Boundary Idempotents and 2-precluster-tilting categories
aa r X i v : . [ m a t h . R T ] F e b BOUNDARY IDEMPOTENTS AND -PRECLUSTER-TILTINGCATEGORIES JORDAN MCMAHON
Abstract.
The homological theory of Auslander–Platzeck–Todorov on idem-potent ideals laid much of the groundwork for higher Auslander–Reiten theory,providing the key technical lemmas for both higher Auslander correspondenceas well as the construction of higher Nakayama algebras, among other results.Given a finite-dimensional algebra A and idempotent e , we expand on a crite-rion of Jasso-K¨ulshammer in order to determine a correspondence between the2-precluster-tilting subcategories of mod( A ) and mod( A/ h e i ). This is then ap-plied in the context of generalising dimer algebras on surfaces with boundaryidempotent. Introduction
Higher cluster-tilting subcategories were introduced and studied by Iyama [3,4]. They remain only partially understood, and not easy to find in general. Ageneralisation to this are higher precluster-tilting subcategories, as introduced byIyama and Solberg [5]. These higher precluster-tilting subcategories are a weakerversion of higher cluster-tilting subcategories, but which are of interest in theirown right.A particular class of algebras where we would expect to find higher (pre)cluster-tilting subcategories are from a higher versions of dimer algebras on a disc thesense of [2]. In recent work [8], we were able to find higher precluster-tiltingsubcategories, as long as the boundary was omitted. In this work we give aninductive criterion to construct higher precluster-tilting subcategories, and showhow this can be applied to boundary idempotents. This criterion is motivated bythe following example.
Example 1.1.
Given a semisimple algebra A with two vertices 1 and 3, we mayadd a vertex 2 together with arrows and relations, to produce a (pre)cluster-tiltingsubcategory in the following ways: we can form an Auslander algebra of type A :21 3 which has a cluster-tilting subcategory given by S , S and the projective-injectives.Alternatively, we can form the (self-injective) preprojective algebra Π( A ), ob-tained from A be adding arrows α : j → i for α : i → j ∈ Q with admissible ideal I generated by P α ∈ Q αα − αα. S , S and the projective-injectives.This inductive criterion is based on a result of Jasso–K¨ulshammer (Lemma 3.1),who needed an inductive approach to define higher Nakayama algebras, and showthat they have higher cluster-tilting subcategories. Theorem 1.2.
Let A be a finite-dimensional algebra and e an idempotent of A and ˜ C ⊆ mod( A ) a -precluster-tilting subcategory. Let C ∼ = ˜
C ∩ mod( A/ h e i ) . Ifalso(i) Hom A ( A/ h e i , ˜ C ) ⊆ ˜ C .(ii) A/ h e i ⊗ A ˜ C ⊆ ˜ C .Then C ⊆ mod( A/ h e i ) is -precluster tilting. Conversely we may (inductively) construct 2-precluster-tilting subcategories.
Theorem 1.3.
Let A be a finite-dimensional algebra, e an idempotent of A and C ⊆ mod( A/ h e i ) a -precluster-tilting subcategory. If also(i) Ext A ( DA, A ) = 0 .(ii)
Ae, D ( eA ) are projective-injective A -modules.(iii) There is an equality of sets { X ∈ C| Ext A ( X, J ) = 0 ∀ J ∈ inj( A/ h − e i ) } = { ( τ − ) A P | P ∈ proj( A ) \ proj( A/ h e i ) } . (iv) There is an equality of sets { X ∈ C| Ext A ( A/ h − e i , X ) = 0 } = { ( τ ) A I | I ∈ inj( A ) \ inj( A/ h e i ) } . Then
C ∪ proj( A ) ∪ inj( A ) =: ˜ C ⊆ mod( A ) is -precluster tilting. The final two conditions have a combinatorial meaning in terms of the relationsin the algebra. For a 2-(internally)-Calabi Yau algebras, for example the prepro-jective algebra above, we expect significant simplifications of the above conditions.Likewise if proj . dim( S ) = inj . dim( S ) = 1, which is often the case for a simplemodule over a higher Nakayama algebra [7].2. Background and Notation
Consider a finite-dimensional algebra A over a field K , and fix a positive integer d . We will assume that A is of the form KQ/I , where KQ is the path algebra oversome quiver Q and I is an admissible ideal of KQ . For two arrows in Q , α : i → j and β : j → k , we denote their composition as βα : i → k . Let A op denote theopposite algebra of A . An A -module will mean a finitely-generated left A -module;by mod( A ) we denote the category of A -modules. The functor D = Hom K ( − , K )defines a duality; by ⊗ we mean ⊗ K and we denote the syzygy by Ω. Denote by ν := DA ⊗ A − ∼ = D Hom A ( − , A ) the Nakayama functor in mod( A ). Let add( M )be the full subcategory of mod( A ) composed of all A -modules isomorphic to directsummands of finite direct sums of copies of M .2.1. Higher precluster-tilting subcategories.Definition 2.1. [4, Definition 2.2] For a finite-dimensional algebra A , a module M ∈ mod( A ) is a d -cluster-tilting module if it satisfies the following conditions:add( M ) = { X ∈ mod( A ) | Ext iA ( M, X ) = 0 ∀ < i < d } . add( M ) = { X ∈ mod( A ) | Ext iA ( X, M ) = 0 ∀ < i < d } . In this case add( M ) is a d -cluster-tilting subcategory of mod( A ).Define τ d := τ Ω d − to be the d -Auslander–Reiten translation and τ − d := τ − Ω − ( d − to be the inverse d -Auslander–Reiten translation . Definition 2.2. [5, Definition 3.2] For a finite-dimensional algebra A , a module M ∈ mod( A ) is d -precluster tilting if it satisfies the following conditions:(P1) The module M is a generator-cogenerator for mod( A ).(P2) We have τ d M ∈ add( M ) and τ − d M ∈ add( M ).(P3) There is an equality Ext iA ( M, M ) = 0 for all 0 < i < d .For a d -precluster-tilting module M , the subcategory add( M ) ⊆ mod( A ) iscalled a d -precluster-tilting subcategory . Proposition 2.3. [4, Theorem 1.5]
We have the following • If Ext iA ( M, A ) = 0 for all < i < d , then Ext iA ( M, N ) ∼ = D Ext d − iA ( N, τ d M ) for all M ∈ mod( A ) and all < i < d . • If Ext iA ( DA, N ) = 0 for all < i < d , then Ext iA ( M, N ) ∼ = D Ext d − iA ( τ − d N, M ) for all N ∈ mod( A ) for all < i < d . Homological theory of idempotent ideals.
Now we review the some ho-mological theory of idempotent ideals, as introduced by Auslander, Platzeck andTodorov. Throughout this section we will let F := Hom A ( A/ h e i , − ). Proposition 2.4. [1, Proposition 1.1]
Let N be an A -module, and let ≤ d ≤ ∞ .Then the following are equivalent:(i) Ext iA ( A/ h e i , N ) = 0 for all i such that < i < d .(ii) Let M be in mod( A/ h e i ) . Then there are isomorphisms: Ext iA/ h e i ( M, F N )) → Ext iA ( M, N ) for all < i < d . JORDAN MCMAHON
A third equivalent condition was incorrectly stated in the original article. Theresult we will need instead is the following:
Corollary 2.5.
Let N be an A -module, and let → N → I → I → · · · → I d be the beginning of a minimal injective coresolution of N and let < i < d . Theneach equivalent condition of Proposition 2.4 implies → F N → F I → · · · → F I d is the beginning of an injective coresolution of F N in mod( A/ h e i ) .Proof. Suppose that Ext iA ( A/ h e i , N ) = 0 for all i such that 0 < i < d , and let C j := coker( I j − → I j ) . Then we have an exact sequence:0 → F N → · · · →
F I d − → F C d − → iA ( A/ h e i , N ) = 0 for all 0 < i < d . Moreover, the exact sequences0 → F C d − → F I d − → F C d → F C d → F I d combine to give the result. (cid:3) We note that the resulting injective coresolution is not necessarily minimal.There is now a characterisation
Proposition 2.6. [1, Proposition 1.3]
Let A be a finite-dimensional algebra and e an idempotent of A . Then the following are equivalent(i) There are isomorphsims Ext iA/ h e i ( M, N ) → Ext iA ( M, N ) for all M, N ∈ mod( A/ h e i ) and all ≤ i < d .(ii) Ext iA ( A/ h e i , N ) = 0 for all N ∈ mod( A/ h e i ) for all i such that < i < d .(iii) Ext iA ( A/ h e i , I ) = 0 for all I ∈ inj( A/ h e i ) for all i such that < i < d . In this case, the ideal h e i is said to be ( d − -idempotent . A related useful resultis the following. For a positive integer d , we define I d to be the full subcategoryof mod( A ) consisting of the A -modules M having an injective resolution0 → M → I → I → · · · with I j ∈ add( I ) for all 0 ≤ i ≤ d . Proposition 2.7. [1, Proposition 2.6]
Let A be a finite-dimensional algebra, e anidempotent of A and I = D (1 − e ) A and ≤ d < ∞ . Then the following areequivalent(i) N ∈ I d .(ii) Ext iA ( M, N ) = 0 for all M ∈ mod( A/ h e i ) for all i such that ≤ i < d .(iii) Ext iA ( A/ h e i , N ) = 0 for all ≤ i < d . Main results.Theorem 1.2.
Let A be a finite-dimensional algebra and e an idempotent of A and ˜ C ⊆ mod( A ) a -precluster-tilting subcategory. If also(i) Hom A ( A/ h e i , ˜ C ) ⊆ ˜ C (ii) A/ h e i ⊗ A ˜ C ⊆ ˜ C .Then C := ˜ C ∩ mod( A/ h e i ) ⊆ mod( A/ h e i ) is -precluster tilting.Proof. Suppose ˜
C ⊆ mod( A ) is 2-precluster-tilting. By assumption (i), we haveinj( A/ h e i ) ∈ C and by assumption (ii) proj( A/ h e i ) ∈ C . So condition (P1) issatisfied. Secondly, Proposition 2.6(iii) implies that h e i is 1-idempotent, and hencethat Ext A/ h e i ( M, N ) ∼ = Ext A ( M, N ) = 0 for all
M, N ∈ C . Hence condition (P3)is also satisfied.Finally, let N ∈ C be a non-injective A -module and let0 → N → ˜ I → ˜ I → ˜ I be the beginning of a minimal injective coresolution of N in mod( A ). It followsthat ( τ − ) A N = coker( ˜ P → ˜ P ). Let I j = Hom A ( A/ h e i , ˜ I j ). Then Corollary 2.5implies that 0 → N → I → I → I is the beginning of an injective coresolutionof N in mod( A/ h e i ). Since I is necessarily minimal, the only case where non-minimality may arise is a trivial map to a summand of I . It follows ( τ − ) A/ h e i N is a summand of coker( P → P ) = Hom A ( A/ h e i , ( τ − ) A N ) ∈ C by assumption.Hence ( τ − ) A N ∈ C and dually C is closed under ( τ ) A . So condition (P2) holds,and C ⊆ mod( A/ h e i ) is a 2-precluster-tilting subcategory. (cid:3) We need a technical result:
Lemma 2.8.
Let A be a finite-dimensional algebra such that Ae, D ( eA ) ∈ proj - inj . ( A ) .Then for any M ∈ mod( A ) with minimal injective resolution → M → I → I → I , then I ∈ add( D ( eA )) ⇐⇒ Ext A ( A/ h − e i , M ) = 0 . Proof.
First note Ext A ( A/ h − e i , M ) ∼ = Hom A ( A/ h − e i , Ω − ( M )). Now anymorphism A/ h − e i → Ω − ( M ) that factors through an injective must fac-tor through an injective summand of Ω − ( M ). This is impossible, since anysuch summand is in add( D ( eA )) and therefore projective by assumption. HenceHom A ( A/ h − e i , Ω − ( A )) ∼ = Hom A ( A/ h − e i , Ω − ( A )) . So Ext A ( A/ h − e i , M ) ∼ =Hom A ( A/ h − e i , Ω − ( M )) ∼ = Hom A ( A/ h − e i , Ω − ( M )) and the result followsfrom Proposition 2.7. (cid:3) Theorem 1.3.
Let A be a finite-dimensional algebra, e an idempotent of A and C ⊆ mod( A/ h e i ) a -precluster-tilting subcategory. If also JORDAN MCMAHON (i)
Ext A ( DA, A ) = 0 .(ii)
Ae, D ( eA ) ∈ proj - inj . ( A ) .(iii) There is an equality of sets { X ∈ C| Ext A ( X, J ) = 0 ∀ J ∈ inj( A/ h − e i ) } = { ( τ − ) A P | P ∈ proj( A ) \ proj( A/ h e i ) } . (iv) There is an equality of sets { X ∈ C| Ext A ( A/ h − e i , X ) = 0 } = { ( τ ) A I | I ∈ inj( A ) \ inj( A/ h e i ) } . Then
C ∪ proj( A ) ∪ inj( A ) =: ˜ C ⊆ mod( A ) is -precluster tilting.Proof. Suppose that
C ⊆ mod( A/ h e i ) is 2-precluster-tilting. We have that ˜ C is agenerator-cogenerator by construction. Since Ext A ( DA, A ) = 0, Proposition 2.3implies for any N ∈ C ∪ proj( A ) the calculationExt A ( DA, N ) ∼ = D Ext ( N, ( τ ) A DA )) ∼ = 0 , since ( τ ) A ( DA ) ∈ C . Dually Ext A ( M, A ) = 0 for all M ∈ C . By constructionExt A ( P, I ) = 0 for any P ∈ proj( A/ h e i ) and I ∈ inj( A/ h e i ) (there are no arrowsin the quiver of A from a sink in the quiver of A/ h e i to a source in the quiverof A/ h e i ). So Proposition 2.6 implies 0 = Ext A/ h e i ( M, N ) = Ext A ( M, N ) for all
M, N ∈ C . Hence condition (P3) is also satisfied.Finally, we have to show closure under ( τ − ) A , we do this for a given X ∈ C withminimal injective resolution in mod( A/ h e i ): 0 → X → I → I → I , where weset ˜ J , ˜ J ′ to be injective A -modules such that Hom A ( A/ h e i , ˜ J ) = 0.(a) 0 → X → ˜ I → ˜ I ( ⊕ ˜ J ) → ˜ I ⊕ ˜ J ′ is a minimal injective resolution of X inmod( A ): then by assumption ( τ − d ) A X ∈ inj( A ) ∈ ˜ C .(b) 0 → X → ˜ I → ˜ I ( ⊕ ˜ J ) → ˜ I is an injective resolution of X in mod( A ): thensimply ( τ − d ) A X ∼ = ( τ − d ) A/ h e i X ∈ ˜ C .(c) 0 → X → ˜ I → ˜ J is an injective resolution of X in mod( A ): then X ∈ inj( A/ h e i ) and ( τ − d ) A X ∈ proj( A/ h e i ) ∈ ˜ C .Hence ˜ C is closed under ( τ − d ) A , and dually also under ( τ ) A . So ˜ C ⊆ mod( A ) is a2-precluster-tilting subcategory. (cid:3) Examples
In this section we will consider algebras with vertices labelled by subsets of { , , . . . , n } . There is a canonical way of constructing the algebra. Let Q be aset of ( d + 1)-subsets of { , , . . . , n } . For X, Y ∈ Q , define Q by adding arrows α i ( X ) : X → Y wherever X \ { i } = Y \ { i + 1 } for some i ∈ X . Let I be theadmissible ideal of KQ generated by the elements α j ( α i ( I )) − α i ( α j ( I )) , which range over all X ∈ Q . By convention, α i ( X ) = 0 whenever X or X ∪ { i +1 } \ { i } is not a member of Q . Hence there are zero relations included in the ideal I .3.1. Higher Nakayama algebras.
One of the motivating examples comes fromhigher Nakayama algebras [6]. In order to define higher Nakayama algebras anddefine higher cluster-tilting subcategories, Jasso and K¨ulshammer make use of thefollowing result, which motivates our main results.
Lemma 3.1. [6, Lemma 1.20]
Let A be a finite-dimensional algebra and ˜ C a d -cluster-tilting subcategory of mod( A ) . Let e ∈ A be an idempotent such that thefollowing conditions are satisfied: • All the projective and all the injective A/ h e i -modules belong to C . • Every indecomposable M ∈ C which does not lie in mod( A/ h e i ) is projective-injective.Then h e i is a ( d − -idempotent ideal and C ⊆ mod( A/ h e i ) is d -cluster tilting. Using this result, Jasso and K¨ulshammer are able to inductively define highercluster-tilting subcategories. For example, the following higher Nakayama algebra A has quiver01 02 03 0412 13 1423 24 2534 35 3601 02 03 0412 13 14and relations indictated by the dotted arrows. We may easily apply Theorems 1.2and 1.3 to the idempotent e , since S has projective dimension and injectivedimension 1.3.2. Boundary idempotents.
Scott found a cluster structure on the Grassman-nian C [Gr( k, n )], with clusters given by non-crossing k -subsets of [ n ]. On the otherhand, Oppermann and Thomas [9] generalised the cluster structure of triangula-tions of convex polygons to cyclic polytopes. Combinatorially, a triangulation ofa cyclic polytope is given by maximal-by-size collections of non-intertwining sub-sets, where, given two k -subsets I = { i , i , . . . , i l } and J = { j , j , . . . , j l } , then I intertwines J if i < j < i < · · · < i l < j l . While no cluster algebra is formed, such triangulations are related to the repre-sentation theory of higher Auslander algebras of Dynkin type A . Two k -subsets JORDAN MCMAHON I and J are said to be non-crossing if there do not exist elements s < t < u < v (ordered cyclically) where s, u ∈ I − J , and t, v ∈ J − I .Oppermann and Thomas [9] were able to describe higher Auslander algebrasof type A by maximal collections of non-intertwining subsets, and also to tri-angulations of cyclic polytopes. In [8], we extended this to tensor products ofhigher Auslander algebras of type A by introducing maximal collections of non- l -intertwining subsets. Critically, we were only able to find higher precluster-tiltingsubcategories in general.Baur, King and Marsh [2] studied dimer algebras on a disc, which are related tomaximal non-crossing collections (and tensor products of type A quivers). In theirwork, boundary idempotents (given by consecutive subsets) play a key role. Thecriterion in Theorem 1.3 gives us hope that we may inductively add these boundaryidempotents back in, when we consider non-intertwining and non- l -intertwiningcollections. This is illustrated in the following example: Example 3.2.
Consider the algebra A , it can be checked that the idempotent e = E , , , , satisfies the conditions for Theorem 1.2 and 1.3. A/ h e i canbe described by a maximal non-intertwining collection of 3-subsets of { , , . . . , } (135,136,146) as well as a semisimple algebra (256,347). So mod( A/ h e i ) containsa 2-cluster-tilting subcategory, hence mod( A ) also contains a (pre)cluster-tiltingsubcategory. 256 236 367 347136125 135 146 147145 References
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