relation contains the interval . In particular, this means all marked points in the interval are circles. The generic diagram is shown below, where it is possible that or their order is flipped.
s(Y^{\prime})$$t(Y^{\prime})$$s(X^{\prime})$$t(X^{\prime})
Claim 1: . Indeed, assume otherwise and recall that and are Hom orthogonal. Since is singly left mutation compatible, it follows that . There are then two possibilities.
Suppose first that there exists such that , so there is an exact sequence . It follows from Proposition 3.15 that and either or . We observe that if then by Lemma 3.13, there is a map , a contradiction. Thus we have , and hence , as shown below where is the dotted green arc.
s(Y^{\prime})$$t(Y^{\prime})$$s(X^{\prime})$$t(X^{\prime})$$t(S)
We observe that since is a brick and contains a relation if . Now, if is a circle, then there is a map by Lemma 3.13. Otherwise, is a square and by Lemma 3.10 there is a map , a contradiction. This shows that there does not exist such that .
It follows that there must exist such that , so there is an exact sequence . Thus and either or by Proposition 3.15.
Assume first that . Then since , it follows from Lemma 3.10 that is a square. Since every marked point in the interval is a circle, this means , as shown below where is the dotted green arc.
s(Y^{\prime})$$t(Y^{\prime})$$s(X^{\prime})$$t(X^{\prime})$$t(S)
We observe that in this case, there is a nonzero map which is neither mono nor epi by Lemma 3.13, a contradiction. We conclude that . Thus since , we must have that . This means is a circle so as before, there is a map which is neither mono nor epi, a contradiction. This finishes the proof of Claim 1.
Claim 2: There does not exist such that . Indeed, assume otherwise, so there is an exact sequence . Thus and either or . There are then three possibilities.
Suppose first that . If , then there is a map which is neither mono nor epi, a contradiction. Thus by Proposition 3.15, we have . As before, we can conclude that , as pictured below, where is the dotted green arc.
s(Y^{\prime})$$t(Y^{\prime})$$s(X^{\prime})$$t(X^{\prime})$$s(Y)
As before if is a circle, then there is a map which is neither mono nor epi. Likewise, if is a square, then Lemma 3.11 implies that there is a map , a contradiction. We conclude that .
Now suppose there exists such that , so there is an exact sequence . Thus as before, and either or . If , then as before, there is a map , a contradiction. Thus we have . Since there is an exact sequence , we also know shares an endpoint with . If , then either or (the latter is only possible if ). In either case, we see that , contradicting the fact that is a brick. Thus we have , contradicting the fact that . This shows there does not exist such that .
The last possibility is that there exists such that , so as before there is an exact sequence . This means and either or . Recall that as before must share an endpoint with as well.
First, suppose . Since there is a map , Lemma 3.10 implies is a square (and hence is in the interval ), as is shown below where is the dashed red arc and is the dotted black arc.
s(Y^{\prime})$$s(S)$$t(X^{\prime})$$t(S)
We observe that by Lemma 3.13 there is a morphism which is neither mono nor epi, a contradiction. If and , then the picture is as below, where is the dotted green arc and is the dashed red arc.
s(Y^{\prime})$$s(X^{\prime})$$t(Y^{\prime})$$t(X^{\prime})
We now know that since is a brick, but this is impossible since contains a relation. It follows that and , and as before there is a map , a contradiction. We conclude that there cannot exist such that . This finishes the proof of Claim 2.
Together, Claim 1 and Claim 2 imply that the semibrick pair must be singly left mutation compatible. By Proposition 2.14 and Corollary 3.17, this completes the proof. β
4. The Classification for Gentle Algebras
The goal of this section is to prove the central theorem of this paper (Theorem C in the introduction). In particular, this disproves Conjecture 1.11 from [HI].
Theorem 4.1**.**
Let be a -tilting finite gentle algebra such that contains no loops or 2-cycles. Then has the pairwise 2-simple minded compatibility property if and only if every vertex of has degree at most 2.
Proof.
Let be -tilting finite gentle algebra such that contains no loops or 2-cycles. We can assume without loss of generality that is connected.
If the degree of every vertex of is at most 2, then and is a Nakayama-like algebra. Therefore, by Theorem 3.18, has the pairwise 2-simple minded compatibility property.
Thus suppose has a vertex of degree at least 3. Since is -tilting finite, contains no multiple edges. This means contains one of the following as a subquiver.
1423\gamma_{1}$$\gamma_{4}$$\gamma_{2}$$Q_{1}1423
We will show that if contains as a subquiver, then it does not have the 2-simple minded compatibility property. The argument for is similar, using if there is no arrow and if there is an arrow .
Suppose contains as a subquiver. If does not contain an arrow , let . We claim that is a singly left mutation compatible semibrick pair and that each subset of of size 2 is a mutation compatible semibrick pair.
Indeed, it is clear that 1 and are Hom orthogonal, that and are Hom orthogonal, and that .
To see that , it is enough to consider the wide subcategory , which by a result of Jasso (see [Jas15], [DIR*+*, Sec. 4]) is equivalent for some algebra with two simple objects. Let be the quiver of . Clearly, contains no loop at , as then would as well. Likewise, only contains a single arrow , otherwise (and hence ) would be -tilting infinite. We conclude that . Moreover, this also shows that and the map is a left minimal -approximation.
Now suppose , so there is a non-split exact sequence
[TABLE]
It follows from [DIR*+*, Lem. 4.26] that a brick (and hence is indecomposable). Now, since is a relation, there must be an arrow with source in and target in . As we have excluded the existence of an arrow , the only possibility is that there exists an arrow . However, if this arrow exists then must be a relation, or else would be -tilting infinite. We conclude that .
We have shown that is a singly left mutation compatible semibrick pair. Moreover, we have that is mutation compatible since it is a semibrick, is mutation compatible since mutating at yields , and is mutation compatible since mutating at yields .
Now consider the semibrick pair
[TABLE]
We observe that in the first case, the minimal -approximation is neither mono nor epi, and hence is not singly left mutation compatible. Thus does not have the pairwise 2-simple minded compatibility property by Corollary 2.10 The second case is nearly identical.
Now suppose that does contain an arrow and let . By arguments analogous to before, we then have that is a singly left mutation compatible semibrick pair, each subset of of size 2 is a mutation compatible semibrick pair, and is not singly left mutation compatible. β
Remark 4.2**.**
- (1)
The counterexamples appearing in the proof of Theorem 4.1 also show that there exist maximal semibrick pairs (in the sense that they are not direct summands of any larger semibrick pairs) which are not 2-simple minded collections. 2. (2)
The number of objects in a 2-simple-minded collection is the number of (isoclasses of) simple modules (see [KY14, Cor. 5.5]). It remains an open question whether there exist βmaximal but not completableβ semibrick pairs with at least this many objects. This will be partially answered in forthcoming work by Emily Barnard and the first author.
Remark 4.3**.**
It is less clear what the statement of Theorem 4.1 should be if we allow to contain loops or 2-cycles. For example, consider the quivers
1\leftrightarrows 2\leftrightarrows 3$$Q_{1}$$1\leftrightarrows 2\leftrightarrows 3\leftarrow 4$$Q_{2}
and the ideals and generated by all 2-cycles. We can verify directly that has the 2-simple minded compatibility property. Moreover, we can see that does not using the counterexample However, both of these algebras are -tilting finite and gentle.
5. Application: Eilenberg-MacLane Spaces for Picture Groups
We begin this section by recalling the definition of the picture group of . We then give a brief reminder of the relationship between the picture group and the so called -cluster morphism category of . This relationship is of particular interest when has the 2-simple minded pairwise compatibility property.
5.1. Torsion Lattices and Picture Groups
We first give a brief overview of the lattice structure of the poset .
Theorem 5.1**.**
- (a)
[IRTT15, Prop 1.3]** The poset is a lattice. That is:
- β’
For every , there exists a unique torsion class such that whenever . The torsion class is called the join of and .
- β’
For every , there exists a unique torsion class such that whenever . The torsion class is called the meet of and . 2. (b)
[BCZ19, Thm. 2.2.6]** Let be an arrow in (that is, and there does not exist such that ). Then there exists a unique brick such that , called the brick label of the arrow. 3. (c)
[DIR*+*, Thm. 4.16]** The lattice is polygonal. That is,
- β’
Given arrows and in , the interval is a polygon (i.e., the interval consists of two disjoint, nonempty strings).
- β’
Given arrows and in , the interval is a polygon (i.e., the interval consists of two disjoint, nonempty strings). 4. (d)
[Asa20, Thm 2.12]** Let . Let be the set of all bricks labeling arrows of the form in and let be the set of all bricks labeling arrows of the form in . Then is a 2-simple minded collection. Moreover, all 2-simple minded collections occur in this way. 5. (e)
[DIR*+*, Prop 4.21]** Every polygon in is of the form shown below,
S_{1}$$S_{2}$$T_{1}$$T^{\prime}_{1}$$T_{k}$$T^{\prime}_{l}$$S_{2}$$S_{1}$$\cdots$$\cdots
where , is a wide subcategory, and is a nonrepeating set of all (isoclasses of) bricks in 333It is possible that , that is and are the only bricks in .. Moreover, every semibrick of size 2 defines a polygon in in this way.
In the above picture, we refer to and as the sides of the polygon . This leads to the definition of the picture group of , first given for representation finite hereditary algebras by the second author, Todorov, and Weyman in [ITW].
Definition 5.2**.**
[HI, Def-Thm. 4.3, Prop. 4.4] The picture group of , denoted , is the group with the following (equivalent) presentations.
- (a)
has one generator for every brick . For every polygon in , we have the polygon relation
[TABLE]
where and the sides of . 2. (b)
has one generator for every brick and one generator for every torsion class . There is a relation
[TABLE]
and for every arrow in , there is a relation
[TABLE]
An interesting problem is to compute the cohomology of picture groups. In many cases, this can be done by computing the cohomology of a topological space which has the picture group as its fundamental group (see [ITW, Sec. 3-5], [IT, Sec. 4]). This topological space is the classifying space of a small category known as the -cluster morphism category of , defined by Buan and Marsh in [BM19] to generalize a construction of the second author and Todorov in [IT].
In order to make this paper self-contained, we give the definition of the -cluster morphism category below. Readers interested in more information about this category and information about its classifying space are referred to [BM19], [BM], [ITW, Sec. 3], [IT], and [HI, Sec. 2].
Definition 5.3**.**
[BM19, Sec. 1] The -cluster morphism category of , denoted is defined as follows:
- (1)
The objects of are the wide subcategories of ; i.e., . 2. (2)
Let . Then
[TABLE] 3. (3)
Let and . Then is the unique element of satisfying:
- (a)
is a direct summand of . 2. (b)
. 3. (c)
, where is computed in the category and is computed in the category .
We denote by the classifying space of the -cluster morphism category of . This is a simplial complex whose vertices correspond to objects of the category, whose 1-simplices correspond to non-identity morphisms, and whose -simplices correspond to sequences of composable morphisms. A more formal definition can be found in [IT, Sec. 3.4]. We recall that a topological space is a called an Eilenberg-MacLane space, or , for the group if the cohomology (with arbitrary coefficients) of is isomorphic to the cohomology of . We use only the following facts about .
Theorem 5.4**.**
- (a)
[HI, Thm. 4.8]** The fundamental group of is isomorphic to the picture group of . That is, . 2. (b)
[Igu, Prop 3.4, Prop 3.7]** **[HI, Thm 2.12, Lem. 2.14]** Suppose there is a faithful group functor for some group , considered as a groupoid with one object. If has the pairwise 2-simple minded compatibility property, then is a .
5.2. Faithful Group Functors
The aim of this section is to construct a faithful group functor in as general of a setting as possible. In particular, this will include the case that is gentle or Nakayama-like. We do so by showing that such a satisfies the hypotheses of the following theorem.
Theorem 5.5**.**
[HI, Thm. 4.13]** Let be a -tilting finite algebra such that
- (a)
For , we have . 2. (b)
For , we have .
Then there is a faithful functor , where is considered as a groupoid with one object.
Our technique for verifying the hypotheses of Theorem 5.5 is similar to that in [HI, Sec. 4.3] used for Nakayama algebras. That is, we proceed by mapping to another group where we know the generators are nontrivial. As the brick algebra defined in [HI] is in general not associative, we instead map to the group of units of the power series 0-Hall algebra of , defined below in Definition 5.7. We begin by recalling the definition of the Hall polynomials of .
Definition-Theorem 5.6**.**
[Rin90, Thm. 1] Write where is a quiver (or more generally, a species) and is an admissible ideal. We identify with for any field by identifying the Auslander-Reiten quivers of and . For , let be the number of submodules such that and when , the finite field with elements. Then the function is polynomial in . The polynomials are called the Hall polynomials of .
We are now ready to define our algebra.
Definition 5.7**.**
The power series 0-Hall algebra of , denoted , is defined as follows.
- β’
The elements of are formal power series over of isoclasses of . That is, elements are of the form , where is a skeleton of and the coefficients are integers.
- β’
Multiplication is defined by .
This is an associative algebra with multiplicative identity .
In order to construct a morphism , we will need the following.
Lemma 5.8**.**
Let .
- (a)
If , then the coefficient of in is 0. 2. (b)
If , then the coefficient of in is 1 if and only if there is an exact sequence . 3. (c)
If , then the coefficient of in is 0.
Proof.
(a) This is clear since cannot possibly contain a submodule such that and unless this equality holds.
(b) Suppose there exists an exact sequence . Now let such that . As , it follows that . Since is a brick, this implies . We conclude that and hence the coefficient of in is 1. The converse follows directly from the definition of the Hall polynomials.
(c) Choose an exact sequence . Since is indecomposable, it follows that is a nonzero morphism . It is clear that for every , the morphism given by defines a distinct submodule with and . We claim these are the only such submodules. Indeed, let and . Since is a brick, is invertible. Thus the image of the morphism given by is the same as that given by . This shows that . We conclude that the coefficient of in is 0. β
We now propose our group homomorphism.
Theorem 5.9**.**
Assume that for all we have one of the following.
- β’
* and are both -stones. That is, and .*
- β’
* and are Ext orthogonal.*
Then there is a group homomorphism given by .
Proof.
We need only show the proposed map preserves the relations of . Let and let be the corresponding polygon in . If and are both -stones, then by [DIR*+*, Prop 4.33], there are only four possibilities for the polygon .
- (1)
and are Ext orthogonal and the sides of are and . 2. (2)
, and the sides of are and where is exact. 3. (3)
, and the sides of are and where is exact. 4. (4)
, and the sides of are and where and are exact.
It can be verified in all four cases that the morphism preserves the relation corresponding to using Lemma 5.8. For example, in case (2), we have
[TABLE]
Likewise, if and are Ext orthogonal, then and is as in Case (1) above. β
The assumption that all pairs of Hom-orthogonal bricks are either -stones or are Ext orthogonal is made since the possible polygons which can occur in are well understood in these cases. Generalizing this proof would require a classification of all 2-vertex -tilting finite algebras, which does not currently exist. We have, however, verified that the theorem still holds for the polygons corresponding to the Dynkin diagrams and , which gives us hope the following conjecture is true.
Conjecture 5.10**.**
Let be an arbitrary -tilting finite algebra. Then there is a group homomorphism given by .
Nevertheless, we have the following.
Proposition 5.11**.**
- (1)
Let be a Nakayama-like algebra. Then satisfies the hypotheses of Theorem 5.9, and therefore the map is a group homomorphism. 2. (2)
Let be a -tilting finite gentle algebra such that the quiver of has no loops or 2-cycles. Then satisfies the hypotheses of Theorem 5.9, and therefore the map is a group homomorphism.
Proof.
(1) Let . It is clear in this case that the endomorphism rings of and must be isomorphic to the field . Thus if neither nor have nontrivial self-extensions, we are done. Thus suppose has a nontrivial self-extension. This implies that is a Nakayama algebra (Corollary 3.16). The result then follows immediately from [HI, Lem. 3.8].
(2) Let and suppose there is a nonsplit exact sequence . We observe that must be indecomposable since it has length 2 in the wide subcategory . Identifying with its string, it then follows from [BDM*+*19, Thm. 8.5] and the fact that has no loops that there is an arrow such that is a string for some . It follows that is a string for all , so is representation infinite and hence -tilting infinite by Theorem 1.8, a contradiction. β
When the map is a group homomorphism, we can prove the following result about the generators of .
Lemma 5.12**.**
Suppose the map is a group homomorphism. Then
- (a)
For , we have . 2. (b)
For , we have .
In particular, there exists a faithful group functor as claimed in Theorem 5.5.
Proof.
The proof is similar to that of [HI, Lem. 4.12].
(a) Let . Then it is clear that and that neither of these are equal to the identity.
(b) Let and let label minimal length paths in . Assume that (and hence ). If , let be of minimal length. By applying the morphism , we then have
[TABLE]
By Lemma 5.8, this implies that occurs in both and . Thus , a contradiction. We conclude that and hence . β
From this along with Theorems 3.3, 4.1, 5.4(b), we conclude our final main result (Theorem D in the introduction).
Corollary 5.13**.**
- (a)
Let be a Nakayama-like algebra. Then the classifying space of the -cluster morphism category of is a . 2. (b)
Let be a -tilting finite gentle algebra whose quiver contains no loops or 2-cycles. Then the classifying space of the -cluster morphism category of is a if the degree of every vertex of is at most 2.
Future Work
The fact that there are -tilting finite algebras which do not have the pairwise 2-simple minded compatibility property opens up several interesting questions for research. First and foremost, we would like to better understand what it is that causes an algebra to fail to have this property. For example, consider the algebra which is cluster tilted of type , shown below.
1423
The only counterexample to the 2-simple minded compatibility property we could find for this algebra is the collection . This seems to indicate the algebra βalmostβ has the pairwise 2-simple minded compatibility property and hence should either be a or βalmostβ be a in some imprecise sense. More precisely, we are interested in the following.
Question 1**.**
- (1)
Is a ? If not, can we obtain from (by adding/deleting/pasting cells, etc.) a cube complex that is a ? 2. (2)
How does the cohomology of compare to that of ? Is there a way to compute the cohomology of without first finding a ?
We also plan to return to algebras of the form , as defined in Remark 3.2. In particular, we are interested in the following.
Question 2**.**
If is -tilting finite, does it have the 2-simple minded compatibility property?
A positive answer to this question would show that all algebras of the form which are -tilting finite have the 2-simple minded compatibility property. In particular, this would show that in order for an arbitrary -tilting finite algebra to fail to have the property, its quiver would need to contain a vertex of degree at least 3.
Along the same lines, we wish to further study (-tilting finite) gentle algebras whose quivers contain loops and 2-cycles. In particular, we wish to answer the following.
Question 3**.**
Let be an arbitrary -tilting finite gentle algebra. When does have the 2-simple minded compatibility property?
Remark 4.3 shows that the answer to this question is not that has the property if and only if every vertex of has degree at most 2. It also shows we cannot replace the requirement that every vertex has degree at most 2 with the requirement that every vertex is connected by an arrow to at most 2 other vertices.
Finally, we wish to resolve Conjecture 5.10. A proof of this conjecture either requires finding a classification of all 2-vertex -tilting finite algebras or finding a new technique that does not depend on complete knowledge of which polygons can occur in .
Acknowledgements
The first author is thankful to Emily Barnard, Corey Bregman, and Job Rock for meaningful conversations and suggestions. Both authors would like to thank Gordana Todorov for support and suggestions. The authors also thank an anonymous referee for thoughtful and thorough comments and suggestions.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[Aih 13] Takuma Aihara, Tilting-connected symmetric algebras , Algebr. Represent. Theory 16 (2013), no. 3, 873β894.
- 3[AIR 14] Takahide Adachi, Osamu Iyama, and Idun Reiten, Ο π \tau -tilting theory , Compos. Math. 150 (2014), no. 3, 415β452.
- 4[APS] Claire Amiot, Pierre-Guy Plamondon, and Sibylle Schroll, A complete derived invariant for gentle algebras via winding numbers and Arf invariants , ar Xiv:1904.02555.
- 5[Asa 20] Sota Asai, Semibricks , Int. Math. Res. Not. IMRN 2020 (2020), no. 16, 4993β5054.
- 6[BCZ 19] Emily Barnard, Andrew T. Carroll, and Shijie Zhu, Minimal inclusions of torsion classes , Algebraic Combin. 2 (2019), no. 5, 879β901.
- 7[BDM + 19] Thomas BrΓΌstle, Giullaume Douville, Kaveh Mousavand, Hugh Thomas, and Emine Yildirim, On the combinatorics of gentle algebras , Canad. J. Math. (2019).
- 8[BM] Aslak Bakke Buan and Bethany R. Marsh, Ο π \tau -exceptional sequences , ar Xiv:1802.01169.
