Mediterr. J. Math. (2026) 23:132 https://doi.org/10.1007/s00009-026-03126-y c⃝The Author(s) 2026 UMP Monomial Algebras: Combinatorial and Homological Consequences Jhony Caranguay-Mainguez , Andr´es Franco , David Reynoso-Mercado and Pedro Rizzo Abstract. In this paper, we apply the techniques developed in [4] to present several consequences of studying UMP algebras and the ramifications graph of a monomial bound quiver algebra. Specifically, we prove that every weakly connected component of the ramifications graph of a UMP monomial algebra is unilaterally connected. Furthermore, using the main result characterizing UMP algebras in the monomial context, we prove that the class of UMP algebras is equivalent to the class of special multiserial algebras when the algebra is a quadratic monomial algebra. Based on this equivalence and the classification of Chen–Shen– Zhou on Gorenstein projective modules in [5], we extend their results to the class of monomial special multiserial UMP algebras, where we use the analysis of homological properties on quadratic monomial algebras given by these authors.

Mathematics Subject Classification. 16G20, 05E10.

Keywords. Monomial algebras, special multiserial algebras, UMP algebras, gorenstein projective modules.

1. Introduction

Gentle algebras [1], almost gentle algebras [11], and SAG and SUMP algebras [7] are examples of classes of bound quiver algebras satisfying the unique maximal path property, briefly UMP algebras. That is, a UMP algebra is a bound quiver algebra such that two different maximal paths have no common arrows (are disjoint). The class of UMP algebras is introduced in [4] in response to the problem proposed in [7] about the characterization of SUMP algebras. The authors in [7] define SUMP algebras as bound quiver algebras verifying a very special condition called the unique maximal path property. A complete classification is presented in [4] for special multiserial and locally monomial algebras (see Sect. 2 below or [4] for the definition). In the context of bound quiver algebras, the unique maximal path property appears to be 0123456789().: V,-vol

Page 2 of 24 J. Caranguay-Mainguez et al.

MJOM

crucial for the study of derived equivalences (see, e.g., [2,12]). This property, along with those introduced above for certain derived categories, plays a significant role in classification problems of certain objects in important categories, as demonstrated in [6,9,11].

In this paper, we present a collection of examples and non-examples of UMP algebras and explore the properties and implications of the combinatorial tools introduced in [4] for a monomial bound quiver algebra A. Indeed, we establish interesting connections between the mentioned tools and key combinatorial aspects of monomial bound quiver algebras. In this specific case, the set of ω-relations (see [4, Def. 4.4]) coincides with the minimal set of relations of the algebra (see [4, Remark 4.2(iii)]). Consequently, the main classification result for UMP algebras is presented in a simplified and manageable version (see Sect. 3), which directly impacts the study of their characterizations and homological properties. More precisely, by applying the techniques from [4] and its main classification theorem for monomial algebras (see Theorem 3.1), we characterize self-injective Nakayama algebras that are also UMP algebras. Furthermore, we prove that special multiserial algebras coincide with UMP algebras in the case of quadratic monomial algebras. This result motivated us to study the category of Gorenstein projective modules over monomial special multiserial UMP algebras. Specifically, the tools and results developed by Chen–Shen–Zhou in [5] regarding Gorenstein projective modules for monomial algebras have a significant impact when applied to quadratic monomial algebras. For instance, they derive important classification results for various homological properties, such as a generalization of the Geiss–Reiten theorem for gentle algebras (see [8]), using combinatorial expressions linked to the relations. Given that, in the quadratic monomial case, special multiserial algebras coincide with UMP algebras and that Theorem 3.1 provides a complete characterization of this class in terms of relations, it is natural to extend the homological results from [5] to the monomial special multiserial UMP case, as presented in Sect. 4.

This article is organized as follows. In Sect. 2, we begin by fixing some notations and present some background material about bound quiver algebras and UMP algebras. Then, we state some results on UMP monomial algebras. Later, we introduce some examples showing the tools and techniques introduced in [4], finalizing with a comparative table exhibiting some properties of each example presented. In the last subsection of Sect. 2, we analyze some properties on the bound quiver algebras derived from its ramifications graph and vice versa. We will introduce here a new class of bound quiver algebras satisfying the property of the fully connected components (see Definition 2.2), which contains the class of locally monomial and special multiserial algebras as well as the class of monomial and UMP algebras. In Sect. 3, we present the classification of some classes of UMP algebras which are relevant in representation theory of algebras such as Nakayama algebras and quadratic monomial algebras. Finally, in Sect. 4, we introduce the necessary preliminaries and tools to study the category of Gorenstein projective modules for monomial special multiserial UMP algebras, extending some classification results presented in [5] for quadratic monomial algebras. Specifically,

MJOM

UMP Monomial Algebras: Combinatorial Page 3 of 24

132

we provide a classification of perfect paths on UMP algebras (Theorem 4.1) and, by generalizing the quiver of relations from [5] (Definition 4.2), we extend the main homological results to our context of UMP algebras (Theorem 4.2, Propositions 4.1 and 4.2). We conclude this section with several examples demonstrating the application of these new tools and compare them with their counterparts in [5].

2. UMP Algebras: The Monomial Case

For convenience of the reader, we recall here some introductory material about bound quiver algebras and the preliminary material about UMP algebras introduced in [4].

Let A = kQ/I be a finite-dimensional algebra over an algebraically closed field k, where I is an admissible ideal of kQ and Q is a finite and connected quiver. Let Q0 (resp. by Q1) be the set of vertices (resp. the set of arrows) of Q. Also, s(α) (resp. t(α)) denotes the vertex of Q0 where the arrow α starts (resp. ends). We denote by R a minimal set of relations such that I = ⟨R⟩.

Each trivial path at vertex i ∈Q0 is denoted by ei and P(Q) denotes the set of all paths in Q. If u, v ∈P(Q), we say that u divides v (or u is a divisor of v, or u is a factor of v, or v factors through u) if and only if u is a subpath of v. We denote this relation by u | v. We also say that two paths are disjoint if they have no common non-trivial divisors. Let m ∈P(Q). We say that m + I is a maximal path of A = kQ/I if m /∈I and for every arrow α ∈Q1 we have αm ∈I and mα ∈I. We denote by M the set of maximal paths of A. We also say that two maximal paths m + I and m′ + I are disjoint if for every pair of representatives w of m + I and w′ of m′ + I, with w, w′ ∈P(Q), we have that w and w′ are disjoint paths.

Definition 2.1. Let A = kQ/I be a bound quiver algebra over an algebraically closed field k. The algebra A is a UMP algebra if it satisfies the Unique Maximal Path property, i.e., every arrow in Q1 can be extended to a unique maximal path of kQ/I.

Given w ∈P(Q), we write w = w0w1 · · · wlw for the factorization of w in terms of arrows w0, w1, . . . , wlw ∈Q1. In this case, we set s(w) := s(w0) and t(w) := t(wlw). Notice that the length of w is lw + 1. A path is said to be repetition-free if it has no repeated arrows as factors. An important tool introduced in [4] is the ramifications graph. Before defining it, however, we need to introduce the following prerequisite concept (see [4, Definition 3.2]). For each arrow a ∈Q1, we define a path ωa in P(Q) as follows.

(i) If Q is a cyclic quiver, we fix a repetition-free path of the form α0 · · · αk in P(Q), where α0, . . . , αk are all the arrows of Q (see e.g., Fig. 1). For each arrow a ∈Q1, we define ωa := α0 · · · αk.

(ii) If Q is not a cyclic quiver, we define ωa in the following fashion: we define Φa as the set of all paths ω satisfying the following conditions.

Page 4 of 24 J. Caranguay-Mainguez et al.

MJOM

Figure 1. A cyclic quiver Q (a) a is a subpath of ω.

(b) If lω > 0, then it holds that |{α ∈Q1 | s(α) = i}| = 1 and |{α ∈Q1 | t(α) = i}| = 1 for every vertex i ∈{t(ωj) : 0 ≤j < lω}.

Then, we define ωa as the path of maximal length in Φa. Roughly speaking, ωa is the longest repetition-free path in Q that contains the arrow a and has no ramification vertices (vertices with at least two incoming or outgoing arrows). Now, following the notations of Definition 3.3 in [4], GQ,I := (V, E) denotes the (oriented) ramifications graph associated with (Q, I), where the set of vertices is V = {ωa | a ∈Q1}, and for any pair a, b ∈Q1, there exists a directed edge δ ∈E from ωa to ωb if and only if ωa̸ = ωb, t(ωa) = s(ωb) and ωla a ω0 b /∈I. The set of weakly connected components of GQ,I is denoted by DQ,I, where a weakly connected component of GQ,I is a subgraph whose underlying undirected graph is a connected component of GQ,I. Here, the underlying undirected graph is the graph obtained by ignoring the orientations of edges in GQ,I.

For each N ∈DQ,I, we denote by QN the subquiver of Q = (Q0, Q1, s, t) defined by the paths ωa which are vertices in the component N. More precisely, QN = ((QN)0, (QN)1, sN, tN) where the set of arrows is defined by (QN)1 := {α ∈Q1 | ωα is a vertex in N}, the set of vertices is defined by (QN)0 := {i ∈Q0 | i ∈{s(α), t(α)} for some α ∈(QN)1}, and sN := s|(QN)1 and tN := t|(QN)1. From the subalgebra kQN of kQ, we define IN as the induced ideal I ∩kQN of kQN. With this, we define AN as the algebra kQN/IN and MN as the set of maximal paths in AN. An important result, proved in [4, Proposition 7], is that IN is an admissible ideal of kQN, for all N ∈DQ,I. In addition, it is not difficult to prove that if A = kQ/I is a monomial algebra, with R a minimal set of zero relations generating I, then AN = kQN/IN is a monomial algebra for all N ∈DQ,I,

MJOM

UMP Monomial Algebras: Combinatorial Page 5 of 24

132

where RN := R ∩P(QN) is a minimal set of zero relations which generates IN.

According to Remark 10 (i) in [4], if u + I is a path in A, there exists a unique weakly connected component of GQ,I, denoted by N(u), such that u ∈P(QN(u)).

From [4, Definition 15], a bound quiver algebra A = kQ/I is called a locally monomial algebra if AN = kQN/IN is a monomial algebra for all N ∈DQ,I. As pointed out above, the class of monomial algebras is contained in the class of locally monomial algebras.

Following [4, Remark 10], for each N ∈DQ,I we can define a function fN : MN →A by m+IN →m+I. Also, we can define an equivalence relation “∼ω” on the set of arrows Q1 by a ∼ω b if and only if ωa + I = ωb + I, for any a, b ∈Q1. Under these notations, we will prove a “monomial” version of Theorem 11 in [4].

Proposition 2.1. Let A = kQ/I be a bound quiver algebra and let N ∈DQ,I. Then, the following statements hold.

(1) For each N ∈DQ,I, the function fN : MN →A is injective.

(2) If A is a monomial algebra, then M =

N∈DQ,I fN(MN).

Proof.

(1) Suppose that there exist maximal paths m + IN and n + IN in

MN, with m, n ∈P(QN), such that fN(m + IN) = fN(n + IN). Then, m + I = n + I and, hence, m −n ∈I. Also, since m −n ∈kQN and IN = I ∩kQN, this implies that m + IN = n + IN.

(2) In [4, Theorem 11], it is proved that M =

N∈DQ,I fN(MN). Now, we will prove that this union is disjoint. Suppose that there are at least two weakly connected components M and N of GQ,I such that fN(MM) ∩ fN(MN)̸ = ∅. Then, there are paths m in P(QM) and n in P(QN) such that m + IM and n + IN are maximal paths in AM in AN, respectively, and m−n ∈I. Since A is monomial algebra, then I is generated by zero relations and, hence, we have that m = n. So, M = N(m) = N(n) = N. □ As an immediate consequence of Proposition 2.1, we recover the UMP classification in the “local case”, which is a special instance of the broader result presented in [4].

Corollary 2.1. Let A = kQ/I be a monomial algebra. Then,

(1) A is a UMP algebra if and only if AN is a UMP algebra, for each

N ∈DQ,I.

(2) |M| = 

N∈DQ,I |MN|.

Example 2.1. Consider the quiver Q given by

1

f 2 b

3

a  d 4 e 5

6

c 

Page 6 of 24 J. Caranguay-Mainguez et al.

MJOM

bounded by the set of relations R = {cabca, de, fb}. Then, we have M = {f, abcabcd, e} and ωf = f, ωa = a, ωb = bc, ωd = de. It follows that the ramifications graph GQ,I is composed of two weakly connected components, N : ωf and N ′ : ωa ωb ωd and thus QN := 1 f 2 IN = ⟨0⟩, MN = {f} QN′ :=

2

b

3

a  d 4 e 5

6

c  IN′ = ⟨cabca, de⟩, MN′ = {abcabcd, e}.

Example 2.2. Let Q be the quiver given by

1

a  b  2 c  bounded by the set of relations R = {ab, ca, a2 −bc}. Thus, we have M = {a2(= bc), cb} and ωa = a, ωb = ωc = bc. Denoting by N and N ′ the weakly connected components defined by the paths ωa and ωb, respectively, we have that: N : ωa, N ′ : ωb and QN :=

1

a  IN = ⟨a3⟩, MN = {a2} QN′ := 1 b  2 c  IN′ = ⟨bcb, cbc⟩, MN′ = {bc, cb}.

Example 2.3. Consider the quiver Q given by Q :

4

f >>>>>>>>

1

a

b

c 2 d 3 e  g >>>>>>>>

6

5

h  ,bounded by the set of relations R = {ab −ba, ef −gh, a2, b2, ac, bc, de}. Therefore, we obtain that M = {ab(= ba), cdg, ef(= gh)} and ωa = a, ωb = b, ωc = cd, ωe = ef, ωg = gh. Denoting by N, N ′, and N ′′ the weakly connected components we have that: N :

ωc ωg , N ′ : ωe, N ′′ : ωa

ωb

.

The quiver, the ideal, and the maximal paths in each case are given by QN :=

1

c 2 d 3 g 5 h 6 IN = ⟨dgh⟩, MN = {cdg, gh} QN′ := 3 e 4 f 6 IN′ = ⟨0⟩, MN′ = {ef}

MJOM

UMP Monomial Algebras: Combinatorial Page 7 of 24

132

Table 1. Example\properties Example\properties Special multiserial Monomial Locally monomial UMP

2.1

× ✓ ✓ ✓

2.2

✓ × ✓ ×

2.3

✓ × × ×

2.4

✓ × ✓ ✓ QN′′ :=

1

a

b

IN′′ = ⟨ab −ba, a2, b2⟩, MN′′ = {ab}.

Example 2.4. Consider the quiver Q given by

2

b ======== Q :

1

c ======== a 

4

e 5

3

d 

,

bounded by the set of relations R = {be, de, ab −cd}. In this case, we obtain that ωa = ωb = ab, ωc = ωd = cd and ωe = e. Denoting by N, N ′, and N ′′ the weakly connected components we have:

QN := 1 c 3 d 4 , IN = ⟨0⟩, QN′ := 1 a 2 b 4 , IN′ = ⟨0⟩ QN′′ := 4 e 5 IN′′ = ⟨0⟩ In the following table, we present some properties for each algebra corresponding to the previous examples. Recall that the bound quiver algebra A = KQ/I is a special multiserial algebra if the following property holds: for every arrow α in Q1, there is at most one arrow β in Q1 such that αβ /∈I and at most one arrow γ in Q1 such that γα /∈I [10, Definition 2.2].

Page 8 of 24 J. Caranguay-Mainguez et al.

MJOM

2.1. Algebras Whose Ramifications Graph is Unilaterally Connected

A monomial algebra A = kQ/I which is also a UMP algebra satisfies an important property related to the ramifications graph GQ,I associated with A, as stated in the following proposition. In this article, we will say that an oriented graph G = (V, E) is unilaterally connected if for any pair of vertices i, j ∈V , it contains an oriented path from i to j or from j to i. Theorem 2.1. Let A = kQ/I be a monomial algebra. If A is a UMP algebra, then every weakly connected component of GQ,I is unilaterally connected. Proof. Let N ∈DQ,I be a weakly connected component and, by contradiction, suppose that N is not unilaterally connected. Then, there exist a, b ∈Q1 such that ωa and ωb are two vertices in GQ,I which are not connected by any oriented path in N. Note that they are connected by a path in the undirected underlying graph of N. Thus, there exists a sequence ωa1, ωa2, . . . , ωan of vertices in N, with ai ∈Q1, a1 = a and an = b, such that there exists an edge linking ωai and ωai+1, for each i ∈{1, . . . , n −1}. Let k ∈Z+ be the least integer such that ωa and ωak are not unilaterally connected. Hence, k > 2 and there is an oriented path δ in N from ωa to ωak−1 or vice versa. Suppose, without loss of generality, that δ is an oriented path from ωa to ωak−1, whose factorization by edges in E is δ = δ1 · · · δd. Let bi be an arrow such that ωbi = t(δi) for 1 ≤i ≤d −1, and we set b0 := a and bd := ak−1. Furthermore, since N is not a unilaterally connected graph, there exists a unique edge ϵ in E from ωak to ωbd. Thus, ω lak ak ω0 bd /∈I and ω lbd−1 bd−1 ω0 bd /∈I.

Due to the fact that A is a UMP monomial algebra, there is a unique maximal path m + I ∈M such that m contains ω lak ak ω0 bd and ω lbd−1 bd−1 ω0 bd as factors. It follows that m = xω lbd−1 bd−1 ω0 bdyω lak ak ω0 bdz or m = xω lak ak ω0 bdyω lbd−1 bd−1 ω0 bdz, for some x, y, z ∈P(Q).

Suppose that m = xω lbd−1 bd−1 ω0 bdyω lak ak ω0 bdz. Since ωbd and ωak satisfy the conditions in (iib) for the arrows ak−1 and ak, respectively, we obtain that ωbd ∈Eω0 bdy and ωak ∈Tyω lak ak . Now, since ω0 bdyω lak ak /∈I, we conclude that there is a path from ωbd to ωak, which is a contradiction with the choice of k. Hence, m = xω lak ak ω0 bdyω lbd−1 bd−1 ω0 bdz. This implies that the set of nonzero paths in A of the form ω lak ak uω lbj bj , with j < d, is non-empty. Let j ∈Z be the least integer with the latter condition. Then, j > 0. On the contrary, ωlk akuω lb0 b0 induce a path in N from ωak to ωa = ωb0, which contradicts the choice of ωak. Since j̸ = d, we have by conditions in (iib) that ωbd ∈Eω0 bdu and ωbj ∈T uω lbj bj . Hence, ω0 bj divides both ω lak ak uω lbj bj and ω lbj−1 bj−1 ω0 bj. Furthermore, observe that the choice of j implies that ulu̸ = w lbj−1 bj−1 . Now, since A is a UMP algebra, there exists a unique maximal path m′ such that either m′ = x′ω lak ak uω lbj bj y′ω lbj−1 bj−1 ω0 bjz′ or m′ = x′ω lbj−1 bj−1 ω0 bjy′ω lak ak uω lbj bj z′, for some x′, y′, z′ ∈P(Q). However, the former case cannot occur since

MJOM

UMP Monomial Algebras: Combinatorial Page 9 of 24

132

our assumption on j implies that ω lak ak uω lbj bj y′ω lbj−1 bj−1 ∈I. Therefore, m′ = x′ω lbj−1 bj−1 ω0 bjy′ω lak ak uω lbj bj z′.

Now, since ωbj̸ = ωak, we have that ωak ∈Ty′ω lk ak . Hence, m′ defines an oriented path in N from ωbj to ωak, which contradicts the choice of the arrow ak. This completes the proof.

□ The property of the ramifications graph GQ,I, in which every weakly connected component is also unilaterally connected, plays an important role for several classes of algebras as in the following discussion. We highlight this property in the next definition.

Definition 2.2. Let A = kQ/I be a bound quiver algebra. We say that A has fully connected components if each weakly connected component of GQ,I is a unilaterally connected graph.

Theorem 2.1 implies that every monomial UMP algebra has fully connected components. However, the converse is not true. That is, there exist monomial algebras having fully connected components which are not UMP algebras. Indeed, consider the monomial algebra A defined by the quiver Q :

1

a  2 b  c 3 , bounded by the set of relations R = {aba, bab}. Then, A is not a UMP algebra since the set of maximal paths is given by M = {bac, ab}. Nevertheless, ωa = ωb = ba, ωc = c and GQ,I has a unique weakly connected component given by N : ωa ωc which is a unilaterally connected graph.

However, note that if A is a special multiserial algebra, then A has fully connected components with an additional feature. More precisely, in this case every weakly connected component has one of the following forms, for certain n ∈N:

N := ω0 ω1 · · · ωn N := ω0 ω1 · · · ωn  We distinguish this additional graph feature by saying that the corresponding component is of Nakayama type. We use this name by its similarity with the classification of finite dimensional Nakayama algebras. It is worth mentioning that this feature is essential in the classification of the special multiserial UMP algebras. Find more details in [4].

A natural question is whether the features of the graph GQ,I determine specific properties of the corresponding algebra A. In this regard, we can announce the following.

Page 10 of 24 J. Caranguay-Mainguez et al.

MJOM

Proposition 2.2. Let A = kQ/I be a bound quiver algebra. If each weakly connected component of GQ,I is of Nakayama type and none of its vertices corresponds to a cycle in Q, then A is a special multiserial algebra. Proof. Let a, b, c be arrows in Q1, with b̸ = c and s(b) = s(c) = t(a). Suppose, by contradiction, that ab /∈I and ac /∈I. In this case, ωla a = a, ω0 b = b and ω0 c = c. Since each weakly connected component of GQ,I is of Nakayama type, then there exists at most one edge in GQ,I starting at ωa. Hence, either ωa = ωb or ωa = ωc. Suppose, without loss of generality, that ωa = ωb. However, this is impossible because t(ωa) = t(a) = s(b) = s(ωb) = s(ωa) and, therefore, ω is a cycle, which is a contradiction to the hypothesis. A similar reasoning applies in the case in which b̸ = c and s(a) = t(b) = t(c), for some arrows a, b, c in Q1.

□ Remark 2.1. The hypothesis that none of the vertices of GQ,I correspond to a cycle in Q is essential for obtaining that A is a special multiserial algebra. Indeed, the following quiver with I = ⟨abc⟩defines the algebra A = kQ/I which is not a special multiserial algebra, and the unique weakly connected component in GQ,I is of Nakayama type.

Q : 2 c >>>>>>>>

1

b 

3

a  d 4 GQ,I : ωa ωd

3. UMP Monomial Algebras: Special Cases of Classification

In this section, we will discuss the classification of some classes of UMP monomial algebras which are relevant in representation theory such as Nakayama algebras and quadratic monomial special multiserial algebras. For the sake of completeness, we present the monomial version of the main theorem in [4] (see [4, Corollary 4.1]) which we will apply in the two cases of classification mentioned above.

Theorem 3.1. Let A = KQ/I be a monomial special multiserial algebra and let R be a minimal set of relations such that I = ⟨R⟩. Then, A is a UMP algebra if and only if for any zero relation r ∈R with length greater than two, there exists a path u, such that t(u) = s(u) and R ∩{subpaths of u(p) : p ∈Z+} = {r}, where u(p) denotes the composition of the path u with itself p-times.

3.1. Nakayama Algebras

In this subsection, we classify the self-injective Nakayama algebras which are also UMP algebras.

Let A = kQ/I be a bound quiver algebra. It is well known that A is a self-injective Nakayama algebra if and only if A = N m n (k), for some positive

MJOM

UMP Monomial Algebras: Combinatorial Page 11 of 24

132

integers m and n (see [13, Thm. 6.15, p. 384]). Here, N m n (k) = kΔn/In,m is the associated bound quiver algebra to the cyclic quiver: Δn := 1 α1 2 α2 3 α3 · · · αi−1 i αi · · · n −1 αn−1 n αn  and In,m is the admissible ideal of the path algebra kΔn generated by all compositions of m + 1 consecutive arrows in Δn. As a direct consequence of Theorem 3.1 we obtain the following proposition.

Proposition 3.1. The self-injective Nakayama algebra A = N m n (k) is a UMP algebra if and only if A = N 1 n(k).

Nevertheless, following the classification Theorem in [13, Thm. 10.3,

p. 102], there exist non-self-injective Nakayama algebras A which are UMP

algebras. For example, consider the quiver Q :

3

c 

1

a 2 b >>>>>>>> with I = ⟨abc⟩. The bound quiver algebra A = kQ/I is a Nakayama algebra which is a UMP algebra by Theorem 3.1 and is not a self-injective Nakayama algebra by Proposition 3.1.

3.2. Quadratic Monomial Special Multiserial Algebras

As an immediate consequence of Theorem 3.1, any quadratic monomial special multiserial algebra is a UMP algebra, which was firstly proved in [7, Lemma 12]. The following result establishes that the converse also holds. Theorem 3.2. Let A = kQ/I be a quadratic monomial algebra and let R be a minimal set of relations such that I = ⟨R⟩. Then A is a UMP algebra if and only if A is a special multiserial algebra.

Proof. If A is a special multiserial algebra, then by Theorem 3.1, A is a UMP algebra. Conversely, suppose that A is a UMP algebra and, by contradiction, suppose also that A is not a special multiserial algebra. Then, we have two cases:

(1) There exist α, β, γ ∈Q1, with β̸ = γ and s(β) = s(γ) = t(α), such that

αβ /∈I and αγ /∈I.

(2) There exist α, β, γ ∈Q1, with α̸ = β and t(α) = t(β) = s(γ), such that

αγ /∈I and βγ /∈I.

In the case (1), since A is a UMP algebra, there exists a maximal path of the form either xαβx′αγx′′ or xαγx′αβx′′, for certain x, x′, x′′ ∈P(Q). In the former situation, βx′α is a cycle, and hence there exists r ∈R such that r | (βx′α)(l), for some l ∈Z+, because I is an admissible ideal. Due to the fact that A is a quadratic monomial algebra, the length of r is two.

Page 12 of 24 J. Caranguay-Mainguez et al.

MJOM

Now, since αβ /∈I, we have r | βx′α, which contradicts the maximality of the path xαβx′αγx′′. An analogous reasoning applies for the latter situation. Similarly, for the case (2) we also obtain a contradiction. Thus, A is a special multiserial algebra.

□

4. Homological Consequences

Inspired by Theorems 3.1 and 3.2 and the study of Gorenstein projective modules over quadratic monomial algebras presented in [5], this section is dedicated to extending the tools and results of [5] to the UMP context. Our main findings are presented in Sect. 4.3, with preceding subsections introducing essential tools and results from a UMP perspective, thus demonstrating the naturality of our generalizations.

4.1. About Minimal Sets of Relations

We start by providing an essential background for understanding the main results and subsequent computations. For each vertex i ∈Q0, we define the sets:

i+ = {α ∈Q1 | s(α) = i} and i−= {α ∈Q1 | t(α) = i}.

(1)

Let A = kQ/I be a monomial algebra, and R be a minimal set of relations such that I = ⟨R⟩. For a nonzero path p, i.e., p does not contain a subpath in R, we denote by Ap (resp. pA) the left (resp. the right) ideal generated by p. In this case, the basis of Ap (resp. pA) are nonzero paths q such that q = rp (resp. q = pr), for some path r.

For any subset S of P(Q) we say that a path p in S is left-minimal (resp. right-minimal) in S provided that there is no path q ∈S such that p = p′q (resp. p = qp′) for some non-trivial path p′. Following [5], we define R(p) (resp. L(p)) the set of right-minimal (resp. left-minimal) paths in the set {nonzero paths q | s(q) = t(p) and pq = 0} (resp. {nonzero paths q | t(q) = s(p) and qp = 0}) Remark 4.1. If A is a monomial and special multiserial algebra, we have the following classification of the sets R(p) and L(p), for any nonzero path p: If q ∈R(p) (resp. q ∈L(p)) then q is either an arrow or the unique path of length greater than 1.

In fact, if q ∈R(p) is not an arrow, then plpq0̸ = 0. Consequently, for any r ∈R(p) with r distinct from q, it follows that plpr0 = 0, since A is a special multiserial algebra. Thus, r = r0. A similar argument applies to L(p). Remark 4.2. An immediate consequence from Theorem 3.1 for the minimal set of relations R of a monomial, special multiserial UMP algebra A is as follows: If p1, p2, p3 are nonzero paths such that p1p2, p2p3 ∈R are relations with length greater than two, then p1p2 = p2p3 as paths in Q.

MJOM

UMP Monomial Algebras: Combinatorial Page 13 of 24

132

Lemma 4.1. Let A = kQ/I be a monomial special multiserial UMP algebra, and let R be a minimal set of relations such that I = ⟨R⟩. Consider p ∈R for which there exists a cycle u = u0 · · · ulu, with lu ≥1, such that R ∩ {subpaths of u(l) : l ∈Z+} = {p}.

(1) Let q and r be nonzero paths with t(r) = s(q) (resp. s(r) = t(q)) and

such that rq = 0 (resp. qr = 0). If q is disjoint from u and, for some l, r | u(l+1), then rlrq0 ∈R (resp. qlqr0 ∈R).

(2) If q is a nonzero path sharing a subpath with u(m) for some m, then q

is a subpath of u(m).

Proof. First, note that lp ≥2 by Remark 4.2, and by Theorem 3.1, p is the unique relation appearing in the cycle u. Thus, for the first claim, if rq = 0, we have rlruj̸ = 0 for some 0 ≤j ≤lu. Since A is special multiserial, it follows that rlrq0 ∈R. The other case follows similarly. For the second claim, suppose that q = q′vq′′, where v a nonzero subpath of u(m). Assume that q′ and q′′ are non-trivial paths such that q′ ∤u and q′′ ∤u. Since lp ≥2 and A is an UMP algebra, it follows that vlvuj̸ = 0, for some 1 ≤j ≤lu. Consequently, vlv(q′′)0 = 0, as A is special multiserial. Then, q = 0, which contradicts the hypothesis. Thus, q′′ must be a trivial path. By a similar argument, q′ is also a trivial path. Hence, q = v and q is a subpath of u(m).

□ Now, if A is a monomial, special multiserial UMP algebra, we can provide a more detailed description of the sets R(p) and L(p), for any nonzero path p (cf. Remark 4.1).

Lemma 4.2. Let A = kQ/I be a monomial special multiserial UMP algebra, and let R be a minimal set of relations such that I = ⟨R⟩. For a nonzero path p, we have q ∈R(p) (resp. q ∈L(p)) if and only if rq ∈R (resp. qr ∈R), for some non-trivial path r with p = p′r (resp. p = rp′). Proof. Suppose that q ∈R(p), that is, s(q) = t(p), pq = 0 and there no exists q′ ∈R(p) such that q = q′q′′, for some non-trivial path q′′. Since pq = 0, we have that pq = p′rq′ where r ∈R. In particular, by p and q be nonzero paths, r ∤p and r ∤q, i.e., q = r2q′ and p = p′r1, with r1, r2 proper subpaths of r. In consequence, pr2 = p′r = 0 but q = r2q′, which implies that q′ is a trivial path. Thus, q = r2, which proves this part of the lemma. Conversely, suppose that q is a nonzero path such that rq ∈R, for some path r with p = p′r. With loss of generality, we can suppose that q is of the form q = q′q′′, with pq′ = 0. We analyze two cases: when the length of rq is equal to two and when it is greater than two. If the length of rq is equal to two, then q must be an arrow. Since p is a nonzero path, q′ is non-trivial. Therefore, q = q′ and q ∈R(p). If the length of rq is greater than two, by Theorem 3.1, there exists a cycle u such that rq satisfies the conclusion. Now, if pq′ = 0, then we can write pq′ = r1sr2 for some s ∈R. Since p and q′ share a subpath with some power of u, Lemma 4.1 implies that they are completely contained within this power. Consequently, s = rq and thus q ∈R(p). □

Page 14 of 24 J. Caranguay-Mainguez et al.

MJOM

A motivation for studying the sets R(p) and L(p) for any nonzero path p lies in the well-known general classification of modules over a monomial algebra A.

Lemma 4.3. Let p be a nonzero and non-trivial path in A. Then, there exists the following exact sequence of left A-modules:

0 →

 q∈L(p) Aq ι −→Aes(p) πp −→Ap →0

(2)

where ι is the inclusion map and πp is the projective cover of Ap, where πp(es(p)) = p. Similarly, we have the following exact sequence of right Amodules:

0 →

 q∈R(p) qA ι −→Aet(p) π′ p −→pA →0

(3)

where π′ p is the projective cover of pA, where π′ p(et(p)) = p.

Proof. Cf. the first paragraph of [14, p. 162].

□ Lemma 4.4. Let M be a left A-module that fits into an exact sequence of A-modules:

0 →M →P →Q

with P, Q projective. Then M is isomorphic to a direct sum  p Ap(Λ(p)), where p runs through all nonzero paths in A and each Λ(p) is some index set. Proof. [14, Theorem I].

□

4.2. Perfect Pairs on Monomial UMP Algebras

In [5], perfect pairs play a crucial role in the study of Gorenstein projective modules over monomial algebras. In this section, we study properties and derive results for the special case of monomial special multiserial algebras, with a particular focus on the subclass of UMP algebras. For the sake of completeness, we recall the definition of a perfect pair, with the necessary modifications for our specific context.

Definition 4.1. Let A = kQ/I be a monomial algebra. A pair (p, q) of nonzero paths in A is perfect if the following conditions are satisfied: (P1) both nonzero paths p, q are non-trivial, satisfying t(p) = s(q) and pq = 0 in A; (P2) if q′ is a nonzero path, with s(q′) = t(p) and pq′ = 0 then, q′ = qq′′, for some path q′′, that is, R(p) = {q}; (P3) if p′ is a nonzero path with t(p′) = s(q) and p′q = 0 then, p′ = p′′p form some path p′′, that is, L(q) = {p}.

Lemma 4.5. Let A = kQ/I be a monomial special multiserial algebra, with R minimal set of relations. For a perfect pair (p, q) with lpq > 1, we have that the sets (cf. (1)) |t(α)−| = 1 = |s(β)+|, for any arrows α dividing p and β dividing q.

MJOM

UMP Monomial Algebras: Combinatorial Page 15 of 24

132

Proof. Let A be a a monomial special multiserial algebra. Let α be an arrow that divides p. Suppose that t(α) = t(p). If αq0 ∈R, we have αq = δγ, where δ = αq0 ∈R. Since (p, q) is a perfect pair, α = p implying pq0 ∈R. This leads to q0 = qq′, that is, q = q0, which contradicts the condition lpq > 1. Therefore, αq0 /∈R. Now, if there exists γ ∈t(α)−, with γ̸ = α, then γq0 ∈R, since A is a special multiserial algebra. This implies that γq = δγ′, where δ = γq0 ∈R. Therefore, γ = p′p, i.e., γ = p, which is impossible. Consequently, |t(α)−| = 1.

Now, suppose that t(α)̸ = t(p). Then, p = p′′αp′, for some paths p′′ and p′, where α(p′)0 /∈R, since p is nonzero. If there exists γ ∈t(α)−, with γ̸ = α, then γ(p′)0 ∈R, since A is a special multiserial algebra. However, since (p, q) is a perfect pair, we have γp′ = p′′′p, which leads to a contradiction. Therefore, |t(α)−| = 1.

The proof for the case |s(β)+| = 1 follows by an analogous argument. □ Corollary 4.1. Let A = kQ/I be a monomial special multiserial algebra, with R a minimal set of relations. Consider paths p1, p2 and p3 such that lp2 ≥1 and both (p1, p2) and (p2, p3) are perfect pairs. Then, for each vertex i within the path p2, there exists a unique incoming arrow and a unique outgoing arrow.

Proof. It immediately follows from Lemma 4.5.

□ Remark 4.3. For a monomial special multiserial algebra A, if (α, β) is a perfect pair, where α and β are arrows in A, then |t(α)−| ≤2 and |s(β)+| ≤2. In fact, for any γ ∈t(α)−, we have two cases: if γβ ∈R, it follows γ = α′α, implying γ = α, since (α, β) is a perfect pair. On contrary, if γβ /∈R, this implies γ̸ = α, since A is special multiserial. The proof for the case |s(β)+| ≤2 is analogous.

By [5, Theorem 4.1], perfect paths in a monomial algebra A are uniquely associated with Gorenstein projective nonprojective modules of A. This motivates the classification of perfect paths for specific classes of monomial algebras. For completeness, we recall the definition of a perfect path (cf. [5, Definition 3.7]). A nonzero path p in a monomial algebra A is called a perfect path if there exists a sequence p = p1, . . . , pn, pn+1 = p of nonzero paths such that (pi, pi+1) is a perfect pair for all 1 ≤i ≤n. If the paths pi in the sequence are pairwise distinct, we call the sequence p = p1, . . . , pn, pn+1 = p a relation cycle of p.

The following result describes perfect paths when the monomial algebra A is also special multiserial and UMP.

Theorem 4.1. Let A = kQ/I be a monomial special multiserial UMP algebra, with R a minimal set of relations. Then any perfect path in A, if it exists, must take one of the following forms:

(1) arrows, or

(2) power of a cycle u = u0u1 · · · ulu where

• there exists an integer l ≥2 such that u(l) ∈R.

Page 16 of 24 J. Caranguay-Mainguez et al.

MJOM

• βu0 /∈R and uluα /∈R, for all arrows α, β in the quiver Q.

Proof. Let p be a perfect path. By definition there exists a sequence p = p1, . . . , pn, pn+1 = p of nonzero paths such that (pi, pi+1) is a perfect pair for all 1 ≤i ≤n. Suppose there exists i ∈{1, . . . , n} such that pi has length greater than or equal to two, i.e., pi is not an arrow. Then pi−1pi and pipi+1 are relations of length greater than two. By Remark 4.2, it follows that pi−1pi = pipi+1. This implies that t(pi+1) = t(pi) = s(pi+1) and s(pi−1) = s(pi) = t(pi−1). Consequently, pi+1 and pi−1 are powers of some cycles. Since (pi, pi+1) is a perfect pair, p(2) i+1 /∈R. However, p lpi i pi+1 /∈R as pi has length greater than or equal to two. This contradict the fact that A is a special multiserial algebra. Therefore, pi+1 must have length greater than 1. By similar arguments, we can show that each pj, for j ∈{1, . . . , n}, is a power of some cycle and lpj ≥1. Since each path in the sequence starts and ends at the same vertex, there exists a cycle u such that pipi+1 = u(l), for some integer l ≥2. Finally, observe that for any i ∈{1, . . . , n}, R(pi) = {pi+1} and L(pi+1) = {pi}, since (pi, pi+1) is a perfect pair. This implies that βu0 /∈R and uluα /∈R, for all arrows α and β in the quiver Q, completing the proof. □

4.3. Main Findings

In [5, Section 5], the authors provide a complete classification of the category of Gorenstein projective nonprojective modules over quadratic monomial algebras, along with significant consequences. Their primary tool is the concept of quiver of relations. Motivated by our characterization result, Theorem 3.2, this subsection aims to extend and generalize the main results and consequences of [5] to the setting of monomial special multiserial UMP algebras. As a first step toward this goal, we introduce a generalized definition of the quiver of relations.

Definition 4.2. Let A = kQ/I be a monomial algebra, with R being a minimal set of relations. The left-minimal paths relations quiver QL A is defined as follows:

• The vertices of QL

A are the nontrivial paths p ∈Q, with p /∈I, that satisfy one of the following conditions:

(1) There exists q, a nonzero path such that t(q) = s(p) and qp ∈R.

(2) There exists q, a nonzero path such that s(q) = t(p) and pq ∈R.

(3) p is an arrow that satisfies neither of the two preceding conditions.

• An arrow QL

A is of the form [pq] : q →p, where s(q) = t(p) and p ∈L(q). Example 4.3 in Sect. 4.4 illustrates the construction of QL A.

Remark 4.4.

(1) If p is an arrow satisfying condition (3) in Definition 4.2,

then p is an isolated vertex in QL A. To see this, suppose p were not isolated. Then there would exist a vertex q ∈QL A such that p ∈L(q) or q ∈L(p). By Lemma 4.2, this implies that pq′ ∈R for some q′ subpath of q or qp ∈R, respectively, which contradicts condition (3).

MJOM

UMP Monomial Algebras: Combinatorial Page 17 of 24

132

(2) For a quadratic monomial algebra A, the quiver QL

A coincides with the relation quiver defined in [5, Definition 5.2]. Therefore, QL A generalizes the relation quiver to monomial algebras.

Let C be a connected component of QL A. We call C perfect if it is a basic cycle and acyclic if it contains no oriented cycles or loops. By [5, Lemma 5.3(2)], under the notation of Definition 4.2, for a quadratic monomial algebra A, (p, q) is a perfect pair if and only if [pq] is the unique arrow in the relation quiver with source q and the unique arrow in the relation quiver with target p. However, as demonstrated by Example 4.4, this characterization does not hold for monomial bound quiver algebras in general.

Theorem 4.2 below provides an extended characterization of QL A for monomial special multiserial UMP algebras A.

Theorem 4.2. Let A = kQ/I be a monomial special multiserial UMP algebra and let p be a nonzero path. Then:

(1) For any nonzero path q, with s(q) = t(p), (p, q) is a perfect pair if and

only if there exists an arrow [pq] : q →p that is both the unique arrow in QL A starting at q and the unique arrow ending at p, and only one of the following conditions hold:

(a) there is no other arrow ending in a path p′, where p = rp′ and r a nontrivial path; (b) there exists a cycle u, such that pq = u(l) for some l ≥2.

(2) the path p is perfect if and only if its corresponding vertex in QL

A belongs to a perfect component.

Proof.

(1) Suppose (p, q) is a perfect pair. Then pq ∈R, so there exists

an arrow [pq] : q →p in QL A. By Definition 4.1, L(q) = {p} obtaining that [pq] is the unique arrow with source q. Now, suppose there exist nontrivial paths p′, q′ such that p = rp′ and p′ ∈L(q′), i.e., there exists an arrow [p′q′] : q′ →p′ in QL A. By Lemma 4.2, p′q′′ ∈R, for some subpath q′′ of q′. Consequently, pq′′ = 0. Since R(p) = {q} by Definition 4.1, it follows that q′′ = qγ for some path γ. We consider two cases. First, suppose that γ is trivial. Then q′′ = q and thus p′q ∈R, i.e., p′ ∈L(q) = {p}. Thus, [pq] in QL A is the unique arrow ending in p and there is no other arrow whose target is the path p′, where p = rp′ and r is a nontrivial path. Now, suppose that γ is nontrivial. Then, p′q′′ generates a relation of length greater than two. Since p′q̸ = 0 (otherwise there would be an arrow from q to p′), p′ is a proper subpath of p and hence pq is a relation of length greater than two. Thus, pq = αp′q and p′q′′ = p′qγ, for some nonzero paths α and γ. By Remark 4.2, it follows that pq = p′q′′. Consequently, t(γ) = t(q) = s(γ). Since γ is non-trivial it is a cycle. Therefore, by Theorem 3.1, pq = u(l), for some cycle u. Conversely, suppose there exists a unique arrow [pq] : q →p in QL A and there is no other arrow ending in a path p′, where p = rp′ and r a nontrivial path. Then, p ∈L(q), so pq = 0. Now, consider a nonzero path p′ with p′ ∈R(q). By Lemma 4.2, p′q′ ∈R, for some subpath q′

Page 18 of 24 J. Caranguay-Mainguez et al.

MJOM

of q, with q = q′r′ for some path r′. By Definition 4.2, there exists an arrow from q to p′. Our hypothesis implies that p′ = p, i.e., L(q) = {p}. Moreover, for any q′ ∈R(p) we have p′q′ ∈R with p = rp′, by Lemma

4.2. In particular, p′ ∈L(q′). Thus, our hypothesis again implies that

p′ = p. Consequently, pq′ ∈R, so [pq′] is an arrow in QL A. However, the unique arrow ending in p is [pq], which implies that R(p) = {q}, i.e., (p, q) is a perfect pair. The case where pq = u(l) for some cycle u and l ≥2 can be handled similarly, also showing that (p, q) is a perfect pair.

(2) Suppose p is a perfect path. By Theorem 4.1, p is either an arrow or

a cycle. From part (1) above, p is a path within a perfect component of QL A. Conversely, let p be a path belonging to a perfect component of QL A. Then, there exist paths p0, p1, . . . , pn such that [pipi+1] is the unique arrow with source pi+1 and target pi, for 0 ≤i ≤n −1. In particular, pipi+1 ∈R. Suppose there exists a nontrivial path rj such that pj = rjp′ j and there is another arrow ending at p′ j, for some 0 ≤ j ≤n. Then, pjpj+1 and pj−1pj are relations of length greater than two. Thus, pj+1 and pj−1 are powers of cycles with length greater than one. Consequently, pipi+1 coincides with a power of some cycle u, for all 0 ≤i ≤n −1. From part (1) above, we conclude that (pi, pi+1) are perfect pairs for each 0 ≤i ≤n −1, that is, p is a perfect path. □ Recall that a vertex j in a quiver is bounded if any path starting at j has an upper bound (in terms of the its length).

Lemma 4.6. Let A = kQ/I be a monomial special multiserial UMP algebra and let p be a nonzero path. Suppose that p corresponds to a vertex in QL A.

Then:

(1) The A-module Ap is a Gorenstein projective nonprojective module if and

only if the vertex of p in QL A belongs to a perfect component.

(2) The A-module Ap has finite projective dimension if and only if the vertex

of p in QL A is bounded.

(3) If the vertex of p in QL

A is not bounded and is not within a perfect component, then for each d ≥0 the syzygy module Ωd(Ap) is not Gorenstein projective module.

Proof.

(1) If the vertex corresponding to p is in a perfect component of

QL A, then p is perfect by Theorem 4.2. By [5, Proposition 4.4(4)], the Amodule Ap is Gorenstein projective nonprojective. Conversely, if Ap is a Gorenstein projective nonprojective A-module, then by [5, Proposition 4.4(2)] there exists a unique perfect path q such that L(p) = {q}. Thus, there exists an arrow [qp] in QL A. By Theorem 4.2, q belongs to a perfect component of QL A, and therefore so does p.

(2) It suffices to note that Ω(Ap) ∼=

 q∈L(p) Aq (see Lemma 4.3) and recall the definition of the quiver QL A.

(3) Suppose the vertex corresponding to p in QL

A is unbounded and not in any perfect component. Assume, by contradiction, there is a d ≥1

MJOM

UMP Monomial Algebras: Combinatorial Page 19 of 24

132

such that its syzygy A-module Ωd(Ap) is Gorenstein projective. By part

(2) above, Ωd(Ap) has infinite projective dimension and thus, Ωd(Ap) is

not projective. [5, Proposition 4.4(4)] then implies of the existence of a path q such that Aq ≃Ωd(Ap) is Gorenstein projective nonprojective. Consequently, there is a path of length d from p to q in QL A. By part (1) above, q is in some perfect component, which implies that p belongs to the same perfect component, contradicting our assumption. □ The following lemma provides a well-known characterization of Gorenstein algebras (see, e.g., [3, Theorem 3.2]) and is included here for completeness. Before stating the lemma, we recall some definitions from [5, pp. 1117–1118]. An algebra A is CM free if every Gorenstein projective A-module is projective. We call A d-Gorenstein (for d ≥0) if the injective dimension of A as a (bimodule) A-module is at most d. The 0-Gorenstein algebras are precisely the self-injective algebras and 1-Gorenstein algebras are simply called Gorenstein algebras. The following result is well known (cf. [3, Theorem 3.2]). Lemma 4.7. Let A be a finite-dimensional algebra and let d ≥0. Then the algebra A is d-Gorenstein if and only if, for each A-module M, the module Ωd(M) is Gorenstein projective.

Proposition 4.1. Let A = kQ/I be a monomial special multiserial UMP algebra. Denote by d the length of the longest path in all the acyclic components of QL A. Then,

(1) The algebra A is CM free if and only if the left-minimal paths relations

quiver QL A has no perfect components.

(2) The algebra A is Gorenstein if and only if each connected component of

the quiver QL A is either perfect or acyclic. In this case, the algebra A is (d + 2)-Gorenstein.

(3) The algebra A has finite global dimension if and only if each component

of the relation quiver QL A is acyclic.

Proof.

(1) This is an immediate consequence of Theorem 4.2 and correspon-

dence Theorem in [5, Theorem 4.1].

(2) Suppose that A is a Gorenstein algebra and there exists a path p whose

corresponding vertex in QL A is unbounded and not in any perfect component. By Lemma 4.6, each syzygy Ωd(Ap) is not Gorenstein projective. Then, by Lemma 4.7, A is not a Gorenstein algebra, which contradicting our hypothesis. Conversely, consider a path p whose corresponding vertex in QL A, belongs to a perfect or acyclic component. By Lemma 4.6, Ap is either Gorenstein projective or has projective dimension at most d. Consequently, Ωd(Ap) is Gorenstein projective. Now, if q is a path such that q = pq′, for some nontrivial path q′, there exists an isomorphism Ap ∼= Aq mapping x to xq′. Thus, Ωd(Aq) is Gorenstein projective for any nonzero path q. In general, by Lemma 4.3, for any A-module M, Ω2(M) is isomorphic to a direct summand of Ap, for some nonzero path

p. As consequence of latter reasoning, Ωd+2(M) is Gorenstein projective.

That is, the algebra A is (d + 2)-Gorenstein.

Page 20 of 24 J. Caranguay-Mainguez et al.

MJOM

(3) By Lemma 4.7, the algebra A has finite global dimension if and only if

is Gorenstein and CM free. Thus, the conclusion follows directly from

(1) and (2) above.

□ Following [5, Proposition 5.9], we aim to characterize the (stable) category of Gorenstein projective modules over a special multiserial, monomial, and UMP algebra A. We first recall some notation from [5, p. 1130]. Two perfect paths p and q in a monomial algebra A are said to overlap if they satisfy one of the following conditions:

(O1) p = q, and p = p′x and q = xq′ for some nontrivial paths x, p′ and q′ such that the path p′xq′ is nonzero.

(O2) p̸ = q, and p = p′x and q = xq′ for some nontrivial path x such that the path p′xq′ is nonzero.

Proposition 4.2. Let A = kQ/I be a monomial special multiserial UMP algebra. Let d1, d2, . . . , dm be the lengths of the relation cycles in A, where relation cycles are identified up to cyclic permutation. Then there is a triangulated equivalence A-Gproj ≃Td1 × · · · × Tdm if and only if the only perfect paths in A are arrows or at most quadratic powers of loops. Proof. By [5, Proposition 5.9], such a triangulated equivalence exists if and only if A admits no overlaps. Theorem 4.1 states that perfect paths are either arrows or powers of cycles. Thus, we only need to characterize the perfect paths that are cycles and admit overlaps, since arrows clearly do not overlap. By Theorem 4.1, suppose there is a cycle u such that u(l) ∈R, for some l ≥2, and βulu /∈R, for all β ∈Q1. If lu ≥1, u(1) is a perfect path. Thus, defining p = u(1) and q = u1 · · · uluu0, p′ = u0 = q′ and x = u1 · · · ulu, we have an overlap, since p′xq′ = u(1)u0̸ = 0 is a proper subpath of u(2). Now, suppose that u is a loop. Overlaps occur only when l ≥4. In this case, u(2) is a perfect path. Defining p = u(2) = q and p′ = x = q′ = u(1), we have an overlap, since p′xq′ = u(3)̸ = 0. If l = 2, then the only perfect path defined by u is u(1), which does not produce an overlap. If l = 3, then the only perfect paths are u(1) and u(2), and neither produces an overlap, which completes the proof. □

4.4. Examples and Discussion

Examples 4.1 and 4.2 show that the CM-free property is not “local”, that is, if A is an algebra with this property, for a weakly connected component N, the algebra AN does not inherit this property and viceversa. In particular, Theorem 4.2 and Lemma 4.6 may not extend to the more general context of locally monomial special multiserial UMP algebras (as stated in the classification theorem of [4, Theorem 4.4]).

Example 4.1. Consider A = kQ/I, where Q is defined as in Fig. 2 and its ideal is I = ⟨α2α1, α1α2⟩.

The algebra A is not CM free, since α1 and α2 are perfect paths, and CM-free algebras have no perfect paths (by [5, Corollary 4.2]).

MJOM

UMP Monomial Algebras: Combinatorial Page 21 of 24

132

Figure 2. Quiver example 4.1 Figure 3. Quiver of Example 4.2 Figure 4. Component QNα1 Now, ωαi = αi y ωβi = βi, with i = 1, 2. Consequently, its weakly connected components are Ni = ωαi −→ωβi and corresponding quiver is given by 2 α1 −→3 β1 −→4 and 3 α2 −→2 β2 −→1 whose ideal is INi = ⟨0⟩. Thus, ANi has no perfect paths and hence ANi are CM free, for i = 1, 2 (by [5, Corollary 4.2]).

Example 4.2. Consider A = kQ/I where Q is the quiver in the Fig. 3. The ideal is I = ⟨α4α1α2α3, α2α3α4α1, βα4⟩. The algebra A is CM free, as it has no perfect paths. Since ωα1 = α2α3α4α1 y ωβ = β the quiver of the component is Fig. 4, whose ideal is INα1 = ⟨α4α1α2α3, α2α3α4α1⟩. Thus, the algebra ANα1 has perfect paths α4α1 and α2α3, and hence is not CM free.

Example 4.3 illustrates the construction of our quiver QL A introduced in Definition 4.2.

Example 4.3. Consider the algebra A = kQ/I whose quiver is defined in Fig. 5 and its ideal is I = ⟨(α2α3α1)(2), βα3, γ1γ2γ3γ1γ2, γ2β⟩. The quiver QL A is given by the components in Fig. 6.

Page 22 of 24 J. Caranguay-Mainguez et al.

MJOM

Figure 5. Quiver of Example 4.3 Figure 6. Quiver QL A of Example 4.3 Example 4.4 illustrates the construction of QL A and highlights a key difference between this quiver and the relation quiver defined in [5]. Example 4.4. Consider the quiver in Fig. 5 with relations R = {γ1γ2γ3, γ2β, α2α3}. The quiver QL A looks like:

β γ2 α3 α2 γ3 γ1γ2 γ2γ3 γ1 α1.

Clearly, (γ2, β), (α2, α3), (γ1, γ2γ3) are perfect pairs. However, since R(γ1γ2) = {γ3, β} we have that the pair (γ3, γ1γ2) is not perfect. Acknowledgements The authors thank the anonymous referee for their valuable suggestions and corrections, which improved the quality of this article. Author contributions All authors contributed equally to this work. Funding Open Access funding provided by Colombia Consortium The first and fourth authors were partially supported by CODI (Universidad de Antioquia, UdeA) by project numbers 2020-33305 and 2023-62291, respectively. The third author gratefully acknowledges the funding provided by CODI (Universidad de Antioquia, UdeA) through its postdoctoral program, project number 2022-52654.

MJOM

UMP Monomial Algebras: Combinatorial Page 23 of 24

132

Data Availability Statement No datasets were generated or analyzed during the current study.

Declarations Conflict of Interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References [1] Assem, I., Skowro´nski, A.: Iterated tilted algebras of type An. Math. Z. 195(2), 269–290 (1987) [2] Avella-Alaminos, D., Geiss, C.: Combinatorial derived invariants for gentle algebras. J. Pure Appl. Algebra 212(1), 228–243 (2008) [3] Avramov, L., Martsinkovsky, A.: Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. Lond. Math. Soc. 85(2), 393–440

(2002)

[4] Caranguay, J., Franco, A., Reynoso-Mercado, D., Rizzo, P.: About UMP algebras and a special classification case (2025). arXiv:2309.11624v3 [5] Chen, X., Shen, D., Zhou, G.: The Gorenstein-projective modules over a monomial algebra. Proc. R. Soc. Edinb. Sect. A Math. 148(6), 1115–1134 (2018) [6] Fonseca, M.: Derived tame quadratic string algebras. Commun. Algebra 50(4), 1662–1684 (2022). https://doi.org/10.1080/00927872.2021.1986832 [7] Franco, A., Giraldo, H., Rizzo, P.: String and band complexes over string almost gentle algebras. Appl. Categ. Struct. 30, 417–452 (2022). https://doi.org/10. 1007/s10485-021-09661-x [8] Geiss, Ch., Reiten, I.: Gentle algebras are Gorenstein. In: Representations of Algebras and Related Topics, vol. 45, pp. 129–133 (2005) [9] Giraldo, H., Rueda-Robayo, R., V´elez-Marulanda, J.A.: On Auslander–Reiten components of string complexes for a certain class of symmetric special biserial algebras. Beitr¨age Algebra Geom. 63(4), 707–722 (2022). https://doi.org/10. 1007/s13366-021-00607-x [10] Green, E.L., Schroll, S.: Multiserial and special multiserial algebras and their representations. Adv. Math. 302, 1111–1136 (2016). https://doi.org/10.1016/ j.aim.2016.07.006

Page 24 of 24 J. Caranguay-Mainguez et al.

MJOM

[11] Green, E.L., Schroll, S.: Almost gentle algebras and their trivial extensions. Proc. Edinb. Math. Soc. 62(2), 489–504 (2018). https://doi.org/10.1017/ S001309151800055X [12] Hu, W., Xi, C.: Milnor squares of algebras, I: derived equivalences (2017). arXiv:1704.04914 [13] Skowro´nski, A., Yamagata, K.: Frobenius Algebras, vol. 1. European Mathematical Society (2011). https://doi.org/10.4171/102 [14] Zimmermann, H.B.: Predicting syzygies over monomial relations algebras. Manuscr. Math. 70, 157–182 (1991) Jhony Caranguay-Mainguez, Andr´es Franco, David Reynoso-Mercado and Pedro Rizzo Instituto de Matem´aticas Universidad de Antioquia Medell´ın Colombia e-mail: david.reynoso@udea.edu.co Jhony Caranguay-Mainguez e-mail: jhony.caranguay@udea.edu.co Andr´es Franco e-mail: andres.francol@udea.edu.co Pedro Rizzo e-mail: pedro.hernandez@udea.edu.co Received: January 18, 2026.

Revised: May 2, 2026.

Accepted: May 4, 2026.