Classification results for n-hereditary monomial algebras
aa r X i v : . [ m a t h . R T ] J a n CLASSIFICATION RESULTS FOR n -HEREDITARY MONOMIALALGEBRAS MADS HUSTAD SANDØY AND LOUIS-PHILIPPE THIBAULT
Abstract.
We classify n -hereditary monomial algebras in three natural contexts: First, wegive a classification of the n -hereditary truncated path algebras. We show that they are exactlythe n -representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n -hereditary quadratic monomial algebras. In the case n = 2 , we prove that there are only twoexamples, provided that the preprojective algebra is a planar quiver with potential. The firstone is a Nakayama algebra and the second one is obtained by mutating A ⊗ k A , where A is the Dynkin quiver of type A with bipartite orientation. In the case n ≥ , we show thatthe only n -representation finite algebras are the n -representation-finite Nakayama algebras withquadratic relations. Contents
1. Introduction 1Setup 32. Preliminaries 32.1. n -hereditary algebras 3Notation for derivatives 5Description of the n -preprojective algebra of a Koszul n -hereditary algebra 52.2. Monomial algebras 52.3. Computing Ext ℓ Λ e (Λ , Λ e )
73. Classification of n -hereditary truncated path algebras 83.1. Vanishing-of- Ext condition for monomial algebras 83.2. n -hereditary truncated path algebras 94. Classification of n -hereditary quadratic monomial algebras 114.1. The case n = 2 n ≥ Introduction
Auslander–Reiten theory has proven to be a central tool in the study of the representationtheory of Artin algebras [ARS97]. In 2004, Iyama introduced a generalisation of some of the keyconcepts to a ‘higher-dimensional’ paradigm [Iya07a, Iya07b]. To put it in his own words, “inthese Auslander–Reiten theories, the number ‘2’ is quite symbolic”. For example, the Auslandercorrespondence establishes a bijection between finite-dimensional representation-finite algebrasand finite-dimensional algebras of global dimension at most and dominant dimension at least Mathematics Subject Classification.
Key words and phrases. n -hereditary algebra, n -representation-finite algebra, n -representation-infinite algebra,preprojective algebra, Jacobian algebra, selfinjective algebra, Calabi–Yau algebra, Auslander–Reiten theory. [Aus71]. This realisation was the starting point of very fruitful research which has had ap-plications in representation theory, commutative algebra, as well as commutative and categori-cal algebraic geometry (e.g. [Iya11, IO11, MM11, IO13, HIO14, HIMO14, IW14, AIR15, IJ17,DJW19, JK19, BHon]).Auslander–Reiten theory is particularly nice over finite-dimensional hereditary algebras Λ . Forexample, there is a trichotomy in the representation theory of these algebras into preprojective,regular and preinjective modules. Moreover, their preprojective algebra Π = T Λ Ext ( D Λ , Λ) provides very useful information [BGL87]. This motivated the study of the so-called n -hereditaryalgebras, which consist of the n -representation-finite (henceforth abbreviated as n -RF) [Iya07b,HI11a, HI11b, Iya11, IO11, IO13] and n -representation-infinite (henceforth n -RI) [HIO14] alge-bras. These are finite-dimensional algebras of global dimension n which enjoy properties analo-gous to hereditary algebras in the classical theory. There is also a natural generalisation of thepreprojective algebra over these algebras.Many instances of n -hereditary algebras were discovered over the years (e.g. [HI11b, IO13,AIR15, Pet19, Pas20, BHon]). For example, algebras of higher type A and type ˜ A are n -RF and n -RI, respectively [IO11, HIO14]. The defining properties of n -hereditary algebras are ratherstrong, so classes of examples should be expected to be somewhat special. However, it seemsthat we are still in an early stage, and that many more classes of examples and classificationresults have yet to be discovered. Such results would allow an even better understanding of therole of these algebras.The aim of this paper is to study characteristics of certain n -hereditary monomial algebras.On many occasions, we use the fact that n -hereditary algebras Λ enjoy the property that Ext j Λ e (Λ , Λ e ) = 0 for all < j < n [IO13], which we refer to as the vanishing-of- Ext condi-tion. Since monomial algebras have a nice bimodule resolution, provided by Bardzell [Bar97],we have good control over these extension groups. Using that fact and a classification of the n -representation-finite Nakayama algebras by Vaso [Vas19], we obtain the following result fortruncated path algebras. Theorem A (Proposition 3.6, Theorem 3.7) . Let
Λ = kQ/ J ℓ be a truncated path algebra, where ℓ ≥ , Q is a finite quiver and J is the arrow ideal. Let A m be the linearly oriented Dynkinquiver of type A with m vertices.(1) If Q is acyclic and Ext j Λ e (Λ , Λ e ) = 0 for all < j < gl . dim Λ , then Q = A m , for some m .(2) The following are equivalent:(a) Λ is n -hereditary;(b) Λ ∼ = k A m / J ℓ , for some m , and ℓ | m − or ℓ = 2 .In this case, n = 2 m − ℓ and Λ is an n -representation-finite Nakayama algebra. We note that the vanishing-of-
Ext condition already allows us to reduce the number of casesby quite a bit.Next, we move to the study of quadratic monomial algebras. Our main results are given asfollows.
Theorem B (Theorem 4.1, Corollary 4.20, Theorem 4.26) . Let
Λ = kQ/I be a quadratic mono-mial algebra of global dimension n .(1) Suppose that n = 2 .(a) If Ext e (Λ , Λ e ) = 0 , then Q is an ( r, s ) -star quiver (Definition 4.19). LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 3 (b) If Λ is n -hereditary and the preprojective algebra Π(Λ) is a planar QP, then Λ isgiven by one of the following two -RF algebras: (1.1) where the dotted arcs denote relations. Note that the first algebra is the Nakayamaalgebra k A / J .(2) Suppose that n ≥ and Λ is n -RF. Then Λ ∼ = k A n +1 / J . Perhaps surprisingly, we see that the class of -RF quadratic monomial algebras is richer thanthose in higher global dimension. In the n = 2 case, we assumed that the preprojective algebrawas a planar quiver with potential. There are examples of other -RF quadratic monomialalgebras where this property is not satisfied, see Example 4.24. This assumption appears often,at least implicitly, in different results aimed at understanding some selfinjective Jacobian algebrasand -RF algebras (e.g. [HI11b, Pet19, Pas20]). Note that all examples covered in the previoustheorem were already known to be n -RF. The algebra corresponding to the (4 , -star above isa cut of Π( A bip3 ⊗ k A bip3 ) , where A bip3 is the Dynkin quiver of type A with bipartite orientationand Π denotes the higher preprojective algebra. Acknowledgements.
We thank Martin Herschend for pointing out a mistake in the statement ofProposition 3.6 when the results were first announced at the fd-seminar in June 2020, and wethank Steffen Oppermann and Øyvind Solberg for carefully reading the manuscript.
Setup.
Let k be an algebraically closed field. The k -dual Hom k ( − , k ) is denoted by D . Unlessspecified otherwise, all modules are left modules. The idempotent associated to a vertex i isdenoted by e i . If a and b are arrows in a quiver, then ab denotes the path b followed by a .The head of an arrow a : i → j is denoted by h ( a ) and equals j , and the tail is denoted by t ( a ) and equals i . These extend to paths p = p ℓ p ℓ − · · · p by letting h ( p ) = h ( p ℓ ) and t ( p ) = t ( p ) .Moreover, the length of a path p = p ℓ p ℓ − · · · p is ℓ and this is denoted by L ( p ) . The syzygyof a module N is the kernel of the projective cover of N and this is denoted by Ω N . If Λ isa k -algebra, then mod Λ denotes the category of finitely generated left modules and D b ( mod Λ) the bounded derived category. When Λ = kQ/I is a basic algebra, we always assume that Q isa connected quiver. 2. Preliminaries n -hereditary algebras. Let Λ be a finite-dimensional algebra of global dimension n . Let S := D Λ L ⊗ Λ − : D b ( mod Λ) → D b ( mod Λ) be the Serre functor with inverse S − = R Hom Λ ( D Λ , − ) : D b ( mod Λ) → D b ( mod Λ) . Denote by S n the composition S n := S ◦ [ − n ] . Definition 2.1.
We say that Λ is • n -representation-finite ( n -RF) if for any indecomposable projective P ∈ proj Λ , thereexists i ≥ such that S − in ( P ) ∈ inj Λ , the category of finitely generated injective modules. MADS HUSTAD SANDØY AND LOUIS-PHILIPPE THIBAULT • n -representation-infinite ( n -RI) if S − in (Λ) ∈ mod Λ for any i ≥ . • n -hereditary if H j ( S in (Λ)) = 0 for all i, j ∈ Z such that j n Z .These definitions, as written, were given in [HIO14], but the concept of n -RF algebras wasstudied before in [Iya07b, HI11a, HI11b, Iya11, IO11, IO13].We have the following dichotomy. Theorem 2.2 ([HIO14, Theorem 3.4]) . Let Λ be a ring-indecomposable k -algebra. Then Λ is n -hereditary if and only if it is either n -RF or n -RI. Recall that hereditary algebras Λ are formal , that is, for any X ∈ D b ( mod Λ) , there is anisomorphism X ∼ = M i ∈ Z H j ( X )[ − j ] . An important feature of n -hereditary algebras is that a certain generalisation of this propertyholds. This follows from [Iya11, Lemma 5.2]. Proposition 2.3.
Let Λ be an n -hereditary algebra. Then for any i ∈ Z and an indecomposableprojective module P ∈ proj Λ , there exists j ∈ Z such that S in ( P ) ∼ = H nj ( S in ( P ))[ − nj ] . As a consequence, n -hereditary algebras satisfy a condition which is closely related to the vosnex (“vanishing of small negative extensions”) property (see [IO13, Notation 3.5]). Corollary 2.4.
Let Λ be an n -hereditary algebra. Then Ext ℓ Λ ( D Λ , Λ) = 0 (2.1) for all < ℓ < n . We refer to this property as the vanishing-of-
Ext condition.As for classical hereditary algebras, preprojective algebras play an important role.
Definition 2.5.
Let Λ be a finite-dimensional algebra of global dimension n . The ( n + 1) -preprojective algebra Π(Λ) is defined as
Π(Λ) := T Λ Ext n Λ ( D Λ , Λ) ∼ = M ℓ ≥ H ( S − ℓn (Λ)) . Note that
Ext ℓ Λ ( D Λ , Λ) ∼ = Ext ℓ Λ e (Λ , Λ e ) [GI19, Lemma 2.9], a fact that we use often.Preprojective algebras and n -hereditary algebras are connected in the following way. Theorem 2.6.
Let Λ be a finite-dimensional algebra.(1) If Λ is an n -representation-finite algebra. Then Π(Λ) is a selfinjective algebra. Theconverse holds if Λ has global dimension .(2) The following are equivalent.a) Λ is n -representation-infinite;b) Π(Λ) is a bimodule Calabi–Yau algebra of Gorenstein parameter . Here, (1) is due to [IO13, Corollary 3.4 & Corollary 3.8], whereas (2) is an amalgam ofresults from [Kel11, Theorem 4.8], [MM11, Corollary 4.13], [HIO14, Theorem 4.36], and [AIR15,Theorem 3.4]. We refer to the papers for definitions.In the case where Λ is Koszul, we have a good understanding of the construction of preprojec-tive algebras. To present the construction we need certain notions of derivatives which we definebelow, and we note that they are used extensively in this paper, and not just in the context ofKoszul algebras. LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 5 Notation for derivatives.
Let S be a semisimple k -algebra and V be an S -bimodule. Let p = v ℓ ⊗ · · · ⊗ v ∈ V ⊗ S ℓ . We define the linear morphisms δ L m ( p ) := v ℓ − m ⊗ · · · ⊗ v and δ R m ( p ) := v ℓ ⊗ · · · ⊗ v m +1 . for m < ℓ and we let both equal when ℓ = m .Moreover, we define L m ( p ) := v ℓ ⊗ · · · ⊗ v ℓ − m +1 = δ R ℓ − m ( p ) and R m ( p ) = v m ⊗ · · · ⊗ v = δ L ℓ − m ( p ) . The subscript is dropped if m = 1 .We also define linear morphisms associated to elements q ∈ V ⊗ m , for m ≤ ℓ : δ L q ( p ) := (cid:26) a if p = q ⊗ a else and δ R q ( p ) := (cid:26) b if p = b ⊗ q else . Similarly, we define L q ( p ) := (cid:26) b if p = b ⊗ q ⊗ a else and R q ( p ) := (cid:26) a if p = b ⊗ q ⊗ a else . When p = b ⊗ q ⊗ a for some paths a and b , we say that q divides p and denote this by q | p . Description of the n -preprojective algebra of a Koszul n -hereditary algebra. Recallthat if Λ is Koszul, it can be given as a tensor algebra T S V / h M i where, as in the previoussection, S is some semisimple k -algebra, V is an S -bimodule, and M ⊂ V ⊗ S V is a subbimodule[BGS96]. Let then K ℓ := ℓ − \ µ =0 ( V ⊗ µ ⊗ M ⊗ V ⊗ ℓ − µ − ) be the terms appearing in the minimal Koszul resolution of Λ according to [BGS96]. Moreover,given a vector space V , let B ( V ) be a basis. Proposition 2.7 ([GI19, Proposition 3.12], [Thi20, Corollary 3.3]) . Let
Λ = T S V / h M i be afinite-dimensional Koszul algebra of global dimension n . Let { e i | ≤ i ≤ m } be a complete setof primitive orthogonal idempotents in Λ . Let V be the vector space obtained from V by addinga basis element e i a q e j for each element q ∈ B ( e j K n e i ) . Let M be the union of M with the set f M of quadratic relations given by f M := X q ∈B ( K n ) a q δ R p ( q ) + ( − n X q ∈B ( K n ) δ L p ( q ) a q | p ∈ B ( K n − ) . There is an isomorphism of algebras Π ∼ = T S V / h M i . Monomial algebras.
In this subsection, we define monomial algebras and describe certainminimal projective resolutions.
Definition 2.8.
Let
Λ = kQ/I , where Q is a finite quiver and I an admissible ideal. We saythat Λ is a monomial algebra if I can be generated by a finite number of paths.There is a nice description of the minimal projective Λ -bimodule resolution of Λ , due toBardzell [Bar97]. Let M be a minimal set of paths of minimal length which generates I . Givena path p , define the support to be the set of all vertices dividing p . For every directed path T in Q , there is a natural order < on the support of T . Let M ( T ) be the set of relations which divide T . MADS HUSTAD SANDØY AND LOUIS-PHILIPPE THIBAULT
Definition 2.9.
Let p ∈ M ( T ) . We define the left construction associated to p along T byinduction. Let r ∈ M ( T ) be the path (if it exists) in M ( T ) which is minimal with respect to t ( p ) < h ( r ) < h ( p ) . Now assume we have constructed r = p, r , . . . , r j . Let L j +1 = { r ∈ M ( T ) | h ( r j − ) ≤ t ( r ) < h ( r j ) } . If L j +1 = ∅ , let r j +1 be such that t ( r j +1 ) is minimal in L j +1 . Definition 2.10.
Let p ∈ M and ℓ ≥ be an integer. We define AS p ( ℓ ) := { ( r = p, r , . . . , r ℓ − ) | ( r , r , . . . , r ℓ − ) is a sequence of paths associatedto p in the left construction } . For each element ( r , . . . , r ℓ − ) ∈ AS p ( ℓ ) , define p ℓ to be the path from t ( p ) to h ( r ℓ − ) and let AP p ( ℓ ) be the set of all p ℓ . Finally, we define AP( ℓ ) := [ p ∈ M AP p ( ℓ ) , if ℓ ≥ and AP(0) := Q , AP(1) := Q .The vector spaces k AP( ℓ ) are the kQ -bimodules which appear in the minimal resolution wewant to construct. Note that AP(2) = M . If p ∈ AP( ℓ ) , define Sub( p ) := { q ∈ AP( ℓ − | q divides p } . Lemma 2.11 ([Bar97, Lemma 3.3]) . The set
Sub( p ) contains two paths p and p such that t ( p ) = t ( p ) and h ( p ) = h ( p ) . Moreover, if ℓ is odd, then Sub( p ) = { p , p } . We are now ready to define morphisms d ℓ : M p ∈ AP( ℓ ) Λ e h ( p ) ⊗ k e t ( p ) Λ → M p ∈ AP( ℓ − Λ e h ( p ) ⊗ k e t ( p ) Λ , noting that we give our conventions with respect to idempotents, and heads and tails of arrowsand paths in the setup immediately following the introduction. Recall that if p ∈ AP( ℓ ) and q ∈ Sub( p ) , we write p = L q ( p ) q R q ( p ) . By the previous lemma, we have that Sub( p ) = { p , p } if ℓ is odd, in which case p = L p ( p ) p and p = p R p ( p ) . Then we define d ℓ (( e h ( p ) ⊗ e t ( p ) ) p ) := ( ( L p ( p ) e h ( p ) ⊗ e t ( p ) ) p − ( e h ( p ) ⊗ e t ( p ) R p ( p )) p if ℓ is odd P q ∈ Sub( p ) ( L q ( p ) e h ⊗ e t R q ( p )) q if ℓ is even . Here, we use the notation ( −⊗− ) p to denote an element in the p -th component in L p Λ e i ⊗ k e j Λ . Theorem 2.12 ([Bar97, Theorem 4.1]) . The complex · · · d n +1 −−−→ M p ∈ AP( n ) Λ e h ( p ) ⊗ k e t ( p ) Λ d n −→ · · · d −→ M e i ∈ AP(0) Λ e i ⊗ k e i Λ µ −→ Λ → , (2.2) where µ (( e i ⊗ e i ) e i ) = e i , is a minimal projective resolution of Λ as a Λ -bimodule. LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 7 Computing
Ext ℓ Λ e (Λ , Λ e ) . In the next sections, we use on many occasions Corollary 2.4 asan obstruction for certain algebras to be n -hereditary. We therefore explain here how to compute Ext ℓ Λ e (Λ , Λ e ) for ≤ ℓ ≤ n .Let Λ be a basic finite-dimensional algebra. By [Hap89, Section 1.5], Λ has a minimal projectivebimodule resolution of the form P • : · · · d n +1 −−−→ M p ∈B ( E n ( i,j )) Λ e h ( p ) ⊗ k e t ( p ) Λ d n −→ · · · d −→ M e i ∈B ( E ( i,j )) Λ e i ⊗ k e i Λ → , where E ℓ ( i, j ) := Ext ℓ Λ ( S i , S j ) and S i denotes the simple module at vertex i . In the case where Λ is monomial, we have E ℓ ( i, j ) ∼ = e j k AP( ℓ ) e i . Note that, in general, it is hard to determinethe differentials d ℓ .In order to compute Ext ℓ Λ e (Λ , Λ e ) , we apply Hom Λ e ( − , Λ e ) to P • and use the isomorphisms Ψ : Hom Λ e (Λ e j ⊗ k e i Λ , Λ e ) ∼ = e j Λ ⊗ k Λ e i ∼ = Λ e i ⊗ k e j Λ φ φ ( e j ⊗ e i ) e j ⊗ e i e i ⊗ e j to obtain a complex Hom Λ e ( P • , Λ e ) : 0 → M e i ∈B ( E ( i,j )) Λ e i ⊗ k e i Λ ˜ d −→ · · · ˜ d n −→ M p ∈B ( E n ( i,j )) Λ e t ( p ) ⊗ k e h ( p ) Λ → · · · , (2.3)where ˜ d ℓ ( e i ⊗ e j ) = Ψ(Ψ − ( e i ⊗ e j ) ◦ d ℓ ) .Computing Ext ℓ Λ e (Λ , Λ e ) requires the understanding of the morphisms ˜ d ℓ , which we do havein the case where Λ is monomial. In fact, we have ˜ d ℓ (( e t ( p ) ⊗ e h ( p ) ) p ) = (P q ∈ AP( ℓ ) ( e t ( q ) ⊗ e h ( q ) δ R p ( q )) q − P q ∈ AP( ℓ ) ( δ L p ( q ) e t ( q ) ⊗ e h ( q ) ) q if ℓ is odd P q ∈ AP( ℓ ) | p ∈ Sub( q ) ( R p ( q ) e t ( q ) ⊗ e h ( q ) L p ( q )) q if ℓ is even . In further sections, we use these to describe cocycles and coboundaries, allowing us to showthat some
Ext ℓ Λ e (Λ , Λ e ) does not vanish for some algebra, thus preventing them from being n -hereditary. Using Corollary 2.4, we can already give a necessary condition for a monomial algebrato be n -hereditary. This is analogous to results established in [GI19, Proof of Theorem 3.14] and[Thi20, Proof of Theorem 3.6] in the case where Λ is Koszul. Lemma 2.13.
Let Λ be a monomial algebra and define δ ( E ( i, j ) ℓ ) := { w ∈ E ( i, j ) ℓ − | w ∈ Sub( w ′ ) for some w ′ ∈ E ( i, j ) ℓ } . Then E ( i, j ) ℓ − = δ ( E ( i, j ) ℓ ) for all ≤ ℓ ≤ n .Proof. Suppose by contradiction that there exists w ∈ E ( i, j ) ℓ − which does not divide anyelement of E ( i, j ) ℓ , for some ℓ . Then ˜ d ℓ (( e t ( w ) ⊗ e h ( w ) ) w ) = 0 , which means that ( e t ( w ) ⊗ e h ( w ) ) w is an ( ℓ − -cocycle in Hom Λ e ( P • , Λ e ) . However, it is not acoboundary, implying that Ext ℓ − e (Λ , Λ e ) = 0 . This contradicts Corollary 2.4. (cid:3) Corollary 2.14.
Let
Λ = kQ/ h M i be a basic n -hereditary monomial algebra where M is a setof relations given by paths in kQ . Then every arrow in Q is part of a relation in M .Proof. By Lemma 2.13, we have that δ ( M ) = V . (cid:3) MADS HUSTAD SANDØY AND LOUIS-PHILIPPE THIBAULT Classification of n -hereditary truncated path algebras In this section, we assume that
Λ = kQ/I is a monomial finite-dimensional algebra, that is,the ideal I in a presentation of Λ can be chosen to be generated by paths. Moreover, for the restof the text, whenever Λ is assumed to be monomial, we also assume I = h M i with M a minimalset of paths of minimal length.Recall that, by the vanishing-of- Ext condition, we have that
Ext i Λ e (Λ , Λ e ) = 0 for all < i < n for any n -hereditary algebra Λ . We thus seek to understand what knowledge one can obtain fromthis property. As an application, we use this information to classify the truncated path algebras Λ = kQ/ J ℓ , where J is the arrow ideal, which are n -hereditary in the second subsection.3.1. Vanishing-of-
Ext condition for monomial algebras.
In this subsection, we find nec-essary conditions on the quiver and relations of monomial path algebras in order to satisfy thevanishing-of-
Ext condition. To be more precise, we only look into the vanishing of the first
Ext . Recall that, by Lemma 2.13, every arrow has to be part of at least one relation, otherwise
Ext e (Λ , Λ e ) = 0 . This is a first obstruction, which does not require the monomial hypothe-sis. We therefore assume this property for the class of algebras we consider in this subsection.Throughout, we let Λ be a monomial algebra in which every arrow divides at least one relation.The main strategy is to construct cocycle elements which are not coboundaries in the complex(2.3), defined as Hom Λ e ( P • , Λ e ) , where P • is the minimal projective Λ -bimodule resolution of Λ ,described in the preliminaries. We refer to Sections 2.2 and 2.3 for more details and the notation. Proposition 3.1.
Suppose that there exists an arrow a which is the start (resp. the end) ofevery relation it divides and such that t ( a ) (resp. h ( a ) ) is not a source (resp. a sink). Then Ext e (Λ , Λ e ) = 0 .Proof. Assume that there is an arrow a which is the end of every relation r i it divides and suchthat h ( a ) is not a sink. The other case is dual. We consider the element ( ae t ( a ) ⊗ k e h ( a ) ) a ∈ M v ∈ Q Λ e t ( v ) ⊗ k e h ( v ) Λ from complex (2.3). Then ˜ d (( ae t ( a ) ⊗ k e h ( a ) ) a ) = X i ( a R ( r i ) e t ( r i ) ⊗ k e h ( a ) ) r i = 0 , so it is a cocycle in complex (2.3). However, since h ( a ) is not a sink, ( ae t ( a ) ⊗ k e h ( a ) ) a cannot bea coboundary. In fact, let b be an arrow such that h ( a ) = t ( b ) . Then, ˜ d (( e h ( a ) ⊗ k e h ( a ) ) e h ( a ) = ( ae t ( a ) ⊗ k e h ( a ) ) a + ( e h ( a ) ⊗ k e h ( b ) b ) b + . . . This is the only place where ( ae t ( a ) ⊗ k e h ( a ) ) a appears as a summand of an element in the imageof ˜ d . The same is true for ( e h ( a ) ⊗ k e h ( b ) b ) b , which means that this term cannot be cancelledby other elements in the image of ˜ d . Therefore, ( ae t ( a ) ⊗ k e h ( a ) ) a is not a coboundary and Ext e (Λ , Λ e ) = 0 . (cid:3) We say that two relations intersect with each other if there is at least one arrow which dividesboth of them. We have the following corollary.
Corollary 3.2.
Assume that there is a relation r which does not intersect with any other relationand such that t ( r ) and h ( r ) are not both a source and a sink. Then Ext e (Λ , Λ e ) = 0 . Continuing on the same ideas, we explore what happens at sinks and sources. We show thatthe vanishing-of-Ext conditions implies that sinks and sources divide only one arrow.
LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 9 Proposition 3.3.
Assume that there is a vertex i in Q which is a sink (resp. a source), suchthat there is at least two arrows having i as head (resp. as tail). Then Ext e (Λ , Λ e ) = 0 .Proof. We suppose that i is a sink. The other case is dual. Let a and b be two arrows such that h ( a ) = h ( b ) = i . We claim that the element ( ae t ( a ) ⊗ k e i ) a ∈ L v ∈ Q Λ e t ( v ) ⊗ k e h ( v ) Λ is a cocyclein degree . In fact, since h ( a ) is a sink, every relation r containing a is of the form r = aR r ( a ) for some path R r ( a ) . Therefore, ˜ d (( ae t ( a ) ⊗ k e i ) a ) = X a | r ( aR r ( a ) e t ( r ) ⊗ e i ) r = 0 . This is however not a coboundary. In fact, since i is also the head of another arrow, we havethat ˜ d (( e i ⊗ e i ) e i ) = ( ae t ( a ) ⊗ k e i ) a + ( be t ( b ) ⊗ k e i ) b + . . . By the same reasoning as in Proposition 3.1, we conclude that
Ext e (Λ , Λ e ) = 0 . (cid:3) n -hereditary truncated path algebras. We now consider the case of truncated pathalgebras
Λ = kQ/ J ℓ for some ℓ ≥ , where Q is a finite quiver and J is the arrow ideal. Inthis case, the terms in the Bardzell resolution (2.2) are particularly easy to describe. Indeed,the vector space k AP( i ) is generated by all paths of length i · ℓ if i is even and those of length (cid:0) i − · ℓ + 1 (cid:1) if i is odd. Let L ( p ) denote the length of a path p . We use the following results. Theorem 3.4. [DHZL08, Theorem 2]
Let Λ be a truncated path algebra. If N is a non-zero Λ -module with skeleton σ , then the syzygy of N Ω N ∼ = M q σ -critical Λ q. We refer to the paper for the definitions of skeletons σ and of σ -critical paths.We also need the following result regarding extensions of certain kinds of indecomposablemodules. Proposition 3.5 ([Vas19, Proposition 3.1]) . Let Λ be a finite-dimensional algebra. Let N ∈ mod Λ be a non-projective indecomposable module. If Ω N is decomposable, then Ext ( N, Λ) = 0 . Let now A m be the linearly oriented Dynkin quiver of type A with m vertices. Proposition 3.6.
Let Q be a finite acyclic quiver and assume that Q = A m . Let Λ := kQ/ J ℓ for some ℓ ≥ be a truncated path algebra. Then there exists < j < gl . dim Λ such that Ext j Λ e (Λ , Λ e ) = 0 .Proof. By Proposition 3.3, if Q is a Dynkin quiver of type A with a non-linear orientation, then Ext e (Λ , Λ e ) = 0 . Since Q = A m , there exists a vertex i which divides at least arrows. If i iseither a source or a sink, then Ext e (Λ , Λ e ) = 0 by Proposition 3.3 as well.Now suppose that i is the head of at least two arrows and the tail of at least one arrow. Theopposite case is treated similarly. Among the arrows with head i we pick two, say, a r and b s satisfying that a r = b s and that there exist paths T := a r · · · a and T := b s · · · b which aremaximal in the following sense: without loss of generality, we let T be the longest path in kQ ending at i and T the maximal path in kQ ending at i not divided by a r . Note that this usesthat Q is acyclic. In particular, we assume L ( T ) ≤ L ( T ) . Moreover, we let T := c t · · · c bethe longest path in kQ beginning in i . We may also assume that h ( T ) is only a sink to the arrow c t and t ( T i ) is a source to only onearrow for i = 1 , , since otherwise Ext e (Λ , Λ e ) = 0 and we are done.We split the proof into the following cases: C1 : L ( T T ) ≤ ℓ − C2 : L ( T T ) ≥ ℓ a) L ( T ) ≤ ℓ − b) L ( T ) ≥ ℓ i) L ( T ) ≥ ℓ − ii) L ( T ) ≤ ℓ − C1 : If L ( T T ) ≤ ℓ − , then b s does not divide any relation, by the maximality assumption onthe length of T T . As a consequence of Lemma 2.13, Ext e (Λ , Λ e ) = 0 and we are done. C2 : Now suppose that L ( T T ) ≥ ℓ . a) If L ( T ) ≤ ℓ − , then for any relation path p (of length ℓ ) such that b s divides p , we havethat L ( L b s ( p )) ≥ ℓ − s ≥ max(1 , ℓ − r ) , where r := L ( T ) and s := L ( T ) . The first inequality is explained by the maximality assumptionon L ( T T ) . For the second inequality, recall that we have assumed without loss of generalitythat r ≥ s . This means that any path of maximal length starting at i is of length at least max(1 , ℓ − r ) . Therefore, the element ( e t ( b s ) ⊗ k e i a r · · · a max(1 ,r − ℓ +2) ) b s ∈ M v ∈ Q Λ e t ( v ) ⊗ k e h ( v ) Λ is a non-trivial cocycle. It is not a coboundary since the only two Λ -bimodule generators in L i ∈ Q Λ e i ⊗ k e i Λ which map non-trivially via ˜ d to an element in Λ e t ( b s ) ⊗ k e i Λ are ( e t ( b s ) ⊗ k e t ( b s ) ) e t ( bs ) ( e t ( b s ) ⊗ k e i b s ) b s + . . . and ( e i ⊗ k e i ) e i ( b s e t ( b s ) ⊗ k e i ) b s + . . . and they cannot be linearly combined to obtain our cocycle. Thus, Ext e (Λ , Λ e ) = 0 . b.i) We now consider the case where L ( T ) ≥ ℓ . Let j ∈ N ≥ be such that the length of thepaths in k AP( j ) is less than or equal to L ( T ) , but the length of the paths in k AP( j + 1) isstrictly bigger than L ( T ) . If L ( T ) ≥ ℓ − , then < j < gl . dim Λ , since k AP( j + 1) is nonempty, as it contains a path dividing T T . Let T := b s · · · b x be the path in k AP( j ) ending at i and dividing T . Then the element ( e t ( T ) ⊗ k e i a r · · · a r − ℓ +2 ) T ∈ M p ∈ AP( j ) Λ e t ( p ) ⊗ k e h ( p ) Λ is a cocycle. Indeed, for any T ′ ∈ AP( j + 1) which is divided by T , we have L ( L T ( T ′ )) ≥ . Thisis explained by the fact that L ( T ′ ) > L ( T ) and the maximality assumption on the length of T .In fact, if L ( L T ( T ′ )) = 0 , then h ( T ′ ) = i and L ( T T ′ ) > L ( T T ) , contradicting our hypothesis.The element is not a coboundary for a similar reason as above. b.ii) Now, if L ( T ) ≤ ℓ − , then we consider the indecomposable injective module I associatedto the vertex h := h ( T ) . We show that either there are more than one σ -critical paths or there LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 11 is only one σ -critical path q and Λ q is projective. In the former case, we conclude by Theorem3.4 and Proposition 3.5 that Ext e (Λ , Λ e ) ∼ = Ext ( D Λ , Λ) = 0 . In the latter case, we obtain thesame conclusion since proj . dim I = 1 .We call branching points the vertices which divide at least arrows. Let S h be the support ofpaths of length ℓ − in kQ which end in h . Let B h be the set of branching points which are in S h . Since L ( T ) ≤ ℓ − , we have that i ∈ B h .Let S ′ h ⊂ S h be the set of vertices which start the paths of length ℓ − that end in h . Notethat P := L ι ∈S ′ h Λ e m ( ι ) ι is the projective cover of I , where m ( ι ) is the number of paths of length ℓ which ends in h and starts in ι . Because L ( T ) < ℓ − , we have |S ′ h | ≥ , since it contains avertex in T and T . Thus, I is not a projective module.Let x ∈ B h be such that there exist arrows α and β ending in x . Then either paths of theform αp or of the form βq are in the skeleton σ , for p, q ∈ σ . The paths not in σ must then be σ -critical. In fact, they get identified via P ։ I . Thus, every such branching point gives rise to σ -critical paths.Now suppose that there exists a vertex x ∈ B h which is the start of an arrow α not in a pathof length ℓ − ending in h . Then for any skeleton σ , we have that any path of the form αp , for p ∈ σ , is σ -critical, since it goes to via P ։ I .Therefore, in order to have only one σ -critical path, it is necessary that the full subquiver ¯ Q containing all the directed paths connected to the branching points in B h is given byi hIn this case, we have that Ω I ∼ = Λ e i is projective. (cid:3) The n -representation-finite Nakayama algebras were classified by Vaso in [Vas19]. Usinghis classification, we obtain as a corollary of the previous proposition a classification of all n -hereditary algebras of the form Λ = kQ/ J ℓ . Theorem 3.7.
Let
Λ = kQ/ J ℓ for some ℓ ≥ and finite quiver Q . The following are equivalent.(1) Λ is n -hereditary;(2) Λ ∼ = k A m / J ℓ , for some m , and ℓ | m − or ℓ = 2 .In this case, n = 2 m − ℓ and Λ is a Nakayama n -representation-finite algebra.Proof. By [DHZL08, Theorem 5], any truncated path algebra of finite global dimension must havean acyclic quiver. By Proposition 3.6, if Λ is n -hereditary, then its quiver must be A m , since n -hereditary algebras satisfy the property that Ext i Λ e (Λ , Λ e ) = 0 for all < i < n . Therefore, Λ is an n -representation-finite Nakayama algebra. The result thus follows from [Vas19, Theorem3]. (cid:3) Classification of n -hereditary quadratic monomial algebras In this section, we give a partial classification of the n -hereditary quadratic monomial algebras.Let Λ = kQ/I be such an algebra. In the first subsection, we tackle the case n = 2 . With theadditional assumption that the preprojective algebra can be given by a planar selfinjective quiverwith potential, we show that there are only two examples. Then, in the next subsection, we show that provided n ≥ , the only n -hereditary quadratic monomial algebras are the Nakayama onesgiven in the previous section.4.1. The case n = 2 . The goal of this section is to prove the following theorem.
Theorem 4.1.
Let
Λ = kQ/I be a -hereditary quadratic monomial algebra. Assume that Π(Λ) is given by a planar quiver with potential. Then Λ is one of the two bounded quiver algebras givenin (1.1). These algebras are -representation-finite. These two algebras were already known to be -representation-finite. In fact, the first oneappears already in [IO13, Theorem 3.12]. The second one is a cut, a notion defined below, of Π( A bip3 ⊗ k A bip3 ) , where A bip3 is the Dynkin quiver of type A with three vertices and bipartiteorientation.We provide more information on the preprojective algebra Π(Λ) of a -hereditary algebra.It is a Jacobian algebra which is selfinjective in the case when Λ is -representation-finite, and -Calabi–Yau in the case when Λ is -representation-infinite. We give a brief overview of theseuseful facts. They are key in our classification result. Definition 4.2.
Let Q be a quiver and J be the ideal generated by arrows. A potential W isan element in c kQ/ [ \ kQ, kQ ] , where c kQ is the completion of the path algebra with respect to the J -adic topology. Definition 4.3.
Let ( Q, W ) be a quiver with potential. The Jacobian algebra of ( Q, W ) isdefined as P ( Q, W ) := c kQ/ h δ a W | a ∈ Q i . Every -preprojective algebra is a Jacobian algebra. Theorem 4.4. [Kel11, Theorem 6.10]
Let Λ be a finite-dimensional algebra of global dimension . Then there exists a quiver Q Λ and a potential W Λ such that Π(Λ) ∼ = P ( Q Λ , W Λ ) . Let M be a minimal set of relations in Λ . The quiver of Q Λ is given by adding new arrows c ρ : i → j for every relation ρ : j → i in M . The potential W Λ is given by W Λ = X ρ ∈ M ρc ρ . In particular, if Λ is quadratic, then Π(Λ) is quadratic as well.One important assumption for the main result of this section is that
Π(Λ) is a planar quiveralgebra with potential. In fact, we give at the end of this subsection an example of a -hereditaryquadratic monomial algebra whose preprojective algebra does not satisfy this property. Weprovide the definition here. Definition 4.5.
Let Q be a quiver without loops or -cycles. An embedding ǫ : Q → R is amap which is injective on the vertices, sends arrows a : i → j to the open line segment l a from ǫ ( i ) to ǫ ( j ) , and satisfies • ǫ ( i ) l a for every i ∈ Q and a ∈ Q and • l a ∩ l b = ∅ for all a = b ∈ Q .The pair ( Q, ǫ ) is called a plane quiver . A face of ( Q, ǫ ) is a bounded component of R \ ǫ ( Q ) which is an open polygon. Definition 4.6.
Let ( Q, ǫ ) be a plane quiver such that every bounded connected component of R \ ǫ ( Q ) is a face and the arrows bounding every face are cyclically oriented. The potentialinduced from ( Q, ǫ ) is the linear combination W of the bounding cycles of all faces. The quiver LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 13 with potential ( Q, W ) is called the planar QP induced from ( Q, ǫ ) and any quiver with potentialobtained in this way is called a planar QP . Remark . A quiver with potential whose underlying quiver is planar is not necessarily a planarQP. In fact, the planarity has to be compatible with the potential, that is, each face is boundedby an oriented cycle.We can obtain algebras of global dimension at most from Π by using cuts. In fact, let ( Q, W ) be a quiver with potential and C ⊂ Q be a subset. We define a grading g C on Q by setting g C ( a ) := (cid:26) a ∈ C a
6∈ C for each a ∈ Q . Definition 4.8.
A subset
C ⊂ Q is called a cut if W is homogeneous of degree with respectto g C .When C is a cut, there is an induced grading on P ( Q, W ) . We denote by P ( Q, W ) C the degree part with respect to this grading. Definition 4.9.
A cut C is called algebraic if it satisfies the following properties:(1) P ( Q, W ) C is a finite-dimensional k -algebra with global dimension at most two;(2) { δ c W } c ∈C is a minimal set of generators in the ideal h δ c W | c ∈ Ci .All truncated Jacobian algebras P ( Q, W ) C given by algebraic cuts C are cluster equivalent[HI11b, Proposition 7.6]. These are related to -APR tilts [IO11].When Λ is -hereditary, Π enjoys some additional characteristics. Proposition 4.10.
Let Λ be a k -algebra such that gl . dim Λ ≤ . • [HIO14, Theorem 5.6] The following are equivalent.(1)
Π(Λ) = P ( Q, W ) is a bimodule -Calabi–Yau Jacobian algebra of Gorenstein param-eter ;(2) Π(Λ) C is a -representation-infinite algebra for every cut C ⊂ Q . • [HI11b, Proposition 3.9] The following are equivalent.(1)
Π(Λ) = P ( Q, W ) is a finite-dimensional selfinjective Jacobian algebra;(2) Π(Λ) C is a -representation-finite algebra for every cut C ⊂ Q . This characterisation allows us to work with the following exact sequences.
Theorem 4.11.
Let
Π = P ( Q, W ) be a Jacobian algebra. • [Boc08, Proof of Theorem 3.1] Π is -Calabi–Yau if and only if the following complex ofleft Π -modules is exact for every simple module S i : → P i [ a ] −→ M a ∈ Q h ( a )= i P t ( a ) [ δ ( a,b ) W ] −−−−−→ M b ∈ Q t ( b )= i P h ( b ) [ b ] −→ P i → S i → , (4.1) where P j := Π e j and δ ( a,b ) W := δ L a ◦ δ R b W . • [HI11b, Theorem 3.7] Π is selfinjective if and only if it is finite-dimensional and thefollowing complex of left Π -modules is exact for every simple module S i : P i [ a ] −→ M a ∈ Q h ( a )= i P t ( a ) [ δ ( a,b ) W ] −−−−−→ M b ∈ Q t ( b )= i P h ( b ) [ b ] −→ P i → S i → . (4.2) We also need a couple of additional definitions to treat the case when Λ is in addition aquadratic monomial algebra. Let Π be a Jacobian algebra with potential W . We say that Π admits a monomial cut C if Π C is a monomial algebra. Also, an arrow a in the quiver of Π iscalled a border if a is part of exactly one summand of W . It is clear that if Π is quadratic,then W is a sum of cyclic paths of length , since it is homogeneous of degree . We call thosesummands triangles . By ideas similar to Lemma 2.13, every arrow is part of at least one triangle.The following lemma is clear. Lemma 4.12.
A cut C is monomial if and only if the arrows in degree are borders. In particular, the existence of a monomial cut in Π implies that there is at least one borderin each summand. An important step in our classification proof is to show that there is exactlyone border, unless there is only one summand.The following lemma is elementary, but we include a proof for the convenience of the reader. Lemma 4.13.
Let
Π = P ( Q, W ) be a Jacobian algebra which is either selfinjective or Calabi–Yau. The matrix [ δ ( a,b ) W ] in the complexes (4.1) and (4.2) is indecomposable, that is, it is notsimilar to a block matrix.Proof. Suppose by contradiction that the complexes can be written as P i " [ a ′ ][ a ′′ ] −−−−→ M a ∈ Q h ( a )= i P t ( a ) [ δ ( a ′ ,b ′ ) W ] [0][0] [ δ ( a ′′ ,b ′′ ) W ] −−−−−−−−−−−−−−−−−−−→ M b ∈ Q t ( b )= i P h ( b ) h [ b ′ ] [ b ′′ ] i −−−−−−−−→ P i → S i → , for some vectors of arrows [ a ′ ] , [ a ′′ ] , [ b ′ ] , [ b ′′ ] . Then the element ([0] , [ a ′′ ]) ∈ L a ∈ Q h ( a )= i P t ( a ) is a cyclewhich is not a boundary, contradicting the exactness of the complex. (cid:3) Using this, we now show that we can exclude an important class of examples, namely thosecoming from truncated Nakayama algebras.
Lemma 4.14.
Let
Π := P ( Q, W ) be a Jacobian algebra which is either selfinjective or Calabi–Yau. Suppose that there exists a summand W ′ of W in which every arrow is a border. Then Π is given by the quiver (cid:8) ,with potential the one obtained by summing over every cyclic rotation of the complete cycle.Proof. Let W ′ = x n · · · x be the summand in which every arrow is a border. By Lemma 4.13,the matrix [ δ ( a,b ) W ] : M a ∈ Q h ( a )= h ( x n ) P t ( a ) → M b ∈ Q t ( b )= h ( x n ) P t ( b ) , is indecomposable. However, the column [ δ ( a,x ) W ] h ( a )= h ( x n ) LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 15 and the row [ δ ( x n ,b ) W ] t ( b )= h ( x n ) each only have one non-zero element, since x and x n are borders. Thus, [ δ ( a,b ) W ] can only beindecomposable if its dimension is × . This means that x n is the only arrow ending at h ( x n ) and x is the only arrow starting at h ( x n ) . Repeating the argument with x , x , . . . , x n − , wededuce that there is also only one arrow ending and one starting at h ( x i ) , for i = 1 , . . . , n − , aswell. Since the quiver Q is connected, Π must be given by the QP described in the statement. (cid:3) Now assume that Π is the preprojective algebra of a -hereditary algebra. If Π admits amonomial cut, then we show that summands of the potential cannot have two borders either.For this, we need the following proposition. It is shown in [HI11b] in the case where P ( Q, W ) is aselfinjective algebra, but the same proof also works in the case when P ( Q, W ) is a -Calabi–Yaualgebra. Proposition 4.15 ([HI11b, Proposition 3.10]) . Let P ( Q, W ) be a preprojective algebra over a -hereditary algebra. Then every cut C ⊂ Q is algebraic. Lemma 4.16.
Let
Π = P ( Q, W ) be a quadratic Jacobian algebra which is either selfinjective or -Calabi–Yau. Suppose that Π admits a monomial cut. Then there does not exist a triangle W ′ of W which has exactly two borders.Proof. Suppose by contradiction that W ′ = xyz is a triangle of W such that x and y are bothborders, and z is not. Then there is another summand W ′′ = uvz containing z . Let C be themonomial cut on Π , in which we may assume without loss of generality that x is in degree .Then, the grading C ′ obtained from C by putting x in degree and y in degree is also a cut,since x and y are borders which do not appear in other triangles. Now, suppose that v is indegree . Then, in Π C , we have that zu = 0 and yz = 0 . Thus gl . dim Π C ≥ , and C is notalgebraic, contradicting Proposition 4.15. Similarly, if u is in degree , then C ′ is a non algebraiccut. As z cannot be in degree in a monomial cut, W ′′ cannot be put in degree in C . (cid:3) Combining the previous two lemmas, we obtain the following corollary.
Corollary 4.17.
Let
Π = P ( Q, W ) be a quadratic Jacobian algebra which is selfinjective or -Calabi–Yau and admits a monomial cut. Then either every summand of W has exactly oneborder, or Π is the quiver algebra with potential with a unique triangle: (cid:8) Note that the latter case is the preprojective algebra of k A / J , the first example in our maintheorem.The vanishing-of-Ext condition also gives information about the quiver of -hereditary qua-dratic monomial algebras. We have the following corollary to Proposition 3.1. Corollary 4.18.
Let Λ be a quadratic monomial algebra with Ext e (Λ , Λ e ) = 0 . Then for everyrelation ρ = ba , the vertex h ( b ) is a sink and the vertex t ( a ) is a source.Proof. Since gl . dim Λ = 2 , every arrow is either the start or the end of every relation they divide.The result thus follows directly from Proposition 3.1. (cid:3) This leads to the following definition.
Definition 4.19.
Let r, s ∈ Z ≥ . The ( r, s ) -star quiver , denoted by S ( r,s ) , is the quiver with r + s + 1 vertices and a central vertex z which is the head of r arrows and the tail of s arrows. We always denote the arrows i → z by a i and the arrows z → j by b j .We conclude that every quadratic monomial -hereditary algebra is a bound quiver algebraover a star quiver. Corollary 4.20.
Let Λ be a quadratic monomial algebra with Ext e (Λ , Λ e ) = 0 . Then the quiverof Λ is an ( r, s ) -star quiver. In particular, -hereditary quadratic monomial algebras are givenby quotients of ( r, s ) -star quiver algebras.Proof. By Corollary 4.18, every relation is a path of length which starts at a source and endsat a sink. Furthermore, Proposition 3.3 implies that these vertices can only be source and sinkto one arrow. This means that the quiver of Λ is made of paths of length which all intersectat a common middle vertex. (cid:3) Before completing the proof of the main theorem of this section, we explore further somequick restrictions on the relations which are imposed by the vanishing-of-Ext condition. Fromnow on in this section, we let Λ be a bound ( r, s ) -star quiver algebra. For each arrow a such that h ( a ) = z , we define Z a := { b : z → j | ba = 0 } and we define a set Z b similarly for arrows b such that t ( b ) = z . By Lemma 4.16, we have that |Z a | and |Z b | are greater than or equal to , unless ( r, s ) = (1 , . Lemma 4.21.
Let Λ be as above. If there are two distinct arrows a and a ′ such that Z a ⊂ Z a ′ ,then Ext e (Λ , Λ e ) = 0 .Proof. Suppose that the two arrows in the statement are such that h ( a ) = h ( a ′ ) = z , the casewhere t ( a ) = t ( a ′ ) = z being similar. Consider then the element ( e t ( a ′ ) ⊗ k e z a ) a ′ ∈ Λ e t ( a ′ ) ⊗ k e z Λ . This is a cocycle since Z a ⊂ Z a ′ . It is however not a coboundary, by the same principles as insection 3. (cid:3) We obtain the following corollary as a particular case.
Corollary 4.22.
Let Λ be as above and assume that Ext e (Λ , Λ e ) = 0 . Suppose that s ≥ andlet a be an arrow such that h ( a ) = z . Then |Z a | ≤ s − . Similarly, if r ≥ and b is an arrowsuch that t ( b ) = z , then |Z b | ≤ r − . We now show that the upper bound on |Z a | is even smaller if Λ is -RF. Lemma 4.23.
Let Λ be as above and suppose that Λ is -RF. Suppose that s ≥ and let a bean arrow such that h ( a ) = z . Then |Z a | ≤ s − . Similarly, if r ≥ and b is an arrow such that t ( b ) = z , then |Z b | ≤ r − .Proof. We can use a Loewy length ( ℓℓ ) argument as follows. Suppose that there is an arrow a i : i → z such that |Z a i | = s − . Consider the preprojective algebra Π over Λ . We refer to itsdescription below Theorem 4.4. Let P i := Π e i . We show that ℓℓ ( P i ) = 3 , whereas ℓℓ ( P z ) ≥ .Since Π is selfinjective, this contradicts [MV99, Theorem 3.3]. LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 17 Let b j : z → j be the only arrow such that b j a i = 0 . Let ρ = b j a x be a relation in Λ and c ρ be the corresponding arrow in Π . Then c ρ b j a i = − P c ρ ′ b j ′ a i = 0 , where the sum is taken overrelations of the form ρ ′ := b j ′ a x in Λ which are not equal to ρ . The sum is not empty since s ≥ .This shows that ℓℓ ( P i ) = 3 . Now, since a path of the form a ν c ρ b µ is never in Π for any vertices ν, µ and relations ρ , we have ℓℓ ( P z ) ≥ . Here, we have used the fact that |Z b µ | , |Z a ν | ≥ . Theargument is dual for an arrow b : z → i such that |Z b | = r − . (cid:3) In particular, this implies that, if Λ is -RF, then either ( r, s ) = (1 , , or r, s ≥ .We now have plenty of tools to give a full classification of the monomial -hereditary algebraswhose preprojective algebras is a planar quiver with potential. We prove the main theorem ofthis section. Note that, for the previous results of this section, we have not assumed that thepreprojective algebra is a planar QP. We need the hypothesis now. Proof of Theorem 4.1.
By reasons given above, one can easily check that the two bound quiveralgebras described in (1.1) are -RF. Assume that Λ is a -hereditary quadratic monomial algebrawhose preprojective algebra is a planar quiver with potential. We prove that they are the onlyones coming from a planar quiver with potential.By Corollary 4.20, Λ is an ( r, s ) -star quiver. By [Pet19, Proposition 3.15], the planarityassumption allows us to conclude that every arrow in Π(Λ) is contained in at most two summandsof the potential W . Combining this with Lemma 4.16, we see that every arrow in Λ is part ofexactly relations. Therefore, the quiver of Π(Λ) is given by the intersection of oriented triangleswhich all share a common vertex z , thus forming a regular polygon shape. In particular, r = s .In addition, if Λ is -RF, then we have that r, s ≥ by Lemma 4.23, unless ( r, s ) = (1 , . If Λ is -RI, then we also obtain the same conclusion, since in the case r = 2 or r = 3 , the preprojectivealgebra is clearly finite-dimensional. If r = 1 or r = 4 , then we recover the bound quiver algebrasdescribed in (1.1).Assume that r ≥ . We show that Ext e (Λ , Λ e ) ∼ = Ext ( D Λ , Λ) = 0 . Let I m be the injectivemodule associated to a sink m and b : z → m . Also recall that Z ∁ b := Q \ Z b . Then |Z ∁ b | = r − .Let a , . . . , a r − be the arrows in Z ∁ b and define t i := t ( a i ) for i = 1 , . . . , r − . Without loss ofgenerality, we can assume that we ordered the arrows so that |Z a i ∩ Z a i +1 | = 1 for i = 1 , . . . , r − .This is due to the planarity assumption on Π(Λ) . We call b i the arrow in this intersection for i = 1 , . . . , r − and define h i := h ( b i ) . Then the projective resolution of I m is given by → M i =1 ,...,r − P h i → P r − z → M i =1 ,...,r − P t i → . Applying
Hom Λ ( − , Λ e t ) , we obtain a complex → M i =1 ,...,r − e t i Λ e t → ( e z Λ e t ) r − → M i =1 ,...,r − e h i Λ e t → Ext ( I m , Λ e t ) → . This complex is not exact at ( e z Λ e t ) r − since dim k ( L i =1 ,...,r − e t i Λ e t ) = 1 , dim k (( e z Λ e t ) r − ) = r − and dim k ( L i =1 ,...,r − e h i Λ e t ) = r − . The last equality can be explained by the fact that ba = 0 for b ∈ Q if and only if b = b or b .Thus Ext ( D Λ , Λ) = 0 . Note that we could have chosen to take Hom Λ ( − , Λ e t µ ) for any µ = 2 , . . . , r − and still obtain the same conclusion. (cid:3) Example 4.24.
We give an example of a quadratic monomial 2-RF algebra whose -preprojectivealgebra is a non-planar selfinjective quiver with potential. If ( r, s ) = (9 , with arrows a i : i → z for i = 1 , . . . , and b j : z → j , for j = 1 , . . . , one gets an example with relations b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a , b a . This can be seen to be a cut of Π( D ⊗ D ) , where both copies of D are oriented witharrows going out of the central vertex. Since D with this orientation is ℓ -homogeneous, [HI11a,Proposition 1.4] implies that the tensor product is -RF, and hence, this example is also -RF.As the quiver of Π( D ⊗ D ) is a non-planar graph, the example is non-planar, too.We note that by observing that to get a quadratic monomial cut the QP must have “enough”borders, it is not too hard to see that the only possible tensor products of Dynkin diagramsthat have -preprojective algebras with such cuts involve A and D with bipartite orientation.Moreover, A ⊗ D can be checked to not be -RF, and A ⊗ A yields the planar example. Remark . One should note that many natural constructions on algebras that preserve theproperty of being n -hereditary do not necessarily preserve being monomial. For instance, thisincludes tensor products and certain skew-group ring constructions.Moreover, while there exist other non-planar examples, the ones we know of are all fairly largeand are somewhat more complicated than the one mentioned above.4.2. The case n ≥ . We classify all n -representation-finite quadratic monomial algebras ofglobal dimension higher than . Note that we do not assume that the preprojective algebra is aplanar QP. Theorem 4.26.
With the exception of k A n +1 / J , there are no quadratic monomial n -RF alge-bras for n ≥ .Proof. To begin with, we observe that, by Proposition 3.1, every arrow in Λ lies on some maximalpath → → → · · · → i → i + 1 → · · · → n − → n − → n in which every two consecutive arrows are a relation. Also note that must be a source and n a sink.We begin by showing that there cannot exist an arrow in Λ different from the one in the diagramabove leaving a vertex i with i < n − . Indeed, say there was some such arrow a ′ i : i → j . Since Π(Λ) is selfinjective, the projective at i over Π(Λ) cannot have a non-simple socle. Hence, theremust be a commutation relation in
Π(Λ) starting at i of the form α i ra i + Σ k α k r k a i,k with arrows r and r k in Π(Λ) (but not in Λ ) and with α j scalars, and α i , α j = 0 for some j .Indeed, to see that the latter claim must hold, note that if pa i , qa ′ i ∈ soc Π(Λ) e i , then pa i − qa ′ i ∈ I = h ρ j i with { ρ j } a set of minimal relations. In other words, pa i − qa ′ i = Σ j u j ρ j v j where u j , v j can be assumed to be paths up to scalars. Rewriting this as pa i − qa ′ i = Σ j u j ρ j v j = Σ k u k ρ k v k + Σ l u l ρ l a i + Σ m u m ρ m a ′ i with v k = a i , a ′ i , we see thus that if pa i , qa ′ i are non-zero in Π(Λ) and a i = a ′ i , then Σ k u k ρ k v k = pa i − qa ′ i − Σ l u l ρ l a i − Σ m u m ρ m a ′ i = 0 . This also implies that we can set v k = 1 . Moreover, we observe that some ρ j is of the form α i ra i + Σ k α k r k a i,k as stated above since pa i must occur as some term in some u k ρ k . If pa i wasthe only non-zero term in that u k ρ k , we would have pa i = 0 in Π(Λ) , contrary to our assumptions.This establishes the claim.Note that Λ is Koszul, so by Proposition 2.7, we know that such a commutation relation andnew arrows beginning in vertices i + 1 and j in the preprojective correspond exactly to elementsin K n ending with arrows a i : i → i + 1 and a ′ i : i → j , and differing only in the final arrow.However, there can be no such element ending in i + 1 as i + 1 < n is not a sink. LASSIFICATION RESULTS FOR n -HEREDITARY MONOMIAL ALGEBRAS 19 Since Λ is n -RF if and only if Λ op is n -RF, we have also shown that there are no arrows endingin i with < i . Hence, without loss of generality, we can assume that if Λ has quiver differentfrom linearly oriented A n , then there must be an arrow a ′ n − different from a n − starting in n − .Yet, if this was the case, the Π(Λ) -projective at n − would be of Loewy length ≥ whereasthe Π(Λ) -projective at i < n − would be of Loewy length ≤ , as there cannot be any newarrows in Π(Λ) not in Λ out of n − as it is not a sink. This yields a contradiction by the factthat Π(Λ) has homogeneous relations and [MV99, Theorem 3.3]. By using Vaso’s classification([Vas19]) of n -RF algebras that are quotients of Nakayama algebras, we are done. (cid:3) References [AIR15] Claire Amiot, Osamu Iyama, and Idun Reiten. Stable categories of Cohen-Macaulay modules andcluster categories.
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Institutt for matematiske fag, NTNU, 7491 Trondheim, Norway
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