# The size of a stratifying system can be arbitrarily large

aa r X i v : . [ m a t h . R T ] F e b THE SIZE OF AN EXCEPTIONAL SEQUENCE CAN BEARBITRARILY LARGE

HIPOLITO TREFFINGER

In honour of Ibrahim Assem in the occasion of his birthday.

Abstract.

In this short note we construct two families of examples of large excep-tional sequences in module categories of algebras. The ﬁrst examples consists onexceptional sequences of inﬁnite size in the module category of an algebra A . In thesecond family of examples we show that the size of a complete exceptional sequencein the module category of a ﬁnite dimensional algebra A can be arbitrarily large incomparison to the number of isomorphism classes of simple A -modules. We note thatboth families of examples are built using well-established results in higher homologicalalgebra. Introduction

In this paper, A is a basic ﬁnite-dimensional algebra over an algebraically closed ﬁeld K , mod A is the category of ﬁnitely presented (right) A -modules and K ( A ) denotes theGrothendieck group of A .The notion of exceptional sequences in representation theory introduced in [3, 13].Even if the deﬁnition of exceptional sequence can be stated in the module category ofany ﬁnite-dimensional algebra, most of the articles on the subject studied exceptionalsequences in the module category of hereditary algebras. Outside the hereditary case, thenotion that have been mostly studied is the more general notion of stratifying systems,ﬁrstly introduced in [5] (see also [9, 10]). We note that the equivalence of both notionsfor module categories of hereditary algebras was shown in [2] (see also [4]). We now recallthe deﬁnition of exceptional sequences and stratifying systems. Deﬁnition.

Let A be a ﬁnite dimensional k -algebra. A stratifying system of size t inthe category mod A of ﬁnitely generated left A -modules is a pair (Θ , ≤ ) where Θ := { θ i : t ∈ [1 , t ] } is a family of indecomposable objects in mod A and ≤ is a linear order onthe set [1 , t ] := { , . . . , t } such that Hom A ( θ j , θ i ) = 0 if i < j and Ext A ( θ j , θ i ) = 0 if i ≤ j . Moreover, we say that { Θ , ≤} is an exceptional sequence if End A ( θ i ) ∼ = K for all i ∈ { , . . . , t } . Finally, we say that a exceptional sequence of size t is complete if there isno A -module θ t +1 such that { Θ ∪ { θ t +1 } , ≤} is an exceptional sequence where i ≤ t + 1 for all i ∈ [1 , t ] .The numerous works that have followed from the deﬁnition of stratifying systems showthat the existence a stratifying system in the module category of an algebra give plentyof homological information of this category. However, the existence of stratifying systems The author funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. in module categories is a problem that has received less attention. To our knowledge,the only works addressing the existence of exceptional sequences or stratifying systemsoutside the hereditary case are [7] for the Auslander algebras of the algebra K [ x ] /x t , [12]for quotients of type A zig-zag algebras and [1, 11] for arbitrary algebras using techniquesfrom τ -tilting theory.The classical examples of stratifying systems are the so-called canonical stratifyingsystems. These are stratifying systems which are constructed using all indecomposableprojective modules and, as a consequence, the size t of every canonical stratifying systemcoincide with rk ( K ( A )) . Moreover, it was shown in [2] that the size of any exceptionalsequence is bounded by rk ( K ( A )) . However, this is not true in general, since examplesof exceptional sequences in module categories whose size is bigger than the rank of theGrothendieck group of the algebra, see for instance [5, 3.2] and [9, Remark 2.7].In this paper we build two families of examples that suggest that a bound for the sizeof an exceptional sequence in a module in terms of the rank of the Grothendieck groupof the algebra does not exists for non-hereditary algebras. In Section 2, we show theexistence of a family of algebras having an exceptional sequence of inﬁnite size. Then, inSection 3, we show that the size of ﬁnite complete exceptional sequences in mod A cannot be linearly bounded by rk ( K ( A )) .2. Exceptional sequences of infinite size

In this section we follow closely the exposition and notation of [6] to build examplesof exceptional sequences of inﬁnite size. For that, we ﬁx a positive integer d > . Thenthe Beilinson algebra B d is the path algebra of the quiver a ... ( ( a d a ... ( ( a d · · · d a d ... + + a dd d + 1 modulo the ideal of relations generated by the elements of the form a ki a k +1 j − a kj a k +1 i .Also, we recall that the d -Auslander-Reiten translations are deﬁned as τ d := D Ext dA ( − , A ) : mod A → mod Aτ − d := Ext dA op ( D − , A ) : mod A → mod A Moreover, we denote by P d and I d the subcategories P d := add { ( τ − d ) k ( A ) : k ∈ N } and I d := add { ( τ d ) k ( DA ) : k ∈ N } of mod B d . Finally, we denote by P d and I d the setof representatives of every isomorphism class of indecomposable objects in P d and I d respectively. Note that both P d and I d are two sets of Theorem 1.

Let d be a positive integer greater than two. With the notation above, thereis a linear order ≤ in N such that ( P d , ≤ ) is an exceptional sequence in mod B d of inﬁnitesize. Similarly, there is a linear order (cid:22) in N such that ( I d , (cid:22) ) is an exceptional sequencemod B d of inﬁnite size.Proof. We only prove the case of ( P d , ≤ ) , since the proof for ( I d , ≤ ) is similar. First,we note that [6, Proposition 4.10.(a)] states that P d has inﬁnite many elements. More-over, it follows from the deﬁnition of P d and [6, Theorem 4.25] that given two non-isomorphic indecomposable objects X, Y in P d we have that either Hom A ( X, Y ) = 0

HE SIZE OF AN EXCEPTIONAL SEQUENCE CAN BE ARBITRARILY LARGE 3 or Hom A ( Y, X ) = 0 ( cf. [6, Example 4.32] for the case d = 2 ). In particular, andthe existence of an order ≤ in N such that Hom A ( X j , X i ) = 0 if i < j . The factthat End A ( X ) ∼ = K for all X ∈ P d follows from [6, Theorem 4.18] and [6, Theo-rem 4.25]. Finally, [6, Proposition 4.10.(f)] that Ext A ( X i , X j ) = 0 for all i, j ∈ N .Hence Ext A ( X i , X j ) = 0 if i ≤ j . (cid:3) Arbitrarily large complete exceptional sequences

In the previous section we show the existence of exceptional sequences of inﬁnite size.However, one can still consider the problem of ﬁnding a bound for the size of completeexceptional sequence of ﬁnite size in terms of the number of isomorphism classes of simplemodules. In this section we show that, if such a bound exists, it can not be linear.

Theorem 2.

For every positive integer m , there exists an algebra A m and a stratifyingsystem (Θ m , ≤ ) of size t m in mod A m such that t m > m. rk ( K ( A m )) . Before proving our theorem, lets recall that a subcategory M of mod A is said to be d -cluster-tilting if M = { X ∈ mod A : Ext iA ( X, M ) = 0 for all M ∈ M and all ≤ i ≤ d − } = { Y ∈ mod A : Ext iA ( M, Y ) = 0 for all M ∈ M and all ≤ i ≤ d − } . A module M is said to be d -cluster-tilting if M = add M for some A -module M . Proof.

Let H n be the path algebra of linearly oriented A n quiver and let A n be theAuslander algebra of H n . It was shown in [8, Theorem 1.14] that A n has a -cluster-tilting module M = L ti =1 M i . We claim that there exists of an order ≤ in the set [1 , t ] := { , . . . , t } such that ( { M i : i ∈ [1 , t ] } , ≤ ) is an exceptional sequence.We ﬁrst note that it was shown in [8, Theorem 6.12] that the quiver of the endo-morphism algebra End A n ( M ) of M is an acyclic quiver. As a consequence, there ex-ists a total order ≤ in [1 , t ] such that Hom A n ( M i , M j ) = 0 if j < i . Moreover, thesame result implies that End A n ( M i ) ∼ = K for all i ∈ { , . . . , t } . We also have thatExt A n ( M i , M j ) = 0 for all i, j ∈ { , . . . , t } because M is a -cluster-tilting object. Inparticular, Ext A n ( M i , M j ) = 0 if i ≤ j . Then ( { M i : i ∈ [1 , t ] } , ≤ ) is an exceptionalsequence. We note that ( { M i : i ∈ [1 , t ] } , ≤ ) is a complete exceptional sequences because L ti =1 M i is a -cluster-tilting module.Now, it is ease to see that rk ( K ( A n )) = n ( n +1)2 . Moreover, it follows directly form [8,Theorem 6.12] that t is equal to the number of integer points of the -simplex spanned bythe vectors { (0 , , , ( n, , , (0 , n, , (0 , , n ) } , that is, t = n ( n +1)( n +2)6 . Hence the ratio R ( n ) between the size of the stratifying system ( { M i : i ∈ [1 , t ] } , ≤ ) and rk ( K ( A n )) is R ( n ) = n ( n + 1)( n + 2) / n ( n + 1) / n + 23 . Fixing n = 3 m − we have that the size t m of the exceptional sequence (Θ m , ≤ ) :=( { M i : i ∈ [1 , t m ] } , ≤ ) in A m − is t m = 3 m − . rk ( K ( A m − )) = m. rk ( K ( A m − )) + 13 > m. rk ( K ( A m − )) as claimed. (cid:3) HIPOLITO TREFFINGER

Acknowledgements:

The author would like to thank Octavio Mendoza, Corina Saenzand Aran Tattar for their comments and remarks. He also thanks Volodymyr Mazorchukfor pointing out the paper [12] and Gustavo Jasso for providing the arguments that leadto the introduction of Section 2.

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