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SHOD STRING ALGEBRASJennifer Béelanger a & Cecilia Tosar ba Department de Mathématiques , Université de Sherbrooke ,Sherbrooke , Québec , Canadab Departmento de Mathemática IME , Universidade de SãoPaulo , São Paulo , BrazilPublished online: 31 Jan 2013.

To cite this article: Jennifer Béelanger & Cecilia Tosar (2005) SHOD STRING ALGEBRAS,Communications in Algebra, 33:8, 2465-2487, DOI: 10.1081/AGB-200065183

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Couummicarions in Algehra®. 33: 2465- 2487. 2005 Copyri ght © Taylo r & Francis. Inc. ISS N: 0092-7872 print / 1532-4125 o nline

DOl : 10. 108 1/ AGB-200065 183

SHOD STRING ALGEBRAS

Jennifer Belanger

W Taylor & Francis ~ Taylor&..Franc1sGroup

Department de Marh ematiques. Un. iversite de Sherbrooke. Sherbmoke. Quebec, Canada

Cecilia T osar Departmento de Mathem.atica !ME, Universidade de Sclo Paulo, Siio Paulo. Brazil

In this article, we gi•·e a simple combinatorial criterion alloll'ing to ••erW• whether a string algebra is shod 01' not. As a consequence, ll'e classify all algebras ll'ith a discrete deril•ed category which are deril'ed equh·aleut to shod algebras.

1. INTRODUCTION

We consider the artin algebras A, which sat isfy the property that for each indecomposable A-modu le M. either the projective dimension of M is at most one, or the injecti ve dimension of M is at most one. The algebras satisfy ing thi s condition are ca lled shod algebras (Coelho and Lanzilotta , 1999). The class of shod algebras, introd uced by Coelho and Lanzil otta. contains a ll tilted, and more generally, a ll quasi-tilted algebras, a nd has been recently the object of extens ive investigations (see. for instance, Coelho, 200 I; Coelho and Lanzi Iotta , 1999; Kleiner et al., 2002; Reiten and Skowrm1ski , 2003).

The class of quasi-tilted algebras was introduced by Happel , Reiten and Smal¢ in order to give a general tilting theory for abelian categories (Happel et a l. , l996a). Any art in k-algebra A= End 11 (7)"" where '!!C is a loca ll y finite heredita ry abelian k-category with tilting object T, is called a quasi-tilted algebra. The following characterization, also due to Happel , Reiten and Smal¢, has been very useful when dealing with this class of algebras: an a lgeb ra A is quasi-tilted if and o nly it it is shod and of global dimension at most two (Happel et al., 1996a).

We are interested in the problem of cha racterizing the shod algebras in terms of bound quivers. In this art icle, we cons ider shod string algebras. String algebras

Received February I, 2004; Accepted May I, 2004 •communica ted by M. Ferrero. Address correspondence to Jennife r Belanger, Department of Mathematics. Universite de

Sherbrooke. Shcrbrooke. Quebec J IK 2R I, Canada; E-mai l: jenn ifer.be la nger@usherbrooke.ca

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play a n impo rtant ro le in the representation theo ry of a lgebras (see, for insta nce, Butler and Ringel. 1987; W a ld a nd W aschbusch, 1985). Hua rd a nd Liu (2000) have shown that a st ring a lgebra A = KQ / I over a n a lgebra ica ll y closed field k is quasi-tilted if and o nl y if (Q, 1) conta ins no double-ze ro. In o ur case. we give a combinatorial criterion , allowing us to verify whether a string a lgebra is shod o r no t. Our characterization will give the complete classification of the shod string algebras.

We now state our ma in theorem. For the deflnitio ns of consecutive-zeros, of over-lapping zero-relations, a nd of sta rt-tight and end-tight, we refer to subsection 2.3 and sect ion 3.

Theorem 1.1. Let k be an algebraically closed field and let A = kQ/ I be a string algebra. Th en A is shod if and only if (Q, I) satisfies the fo llowing conditions:

( I ) Every consecuti l'e-::ero in ( Q , 1) contains at most two zero-relations. In case there are two, they are overlapping and the path containing these two ::.era-relations is either start-tight or end-tight.

(2) (Q, 1) contains no fu ll subcategorv of the followin g forms or the duals (i') and (ii ' ) of (i) and (ii ) respecti velY.

( i ) 0

-~ . . 0 _____,._ 0 _____,._ 0 _____,._ 0

0

(i i )

0 _____,._ 0 _____,._ 0 _____,._ 0 _____,._ 0

.. ·· ···· 0

( iii ) 0 ... .. .

""" . . . . . . ~ 0 _____,._ 0 -----;,..: 0 _____,._ 0

~- · .

0

Obse rve that the bound quiver (iii ) is self-dual. Since a strictl y shod a lgebra A = KQ / I has global dimension 3, we obtain from

Green et a l. ( 1985) (see also Gauvreau, 1997) that (Q. /) ad mits a path containing two overlapping ze ro-relations. So, as an important consequence of Theorem 1. 1, we obtain tha t if A = kQ/ I is a shod string algebra , then A is strictl y shod if and o nl y if (Q, I) admits a consecutive-ze ro .

In pa rticular, the class of string algebras conta ins the gentle ones. Gentle algebras were introduced by Assem a nd Skowronski ( 1987) in o rder to characterize

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SHOO STRING ALGEBRAS 2467

the a lgebras derived equivalent to a heredita ry a lgebra of type ffi.w Ano ther example of gent le algebras is the class of algebras derived equiva lent to a heredita ry algebra of type IA

11, described by Assem a nd Happel ( 1981). Yossieck (2001) classified the

algebras with di screte derived category and thi s classifi ca tion a lso involves the gentle a lgebras.

As a coro ll a ry to Theorem 1.1 , we present in thi s a rticle a characterization of the shod gentle a lgebras. This cha racterizat io n a llows us to class ify the shod algebras deri ved equi va lent to a hereditary algebra of type IA

11 or ffi.

11 and the shod a lgebras

with discrete derived category not of D ynk in type. The class of a lgebras with di screte deri ved category not of Dynkin type is an exa mple of a class o f a lgebras which are not derived eq uivalent to a quasi-ti lted a lgebra. But many a lgebras in that class are derived equiva lent to shod algeb ra. We end thi s article by showi ng which algebras with discrete derived ca tegory are derived equivalent to shod a lgebras.

Theorem 1.2. Let A be an algebra with discrete deri ved category. The fo llowing conditions are equivalent:

(i) A is derired equivalent to a shod algebra. (i i) A is deri l'ed equivalent to a hereditary algebra of Dynkin type or to A(r. n , m) with

r > 0, n :=:: 3r and m :=:: 2r.

2. PRELIMINARIES

We first establish some terminology and notat ions. Let Q be a finite qui ver. For a n arrow IX of Q, we denote by s(o:) its source a nd by t(a) its target . We associa te to IX its form al inverse o: - 1 wit h source s(a- 1) = t(a) and target t(o:- 1) = s(a). We agree that (o: - 1

) -1 = o: . A path p of length n, with n > 0. from a source a to a target

c is a sequence (a lo: 1, IX2 , . .. , o:11

1c) with IX; an arrow of Q for a ll i, where I :::=:: i :::=:: n, such that s(a 1) =a, t(x,J = c and t(o:;) = s(a;+ 1) for a ll i such tha t I :::=:: i < n. Such a path is briefl y denoted by o: 1a2 · · • IX

11• We denote by Ea the stationary path at a vertex

a. A walk in Q of length n, with n > 0. is a seq uence w = c 1 • • • C11 with c; an arrow or the inverse of a n a rrow such that s(c;+1) = t(c;) for i where I :::=:: i < n and we say that w 1 = c,. · · · c, with I :::=:: r :::=:: t :::=:: n is a subwalk of w. We define s(w) = s(c 1)

and t(w) = t( c,J and we say that w is a walk from s(w) to t(w) . Fina ll y, we define w - 1 = c;;- 1 · · · c1 1. A trivial walk a t a vertex a of Q is the stationary parE".

A wa lk w in Q is called reduced if w is tri vial o r if w = c1 · • • c11

is such that ci+ l i- c;- 1 for a ll i where I :::=:: i < n. A non-trivial reduced wa lk w = c 1 · · · C11 is called a reduced cycle if s(w) = t(w) and c, i- c1 1

•

Let w be a non- tr ivia l reduced walk in Q a nd let w 1, w2 , •.. , w1 be subwalks of w. We say that w1 and w2 point to the same direction in w, if there are pa ths p a nd q of Q such that either w 1 = p a nd w2 = q o r w 1 = p - 1 and w1 = q- 1. Otherwise, we say that w1 and w2 point to opposite directions in w. More gene rall y, we say that w 1 , w1 , ... , w1 point to the same direction in w, if there a re paths, p 1, p2 , ... , p1 of Q such that either w 1 = p 1, w2 = p1 , ... , w1 = p1 o r w 1 = P! 1

, w2 = P21, ••• , wl = P/ 1

-

Let (Q, ! ) be a bo und quiver. A path pin Q is called a zero-path if p belongs to the idea l /. A zero-relation on (Q, /) is a zero-path such that none of its proper subpaths is a ze ro-path . Let w = c 1 • • • c, be a no n-trivia l reduced walk in Q. We say that a subwalk u = c; · · · c;+, is a zero-rel a tio n conta ined in w if u = p o r u = p- 1

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2468 BELANGER AND TOSAR

with p a zero-relation on (Q , 1) . Note that a zero-relation on (Q, 1) may appear many times in a reduced walk. By saying that a reduced cycle w = c 1 • • • c" contains no zero-relation , we mean that , for all i such that I S i S n, there is no ze ro-relation contained in the reduced walk c; · · · C

11c 1 · · · c;_1• A path in Q is a non::.ero path if it

contains no zero-relation. We say that a path pin Q contains rwo overlapping zero­relations if p = p 1 p2p3, where P; are non-trivial nonzero paths such that p 1 p2 and p2p3 are zero-relations.

Throughout thi s article by algebra we mean a basic, connected. finite dimensional algebra over an algebraically closed field k. Thus, an algebra A is isomorphic to kQ / I with ( Q, 1) a finite bound quiver (Gabriel. 1980). We identify finite dimensional right A-modules with finite dimensional representations of (Q, 1). The global dimension of an algebra A is denoted by gl.dim A and for an A­module M. we denote by pdAM and id c~ M the projective dimension and the injective dimension of M, respectivel y.

Finally, we say that an algebra A ~ kQ/ I is triangular, if its ordinary quiver Q contains no oriented cycle.

2.1. Shod Algebras

We recall the definition of shod algebras.

Definition 2.1 (Coelho and Lanzilotta , 1999). An artin algebra A which satisfies the property that. for each indecomposable A-module M. either pd AM s I or id c~M S I , is called a shod algebra (for small homological dimension). We say that an algebra is strictly shod if it is shod but not quasi-tilted.

The following reformulation of a result of Kleiner et a !. (2002) will be very useful.

Theorem 2.2 (Kleiner et a!., 2002, see also Assem and Coelho. 2004). Let A = kQ/ I be a shod algebra, then even · full subcategmy of A is shod.

2.2. String Algebras

We recall the definition of string a lgebras.

Definition 2.3 (But ler and Ringel, 1987). An algebra A is called a st ring algebra if A ~ kQ/ I with (Q. 1) a bound quiver satisfyi ng the following conditions :

(I) The ideal I is generated by a set of paths. (2) Each vertex of Q is the source of at most two arrows and the target of at most

two arrows. (3) For an arrow 7.. there exist at most one arrow /1 and at most one arrow ~ · such

that af3 and }'7. do not belong to I .

In the sequel. whenever we write that A = kQ / I is a string a lgebra, we mean that the bound quiver (Q, I) satisfies the above conditions.

We now give the constructions of string modules and band modules. Let A = kQ/ I be a string a lgebra.

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SHOD STRING ALGEBRAS 2469

A reduced walk in Q is a string if it contains no zero-relations. Let w be a string. We define the string module M(w) as follows. First, if w = E

0 is the trivial

path at a, then M(w) is defined to be the simple A-module S(a) at a. Second, assume that w = c 1 • • • C

11 is a non-trivial string (so that c; is an arrow or the in verse of an

arrow for all i where 1 S i S n). Fori such that I S i S n + 1, we define U; = k and, for i such that 1 s i s n, let U,, be the identity map from U; to Ui+ l if c; is an arrow and otherwise, let U,, be the identity map from Ui+ l to U;. The string module M(w) is defined as follows: for a vertex a in Q, M(wL is the direct sum of all the vector spaces U; such that s(c;) =a together, in case r( cJ =a, with the vector space U

11+1;

if a does not appear in w, M(w) " = 0. For a n arrow 'Y. in Q, if a appears in w, then M( w), is the direct sum of the maps Uc such that c; = a or c; = a- 1

; otherwise M(w), is the ze ro map. '

A reduced cycle w = c 1 • • • C11

in Q is a band if w it is not a power of a reduced cycle of smaller length and none of its powers contains a zero-relation . Let (p be an indecomposable automorphism of a k- vector space V. Fori such that I S iS n, we define V(i) = V. Fori such that I s i s n - l, l et./~ , be the identity map from V(i) to V(i + I) , if c; is an arrow and otherwise, the ide nti ty map from V(i + I) to V(i). We finall y let{,. = cf; : V(n) ~ V(l) , if c, is a n arrow and otherwise./~ = cf; - 1

: V(l) ~ V(n ). The · b~ nd module N(w , rf;) is defined as follows: for a vertex "a in Q, N(w , c/> )a is the direct sum of the vector spaces V(i) such that s(c;) =a; if a does not appear in w, N(w, (P)a = 0. For an arrow a in Q, if a appears in w, then N(w, ¢)a is the direct sum of the maps f> such that c; =a or c; = a - 1

; otherwise N( w , cf; ), is the ze ro map. It is shown i~ ' Butler and Ringel ( 1987) that, if A is a string algebra , then every

indecomposable A-module is either a string mod ule or a band module.

2.3. Consecutive-Zeros

The following definition is pa rticularly useful for o ur classification of the shod string algebras.

Definition 2.4. Let (Q, !) be a bound quiver. Let w be a reduced walk in (Q, !) conta ining at least two zero-relations such that all zero-relations point to the same direction in w. Then any subwalk of w containing at least two zero-relations is called a con secutive-zero.

Note that , for instance, we can get a consecutive-zero from a red uced cycle conta ining exactly one zero-relation. Also, a path containing two overlapping zero-relations is a consecutive-ze ro .

No tice that the definition of consecuti ve-zero is a generalization of that of double-zero (Huard , 1998, see a lso Assem, 1982). Thus, we may reformulate the characterization of quasi-tilted string algebras due to Hua rd and Liu (2000) as follows.

Theorem 2.5 (Huard and Liu , 2000). Let A = kQ/ I be a string algebra. Then, A is quasi-tilted if and only if ( Q, ! ) contains no consecutive-zero.

2.4. Derived Categories

For an a lgebra A, we denote by Db(modA), or briefly by Db(A) , the derived category of bounded complexes of fini te dimensional A-modules. This category has

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the structure of a triangulated category. We say that two a lgebras A and B are derived equivalent if Db(A) and D"( B) are equivalent as tri angulated categories, and we denote it by Db(A) ~ D"(B). We also say that A and Bare rilring-cotilting equ ivalent if there exist a sequence of algebras A= A0 , .. . , A111 =Band a sequence of modules T~il , with 0 .::; i .::; m- I, such that Ai+l = End(T~il) and T

1;i) is either a

ti lting or a cot'ilting Ai-module . It is well-known that if A and ' Bare tilting-cotilting equivalent a lgebras then they are derived equivalent. In general , the conve rse is not true. For details about these notions we refer to Happel (1988).

Now, if A is derived equivalent to B, with B a quasi-tilted algebra, then A and Bare tilting-cotilting equivalent (Happel et al., 1996a). Following Happel et al. ( 1996a), we say that A is a piecewise heredirarv algebra if A is derived equivalent to a quasi-tilted algebra. Also , in Happel et a!. ( 1996a), the a uthors prove that if A is a piecewise hereditary algebra then A is triangular. Happel (200 1) shows that a piecewise hereditary a lgebra is derived equivalent ei ther to a hereditary algebra or to a canonical a lgebra. An a lgebra A derived eq uivalent to a (hereditary) path algebra k 6. is called derived hereditary ol t)'IJe 6. .

3. SHOD STRING ALGEBRAS

In this section , we classify the shod string a lgebras.

lemma 3.1. Let A = kQ/ I be a string algebra. If (Q. !) admits a consecuti ve-zero u containing no overlapping zero-relations, th en A is not shod.

Proof Since u is a consecutive-zero in (Q . ! ), u contains at least two zero- relations. M o reover. u contains no overlapping ze ro-relations. Then , there exists w = z1 v::: 2 , a subwa lk of u such that w contains two zero-relations z1 and z2 and v a string, a nd where z1 a nd ::: 2 point to the sa me direction in w. Without loss of generality, we ma y assume that w = 1. 1 • • • :J.sv/3 1 • · · {31 where :xi and [Jj are arrows for I .::; i .::; s a nd for I .::; j .::; t and where z1 = :x 1 • • • 1._,. and z2 = f3 1 • · · {3 1 are the two ze ro-relations contained in w. Notice that :xsv and vf3

1 are strings, since u contains no overlapping

zero-relations. We cons ider the string module M( v). There is a path ::: 2 such that v{J 1

is a string and vz 2 admits exactly one zero-relation which contains ::: 2 . Then , from Huard and Liu (1999, 2.3), pdAM(v) > I. Moreover, there is a pain ::: 1 such that o:.,v is a st ring and z1 v admits exactly one zero-relation which contains ::: 1 • Hence, from Huard and Liu ( 1999. 2.3) idAM(v) > I. Therefore A is not shod. 0

The following definition will be useful to express the next lemma.

Definition 3.2. Let (Q, !) be a bound quiver. Let p = p 1p2 fJ., be a path in Q containing two overlapping zero-relations p 1 p2 and p2p 3• If p 1 (or p3) is an arrow, we say that p is start-tight (or end-tight. respectively).

Example 3.3 . Let (Q, !) be the following bound quiver :

... ,·.·.··· ... .... .. · ··

0~0~0~0~0~0, Q 1 0:2 0:3 0:4 o:s

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SHOD STRING ALGEBRAS 2471

where a dotted line on a path means that the composition of the arrows on this path is a generator of the idea I.

First , we notice that the path o: 1 o:2e<3o:4 contains two overlapping zero-relations, namely o: 1 e<":x3 and :x2o:3o:4 , and that :x 1 o:2o:3o:4 is start-tight and end-tight. Also, we observe that e<2o: 3o:4 :x5 is start-tight and end-tight. However, the path o: 1o:2o:3o:4 o: 5 contains the overlapping zero- relations o: 1 :x2o: 3 and o:3et.4 o: 5 , so :x 1 o:2o:3o:4 Ct. 5 is neither start-tight nor end-tight.

lemma 3.4. Let A = kQ/ I be a string algebra. If A is shod, then any path in (Q, 1) containing t1vo overlapping zero-relations is either starr-right or end-tight.

Proof Assume that there is a path 7. 1 • • • o: , in (Q, ! ) containing two overlappi ng zero-relat ions , o: 1 • • • :x, and o:, · · · o:, such that I < r S s < t , wh ich is neither start-tight nor end-tight. Then, r > 2 and 1 > s + I. Notice that o:, · · · o:, , o:, · · · o:p,+1 and o:,_17., · · · et., are strings . Consider the string module M(rx, · · · :xJ . There is a path o:,+ 1 • • • o:, such that 7., · · · rxp,+ 1 is a string and such that o:, · · · :xp,+ 1 • • • o:, admits exactly one zero-relation which contains o:,+1 • • • o:,. Then, from Huard and Liu (1999, 2.3), we have pd AM(o:, · · · o:J > l. Moreover, there is a path o: 1 · • • o:,_1 such that et.,_ 1o:, · · · o:, is a string and such that o: 1 • • • o:,_ 17., · · · o:, admits exactly one zero-relation which con tains rx 1 · • · rx,_ 1• Hence, from Huard and Liu ( 1999, 2.3), rd AM(o:, · · · 7., ) > l. Therefore, A is not shod. 0

lemma 3.5. Let A = kQ/ I he a string algebra. Then A is not shod if A contains a full subcategory of the following form (or its dual):

0

~-~~11 · . . ~ 0'1 0:2 . 0:3

·. 0--+- 0--+- 0--+- o.

~ . · ..... 0

Proof Assume there exists a fu ll subcategory B = kQ'/ 1' of A where (Q', ! ') is the bound quiver above. Observe that o:;- 1 /32 and o: 1 /32 are strings in B. Consider the string B-module M(/32). First, there is a path o:2o:3 such that o:} l /3 2 is a string and such that o:_l 1rx] 1 {3 2 admits exactly one zero-relation which contains o:2o: 3. Then , from Hua rd and Liu (1999, 2.3), pd 8 M(f32 ) > I. Moreover, there is a path /3 1a 1 such that aJ32 is a string and such that {1 1 o: 1 {32 admits exactly one zero-relation which contains {3 1et. 1 • Hence, from Huard and Liu (1999, 2.3), id 8 M(fJ2 ) > I. Therefore, B is not shod and it follows from 2.2 that A is not shod. 0

lemma 3.6. Let A = kQ/ I he a string algebra. Then A is not shod il A contains a full subcategory of the following form (or its dual):

. . at . -Cc2 o:s a:4· ·.

0 --+- 0 ____,._ 0 ____,._ 0 ____,._ 0 .

.... .. 0

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Proof. Assume there ex ists a full subcategory B = kQ' / 1' of A, where (Q', ! ' ) is the bound q ui ve r above. Observe that {3 j i'l.2 and {3 ji'X2 'l.3 are strings in B. Consider the st ring B-module M({J j i'l.2) . F irst, there is a path 'l. 3'X4 such that {3j ia2 'l.3 is a string and such that a2cr.3a.J admits exactl y one zero-rela ti o n which contains 'l.3 et.4 . Then, from Huard a nd Liu ( 1999. 2.3), pd 8 M({Jjiet.2) > I. Moreover, there is a non-ze ro path Ct.i such that et.i(Ji a nd Ct.iet. 2 are ze ro paths. Hence. f rom Hua rd and Liu ( 1999, 2.4). id 8 M(fj j i 7.2) > I . Therefore, B is not shod and it fo llows fro m 2.2 that A is not shod. 0

Lemma 3. 7. Let A = kQ/ I be a string algebra. Th en A is not shod if A contains a fill/ subcategory of the fo liO\ ring form:

~· . !31 ~· · ·

...... 0

Proof Assume there ex ists a full subcategory B = kQ' I ! ' of A where ( Q', r) is the bound quiver above. Observe that (3 ji et. 2{3] i is a st ring in B. Consider the string B­mod ule M({J ji et.2{J).i). F irst , there is a nonzero path 7. .1 such that et.2cr. 3 a nd (32'1. 3 are ze ro paths. Then , from Huard a nd Liu ( 1999, 2.4), pd 8 M({Jjiet.2{3). i) > I. Moreover, there is a nonzero path Ct.i such that 'Y.i(Ji and cr.icr.2 are zero paths. Hence, fro m Hua rd and Liu (1999. 2.4) . id 8 M({3 ji 'Y. 2 fj2 i) > I. Therefore, B is not shod and it follows from 2.2 that A is not shod. 0

We now give the proof of the Theorem 1.1 sta ted in the introd uction.

Proof of the Theorem 1.1 . First, suppose that (Q, !) does not sa ti sfy the conditio n ( I). then there ex ists a consecutive-zero u in ( Q, /) containing more tha n two zero-relations o r there exists a consecutive-zero u in (Q, /) contain ing exactl y two zero-relat io ns which a re not overlapping or where the path con ta in in g these two ze ro- relat ions is neither start-tight nor end-tight . Therefo re, in both cases, either u conta ins two zero-relations which are not overlapping o r u contains a pa th having two overl apping zero-relati o ns which is nei ther sta rt-ti ght nor end-tight. Hence, it follows from 3.1 and 3.4 that A is no t shod . Finally. suppose tha t (Q, /) does not sat isfy the cond ition (2), then it follows from 3.5, 3.6 and 3.7 tha t A is not shod.

Conversely, ass ume th a t (Q, I) sa ti sfi es the conditio ns ( I) a nd (2). and suppose to the contra ry that A is not shod. Then , there ex ists an indecomposable A-module M such that pdAM > I a nd id 11 M > I. M oreover, M is either a string module o r a band module (Butler a nd Ringel, 1987).

First , suppose that M=M(w) is a string modu le. Let cv = p j iqi· ·· p ;; iq," where P; and qj are paths that a re no n- trivi a l fo r I < i ::::: n and I ::::: j < n.

(a) First, we conside r the case where Pi is non-tri via l a nd q" is tri via l. I t follows from Huard a nd Li u (2000, 2.1) that one of the following ho lds:

(A) There is a non-trivial path :i = cc; , with Ct. an arrow, such that Pi : i IS a zero pa th. while P i Ct. is a nonzero path;

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SHOD STRING ALGEBRAS 2473

( B) There is a non-trivial path ;.1, with I < j _:::: 11. such that p1z1 and q1_1z1

are both ze ro paths;

(C) There is a zero path Z11+1 = f3z;,+l' with (Jan arrow, such that wzn+l is a reduced walk.

Observe that we can write w = q0p ] 1 · · · p-;; 1

, where q0 is a trivial path. It follows from Huard and Liu (2000, 2.1) that one of the following holds:

(A ' ) There is a zero path Yo = yb-;, with )' an arrow, such that y0w is a reduced walk ;

(B;) There is a non-trivial path Y; with l _:::: i < n, such that Y ;{j; and Y ;P; are zero paths;

(C') There is a non-trivial path "" = y;,h , with han arrow, such that y"p" is a zero path, while hp" is a nonzero path.

• If the cases (A) and (A') hold , then f'Y. is a zero-relation, because p 1 a. is a nonzero path. If y;l is a zero path, then the reduced walk y0 a. contains a consecutive-zero having two zero-relations which are not overlapping. Now, suppose that Yb is a nonzero path, then the reduced wa lk y0::x contains two overlapping zero-rela tions. If z1 is a zero path, then the reduced walk y0 z1 contains a consecutive-zero having two zero­relations which are not overlapping. lf z

1 is a nonzero path, then (Q, I)

contains a full subcategory of the form (i').

• If the cases (A) and (B;) hold , then the reduced walk Y;P;q;_1

1 • • • q] 1 p 1z 1 contains a consecutive-zero having two zero-relations which, from (I), must be overlapping. Hence, i = l. Since (Q, I) satisfies (I ). the path containing these two overlapping zero-relations contained in y 1p 1z 1 is either start-tight or end-tight. Let u 1u2 u3 be this path. If z 1

is a ze ro path, then the reduced walk y 1p 1z1 contains a consecutive-zero having two zero-relations which are not overlapping. Suppose z1 is a non­zero path. First, the path u 1u2u 3 cannot be end-tight, because p 1 ::x is a nonze ro path. Hence, u 1u2 u3 is start-tight and u 1 is an arrow. Let ::x 1 be this arrow. Since ::x 1u2 u3 is contained in Y 1p 1z1, we infer that y 1 is a path with terminal arrow ::x 1• Recall that y 1 q1 is a zero path. Suppose that ::x 1 q 1 is a nonzero path. If y 1 is a zero path, then the reduced walk y 1 p 1 z1 contains a consecutive-zero hav ing two zero-relations, which are not overlapping. If y

1 is a nonzero path, then (Q, I) contains a full subcategory of the form (i).

Now, suppose that :x 1 q1 is a zero path . Since the path :x 1 u2u3 cannot also be end-tight, then (Q, I) contains a full subcategory of the form (ii).

• If the cases (A) and (C') hold, then the reduced walk Y11 fJ 11 (j,~~~ · · · q] 1 p 1 z1 contains a consecutive-zero having two zero-relations which, from (1), must be overlapping. Hence, n =I and y 1p 1z1 contains two overlapping zero-relations, but the path containing these two overlapping zero-relations is neither start-tight nor end-tight, because bp 1

and p 1:x are nonzero paths.

• Suppose that the cases (B) and (A ' ) hold, then the reduced walk YoPI 1 q 1 • • • Pj~ 1 q1_ 1 z1 contains a consecutive-ze ro having two zero-relations which are not overlapping, because j > I .

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2474 BELANGER AND TOSAR

• If the cases (B;) and (B;) hold . If j :S i , then the reduced walk v;p;q;-_1

1 .. • q·;t p1z1 conta ins a co nsecutive-zero having two zero-rel a tions which, fro m ( 1), must be overl apping. Hence, i = j . F rom ( 1), the pa th conta ining these two overla pping zero-rela tions contained in Y;P;Z; is either sta rt-tight o r end-tight. Let u1 u2u3 be this pa th .

• Suppose first tha t u 1u2u3 is sta rt-tight , then u 1 is an a rrow. Let a 1 be thi s a rro w. Recall that -";q; is a zero pa th . Su ppose et. 1 q; is a no n­zero pa th . If Y; is a zero pa th , then the reduced wa lk Y; P;-::; conta ins a consecuti ve-ze ro having two zero- relations which are no t overl a ppin g. If Y; is a no nze ro pa th , then (Q. / ) conta in s a full subca tegory of the fo rm (i). Now, suppose tha t ::t. 1 q; is a zero path. If a 1 u2u 1 is no t a lso end-tight. then (Q. /) conta ins a full subca tegory o f the fo rm (ii ). Suppose tha t a 1u2u 1 is a lso end-tight. then u3 is an a rrow. Let ::t., be thi s a rrow. Reca ll tha t q i-1-::; is a ze ro path . If q,_ 1et., is a ze ro pa th , then (Q. /) co nta ins a full subca tegory of the form (iii ). Suppose tha t q;_1et. , is a nonze ro path . If Z; is a ze ro path, then the reduced wa lk Y;P;-:: ; co nta ins a consecuti ve-ze ro having two ze ro­rela tio ns which a re no t overl apping. If -::; is a nonze ro path , then (Q, /) con ta ins a full subca tegory of the fo rm (i').

Now. suppose tha t u 1u2 u, is end-tight , then u 1 is an arrow. Let et. , be thi s a rrow. Reca ll that qi-1-; i is a ze ro pa th . Suppose that q;_ 1Y. , is a no nze ro pa th . If -:: ; is a zero pa th , then Y;P;-::; conta ins a consecuti ve-zero having two ze ro-re la tio ns which a re no t overla pping. If Z; is a no nze ro path . then (Q. / ) conta ins a full subca tegory of the fo rm (i' ). N ow, suppose tha t qi-J 'Y., is a ze ro pa th . If lt 1u2u., is no t a lso sta rt-tight , then (Q./) conta ins a full subca tegory of the fo rm (ii ' ). Suppose tha t u 1 u 2Y. , is a lso sta rt-tight . Reca ll th a t -";Cf; is a zero pa th. If Y. 1 q; is a zero path, then (Q, /) conta ins a full subca tegory o f the fo rm (iii ). Suppose that Y. 1 q; is a no nze ro pa th. If r ; is a zero pa th , then Y; P;: ; co nta ins a consecutive-ze ro hav ing two zero-re la tio ns which a re no t overl a pping. If Y; is a no nzero pa th , then (Q. /) conta ins a full subcategory of the fo rm (i).

Now, consider the case where j > i. Then the reduced wa lk Y; Cf;Pi). 1 • • • Pj~ 1 q1_1 z J conta ins a co nsecutive-zero having two ze ro-re la tio ns which. from ( I), must be overl apping. Hence, i = j- I. F rom ( I), the pa th conta ining these two overla pping ze ro-rela tio ns contained in _r;q ;-:: ;+J

is either sta rt-tight o r end-tight. Let u 1u2u3 be thi s pa th. Suppose first tha t u 1u2u3 is sta rt-tight. then u 1 is an arrow. Let Y. 1 be

thi s a rrow. Reca ll that Y; P; is a ze ro pa th . Suppose a 1 p; is a no nzero path. If Y; is a ze ro pat h. then the reduced wa lk r ;q ;::. ;+ J conta ins a consecuti ve­zero having two ze ro-rela tio ns which are no t ove rl apping. If Y; is a no nzero pa th , then (Q, ! ) conta ins a full subcatego ry o f the fo rm (i). N ow, suppose tha t u. 1 P; is a zero pa th . If Y. 1 u2u 1 is no t a lso end-tight. then (Q. /) co nta ins a full subca tegory of the fo rm (ii). Suppose tha t a 1 u 2u3 is a lso end-tight , then u3 is an arrow. Let a, be this arrow. Reca ll tha t P ;+ 1z.;+ I is a zero pa th . If P ;+ 1CI., is a zero pa th . then (Q, /)conta ins a full subca tegory of the fo rm (iii ). Suppose that P i+ I u., is a nonzero path . If Z;+ I is a zero pa th , then the reduced walk .)'; q ;Z;+ J conta ins a consecutive-ze ro having two zero-relat io ns

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SHOD STRING ALGEBRAS 2475

which are not overlapping. If Z;+J is a nonzero path, then (Q , !) contains a full subcategory of the form (i' ).

Now, suppose that u 1u2u3 is end-tight. then u3 is an arrow. Let ('/. 1

be this arrow. Recall that P;+ J Z;+ J is a zero path. Suppose that Pi+ I (J. 1 is a nonzero path. [f Z;+J is a zero path , then Y;q;Z;+J contains a consecutive­zero having two ze ro-relations which are not overlapping. [f Z;+J is a nonzero path, then (Q, !) contains a full subca tegory of the form (i'). Now, suppose tha t P;+ 1 (J. , is a zero path . [f u 1 u1(J. , is not also start-tight , then (Q, ! ) contains a full subcategory of the form (ii'). Suppose that u 1 Ll l'J., is also start-tight. Recall that -'";P; is a zero path. If (J.JP; is a zero path, then (Q . ! ) conta ins a full subca tegory of the form (iii). Suppose that (J.JP; is a nonzero path. If r ; is a zero path , then Y;q;Z;+ J contains a consecutive­ze ro having two zero-relations which are not overlapping. If Y; is a nonze ro path, then (Q , !) contains a full subcategory of the form (i).

• If the cases (B) and ( C') hold , then the reduced walk y,p, q,~~ ~ ... q·/ p1z1 contains a consecutive-zero having two zero-relations which, from (1), must be overlapping. Hence, n = j. From ( 1), the path containing these two overlapping zero-relations contained in y, p, z, is either start-tight or end-tight. Let u1u2u3 be this path. [f y, is a zero path , then y,p, z, contains a consecutive-ze ro havi ng two ze ro-rel a tions which are not overlapping. Suppose that v, is a nonzero path. Since bp, is a nonzero path , the path u 1u2u3 cannot be start-tight. Hence, it is end-tight and u3 is an arrow. Let IJ., be this arrow. Recall that q,_ 1 z, is a zero path. If q, _

1(J.

1 is a zero path , then (Q, !) contains a full subcategory of the form

(ii'), beca use bp, is a nonze ro path. Suppose that q, _(J. , is a nonzero path. If ~, is a zero path, then y, p ,z, contains a consecutive-zero having two zero- relations which are not overlapping. If z, is a nonze ro path , then (Q. /) contains a full subcategory of the form (i').

• If the cases (C) and (A ' ) hold , then the reduced walk y0 p j 1 q1 • • • p-;; 1 z,+J contains a consecutive-zero having two zero-relations which are not overlapping.

• lf the cases (C) and ( s;) hold , then the reduced walk Y;q;p-;+1

1 • •• p-;; 1 z,+ 1 contains a consecutive-zero having two ze ro-relations

which are not overlapping, because i < n.

• If the cases (C) and (C') hold , then b/3 is a zero-relation , because c5p, is a nonzero path. If z;,+ l is a zero path, then the reduced walk c5{3 ::_;,+

1 contains a consecutive-zero having two zero-relations which a re

not overlapping. If z;,+1 is a nonze ro path , then bf3z;,+1

contains two overlapping zero-relations. If y, is a zero path, then the reduced walk y,~,+ l conta ins a consecutive-ze ro having two zero-relations which are not overlapping. If y, is nonzero path, then (Q, ! ) contains a full subcategory of the form (i).

(b) The case where p1

is trivial and q, is non-trivial is clearly the dual of the case where p

1 is non-trivial and q, is trivial done in (a).

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2476 BELANGER AND TOSAR

(c) Now, we consider the case where p 1 and q" are both trivial paths. It follows from Huard and Liu (2000, 2.1) that one of the fo llowing holds:

(D) There is a zero path ::: 1 = a::: '1, with rx an a rrow, such that zj 1w is a reduced walk :

(B) There is a non-trivial path ::: j . with I < j.:::: n. such that pjzj and qj_ 1 z.j are both zero paths;

(C) There is a zero path ~~~+ 1 = /3::: ;,+ 1 , with f3 an a rrow, such that W Z 11+1 is a reduced wa lk.

Observe that we can write w = q1 Pi. 1 • · · p;; 1• It fo llows from Huard and Liu

(2000, 2.1 ) that one of the fo llowing ho lds:

(D' ) There is non-trivial path y 1 = v; y, with y an arrow, such that y 1 q1 IS a zero path , while ~·q 1 is a nonzero path;

(B;) There is a non-trivial path Y; with I < i < n. such that .)'; q ; and Y;P; are zero paths;

( C') There is a non-trivial path r " = y;,c5, with c) an arrow, such that Y11 fJ 11 IS a zero path. whi le bp

11 is a nonze ro path.

Clearly, we on ly have to consider the following cases:

• The case where (D) and (D' ) ho ld . We obta in that ~'CJ. is a zero­relation. because vq1 is a nonzero path. If 2; is a zero path , then the reduced walk y-x:::; con tains a consecut ive-zero having two zero-relations which are not overlapping. If ::: '1 is a nonzero path, then yrxz'1 conta ins two overlapping zero­re lat ions. If y1 is a zero path, then the reduced walk y 1 2 1 contains a consecutive­ze ro having two zero-relations which are not overlapping. If y 1 is a nonze ro path , then (Q, ! ) contains a full subcategory of the form (i).

• The case where (D) and (B;) hold. The reduced walk Y; fJ ;q;-_11 ••• p 2q j 1 2 1

contains a consecut ive-zero having two zero-relations which are not overlapping.

• The case where (D) and (C') hold. The reduced wa lk v" p"q'~~ 1 •.. p2qj 1

::: 1 conta ins a consecut ive-ze ro having two zero-relations which are not overlapping.

• The case where (Bj) and (D' ) hold. The reduced walk y 1 q 1 ••• qj- l2j contains a consecutive-zero having two zero-re lat ions wh ich, from (I), must be overlapping. Hence, j = 2. From (I), the path containing these two overlapp ing ze ro-relations contained in y 1q 12 2 is either start-t ight or end-tight. Let u 1u2u3

be this path . If y 1 is a zero path, then the reduced wa lk y 1c1J22 contains a consecutive-zero having tow zero-relat ions which are not overlapping. Suppose y1 is a nonzero path. Since yq1 is a nonzero path , the path u1 u2u3 cannot be sta rt-tight. Hence, it is end-tight and u3 is an arrow. Let 'X1 be this a rrow. Since u1u2 'X 1 is contained in y 1q12 2 , we obtai n that 2 2 is a path with initial arrow r:t

1•

Recall that p 22 2 is a zero path. Suppose that p2 rx 1 is a nonzero path. If ::: 2 is a zero path , then the reduced walk y 1 q1 2 2 contains a consecutive-zero having two zero-relations which are not overlapp ing. If 22 is not a zero path. then (Q, !) contains a full subcategory of the form (i '). Now, suppose that p2rx1 is a zero path.

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SHOD STRING ALGEBRAS 2477

Since the path ui u2u3 cannot also be sta rt-tight, (Q, /) contains a full subcategory of the form (ii ' ).

• The case where (C) and (D' ) hold. The red uced wa lk y iqipz i ... q"p;;i z"+i contains a consecuti ve-ze ro having two ze ro-relations which are not overlapping.

(d) Now, we consider the case where Pi and q" are both non-trivi al paths. It foll ows from Huard and Liu (2000, 2. 1) that one of the fo llowing holds:

(A) There is a non-trivial path :i = az'i · with a an arrow, such tha t pi zi is a zero path, while Pi a is a non-zero path;

(B) There is a non-trivial path : J, wi th I < j ~ 11. such that PJZJ and qJ- i ZJ a re both zero paths;

(£) There is a non-trivial path : "+ i = [J:;,+ i, with fJ a n arrow, such that q":"+i is a zero path, while q"[J is a non-zero path.

Observe that we can write w = q0p[i · · · q11 p;;~ i , where q0 and p 11+ i are both tri vial paths. It follows from Huard a nd Liu (2000, 2.1) that one of the following holds:

(A ' ) There is a ze ro path y0 = v;1i', with ')' a n arrow, such that y0 w is a reduced walk ;

(B;) There is a non-trivial path -''i with I ~ i ~ 11 , such that yiqi and YiPi are zero paths;

(£') There is a zero path y"+i = y;,+ib, wi th <5 an a rrow, such that wy,~~i is a reduced walk.

Clearly. we only have to consider the following cases:

• The case where (A) a nd (£') hold. The reduced wa lk y"+ i q,~ i ... p 1z1

conta ins a consecutive-zero having two zero-relatio ns which a re not overlapping.

• The case where (B) and (£') hold. The reduced wa lk )'11+1 cC 1 ••• PJZJ

conta in s a consecuti ve-zero havi ng two zero-relations which are not overlapping.

• The case where (£) and (A ' ) hold. The reduced wa lk y0p[ 1 .•• q"::."+ l

contains a consecutive-zero having two zero-rela tions which are not overlapping.

• The case where (£) a nd (B;) ho ld. The reduced walk riqi .. . q11 ::. 11+ 1

contains a consecutive-zero hav ing two zero-relations which , from ( I), must be overlapping. Hence, i = 11. From (1), the path containing these two overlapping zero-relations contained in )'11 Cf11 Z 11 +i is either start-tight or end-tight. Let u 1 u1u 3

be this path. If ::."+1 is a zero path , then the reduced walk y" q" z"+l contains a consecutive-zero having two ze ro-relations which are not overlapping. Suppose that z;,+1 is a nonzero path. The path uiu 1 11 3 cannot be end-tight, beca use q,} is a nonzero path. Hence, it is sta rt-ti ght and u 1 is a n arrow. Let a 1 be this arrow. Since a 111 2 u3 is contained in y

11q

11::.

11+

1, we o bta in that )'11 is a pa th with termina l

arrow et. 1• Reca ll that y"p" is a zero path. Suppose that a 1 p11 is not a zero path. If y" is a zero path, then the reduced walk y"q"::."+ l contains a consecutive­zero having two zero-rel ations which are not overlapping. If y" is not a zero

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2478 BELANGER AND TOSA R

path, then (Q, I) conta ins a full subca tegory of the fo rm (i). N ow, suppose that Y. 1p, is a zero pa th. Since u 1u2u3 ca nn ot al so be end-tight , (Q, !) conta ins a full subca tegory of the fo rm ( ii ).

• T he case where (£) a nd (£') ho ld . Then b{J is a zero-relat io n, beca use q, [3 is a nonzero pa th . If y;,+1 is a zero pa th , t hen the reduced wa lk .<+ 1 ~ · [3 conta ins a consecuti ve-ze ro having two zero-rela tio ns which are not ove rl a pping. Suppose tha t _r;,+l is a no nzero pa th , then y;,+1 ~ ~ [3 co nta in s two overla pping zero-relations. If .::;,+1 is a zero path, then the reduced wa lk Y, + 1 z, 1 conta ins a consecutive-ze ro hav ing two zero- rela tions which a re not ove rlapping. If .:: ;,+1 is a no nzero path, then (Q, /)conta in s a full subca tego ry of the fo rm (i').

Fina ll y. suppose tha t M = M(w. ¢ ) is a ba nd module dete rmined by w a nd an indecom posable a uto morphism ¢ of a finit e-di mensio na l k-vecto r space. T he defi niti o n of ba nds im plies tha t w cannot be a n o ri ented cycle. T hen. we ca n wri te w = p j 1 q1 ••• p;; 1 q, where P; and Cf; a re paths which a re non-tri via l fo r I S i S 11

such tha t t(p 1) = t(q ,J . F ro m Kra use ( 199 1) (see a lso Hua rd and Liu , 2000) , we get the fo llowing case:

(B) T here is non-tri via l path .::1, with I S j S n, such tha t p1::.1 a nd q1_1

.::1 are both ze ro paths, and

(B;) There is a non-tri via l pa th Y; with I S i S 11 , such that Y ;Cf; and Y;P; are both ze ro paths; where q1_ 1 = q, if j = I .

C lea rl y. the proof in thi s case is the same as in (a). T his completes the proo f of the theorem . 0

Example 3 .8 . (a) Consider the a lgebra A = kQ/ I where (Q, /) is the bo und qui ve r shown in the Exam ple 3.4. C lea rl y. A is a string a lgebra . No tice that 'Y. 1 'l. ~ Y. 3 Y.4 'Y. 5 is a consecut ive-zero co nta in ing th ree zero- re la tions. T herefo re A is no t shod. since (Q. I) does no t sa ti sfy the conditio n ( I) stated in 1.1.

(b) Let (Q, I) be the fo llowing bo und qui ver:

0

j) .o 0

I . . t . . 0---;..- 0

. ···· ···· / .. -- ~-­/0 0 ~ 0~ 0~ 0

- ~ 0- 0

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SHOD STRING ALGEBRAS 2479

Then, A = kQ/ 1 is a string algebra . Since there exists a consecutive-zero, A is not quasi-tilted. But, (Q, /) satisfies the conditions (I) and (2) as stated in 1.1. Therefore, A is shod.

As an easy consequence of our classification, we also get the class ification of the shod gentle algebras.

Definition 3.9 (Assem and Skowro!'\ski , 1987; Schroer, 1999). Let A = kQ/1 be a string algebra. Then A is called gentle if (Q, /) is a bound quiver satisfying the following conditions:

(I) Every zero-relation is of length two. (2) For an arrow a, there exist at most one arrow (5 and at most one arrow E such

that rJ.(5 and Elf. belong to /.

Example 3.10. Let A = kQ/ 1 where 0 = (Q, I) is the following bound quiver:

0~0~0~0-

Then , A is a gentle algebra.

Corollary 3.11. Let A = kQ/ 1 be a gentle algebra. Then A is shod if and only if every consecutive-::.ero in (Q , /) contains at most two zero-relations, and in case there are two, ther are overlapping and the path containing them is isomorphic to the bound quiver ® in 3.10.

Proof Since A is a string a lgebra. it suffices to verify that (Q, I) satisfies the conditions (I) and (2) as stated in 1.1. Clearly, A satisfies the condition (2), smce every zero-relation is of length two. D

Corollary 3.12. An algebra A = kQ/ 1 is shod derived hereditary of type iA11

if and only if A is gentle, Q is a tree and evety consecutive- zero in (Q, !) contains at most two zero-relations, and in case there are two, they are overlapping and the path containing them is isomorphic to the bound quiver e in. 3.1 0.

Proof This follows from the classification of algebras derived hereditary of type !A

11 given in Assem and Happel (I 981) and Corollary 3.11. D

Corollary 3.13. An algebra A = kQ/ 1 is shod derived hereditary of type 1A11

if and only if A is gentle and (Q, !) satisfies the followin g three condition s:

(i) Q contains a unique (non-oriented) cycle C. (ii) On C, the number of clockwise oriented relations equals the number of

counterclockwise oriented relations. (iii) Every consecutive-zero in (Q, !) contains at most two zero-relations, and in. case

there are two, they are overlapping and the path containing them is isomorphic to the bound quiver ® in 3. 10.

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2480 BELANGE R AND TOSAR

Proof This follows from the class ificat io n of algebras derived heredita ry of type lA

11 given in Assem a nd Skowro nski ( 1987) a nd Corollary 3.11 . 0

4. DISCRETE DERIVED CATEGORY

In this section , we first classify the shod a lgebras with di screte derived category not o f D ynkin type. Then . after stating some previous results , we show

which a lgebras with discrete deri ved category a re deri ved eq uivalent to a shod

a lgebra. Following Yoss ieck (200 1), if A is a finite dimensional k-algebra. D"( A) is said

to be a discrete deri ved categorv if for every vector n = (n;);Ell. of natura l numbers there are only finite ly ma ny isomorphism classes of indeco mposable objects in D"(A) of homology dimension vector n. From the description of the derived ca tego ry of he redita ry a lgebras in Happel ( 1988). it is clear that every a lgebra deri ved hered ita ry o f D ynkin type has d iscrete derived ca tego ry.

Corollary 4.1. Let A = kQ/ I be an algebra. The fo llowing conditions are equivalent:

(a) A is shod with discrete derived category not of Dynkin type. (b) A is gentle and (Q, ! ) satisfies.

(b I ) Q contains a un ique (non-oriented ) cYcle C. (b2) On C, the numbers of clockwise oriented relations and of counterclockll'ise

oriented relations are different. (b3) EverY consecutive--;;ero in (Q, !) contains at most rwo -;era-relations, and in

case there are two, they are overlapping and the path containing them is isomorphic TO the bound quiver 0 in 3. 10.

Proof In Yoss ieck (2001 ), it is proved that a derived category D"(A) is di screte but not of D ynkin type if a nd only if A = kQ/ I is ge ntle, Q conta ins a unique (non-oriented) cycle C and o n C the numbers of clockwise a nd counterclockwise

o riented rel a ti o ns are different. So, the corollary fo llows from this description a nd Coro ll a ry 3.11 . 0

Observe that the bound quive r (Q, I) of a gentle algebra A = kQ/ 1 conta ining a unique cycle has also some branches, which a re joined to the cycle at a single ve rtex. We ca ll such vertex the root of the branch. A quiver of type IA

11• with 11 > 0,

such that all arrows point to the same direction is ca lled an equioriented linear quiver. So, a branch which is an equioriented linear qui ve r oriented toward the cycle has it s root in a sink .

We now give an example of a class of algebras with discrete derived category not of D ynkin type. Denote by D the set {(r, n , m) . r, n . mE 7l. Jn ::: r > 0 and m::: 0}. For each (r, n. m) ED, the a lgebra i\( r, n , m ) is defined as the quotient of a

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SHOD STRING ALGEBRAS 2481

path a lgebra kQ( r, n , m)/ l(r, n, m) , where Q(r, n , m) is the following quiver:

n-1+-·· ·+- n -r+l O:n-2 Ck'71 -r+l

a nd l (r, n, m) is the ideal generated by the paths IX11

_,.er:11

_ ,.+ 1, • . . , er:11

_ 2a11

_ 1, Cr:11

_ 1a0 .

Recently , Bobinski, Geiss and Skowronski obtained the foll owing result (Bobir1ski et a l. , 2004).

Theorem 4.2 (Bobinski et al. , 2004). For an algebra A the following conditions are equivalent :

(i) D" (A) is discrete not of Dynkin tvpe. (ii) D"(A) ~ D"(A(r, n, m)) for some (r, n, m) E fl.

(iii ) A is tilting-cotilting equivalent to A(r, n, m) for some (r, n , m) E fl.

Moreo\'er, for (r, n. m), (r' , n', m') E fl, we have D"(A(r, n, m) ) ~ D"(A( r', n' , m' )) if and only if (r, n , m ) = (r', n' , m' ).

Remark 4.3. As a consequence of the above theorem, if A is an algebra with discrete derived category, then D"(A) ~ D"(B) if a nd only if A and B are tilting­cotilting equivalent. Observe also, that if Db(A ) is discrete not of D ynkin type, then A is not piecewise hereditary. This follows from the fact that every piecewise hereditary a lgebra is triangular, whereas A(r, n , m) is not triangula r for any

(r, 11, m) ED.

The proof of Theorem 4.2 , given in Bo binsk i et a l. (2004), is done in several steps. We use these steps in the proof of our next result, and for convenience, we present them in the following fo ur lemmata. Note tha t in 4.5 and 4.6 we give more specific resu lts than wha t is given in Bobinski et a l. (2004).

Lemma 4.4. Let A = kQ/ I be a genrle algebra containing a unique cycle C. Then, A is tilting-cotilting equivalent to a gentle algebra A' = kQ' / I' with a unique cycle C' such that :

(i) All external branches of C' are equioriented linear quivers oriented toward C'. (ii) C' is obrained from C by adding arrows inside the strings of C.

(iii) All zero-relations are those C.

Proof See Bobinski et al. (2004, Proposi tion 2.3, Steps a) , b), c) and d)) . D

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2482 BELANGER AND TOSAR

lemma 4.5. Let A = kQ/ I be a gentle algebra containing a unique cycle C and having a full subquiver ~ of the form

+ b1 -- ··· -- bt-1 bt e

/ ·· ... %=~ . · · ·· . . ~ .·· ··· ·· ...

a1 -- . .. -- as - 1 ...,....___ as ...,....___ c

t

with the ve rtices c, a" .. . , a0 = b0 , . .. . b, e, on the cycle, the string w with end-points a , and b1 not being a path, s ::=: 1, t ::=: I , and equioriented linear quivers rooted to the cycle in as and h1 of length k ::=: 0 and I ::=: 0, respectively. Then, A is tilting-cotilting equivalent to a gentle algebra A' = kQ1 / I' where (Q' , !') is obtained from (Q , I) by replacing ~ by the following quiver without zero-relations

where i is the number of arrows of the branch rooted to b1 plus the number of arrows in w oriented from a , to bl' and j is the number of arrows of the branch rooted to a, plus the number of arrows in w oriented from b

1 to a ,.

Proof See Bobinski et al. (2004, Proposition 2.3, Step e). lt is not difficult to check the observation on the numbers i and j. D

lemma 4.6. Let A = kQ/ I be a gentle algebra containing a unique cycle C and having a fit// subquiver ~ of the form

t a--;...b-.;...C = Wo --Wl -- ··· --WL- 1 -- d= w1 -.;...eo ~j

.· .... , .. . -· ·

with the vertices a , b, c = w0 , ... , d = w 1 , e0 , f on the cycle, the string w with end­points c and d no/ being a path, l ::=: 2, and an equioriented linear quiver rooted to the

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SHOD STRING ALGEBRAS 2483

in e11 , p 2:. 0. Th en, A is tilting-cotilting equit•alent to a gentle algebra A'= kQ / I' : ( Q', f') is obtained from ( Q, 1) In replacing 2 br The follml'ing quit·er

__,... b ~ c ~vz __,... · · ·...,... Vt+ l __,.. Vt -+-- Vt-l....,... · · · ....,..._ v1-+----- eo --;.- f,

' t 2:_ I is the number of arrows in w oriented from d to c. and I - 1 2:_ I is the er of arrows in w oriented ji·om c to d, and the branch with root <"11 is inserted ' a string of the crclt'.

f See Bobinski et a!. (2004, Propos ition 2.3, Step g). It is not difficult to check bserva tion on the numbers I and 1. D

na 4.7. Ll't A = kQ/ I be a genTle algehra containing a unique crc/e C = .\u, ' s is u clockwise oriented path wiTh r overlapped ;.era-relations and 11 is a string r clock1 1' ise oriented arro\\'s and r counterclockwise oriemed arrml's, such that ' has at most one branch \·l'hich is an equ iorien ted lin ear quiver of length z oriented d C, und the zem-relations of ( Q. !) are those of's. Then , A is tilting-cotilring ~dent to 1\ (r, r + I , x, y + :::).

f See Bobi ti ski et a l. (2004, Proposition 2.3, Step h). D

We no w give the proof of the Theorem 1. 2 sta ted in the introduction.

f of the Theorem 1.2. i) ==? ii) From 4 .2, A is of Dynkin type or D"(A) ~ (r, n, 111)) for some (r, n. 111) En. If A is of Dynkin type, we have nothing to · D"(A) ~ D"(A (r. n, m)). and assum ing there ex ists a shod a lgeb ra B such th a t ) ~ D"(B), we wi ll prove that n 2:. 3r and m 2:. 2r. Notice that B is strictl y shod . If B is not strictly shod , that is, if B is quasi-then A is piecewise hereditary. But this contradicts 4.3.

Then. B ~ kQ/ I where (Q, I) satisfies the conditi ons of 4.1. Moreover, being B a st rictly shod a lgebra. there ex ists a pair of ove rlapping ·elations. From (b2), there exist zero-rela ti o ns on C. From (b3), C = 1j 1 v1 · · · fJ 11 W11 CJ,~ 1 u" where I' ; and q; arc. each of them. a zero-rela ti on or h containing two overlapping zero-relations, w; and v; a re strings , a nd all ·elations on C are those o f P; and Cf; , for I S i S n.

From 4.4. B is tilting-cotilting equiva lent to a gentl e algeb ra 8 1 = kQ 1 I 11

a unique cycle C1 such that all branches a re eq uiori entcd linea r quivers ed toward C1• C1 is obtained from C by adding arrows inside the strings

and all zero-re latio ns of (Q 1 • / 1 ) are the ze ro -relations of C. So, C 1 = rj 1 u'

1 • • · p

11w;,cc 1 v;, with I'; and Cf ; the zero- re lati ons or paths con tai ning two

tpping ze ro-relations from C, fo r all I SiS n. Observe that the branches must J ted in the midpoints of the zero-relat ions.

1. The algebra 8 1 is tilting-cotilting equiva lent to a gentle a lgebra B~ = kQ~ / 11

1 unique cycle C1 = s 111 1 · · · .1,u, with s; a clockwise oriented zero-re lat ion a nd ;tring with at leas t one a rrow clockwise oriented and at least two arrows erclockwise oriented, for I s i ~ r. a ll the branches are eq uioriented linea r t·s oriented toward C2 , and .1 1, •••• • 1·, are a ll the zero-relations in (Q 2 , 12 ) .

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2484 BELANGER AND TOSAR

Applying 4.5 , in a first step the consecutive zero-relations oriented in opposite directions of each pair (p;, qJ (with I :::= i :::= n) are removed. In a second step. the rema ining consecutive zero-relat ions in opposite directions are removed. Then , we obtain r zero-relations, with the same orientation, left over on the cycle. From (b2), we have that r > 0. We can assume that these zero-relations are clockwise oriented. Each of them was a ze ro-relation of some pa ir (p;, q;) such that P; = 'Y.fil' is a path containing two overlapping zero-relations and q; = i5E is a zero-relation. So. when 4.5 is applied to erase the zero-relations fly and i5E. we replace the following full subquiver of (Q 1 , / 1 )

6 b1 -- ··· --bt-l--- bt -+--- e / .

. . • a - b

0 - 0~ . . ·· . ·· __ .::--:._. ···· . .. _

a1 -- · · · -- as - 1 -+---- as -Co...,___ d 'I t {3 t Q

with the vertices d, c0 , a,, ... , a0 = b0 , ... , b,. e on the cycle, s 2: I. t 2: I , k 2: 0, I 2: 0, p 2: 0. by a quiver of the form

Yi ...,___ · · · +-Y1 ~ e

/ z

~ Xi ~ · · · ~ X1 ...,...___Co...,___ d

{3 Q

with i 2: I , j 2: I. Hence, applying systematically 4.5 , the required a lgebra 8 1 = kQ2/ 12 is obtained.

Claim. The algebra 8 2 is tilting-cotilt ing equ iva lent to a gentle algebra 83 = kQ 3/ !3 with a unique cycle C1 = su. where s is a clockwise oriented path of length r + I and with r overlapping zero-relations. a nd u is a string with at least 2r- I arrows clockwise oriented and at least 2r arrows counterclockwise oriented, all the branches

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SHOD STRING ALGEBRAS 2485

are equioriented linear quivers oriented toward C1, and the zero-rela tions of (Q3 , 13)

are those of s. In order to obtain the a lgebra 8 3, 4.6 must be app lied r- I times. We prove,

by induction, that after h app lications of 4 .6, with 0 :=:: h :=:: r- I, we o btain a cycle c;, = s;,+1 u;,+1sil+2u11+2 · · · s,u , where s;,+ l is a clockwise oriented path of length h + 2 and with h + I overlapping zero- relations, u;,+l is a string with at least 2h + I arrows clockwise oriented and at least 2h + 2 arrows counterclockwise oriented, and for h + 2 :=:: i :=:: r, s;, u; are those defined on C2 . If h = 0, that is, before applying 4.6, C~ is just C2 and the statement of the induction is true. Suppose that the statement is true fo r 0 ::5: h- I ::5: r- 2. Then , c;,_1 = s;, u;, sil+ 1 uil+ 1 • • • s,u , and c;,_1 has the following full subqui ver

.. ·· ~-- . ....."! t··. a ___z_ b- C = Wo - WI . . . Wt-I - d = Wz ---* e --;;.- f,

where rx fJ is the last of the h overlapping zero-rel a tions of s;,, the subqui ver between c and d is u;, a nd p5 = sfl+l · Applying 4.6, the last subquiver is replaced by the following one

_.· ···.-:. ·. . 0

a~ b :__; C ...... ;, . .' v1 ~ · · ·---;.- Vt+ l ---;.- Vt ~ Vt-l- ··· - VI - e ---;.- j

with t ::::_ 2h and l - t ::::_ 2h - I. If there is a branch rooted to the cycle in the vert ice e, that branch is inserted into the cycle. So, we obtain c;, = s;,+ 1 u;,+ l sil+2 tt il+2 · · · s,u,.,

where s;,+ l has h + 1 overlapp ing zero-relatio ns. The clockwise o riented arrows of u;,+ l are obta ined fro m the I - t of u;,, b, a nd those of ufi+J· The counterclock wise or iented arrows of u;,+ l are obta ined fro m the t of u;, . a nd those o f ufi+ J· So, u;,+ l has at least (2h - 1) + 2 = 2h + I a rrows clockwise o ri ented and 2h + 2 arrows counterclockwise oriented

Therefore, after applying 4.6, ,. - I times, <- I = c3 = su with the desired cond itio ns.

Fi na ll y, fro m 4.7, 8 3 is tilting-cotilting equivalent to i\(r, n, m), with 11 ::::_

( r + I)+ (2r- I) = 3r and m ::::_ 2r.

ii ) :::::} i) Hereditary a lgebras are shod, so we only have to prove that if n ::::_ 3r and m ::::_ 2r, then A(r, 11, m) is tilting-cotilting eq uivalent to a shod algebra.

Let B = kQ/ I be a gentle a lgebra such that (Q, !) consists of a unique cycle C = p

1 qj 1

• • · p,q,~ 1 u- 1 v where P; is a path conta ining two overlapping zero-relations and Cf; is a zero-relation for I :=:: i :=:: r. u is a path of length m - 2r a nd v is a path of length 11 - 3r, and the ze ro-relat ions of (Q. ! ) are those of P; and q;. C learly. B is a strictly shod a lgebra with discrete derived category not of D ynkin type.

Applying the d iffe ren ts steps as above, we get that B is tilting-cotilting equiva lent to A( r , 11 , m). 0

Example 4 .8.

(a) The a lgeb ra A= kQ/1 given by the following bound qui ve r (Q, /) is strictly shod with di screte derived category not of Dynkin type. In fact, A is derived

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2486 BELANGER AND TOSAR

equiva lent to A(l, 3, 2).

0 ···.·.·. 0 /-r ··-~

0~---~0-0

(b) Let (Q, !) the follow ing bound quiver:

0

/···~ 0 -----.;>- 0-

Then, A = kQ/ I is derived equ ivalent to A( I , 2, I) , so it is not derived equ ivalent to a shod algebra.

ACKNOWLEDGMENTS

This article is part of both authors Ph.D. thesis done respectively in the University of Sherbrooke- under the direction of Ibrahim Assem and Fran~o i s

Huard-and in the University of Sao Paulo-under the direction of Flavio Coelho. The first author gratefu ll y thanks professors Ibrahim Assem , Fran~ois Huard, and Shiping Liu for their helpful comments. The second author acknowledges the financial support by FAPESP during her Ph.D.

REFERENCES

Assem. I. ( 1982). Tilted a lgebras of type lA , . Comm. Algebra I 0:2121 - 2139. Assem. 1. , Happel, D . ( 198 1 ). Generalized tilted algebras o f type lA ,. Comm. Algebra

9(20) :21 01-2125. Assem. 1. , Skowronski. A. ( 1987). Iterated tilted a lgebras of type /A,. Ma1h. Z. 195:269- 290. Assem. 1. , Coelho. F. U. (2004). End omorphism rings of projecti ve mod ules over laura

a lgebras. J. Algebra Appl. 3( I ):49- 60. Bob insk i, G. , Geiss. Ch .. Skowronsk i. A. (2004). Classficati o n of discrete de ri ved categories.

Ce111. Eur. J. Ma1h. 2( I): 19-49. Butler. M. C. R .. Ringel. C. M. ( 1987). Auslander-Reiten seq uences with few middle terms

a nd applications to string algebras. Comm. Algebra 15( I &2): 145-179. Coel ho, F. U. (200 1). Shod algebras. Resenhas IME-USP 5:25-61. Coel ho. F. U.. Lanzi Iotta , M. ( 1999). Algebras with small homological dimensions.

Manuscripla Ma1h . I 00: 1- 1 I. Gabriel. P. ( 1980). Auslcmder-Reilen Sequences and Represe/1/alion-Finile Algebras. Lecture

o tes in Math. Vol. 831. Berlin: Springer. pp. 1-7 1. Gauvreau , C. ( 1997). A new proof of a theo rem of Green. Happel. a nd Zacharia. Ann. Sci.

Ma1h. Quebec 21 ( I ):83-39. Green. E. L.. Happel. D .. Zacharia . D. ( 1985). Project ive resolution s over a rt in algebras with

ze ro relations. Illin ois J. Ma1h. 29(1 ):180- 190. Happel. D. ( 1988) . Triangulaled Ca1egories in I he Representcllion Theon· oj" Finile Dimensional

Algebras. Londo n Math. Soc. Lecture Note Series. 119. Cambridge Univ. press.

Dow

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] at

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SHOO STRI NG ALGEBRAS 2487

Happel, D. (200 I). A characterizat ion of hereditary categories with tilting o bject. lm•ent. Math. 144:38 1-398.

Happel, D .. Rei ten , 1. , Smal¢. S. 0. ( 1996a). Piecewise hereditary a lgebras. Arch. Math. 66:182-186.

Happel , D. , Reiten, 1. . Smal¢, S. 0. ( 1996b). Tilting in Abelian catego ries and quasitilted algebras. Mem . Amer. Math. Soc. 120(575).

Hua rd , F. ( 1998). Tilted gentle a lgebras. Comm. Algebra 26(1):63-72. Hua rd , F. , Liu, S. ( 1999) . Tilted special biseria l a lgebras. J. Algebra 2 17:679- 700. Huard , F. , Liu, S. (2000). Tilted string a lgebras. J. Pure & Appl. A lgebra 153: 15 1- 164. Kleiner, M. , Skowroriski , A. , Zacharia. D. (2002). On endomorphism a lgebras with sma ll

homol ogical dimensions. J. Math. Soc. Japan 54(3):621 - 648. Kra use, H. (199 1). Maps between tree and band modules. J. Algebra 137: 186- 194. Reiten, 1. , Skowronski, A. (2003). Characterizations of algebras wit h small homological

dimensions. Advances in Ma!h. 179:122- 154. Schroer, J. ( 1999). Mod ules wi th o ut self-extensions over gentle a lgebras. J. Algebra

216: 178- 189. Yossieck , D. (200 I). The a lgebras with discre te derived category ./. Algebra 243( I): 168- 176. Wald , B. , Waschbi.isch, J. ( 1985). Tame biseria l a lgebras . ./. Algebra 95:480-500.

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