indecomposables in derived categories of skewed-gentle algebras

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Indecomposables in Derived Categories of Skewed-Gentle AlgebrasViktor Bekkert a c , Eduardo N. Marcos b & Héctor A. Merklen ba Universidade Federal do Rio Grande do Norte , Departamento de Matemática , CCET ,Campus Universitário , Lagoa Nova, Natal – RN, Brazilb Universidade de São Paulo, Instituto de Matemática e Estatística , São Paulo, Brazilc Universidade Federal de Minas Gerais, ICEx , Departamento de Matemática , Av. AntônioCarlos, 6627, CP 702, CEP, 30123-970, Belo Horizonte – MG, BrazilPublished online: 20 Oct 2011.

To cite this article: Viktor Bekkert , Eduardo N. Marcos & Héctor A. Merklen (2003) Indecomposables in Derived Categories ofSkewed-Gentle Algebras, Communications in Algebra, 31:6, 2615-2654, DOI: 10.1081/AGB-120021885

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Indecomposables in Derived Categories ofSkewed-Gentle Algebras

Viktor Bekkert,1,* Eduardo N. Marcos,2 and

Hector A. Merklen2

1Universidade Federal do Rio Grande do Norte,Departamento de Matematica, CCET,Campus Universitario, Lagoa Nova,

Natal – RN, Brazil2Universidade de Sao Paulo, Instituto de Matematica e Estatıstica,

Sao Paulo, Brazil

ABSTRACT

We give a description of the indecomposable objects in the derivedcategory of a finite-dimensional skewed-gentle algebra.

Key Words: Skewed-gentle algebras; Derived categories.

*Correspondence: Viktor Bekkert, Universidade Federal de Minas Gerais, ICEx,Departamento de Matematica, Av. Antonio Carlos, 6627, CP 702, CEP: 30123-970, Belo Horizonte – MG, Brazil; E-mail: [email protected].

COMMUNICATIONS IN ALGEBRA�

Vol. 31, No. 6, pp. 2615–2654, 2003

2615

DOI: 10.1081/AGB-120021885 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com

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1 INTRODUCTION

In Bekkert and Merklen (2000) we determined all the derivedindecomposables of a gentle algebra and we gave an explicitdescription of them by using a matrix problem studied by Bondarenko.Here, we do the same for the case of algebras A whose skew-groupalgebras AG in the case of char k 6¼ 2 are Morita equivalent to gentlealgebras. For this we use the papers of Geiss and de la Pena (1999),Deng (2000), and Bondarenko and Drozd (1977, 1982) for which thegeneral findings of Reiten and Riedtmann (1985) are basicallyneeded.

Let A be a finite-dimensional algebra of the form kQ= hI i over a fieldk, where I is a set of relations for a quiver Q and A-mod is the category offinitely generated left A-modules, and let Db(A) be the bounded derivedcategory of the category A-mod.

The category Db(A) is known only for a few algebras A. For example,the description of indecomposable objects of Db(A) is well-known forhereditary algebras of finite and tame type (Happel, 1988), for tubularalgebras (Happel and Ringel, 1984), for gentle algebras (Bekkert andMerklen, 2000), for derived tame algebras with radical square zero,and for derived tame local and two-point algebras (Bekkert and Drozd,in preparation).

In this paper we give a description of the indecomposable objectsin the derived category when the algebra is skewed-gentle. Thisclass of algebras was introduced in Geiss and de la Pena (1999).We have found that there is a connection between the derived categoryof a skewed-gentle algebra and a matrix problem presented byBondarenko and Drozd (1977, 1982). We show that the problem offinding the indecomposable objects of the derived category may bereduced to finding the indecomposable objects in that matrixproblem.

The structure of the paper is as follows. In Sec. 2 we fix notationsand show that the problem of finding the indecomposables in Db(A)may be reduced to the problem of finding the indecomposables in acategory p(A), a certain subcategory of the category of bounded projec-tive complexes Cb(A-pro). In Sec. 3 we introduce the category ofS-representations of a linearly ordered poset, Bondarenko-Drozd’smatrix problem. In Sec. 4, a functor is defined which will solve ourproblem. In the final section, the description of the indecomposablesof Db(A) is given.

2616 Bekkert, Marcos, and Merklen

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2 PRELIMINARIES

2.1 Derived Representation Type

Let A be a finite-dimensional algebra over a field k and A-mod be thecategory of left finite-dimensional A-modules. We will follow in generalthe notations and terminology of Ringel (1984) and Happel (1988).

Given A, we denote by D(A) (resp., D�(A) or Db(A)) the derived cate-gory of A-mod (resp., the derived category of right bounded complexes ofA-mod or the derived category of bounded complexes of A-mod); by Cb-(A-pro) (resp., C�(A-pro) or C�,b(A-pro)) the category of bounded pro-jective complexes (resp., of right bounded projective complexes or ofright bounded projective complexes with bounded cohomology (that is,complexes of projective modules with the property that the cohomologygroups are non zero only at a finite number of places)); and by Kb(A-pro)(resp.,K�(A-pro) orK�,b(A-pro)) the corresponding homotopy categories.

We identify the homotopy category Kb(A-pro) with the full subcate-gory of perfect complexes in Db(A). Let us recall that a complex is perfectif it is isomorphic to a bounded complex of finitely generated projectiveA-modules.

We will also use the following notations. By p(A) we denote the fullsubcategory of Cb(A-pro) defined by the projective complexes such thatthe image of every differential map is contained in the radical of the cor-responding projective module. Since any projective complex is the sum ofone complex with this property and two complex where, alternatively,all differential maps are 0’s or isomorphisms (which is, hence, isomorphicto the zero object in the derived category) we can always assume that wereduce ourselves to consider projective complexes of this form.

It is well known that Db(A) is equivalent to K�,b(A-pro) (see,for example, Konig and Zimmermann, 1998, Proposition 6.3.1, andHartshorne, 1966).

Proposition 1 (Hartshorne, 1966). D�(A) is equivalent to K�(A-pro). Theimage of Db(A) under this equivalence is K�,b(A-pro).

Given M� 2Db(A), we denote by P�M� the projective resolution of M�

(see Konig and Zimmermann, 1998) and by Hi(M�) the i-th cohomologymodule.

We call a category C basic if it satisfies the following conditions:

� All its objects are pairwise non-isomorphic.� For each object x there are no non-trivial idempotents in C(x, x).

Skewed-Gentle Algebras 2617

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A full subcategory S�C is called a skeleton of C if it is basic andeach object x2C is isomorphic to a direct summand of a (finite) directsum of some objects of S. It is evident that if C is a category with uniquedirect decomposition property, then it has a skeleton and the last one isunique up to isomorphism. We will denote it by SkC and the set of itsobjects by VerC.

In order to simplify our exposition, let us introduce two easy con-structions, as follows.

For P� 2C�,b(A-pro) nCb(A-pro), let s be the maximal number suchthat Ps 6¼ 0 and Hi(P�)¼ 0 for i� s. Then, a(P�)� denotes the brutal trun-cation of P� below s (see Weibel, 1994), i.e. the complex given by

aðP�Þi ¼ Pi; if i � s;

0; otherwise;

�

@iaðP�Þ� ¼

@iP� ; if i � s;

0; otherwise:

�For P� 6¼ 0� 2Cb(A-pro), let t be the maximal number such that Pi¼ 0for i < t. Then, b(P�)� denotes the (good ) truncation of P� below t (seeWeibel, 1994), i.e., the complex given by

bðP�Þi ¼Pi; if i � t;

Ker @tP� ; if i ¼ t� 1;

0; otherwise;

8><>:@ibðP�Þ� ¼

@iP� ; if i � t;

iKer @tP�; if i ¼ t� 1;

0; otherwise;

8><>:where iKer@t

P�is the obvious inclusion.

Lemma 1. Let M� 2K�,b(A-pro) nKb(A-pro) be an indecomposable. Thenb(a(M�)�)� is also indecomposable in Db(A) and

M� ffi P�bðaðM�Þ�Þ� :

Proof. Obvious. &

Lemma 2. There exist skeletons Sk p(A) and SkKb(A-pro) of p(A) andKb(A-pro), respectively, such that Ver p(A)¼VerKb(A-pro).


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Proof. Obvious. &

Let XðAÞ ¼ fM� 2Ver p(A) jP�bðM�Þ� 62Kb(A-pro)g. Let ffiX be the

equivalence relation on the set XðAÞ defined by M�ffiX N� iff P�bðM�Þ� ffi

P�bðN�Þ� in K�,b(A-pro). We use the notation X(A) for a fixed set of repre-

sentatives of the quotient set XðAÞ over the equivalence relation ffiX.From lemmas 1 and 2 we obtain the following

Corollary 1. There exist skeletons SkDb(A) and Sk p(A) of Db(A)and p(A), respectively, such that VerDb(A)¼Ver p(A)[fb(M�)� jM�

2X(A)g.

Remark 1. If A has finite global dimension, we have X(A)¼; andVerDb(A)¼Ver p(A).

Let T be the translation functor D(A)!D(A). By analogy withDrozd (1979, 1986) we will use the following definitions.

Definition 1. Let k be an algebraically closed field and A be a finite-dimen-sional k-algebra. Then

� A is called derived wild if there exists a complex of A-k hx,yi-bimodules M� such that each Mi is free and of finite rank asright k hx,yi-module and such that the functor M�khx,yi-preservesindecomposability and isomorphism classes.

� A is called derived tame (see Geiss and Krause, 2000) if, for eachcohomology dimension vector (di)i2Z, there exist a localizationR¼ k[x]f with respect to some f2 k[x] and a finite number ofbounded complexes of A-R-bimodules C�

1, . . . , C�n such that each

Cij is free and of finite rank as right R-module and such that every

indecomposable X� 2Db(A) with dimHi(X�)¼ di is isomorphic toC�

j � RS for some j and some simple R-module S.� A is called derived discrete (see Vossieck, 2000) if for every co-

homology dimension vector (di)i2Z, we have up to isomorphism afinite number of indecomposables X� 2Db(A) with dimHi(X�)¼ di.

� A is called derived finite if we have a finite number of indecompo-sables

X �1 ; . . . ;X

�n 2 DbðAÞ

such that every indecomposable object X� 2Db(A) is isomorphic toTi(X �

j ) for some i2Z and some j.


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In the case of an arbitrary field k we will say that a k-algebra A isderived wild (resp., derived tame, derived discrete, derived finite) if thek�-algebra A� k k� is derived wild (resp., derived tame, derived discrete,derived finite), where k� is the algebraic closure of k.

2.2 Quivers and Relations

A quiver Q is a tuple (Q0, Q1, s, e), where Q0 is the set of vertices, Q1

is the set of arrows and s, e are functions s, e :Q1!Q0 which determine,resp., the starting and ending vertex of the arrows.

Given two vertices a and b we define Q1[a, b] as the set of all arrowsfrom a to b.

A path p in Q of length l( p)¼ n� 1 is a sequence a1� � �an of arrowssuch that s(aiþ1)¼ e(ai) for 1� i� n� 1. We set s( p)¼ s(a1) ande( p)¼ e(an). The concatenation p1p2 of paths p1 and p2 is defined (inthe natural way) if and only if e( p1)¼ s( p2). Additionally, for everya2Q0, we introduce 1a, a path (of length 0) with s(1a)¼ e(1a)¼ a. Theset of all paths (resp., all paths of length �m) in Q is denoted by P(Q)(resp., P�m(Q)).

Let k be a field. A relation in Q is a non-zero k-linear combination ofpaths of length at least 2 having the same starting vertex and the sameending vertex. A zero relation in Q is a relation of the form w where wis a path. A commutative relation in Q is a relation of the form u� v whereu and v are paths.

If Q is a quiver, then we denote by kQ the corresponding path alge-bra with basis the set of paths in Q. The multiplication is induced fromthe concatenation of paths.

As usual, if I is a set of relations in Q, let (Q, I ) denote kQ= hI i, thepath algebra modulo the ideal generated by the elements in I. Note thatour algebras have a unit element if and only if the set of vertices Q0 isfinite.

We call a path p in Q a path in (Q, I ) if p 62 hI i. The set of all paths(resp., all paths of length �m) in (Q, I ) is denoted by P(Q, I ) (resp.,P�m(Q, I )). Note that if u� v is a commutative relation, then we identifythe paths u and v. It is clear that if I consists of zero and commutativerelations, then the set of elements of the algebra A¼ kQ=hI i which corres-pond the elements of P(Q, I ) is a bases of A. We warn the reader that weare taking only one element for each commutative relation.

A path w¼w1� � �wn of length l(w)� 1 in (Q, I ) is called maximal in(Q, I ), or simply, maximal if uw and wv are not paths in (Q, I ) for eachu, v2Q�1.


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We will denote by M¼M(Q, I ) the set of maximal paths in (Q, I ).It is clear that if kQ=hI i is finite-dimensional, then any path w2

P(Q, I ) is a subpath of a maximal path in (Q, I ).

2.3 Gentle Algebras

Let Q be a quiver and I a set of relation for Q.

Definition 2. The pair (Q, I ) is said to be special biserial (Skowronskiand Waschbusch, 1983) if the following holds:

� At every vertex of Q at most two arrows stop and at most twoarrows start.

� For each arrow b there is at most one arrow a with e(a)¼ s(b) andab 62 I and at most one arrow c with e(b)¼ s(c) and bc 62 I.

The pair (Q, I ) is said to be gentle (Assem and Skowronski, 1987) if, itis special biserial, and moreover the following holds:

� The set I is generated by zero-relations of length 2.� For each arrow b there is at most one arrow a with e(a)¼ s(b) and

ab2 I and at most one arrow c with e(b)¼ s(c) and bc2 I.

A k-algebra A is special-biserial, or gentle, if it is Morita-equivalent toa factor algebra kQ=hI i, where the pair (Q, I ) is special-biserial or gentle,respectively.

2.4 Skewed-Gentle Algebras

We now define some basic notion and fix some notation. Let Q be aquiver with a fixed distinguished set of vertices which we denote by Sp,and I a set of relations for Q. We call the elements of Sp special vertices,the remaining vertices are called ordinary.

For a triple (Q,Sp, I ) let us consider the pair (Qsp, Isp), whereQsp0 :¼Q0,

Qsp1 :¼Q1[fai j i2Spg, s(ai) :¼ e(ai) :¼ i and Isp :¼ I[fa2i j i2Spg.

Definition 3. A triple (Q, Sp, I ) as above is called skewed-gentle if thecorresponding pair (Qsp, Isp) is gentle.

Let (Q, Sp, I ) be a skewed-gentle triple. We associate to each vertexi2Q0 a set, which we will denote by Q0(i), on the following way: If i is anordinary vertex then Q0(i)¼fig, if i is special then Q0(i)¼f(i, �), (i, þ)g.


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(Q sg, I sg) will denoted the pair defined in the following way:

Qsg0 :¼

[i2Q0

Q0ðiÞ;

Qsg1 ½a; b :¼ fða; a; bÞ j a 2 Q1; a 2 Q0ðsðaÞÞ; b 2 Q0ðeðaÞÞg;

Isg :¼X

b2Q0ðsðbÞÞlbða;a;bÞðb;b;gÞ jab 2 I ;a2Q0ðsðaÞÞ;g2Q0ðeðbÞÞ

8<:9=;;

where lb¼� 1 if b¼ (i, �) for some i2Q0, and lb¼ 1 otherwise.Note that the relations in I sg are zero relations or commutation rela-

tions.

Definition 4. A k-algebra A is called skewed-gentle, if it is Morita-equiva-lent to a factor algebra kQsg=hI sgi, where the triple (Q, Sp, I ) is skewed-gentle.

Remark 2. We use signs lb in the definition of I sg for technical reasons.Consider the algebra B¼ kQsg=hfI sgI sgi, where

fIsgIsg :¼X

b2Q0ðsðbÞÞða; a; bÞðb; b; gÞ j ab 2 I ; a 2 Q0ðsðaÞÞ; g 2 Q0ðeðbÞÞ

8<:9=;:

It is easy to see that the algebras A¼ kQsg=hI sgi and B are isomorphic.

Example 1.

where a¼ (2, þ), b¼ 1, g¼ (2, �), a1¼ (b, a, a), a2¼ (b, a, g), b1¼ (a, b, b)and b2¼ (g, b, b).

Example 2.


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where ai¼ (i, �), bi¼ (i, þ), ci¼ (bi, ai, biþ1), di¼ (bi, ai, aiþ1), fi¼(ai, ai, biþ1) and gi¼ (ai, ai, aiþ1).

Let (Q, Sp, I ) be a skewed-gentle triple. Let us consider a group oforder 2, like G¼fe, g j g2¼ eg, with the following left and right actionon A¼ kQsg=hI sgi: g(i, þ) :¼ (i, þ)g :¼ (i, �), g(i, �) :¼ (i, �)g :¼ (i, þ),gj :¼ jg :¼ j for all i2Sp and all j2Q0nSp, g(a, a, b) :¼ (ga, a, b),(a, a, b)g :¼ (a, a, bg), g(uv) :¼ (gu)v and (uv)g :¼ u(vg) for all a, b2Qsg

0 ,a2Q1 and u, v2P(Qsg, I sg). We denote by Aþ the algebra generatedby the elements of the form (a1, a, a2) and 1a3, where ai 6¼ ( j, �), j2Sp.We remark that in general the units in A and in Aþ are different.

Lemma 3. The algebra Aþ is gentle.

Proof. It is easy to see that Aþffi kQ=hJi, where J¼ I n fab j ab2 I,e(a)2Spg. &

We set Mþ(A) :¼Mþ(Qsg, I sg) :¼M(Aþ)¼M(A)\Aþ.

We define a k-linear map E :A!Aþ by the following rule: E((i, þ)) :¼(i, þ), E((i, �)) :¼ (i, þ), E( j ) :¼ j for all i2Sp and all j2Q0nSp, E((a, a,b)) :¼ (E(a), a, E(b)), which extends to paths by the rule: E(uv) :¼ E(u)E(v)for all a, b2Qsg

0 , a2Q1 and u, v2P(Qsg, I sg).

Lemma 4. Let (Q, Sp, I ) be a skewed-gentle triple and w2P�1(Qsg, I sg).

Then

� If s(w) 62Aþ and e(w)2Aþ, then w¼ gE(w).� If s(w)2Aþ and e(w) 62Aþ, then w¼ E(w)g.� If s(w) 62Aþ and e(w) 62Aþ, then w¼ gE(w)g.

Proof. Evident. &

Corollary 2. Let (Q, Sp, I ) be a skewed-gentle triple and w2P�1(Qsg, I sg).

Then the following hold:

� If w2Aþ, then there exist maximal path m¼m(w)2Aþ and pathso(w), w0 2Aþ such that m¼o(w)ww0.

� If s(w) 62Aþ and e(w)2Aþ, then there exist maximal pathm¼m(w)2Aþ and paths o(w), w0 2Aþ such that m¼o(w)gww0.

� If s(w)2Aþ and e(w) 62Aþ, then there exist maximal pathm¼m(w)2Aþ and paths o(w), w0 2Aþ such that m¼o(w)wgw0.

� If s(w) 62Aþ and e(w) 62Aþ, then there exist maximal pathm¼m(w)2Aþ and paths o(w), w0 2Aþ such that m¼o(w)gwgw0.

In each case, paths m(w), o(w) and w0 are uniquely determined by w.


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Proof. By Lemma 3, Aþ is gentle. Hence the first statement followsfrom Bekkert and Merklen (2000). The others statements followfrom first statement and Lemma 4. For example, if s(w) 62Aþ ande(w) 62Aþ, then m(w)¼m(E(w)), o(w)¼o(E(w)) and w0 is the sameas for E(w). &

2.5 Skew-Group Algebras

(Compare with Geiss and de la Pena, 1999, Sec. 4.) Geiss and de laPena define skewed-gentle k-algebras as (non necessarily basic) algebraswhere certain loops – together with their corresponding vertices – are dis-tinguished to allow for naturally determining a group action of a groupof order two. We find preferrable to, say, create two vertices for eachordinary one, and two arrows for each arrow, and to add some naturalrelations, obtaining in this way a basic algebra which is Morita equivalentto the other one.

Let A be a k-algebra, and G a finite group acting on A via k-linearautomorphisms. The skew-group algebra AG is the vector space

Lg2GA[g]

with multiplication induced by

a½gb½h :¼ agðbÞ½gh:

Let (Q, Sp, I ) be a skewed-gentle triple. For a given special(resp., ordinary) vertice i let us denote by Q0[i ] the set fig (resp.,f(i, �), (i, þ)g). Consider the pair (Q g, I g), where Qg

0 :¼S

i2Q0Q0[i ],

Qg1 :¼f(a, þ), (a, �) j a2Q1g,

sðða;ÞÞ:¼ðsðaÞ;Þ; if sðaÞ62Sp;sðaÞ; if sðaÞ2Sp;

(eðða;ÞÞ:¼

ðeðaÞ;Þ; ifeðaÞ62Sp;eðaÞ; if eðaÞ2Sp

(

and I g :¼f(a, þ)(b, þ), (a, �)(b, �) j ab2 I, e(a) 62Spg[ f(a, þ)(b, �),(a, �)(b, þ) j ab2 I, e(a)2Spg. It follows from Geiss and de la Pena(1999) that algebra B :¼ kQ g=hI gi is gentle. Consider the group G¼fe, g j g2¼ eg which acts on B defined by the rule:

gðði;þÞÞ :¼ ði;�Þ; gð jÞ :¼ j; gðða;þÞÞ :¼ ða;�Þ

for all i2Q0 nSp, j2Sp and a2Q1. It follows from Geiss and de la Pena(1999) that the skewed-gentle algebra kQsg=hI sgi is Morita-equivalent tothe skew-group algebra BG in the case of char k 6¼ 2.


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Consider the group G¼fe, g j g2¼ eg which acting on A¼ kQsg=hI sgidefined by the rule:

gðði;þÞÞ :¼ ði;�Þ; gð jÞ :¼ j; gðða; a; bÞÞ :¼ ðgðaÞ; a; gðbÞÞ:

It is easy to see that the skew-group algebra AG is gentle in the caseof char k 6¼ 2.

Example 3.

It is easy to see that B¼ kQ=hI i is a gentle algebra, Morita equivalent toAG, where A is the skewed-gentle algebra of Example 1 (see Subsection2.4).

Example 4.

It is easy to see that B¼ kQ=hI i is a gentle algebra, Morita equivalentto AG, where A is the skewed-gentle algebra of Example 2 (see Subsection2.4).

3 S-REPRESENTATIONS

(cf. Bekkert and Merklen, 2000.)We explain now the strategy we use from now on, and fix some nota-

tion. In the remaining of the paper we consider A a skewed-gentle algebraand we want to describe the indecomposables of p(A). We reduce thisproblem to a solved problem (Bondarenko and Drozd, 1977, 1982). Weproceed as follows.

We consider Y a linearly ordered set and a fixed subset Y�, in thissubset we assume we have defined an involution. We introduce ak-category S(Y, k). This category is useful since the problem of findingthe indecomposables of it can be reduced to a matrix problem solved inBondarenko (1991, 1992) and Deng (2000). In this paper we show that,when A is skewed-gentle, we can construct a linearly ordered set


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Y¼Y(A) and subset Y� with an involution defined on the subset, suchthat the problem of finding the indecomposables of p (A) is equivalentto finding the indecomposables of a certain subcategory of S(Y, k),which we describe completely.

Having described the strategy which we will use we start introducingthe necessary preliminaries.

In this section k is a field (as usual) and Y is a linearly ordered set(may be infinite) provided with an (fixed) involution � on some subsetY��Y (see Bondarenko and Drozd, 1977, 1982). For each x2Y� suchthat x� ¼ x we introduce a new symbol x� and let Y denote the union ofY with the set of all x�. We extend the order to Y assuming that theinequalities x� < y, x < y� and x� < y� (more precisely, those that havemeaning) hold if and only if x < y.

Given two block matrices B and C (not necessarily square blocks),we say that the horizontal partition of B is compatible with the verticalpartition of C if the number of rows in each Bx is equal to the numberof columns of each Cx – so that we can multiply CB by blocks -, andsimilarly we define what it means that the vertical partition of B iscompatible with the horizontal partition of C.

We define next the category S(Y, k).

Definition 5. The objects of S(Y, k) are ( finite) square block matrices,B ¼ By

x ðx; y 2 YÞ, called S-representations of Y or Y-matrices with allthe entries of all the blocks sitting in k, and verifying the following proper-ties. (Notice that we represent the row x of the blocks of B by Bx, and thecolumn x, by Bx. Notice also that some blocks may be empty.)

� The horizontal and vertical partitions of B are compatible.� If x, y2Y� are such that y¼ x�, then all matrices in Bx have the

same number of rows as the matrices in By (and, consequently, allmatrices in Bx have the same number of columns as the matricesin By ).

� B2¼ 0.

A morphism of S(Y, k) from B to C is a block matrix Tyx ðx; y 2 YÞ

with entries in k such that the following are satisfied:

� The horizontal (resp., vertical ) partition of T is compatible with thevertical (resp., horizontal ) partition of B (resp., C ).

� TC¼BT.� If y 6� x, then Ty

x ¼ 0.� If x� ¼ y, then Tx

x ¼Tyy.


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It follows from the definition that T is invertible if and only if alldiagonal blocks Ti

i are invertible.It is clear that S(Y, k) is an additive k-category. It was shown in

Bondarenko and Drozd (1977, 1982) that finding the indecomposablesof S(Y, k) can be reduced to finding the indecomposables of a matrixproblem introduced and solved in Nazarova and Roiter (1973). Presently,we show that, when A is skewed-gentle, finding the indecomposables ofp(A) is equivalent to finding the indecomposables of certain subcategoryof S(Y, k).

3.1 Bunches of Semi-chains

We recall some definitions and results related to the bunches ofsemi-chains considered by Bondarenko (1991, 1992) and Deng (2000)in a form convenient for our purposes (see also Crawley-Boevey,1989, for an alternative approach). We will use the classification ofindecomposables representations of a bunch of semi-chains given inDeng (2000).

Definition 6. A bunch of semi-chains C¼fI, Ei, Fi,�g is defined by the

following data:

1. A set I of indices.2. Two chains (i.e., linearly ordered sets) Ei and Fi given for each i2 I;

Put E :¼Si2IEi, F :¼S

i2IFi and jCj :¼E[F.An involution � on some subset jCj� � jCj.

We consider the ordering on jCj, which is just the union of all order-ings on Ei and Fi (i.e., a < b means that a and b belong to the same chainEi or Fi and a < b in this chain).

For each x2 jCj� such that x� ¼ x we introduce a new symbol x� andlet Ei (resp., Fi) denote the union of Ei (resp., Fi) with the set of all x�for x2Ei (resp., x2Fi). We extend the partial order to Ei and Fi aswe did above for Y, and we put E :¼ S

i2I Ei, F :¼ Si2I Fi and

j C j ¼ E [ F. We extend the partial order to j C j as we did above for jCj.We recall the definition of the category repC of representations of C

(see Bondarenko, 1991, 1992).

Definition 7. The objects of rep C are sets A¼fA(i) j i2 Ig of blockmatrices A(i)¼ (Ay

x) (x 2 Fi; y 2 Ei) with entries in k, and verifying thefollowing properties. (Notice that we represent the row x of the blocks of


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A(i) by Ax, and the column y, by Ay. Notice also that some blocks may beempty.)

� If x� ¼ y 6¼ x, where x, y2F (resp., x, y2E), then the number ofrows in Ax and Ay (resp., the number of columns in Ax and Ay) isthe same.

� If x� ¼ y 6¼ x, where x2F, y2E, then the number of rows in Ax isequal to the number of columns in Ay.

A morphism S :A!B in repC is a set of pairs of block matricesS¼f(S(i, 1), S(i, 2) j i2 I )g with entries in k, where S(i, 1)¼ (Sy

x)(x; y 2 Fi), S(i, 2)¼ (Sy

x) (x; y 2 Ei), such that:

� The horizontal (resp., vertical) partition of S(i, 1) is compatible withthe horizontal partition of A(i) (resp., B(i)).

� The vertical (resp., horizontal) partition of S(i, 2) is compatible withthe vertical partition of B(i) (resp., A(i)).

� S(i, 1)B(i)¼A(i)S(i, 2) for all i2 I.� If y 6� x in Ei (or Fi), then Sy

x ¼ 0.� If x� ¼ y 6¼ x for x, y2 jCj�, then Sx

x ¼Syy.

From now onwards, we suppose that C is complete, i.e., thatjCj� ¼ jCj.

Definition 8. Let C¼fI, Ei , Fi ,�g be a complete bunch of semi-chains.

We give now the following related definitions.

� Words.

– A C-word is a sequence w¼w1w2� � �wm of elements of jCj suchthat wi

�jwiþ1 for all i < m, where, by definition ujv if and only ifu2Ei, v2Fi for some i2 I or vice versa.

– The reverse for a C-word w¼w1w2� � �wm is the sequencew� :¼w�

m� � �w�2w

�1.

– Call a C-word w symmetric if w� ¼w and asymmetric other-wise.

� Periodic Words.

– A periodic C-word is a sequence w¼ (wi)i2Z of elements of jCjsuch that wi

�jwiþ1 and wiþp¼wi for some p > 0 and all i2Z.The smallest p satisfying these conditions is the period of w.

– The reverse for a periodic C-word w¼ (wi)i2Z is the sequencew� :¼ (ui)i2Z such that ui :¼w ��i for all i2Z.


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– The p translates ( p2Z) for a periodicC-wordw¼ (wi)i2Z are thesequences w[p] :¼ (w[p]i)i2Z such that w[p]i :¼wiþp for all i2Z.

– Call a periodic C-word w¼ (wi)i2Z symmetric if w� ¼w[p] forsome p and asymmetric otherwise.

We remark that in general a periodic C-word is not a C-word.

We will consider two equivalence relations on the set of C-words andperiodic C-words, which will be denoted by ffis and by ffir. By definition,ffis is the equivalence relation on the set of all C-words, given byuffisw, u¼w�; and ffi r is the equivalence relation on the set of all perio-dic C-words which identifies each periodic C-word with its translationsand their reverses.

We denote by Irr0 k[x] the set of all irreducible polynomial f (x) 6¼ xover a field k with leading coefficient 1.

We set Ind0 k[x] :¼f f n j f2 Irr0 k[x], n2Ng.We denote by M the set of the following matrices (n� 0):

01n 1n1nþ1 ð1n0ÞT

� �;

ð1n0ÞT 1nþ1

1n 01n

� �;

1nþ1 1nþ1

1nþ1 Jnþ1

� �;

Jnþ1 1nþ1

1nþ1 1nþ1

� �;

1nþ1 ð1n0ÞT01n 1n

� �;

1n 01nð1n0ÞT 1nþ1

� �;

1nþ1 1nþ1

Jnþ1 1nþ1

� �;

1nþ1 Jnþ1

1nþ1 1nþ1

� �;

Ff ðxÞ 1nþ1

1nþ1 1nþ1

� �;

where

Jnþ1 ¼

0 1 0 � 0 0

0 0 1 � 0 0

0 0 0 � 0 0

� � � � � �0 0 0 � 0 1

0 0 0 � 0 0

0BBBBBBB@

1CCCCCCCA and

Ff ðxÞ ¼

0 1 0 0 � 0 0

0 0 1 0 � 0 0

0 0 0 1 � 0 0

� � � � � � �0 0 0 0 � 0 1

at at�1 at�2 at�3 � a2 a1

0BBBBBBB@

1CCCCCCCA;

and where f (x)¼ xt� a1xt�1� � � � � at2 Ind0 k[x], f (x) 6¼ (x� 1)t.


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For each asymmetric C-word w (resp., pair (w, i), where w is asymmetricC-word and i2f0,1g), Deng (2000) constructed some indecom-posable representation of C, which we will denote by T(w) (resp., T(w, i)),and for each pair (w, f (x)) (resp., (w, M)), where w is an asymetric (resp.,symmetric) periodic C-word and f (x) is an indecomposable polynomialfrom Ind0 k[x] (resp.,M is a matrix from the setM), some indecomposablerepresentation of C, which we will denote by T(w, f (x)) (resp., T(w, M))(see Deng, 2000 for details).

A representation of a bunch of semi-chains C, that is isomorphic tosome T(w) (resp., T(w, i), T(w, f (x)) or T(w, M)) will be called anasymmetric string (resp., dimidiate string, asymetric band or dimidiateband ).

We denote by O (resp., Op) the set of all C-words (resp., all periodicC-words).

We denote by Oa (resp., Os) a fixed set of representatives of asym-metric (resp., symmetric) C-words over the equivalence relation ffis andwe denote by Oap (resp., Osp) a fixed set of representatives of asymmetric(resp., symmetric) periodic C-words over the equivalence relation ffir.

Theorem 1. (Deng, 2000; Bondarenko, 1991, 1992). Let C be a completebunch of semi-chains. Then each indecomposable representation of C is astring (asymmetric or dimidiate) or a band (asymmetric or dimidiate).The representations T(d), where

d 2 Oa qOs f0; 1gqOap Ind0 k½x qOsp M;

constitute an exhaustive list of pairwise non-isomorphic indecomposablerepresentations of the bunch of semi-chains C.

4 THE FUNCTOR

Let A¼ kQsg=hI sgi be a skewed-gentle algebra and let (Q, Sp, I ) bethe corresponding skewed-gentle triple (see Sec. 2). Set P :¼P(Qsg, I sg)and P�1 :¼P�1(Q

sg, I sg). To begin with, we will fix a finite projectivecomplex, P�, of length m, with the property that the images of all differ-ential maps are contained in the radical of the corresponding module(in other words, P� 2 p(A))

P� : Pn!@n

� � � !@nþm�2

Pnþm�1 !@nþm�1

Pnþm; n;m 2 Z:

(cf. for all what follows, Bekkert and Merklen, 2000.)


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We denote by Pi the indecomposable projective corresponding tothe vertex i2Qsg

0 and by p(w) the morphism between two indecom-posable projectives corresponding to the path w in (Qsg, I sg). Letus say that in each P j of the complex P�, the indecomposable Pi

appears, di, j times or, simplifying our notations, that Pdi; ji is the com-

ponent of P j envolving the indecomposable Pi. Thus, we can rewriteour complex as

Mt

i¼1

Pdi;ni �!@

n

� � � �!@nþm�2 Mt

i¼1

Pdi;nþm�1

i �!@nþm�1 Mt

i¼1

Pdi;nþm

i :

As it is well known, each morphism between projectives (these beingfinite direct sums of indecomposables) is given by a block matrix, eachblock giving the morphism component that corresponds to each pair ofindecomposables. In other words, each block matrix corresponds to amorphism Pdr; j

r !Pds; jþ1s . And, as we know, the paths w2P, s(w)¼ r,

e(w)¼ s form a basis of the morphisms space, Hom (Pr, Ps), but in ourparticular case of the category p(A) we can assume that only pathsw2P�1 are involved.

(If w is as indicated, it defines the morphism p(w) from Pr to Ps

consisting in multiplication times w on the right: u 7! v¼ uw. Anyhomomorphism from Pr to Ps is associated then to a linear combinationof paths like w.)

Hence, in order to represent our complex, we need to give a matrix,say, X¼ (Xs; jþ1

r; j ) determining the sequence of morphisms @ j, ( j¼ n, . . . ,nþm� 1) which in turn determine our complex. In particular, we haveto represent the family of morphisms p(w) which appear in@ j :P j!P jþ1. To facilitate to remember, it is now convenient that weuse a formal sum

@ j :X

w2P�1

Xw; jpðwÞ;

where Xw, j denotes the matrix block that expresses the ‘‘multiplicities’’ ofthe morphism p(w) in the component corresponding to Pds; jþ1

s of therestriction of @ j to Pdr; j

r . Let us explain this in a detailed way:Fixed the place j the component of @ j going from Pdr; j

r to Pds; jþ1s is

represented by a matrix Xs; jþ1r; j 2Mat(dr ds ; khp(w1), . . . , p(wt)i), where

wi’s are the parallel non trivial paths from r to s and khp(w1), . . . , p(wt)iis the k-vector space with basis fp(w1), . . . , p(wt)g. It is clear that Xs; jþ1

r; j


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can be writing uniquely as

Xs; jþ1r; j ¼

Xt

i¼1

Xwi ; jpðwiÞ;

where Xwi, j2Mat(dr ds ; k).

(It should be kept in mind that our convention is that the indecom-posable projectives appearing in the domain of our @ j, say, correspond torows, whereas the indecomposable appearing in the co-domain (target)correspond to columns.)

The condition @i@iþ1¼ 0 is equivalent to:Xw12P�1;w22P�1:w¼w1w2

Xw1; jXw2; jþ1 ¼ 0 ð1Þ

for all w2P�2 and all j2Z.Now, let us consider a morphism j�:P�!P 0� between two com-

plexes in p(A). At each place, the morphism j� is a homomorphismfrom the projective, P j to the projective P 0 j; that is, a block matrixbetween the direct sums of indecomposable projectives. By representingthe blocks of j j similarly as how we did with the differentials maps, byfw, j, and the blocks of the differential maps of P 0� by X 0

w, j, we musthave that X

w12P;w22P�1:w¼w1w2

fw1;jX 0

w2;j¼

Xw32P�1;w42P:w¼w3w4

Xw3; jfw4; jþ1 ð2Þ

The preceding ideas and Corollary 2 lead us to the definition of ourposetY¼Y(A). It has to be a product of two posets: the first correspond-ing to the paths and the second to the places j. As a matter of fact, weintroduce a poset Yw¼Yw(A) for each path w2Mþ¼Mþ(A). It is theset of the subpaths u of w such that s(u)¼ s(w), ordered by the lenghtof them. Then our definitions are the following.

Y ¼�

_[[w2Mþ

Yw

� Z;

where the first component is an ordered disjoint union and the secondone is the set of the integers and where we order the two productsantilexicographically. (As for this, we assume to have given some linearordering, fixed, to Mþ.) This means that [u, i ] < [v, j ] if and only if i < j


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or (i¼ j and m(u) < m(v)) or (i¼ j,m(u)¼m(v) and l(u) < l(v)) (seeCorollary 2 for the meaning of the notation m(w)). It should be observedthat it is possible that a (trivial) path u belongs to two different paths fromMþ. If it is so, the two occurrences of u must be regarded, obviously,as different.

Next, we indicate how to define the subset Y��Y and the involution� on Y�. For given u, v2 [_ w2MþYw, we state that u� ¼ v if and only ifeither u 6¼ v and e(u)¼ e(v) or u¼ v and e(u)2Sp. Then we state that[u, i ]� ¼ [v, j ] if and only if i¼ j and u� ¼ v.

Next, to facilitate our exposition, let us introduce maps g1 :Y�!Pand g2 :Y�!Z, which we define according to the following rules:

� g1([u, i ]) :¼ u and g1ð½u; iÞ :¼ ug.

� g2([u, i ]) :¼ i and g2ð½u; iÞ :¼ i.

In order to fulfill all our promises, it will be enough to define afunctor, F, from the category p(A) to the category S(Y,k) and showthat it respects and preserves indecomposable objects. This is what wedo next.

In objects,

FðP�Þyx ¼ Xw;g2ðxÞ; if g2ðyÞ ¼ g2ðxÞ þ 1; g1ðyÞ ¼ g1ðxÞw;w 2 P�1;

0; otherwise;

(

where the block F(P�)x (resp., F(P�)x) has dg1(x),g2(x) rows (resp., columns).

In morphisms,

Fðj�Þyx ¼ fw;g2ðxÞ; if g2ðyÞ ¼ g2ðxÞ; g1ðyÞ ¼ g1ðxÞw;w 2 P;0; otherwise:

�

It follows easily that (F(P�))2¼ 0 and that F(j�) is a morphism ofS(Y,k) for all P� 2Ob p(A) and all j� 2Mor p(A).

Example 5. Let (Q, I, Sp) be the skewed-gentle triple of Example 1 (seeSubsection 2.4) and A¼ kQsg=hI sgi be the corresponding skewed-gentlealgebra. We will use the notations of Example 1. Firstly, we look forthe set Mþ of maximal paths in the algebra Aþ. We see that there is onlyone maximal path, so that Mþ¼fa1b1g, and P�1(Q

sg, I sg)¼fa1, a2, b1,b2, a1b1g. Hence, the poset Y� will be

f1b < a1 < a1b1g Z;


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and the involution � will be given by [1b, j ]� ¼ [a1b1, j ] and [a1, j ]

� ¼ [a1, j ].The poset Y� will be

f1b < fa1; a1g < a1b1g Z:

We see that the differential maps correspond to the formal sums

@j ¼ Xa1;jpða1Þ þ Xa2;jpða2Þ þ Xb1;jpðb1Þ þ Xb2;jpðb2Þ þ Xa2b2;jpða2b2Þ:

Now, let us consider the following projective complex P�:

� � � ! 0 ! P1 ¼ Pb !@1

P2 ¼ P2a � Pb � Pg ! 0 ! � � � ;

where

@1 ¼ 2pða1Þ 0 pða1b1Þ pða2Þð Þ:

Then we have Xa1,1¼ (2 0), Xa1b1,1

¼ (1), Xa2,1¼ (1), Xb1,1

¼ 0, Xb2,1¼ 0,

FðP�Þ½a1;2½1b;1 ¼Xa1,1, FðP�Þ½a1b1;2½1b;1 ¼Xa1b1,1

, FðP�Þ½a1;2½1b;1 ¼ Xa2;1 and FðP�Þyx is a

zero or the empty matrix in other cases.

Let U be the full subcategory of S(Y,k) defined by the objects of ImF. We can prove the following lemma which has, clearly, the corollarythat follows it. Remember that we use the symbol Ver to denote the setof objects of a skeleton of a Krull-Schmidt category.

Lemma 5. Let F and U be as above. Then

� KerF¼ 0.� XffiY in U if and only if XffiY in Im F.

Proof. The proof is similar to the proof of the Lemma 3 in Bekkert andMerklen (2000). &

Corollary 3. Ver Im F¼ (VerS(Y, k))\ Im F¼Ver U.

5 DESCRIPTION OF THE INDECOMPOSABLES

In this Section, k is a field and A denotes a finite-dimensional skewed-gentle algebra of the form kQsg=hI sgi, where (Q, Sp, I ) is the correspond-ing skewed-gentle triple (see Subsection 2.4 for the definition). We will


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identify the algebra Aþ with the algebra kQ=hJi, where J¼ I n fab j ab2 I,e(a)2Spg.

5.1 Generalized Strings and Bands

In this Subsection we use the following notations: P :¼P(Q, J )and P�1:¼P�1(Q, J ). Note that this notations are distinct from thesame given in Sec. 4. Given an arrow a of Q, let us denote by a�1

a formal inverse of a, and let us set s(a�1)¼ e(a) and e(a�1)¼ s(a),and let us extend it, as usual, writing (a�1)�1¼ a. For each pathp¼ a1� � �an we define (a1� � �an)�1¼ a�1

n � � �a�11 , s( p�1)¼ e( p) and

e( p�1)¼ s( p).By a walk w (resp., a generalized walk) of length n > 0 we mean a

sequence w1� � �wn where each wi is either of the form p or p�1, p beinga path of length 1 (i.e., an arrow) (resp., a path of length > 0) in(Q, J ) (= in Aþ) and where s(wiþ1)¼ e(wi) for 1� i < n. Again,s(w)¼ s(w1) and e(w)¼ e(wn). As usual, we consider inverses of a walk(resp., generalized walk). It is clear that passage to inverse is an involu-tory transformation.

If we have a closed walk (resp., closed generalized walk), i.e., ithappens that s(w)¼ e(w) we consider also its, rotations, w[ j ], which arethe walks (generalized walks) wjþ1 � � �wnw1 � � �wj ( j¼ 1, . . . , n� 1).

The product (= concatenation) of two walks (resp., generalizedwalks) w¼w1 � � �wn and w0 ¼w0

1 � � �w0n0 is defined as the walk (resp.,

generalized walk) ww0 ¼w1 � � �wnw01 � � �w0

n0 provided that e(wn)¼ s(w01).

We will consider two equivalence relations on the set of generalizedwalks, which will be denoted by ffis and by ffir. By definition, ffis is theequivalence relation on the set of all generalized walks, generated bystablishing that uffisw, u¼w�1; and ffir is the equivalence relation onthe set of all closed generalized walks which identifies each generalizedwalk with its rotations and their inverses.

By definition, a string is a walk w¼w1 � � �wn such that wiþ1 6¼w�1i , for

1� i < n and such that no subword of w or of w�1 is in J. The set of allstrings will be denoted by St.

With GSt let us denote the set of all generalized walks w¼w1� � �wn

satisfying

� If wi, wiþ12P�1 and e(wi) 62Sp, then wiwiþ12 hJ i.� If w�1

i , w�1iþ1 2P�1 and e(wi) 62Sp, then w�1

iþ1w�1i 2 hJ i.

� If wi, w�1iþ1 2P�1 or w�1

i , wiþ12P�1, and e(wi) 62Sp, thenwiwiþ12St.


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We denote by GSt a fixed set of representatives w of GSt over theequivalence relation ffis and all trivial paths, and its elements will be calledgeneralized strings.

Call a nontrivial generalized string w symmetric if w¼w�1 and asym-metric otherwise. Call a trivial generalized string 1i symmetric if i2Spand asymetric otherwise. We denote by GSts the subset of all symmetricgeneralized strings and put GSta :¼GSt nGSts.

Similarly, we define the generalized bands in the following way.Given the generalized walk w¼w1� � �wn, we put, for 1� i� n,

mwð0Þ¼ 0; mwðiÞ¼ mwði�1Þþ1 ðif wi 2P�1Þor mwðiÞ¼ mwði�1Þ�1 ðotherwiseÞ:

After this, let us consider the set GBa of all closed generalized walksw¼w1� � �wn (i.e., e(wn)¼ s(w1)) such that w22GSt, such thatmw(n)¼ mw(0) and such that they are not themselves powers. We useGBa for a fixed set of representatives of the quotient set of GBa overthe equivalence relation ffir and we call its elements generalized bands.Call a generalized band w symmetric if w¼w�1[t] for some t and asym-metric otherwise. We denote by GBas the subset of all symmetric general-ized bands and put GBaa :¼GBa nGBas.

Remark 3. In practice, we assume that mw(0)� mw(n). One is allowed todo this (inverting w if necessary) because if mw(0)� mw(n), thenmw�1(0)� mw�1(n).

We will need the following

Lemma 6. Let w¼w1� � �wn be a generalized walk. Then

mwðiÞ � mw�1ðn� iÞ ¼ mwðnÞ for any 0 � i � n:

Proof. Straightforward. &

We define an order relation on the set GSt by the following rule:u¼ u1 . . . um < v¼ v1 . . . vn if and only if one of the following conditionshold:

� vi¼ ui for 1� i�m and v�1mþ1 2P�1.

� ui¼ vi for 1� i� n and unþ12P�1.


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� vi¼ ui for i < s (it is always true if s¼ 1), us, vs2P�1 andl(o(us)us) < l(o(vs)vs) (see Corollary 2 for the meaning of thenotation o(ui)).

� vi¼ ui for i < s, u�1s , v�1

s 2P�1 and l(o(u�1s )) < l(o(v�1

s )).

5.2 String and Band Complexes

In this subsection we associate to generalized strings and bandscertain finite projective complexes which, as we shall see, give all theindecomposables in the category p(A).

For each i2Q0 we put

PðiÞ :¼ Pi � Pgi; if i 2 Sp;Pi; otherwise.

�We consider the following left and right action of the group

G¼fe,g j g2¼ eg (see Subsection 2.4) on A� pro : gp(w) :¼ p(gw),p(w)g :¼ p(wg) for all w2P(Qsg,I sg).

For given w¼w1� � �wm2GSt and i¼ 1, . . . ,m, we define two matricesG(w, i) and H(w, i) over kG by the following rule:

� Case wi2P�1. Construction of G(w,i):

If s(wi) 62Sp, we set G(w,i) :¼ (e).

If s(wi)2Sp and i¼ 1, we set G(w,i) :¼ (e g)T.

If s(wi)2Sp, i > 1 and wi�12P�1, we set G(w,i) :¼ (e �g) T.

If s(wi)2Sp, i > 1 and w�1i�1 2P�1, we set G(w,i) :¼ (0 g)T

provided w�1i�1 � � �w�1

1 �wi � � �wm andG(w,i) :¼ (e g)T otherwise.

� Case wi2P�1. Construction of H(w, i):

If e(wi) 62Sp, we set H(w, i) :¼ (e).

If e(wi)2Sp, we set H(w, i) :¼ (e 0) provided i < m, w�1iþ1 2P�1

and w�1i � � �w�1

1 �wiþ1� � �wm, and H(w, i) :¼ (e g) otherwise.

� Case w�1i 2P�1. Construction of G(w, i):

If e(wi) 62Sp, we set G(w, i) :¼ (e).

If e(wi)2Sp and i¼m, we set G(w, i) :¼ (e g)T.


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If e(wi)2Sp, i < m and w�1iþ1 2P�1, we set G(w, i) :¼ (e �g)T.

If e(wi)2Sp, i < m and wiþ12P�1, we set G(w, i) :¼ (e 0)T pro-vided w�1

i � � �w�11 > wiþ1� � �wm, and G(w, i) :¼ (e g)T otherwise.

� Case w�1i 2P�1. Construction of H(w, i):

If s(wi) 62Sp, we set H(w, i) :¼ (e).

If s(wi)2Sp, we set H(w, i) :¼ (0 g) provided i > 1, wi�12P�1

and w�1i�1� � �w�1

1 > wi� � �wm, and H(w, i) :¼ (e g) otherwise.

For given i¼ 1, . . . ,m we define a matrix F(w, i), defining a morphismin A-pro, by the following rule:

� F(w, i) :¼G(w, i)p(wi)H(w, i) if wi2P�1.� F(w, i) :¼G(w, i)p(w�1

i )H(w, i) if w�1i 2P�1.

Definition 9.

� For each nontrivial generalized string w¼w1� � �wn let us define aprojective complex P�

w as follows. For each i2Z, let us define theprojective module at the place i by

Piw ¼

Mn

j¼0

dðmwð jÞ; iÞPðcð jÞÞ;

where c( j ) :¼ e(w j) for 1� j� n and c(0)¼ s(w), and where d is theKronecker-delta.And, for each i2Z, let us define the differential map from the mod-ule at i to the module at iþ 1 by @i

w ¼ (@ijk)0� j,k�n, where

@ijk ¼

Fðw; j þ 1Þ; if wjþ1 2 P�1; mwð jÞ ¼ i and k ¼ j þ 1;Fðw; jÞ; if w�1

j 2 P�1; mwð jÞ ¼ i and k ¼ j � 1;0; otherwise:

8<:� For each trivial generalized string 1a let us denote by P

�1athe follow-

ing projective complex

� � � ! 0 ! P0 ¼ Pa!@0

0 ! � � �


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Example 6. Let (Q, I, Sp) be the skewed-gentle triple from Example 1(see Subsection 2.4) and A¼ kQsg= h I sgi the corresponding skewed-gentle algebra. We will use for A the notations from Example 1. As atypical example, we consider the generalized string

w ¼ w1w2 � � �w6 ¼ ðabÞðaÞðaÞ�1ðbÞ�1ðbÞðaÞ;

which is visualized by the following diagram:

The corresponding projective complex P�w is then visualized by the fol-

lowing diagram:

whereF(w, 1)¼ p(a1b1),F(w, 2)¼ ( p(a1)0),F(w, 3)¼ ( p(a1) p(a2)),F(w, 4)¼( p(b1) 0)

T, F(w, 5)¼ ( p(b1) p(b2))T and F(w, 6)¼ ( p(a1) p(a2)).

It follows from Subsection 5.2 that, if w is an asymmetric generalizedstring, then the projective complex P�

w is indecomposable. Projectivecomplexes isomorphic to such a P�

w will be called asymmetric strings. Ifw is a symmetric generalized string, then the projective complex P�

w

decomposes into the direct sum of two indecomposables projective com-plexes P�

w;0 and P�w;1. Projective complexes isomorphic to a P�

w;i (i¼ 0, 1)will be called dimidiate strings.

Next, we consider the case of generalized asymmetric bands.


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Definition 10. For each asymmetric generalized band w¼w1� � �wn andeach f (x)2 Ind0 k[x] we define a projective complex P�

w;f as follows.First, we consider the matrices G(u,i), H(u,i) and F(u,i) which corres-

pond to the generalized string

u ¼ u1 � � � ununþ1 � � � u2nu2nþ1 � � � u3n ¼ w1 � � �wnw1 � � �wnw1 � � �wn:

For each i2Z, let

Piw;f ¼

Mn�1

j¼0

dðmwð jÞ; iÞPðcð jÞÞ �k kdeg f ðxÞ;

and, also, for each i2Z, @iw;f ¼ (@i

jk)0� j,k�n�1, where

@ijk ¼

Fðu; j þ nþ 1Þ � 1deg f ðxÞ;if wjþ1 2 P�1; mwð jÞ ¼ i and k ¼ j þ 1;

Fðu; j þ nÞ � 1deg f ðxÞ;

if w�1j 2 P�1; mwð jÞ ¼ i and k ¼ j � 1;

ðGðu; 2nÞpðwnÞHðu; nÞÞ � Ff ðxÞ;if wn 2 P�1; mwð jÞ ¼ i; j ¼ n� 1 and k ¼ 0;

ðGðu; nÞpðw�1n ÞHðu; 2nÞÞ � Ff ðxÞ;

if w�1n 2 P�1; mwð jÞ ¼ i; j ¼ 0 and k ¼ n� 1;

0; otherwise:

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:Example 7. Let (Q, I,Sp) be the skewed-gentle triple from Example 1(see Subsection 2.4), k the field Z2 and A¼ kQsg=hI sgi the correspond-ing skewed-gentle algebra. We will use for A the notations fromExample 1. As a typical example, we consider the asymmetric general-ized band

w ¼ w1 � � �w8 ¼ ðbÞðaÞðbÞðaÞðaÞ�1ðabÞ�1ðabÞ�1ðbÞ�1;



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Given the indecomposable polynomial f (x)¼ x2þ xþ 1, the cor-responding projective complex P�

w;f is then visualized by the followingdiagram:

where P¼P2a �P2

g and

M1 ¼ pðb1Þ 0 pðb2Þ 0

0 pðb1Þ 0 pðb2Þ� �T

;

M2 ¼ pða1Þ 0 pða2Þ 0

0 pða1Þ 0 pða2Þ� �

;

M3 ¼ pðb1Þ 0 �pðb2Þ 0

0 pðb1Þ 0 �pðb2Þ� �T

;

M4 ¼ pða1Þ 0 pða2Þ 0

0 pða1Þ 0 pða2Þ� �

;

M5 ¼ 0 0 pða2Þ 0

0 0 0 pða2Þ� �

;

M6 ¼ pða1b1Þ 0

0 pða1b1Þ� �

;

M7 ¼ pða1b1Þ 0

0 pða1b1Þ� �

;

M8 ¼ 0 pðb1Þ 0 0

pðb1Þ pðb1Þ 0 0

� �T

:

It follows from Subsection 5.2 that if w is an asymmetric generalizedband and f (x)2 Ind0 k[x] , then the projective complex P�

w;f ðxÞ is indecom-posable. Projective complexes isomorphic to such a P�

w;f ðxÞ will be calledasymmetric bands.

Finally, we consider the case of a generalized symmetric bandw¼w1� � �wn. It is easy to see that w then has the form

w ¼ w1 � � �wn ¼ a1 � � � ara�1r � � � a�1

1 b1 � � � bsb�1s � � � b�1

1 ;

where e(ar), e(bs)2Sp.


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Definition 11. For each symmetric generalized band

w ¼ w1 � � �wn ¼ a1 � � � ara�1r � � � a�1

1 b1 � � � bsb�1s � � � b�1

1 ;

where e(ar),e(bs)2Sp, and each matrix (see Subsection 3.1)

M2M; M ¼ A CB D

� �2 Mat ðk; ðl þ l0Þ ðmþm0ÞÞ;

we define the projective complex P�w;M as follows.

First we consider the matrices G(u,i), H(u,i) and F(u,i) which corres-pond to the generalized string

u ¼ u1 � � � ununþ1 � � � u2nu2nþ1 � � � u3n ¼ w1 � � �wnw1 � � �wnw1 � � �wn:

For each 0� j< n we set

dð jÞ :¼l; if 0 � j � r� 1;l0; if rþ 1 � j � 2r;m0; if 2rþ 1 � j � 2rþ s� 1;m; if 2rþ sþ 1 � j � n� 1:

8><>:For each i2Z, let Pi :¼Ln�1

j¼0P(i, j ), where

� P(i, j ) :¼ d(mw( j ),i)kd( j )� kP(c( j )) if j 62 fr, 2rþ sg.

� P(i, j ) :¼ d(mw(r), i)((kl�Pc(r))� (kl

0 �kPgc(r))) if j¼ r.� P(i, j ) :¼ d(mw(2rþ s),i)((km

0 �Pc(2rþs))� (km�kPgc(2rþs)))if j¼ 2rþ s.

and, also, for each i2Z, @iw; f ¼ (@i

jk)0� j,k�n�1, where

� @ijk :¼ 1d( j )�F(u, nþ jþ 1) if w jþ12P�1, mw( j )¼ i, k¼ jþ 1

and j 62 fr� 1, r, 2r, 2rþ s� 1, 2rþ sg.� @i

jk :¼ 1d( j )�F(u, nþ j ) if w�1j 2P�1, mw( j )¼ i, k¼ j� 1 and

j 62 fr, rþ 1, 2rþ 1, 2rþ s, 2rþ sþ 1g.� @i

jk :¼ (1l�G(u, nþ r)p(wr) 0)if wr2P�1, mw( j )¼ i, j¼ r� 1 and k¼ jþ 1.

� @ijk :¼ (1l� p(w�1

r )H(u, nþ r) 0)T

if w�1r 2P�1, mw( j )¼ i, j¼ r and k¼ j� 1.

� @ijk :¼ (0 1l0 � p(gwrþ1)H(u, nþ rþ 1))T

if wrþ12P�1, mw( j )¼ i, j¼ r and k¼ jþ 1.


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� @ijk :¼ (0 1l0 �G(u, nþ rþ 1)p(w�1

rþ1g))if w�1

rþ1 2P�1, mw( j )¼ i, j¼ rþ 1 and k¼ j� 1.� @i

jk :¼ (1m0 �G(u, nþ 2rþ s)p(w2rþs) 0)if w2rþs 2P�1, mw( j )¼ i, j¼ 2rþ s� 1 and k¼ jþ 1.

� @ijk :¼ (1m0 � p(w�1

2rþs)H(u, nþ 2rþ s) 0)T

if w�12rþs 2P�1, mw( j )¼ i, j¼ 2rþ s and k¼ j� 1.

� @ijk :¼ (0 1m� p(gw2rþsþ1)H(u, nþ 2rþ sþ 1))T

if w2rþsþ12P�1, mw( j )¼ i, j¼ 2rþ s and k¼ jþ 1.� @i

jk :¼ (0 1m�G(u, nþ 2rþ sþ 1)p(w�12rþsþ1g))

if w�12rþsþ1 2P�1, mw( j )¼ i, j¼ 2rþ sþ 1 and k¼ j� 1.

� @ijk :¼ (1l�G(u, n))Ap(b1)(1m�H(u, nþ 2rþ 1))if b12P�1, mw( j )¼ i, j¼ 0 and k¼ n� 1.

� @ijk :¼ (1m�G(u, nþ 2rþ 1))ATp(b�1

1 )(1l�H(u, n))if b�1

1 2P�1, mw( j )¼ i, j¼ n� 1 and k¼ 0.� @i

jk :¼ (1l�G(u, n))Cp(b1)(1m0 �H(u,2n))if b12P�1, mw( j )¼ i, j¼ 0 and k¼ 2rþ 1.

� @ijk :¼ (1m0 �G(u,2n))CTp(b�1

1 )(1l�H(u, n))if b�1

1 2P�1, mw( j )¼ i, j¼ 2rþ 1 and k¼ 0.� @i

jk :¼ (1l0 �G(u, nþ 2rþ 1))Bp(b1)(1m�H(u, nþ 2rþ 1))if b12P�1, mw( j )¼ i, j¼ 2r and k¼ n� 1.

� @ijk :¼ (1m�G(u, nþ 2rþ 1))BTp(b�1

1 )(1l0 �H(u, nþ 2rþ 1))if b�1

1 2P�1, mw( j )¼ i, j¼ n� 1 and k¼ 2r.� @i

jk :¼ (1l0 �G(u, nþ 2rþ 1))Dp(b1)(1m0 �H(u, 2n))if b12P�1, mw( j )¼ i, j¼ 2r and k¼ 2rþ 1.

� @ijk :¼ (1m0 �G(u,2n))DTp(b�1

1 )(1l0 �H(u, nþ 2rþ 1))if b�1

1 2P�1, mw( j )¼ i, j¼ 2rþ 1 and k¼ 2r.� @i

jk :¼ 0 in the others cases.

Example 8. Let (Q, I, Sp) be the skewed-gentle triple of Example 1 (seeSubsection 2.4) and A¼ kQsg=hI sgi be the corresponding skewed-gentlealgebra. We will use for A the notations from Example 1. As a typicalexample, we consider the symmetric generalize band

w ¼ w1 � � �w6 ¼ ðaÞðaÞ�1ðabÞ�1ðbÞ�1ðbÞðabÞ;



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Given a matrix

M 2 M; M ¼ A CB D

� �2 Mat ðk; ðl þ l0Þ ðmþm0ÞÞ;

the corresponding projective complex P�w;M is then visualized by the fol-

lowing diagram:

where M1¼ATp(a1b1), M2¼BTp(a1b1), M3¼CTp(a1b1), M4¼DTp(a1b1).

It follows from Subsection 5.2 that if w is a symmetric generalizedband and M2M (see Subsection 3.1), then the projective complexP�w;M is indecomposable. Projective complexes isomorphic to such a

P�w;M will be called dimidiate bands.

5.3 Bunch of Semi-chains C(A)

In this subsection we associate to a skewed-gentle algebra A a bunchof semi-chains.

Initially, we recall some definitions and notations from Bondarenkoand Drozd (1977, 1982) in a form convenient for our purposes.

We set

Yð1Þ :¼ Y nY�; Yð2Þ :¼ fx 2 Y� j x� ¼ xg;Yð2Þ :¼ f�xx j x 2 Yð2Þg; Yð3Þ :¼ fx 2 Y� j x < x�g;Yð4Þ :¼ fx 2 Y� j x > x�g; YwðiÞ :¼ fg1ðxÞ j x 2 YðiÞg \Yw;

where g1 is as in Subsection 4.


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Definition 12. We associate to the given skewed-gentle algebra A a com-plete bunch of semi-chains C¼C(A) :¼ fI,Ei,Fi,

�g, where

� I¼ (Mþ Z)[ (Y(1) f� ,þg).� F(w,i)¼fu½i1�, v½i1�, v½i0� j u2Yw(1),v2[ 4

j¼2Yw( j )g.� E(w,i)¼fu½i þ 11þ, v½i þ 11þ, v½i þ 10þ j u2Yw(1),v2[4

j¼2 Yw( j )g.� Fx[ j ],�¼fx½ j0�g, Fx[ j ],þ¼;.� Ex[ j ],þ¼fx½ j0þg, Ex[ j ],�¼;.� u½ij� < v½ik� if and only if either u < v or u¼ v and j > k.� u½ijþ < v½ikþ if and only if either u < v or u¼ v and j < k.� (u½i1þ)� ¼ ðu½i�Þ1þ, ðu½i1�)� ¼ ðu½i�Þ1� and (u½i0þ)� ¼ ðu½i�Þ0�

for all u[i ]2Y�.� (u½i1�)� ¼ u½i0� and (u½i1þ)� ¼ u½i0þ for all u[i ]2Y(1).

Let us define a map f : jCðAÞj ! Y N by the following rules:

f ðx½i1þÞ ¼ðx½i; 2Þ; if x½i 2 Yð1Þ;ðx½i; 3Þ; if x½i 2 Yð2Þ;ðx½i; 4Þ; if x½i 2 Yð3Þ [Yð4Þ;

8<:f ðx½i0þÞ ¼

ðx½i; 2Þ; if x½i 2 Yð2Þ [Yð4Þ;ðx½i; 3Þ; if x½i 2 Yð3Þ;

�f ðx½i0�Þ ¼

ðx½i; 2Þ; if x½i 2 Yð2Þ [Yð3Þ;ðx½i; 3Þ; if x½i 2 Yð4Þ;

�f ðx½i1�Þ ¼ ðx½i; 1Þ; f ðx½i1þÞ ¼ ðx½i; 3Þ; f ðx½i1�Þ ¼ ðx½i; 1Þ:

Now, if T is an indecomposable representation of the bunch of semi-chains C(A), we define an Y-matrix B¼B(T) as follows.

The horizontal and vertical bands of B will be partitioned in a com-patible way into narrower bands Bxk and Bxk, where x2Y and k assumesthe following values: k¼ 1, 2 if x2Y(1), k¼ 1, 2, 3 if x2Y(2)[Y (2) andk¼ 1, 2, 3, 4 if x2Y(3)[Y(4).

The blocks Bysxk are defined as follows:

Bysxk ¼ Tt

r ; if ðx; kÞ ¼ f ðrÞ and ðy; sÞ ¼ f ðtÞ;Bysxk ¼ �Bys

xk; if x 2 Yð2Þ;Bysxk ¼ Bys

xk; if y 2 Yð2Þ; andBysxk ¼ 0 otherwise;

where the sizes of zero blocks are the smallest such that conditions in thedefinition of S-representations are satisfied.


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We define a map l : jC(A)j!P by

lðx½ibaÞ :¼ x

for any a2f� ,þg and any b2f0,1g.

Definition 13. We denote by Oim the set of all C(A)-words w¼w1w2� � �wn

such that the following conditions hold:

� n� 2.� If w12E, then l(w1)[1]2Y(1).� If wn

� 2E, then l(wn)[1]2Y(1).� If wi2E, then l(wi�1

� ) < l(wi), where 1 < i� n.� If wi2F, then l(wi�1

� ) > l(wi), where 1 < i� n.

We set Oima :¼Oim\Oa and Oim

s :¼Oim\Os.

Definition 14. We denote by Oimp the set of all periodic C(A)-words

w¼ (wi)i2Z such that the following conditions hold:

� If wi2E, then l(wi�1� ) < l(wi) for all i2Z.

� If wi2F, then l(wi�1� ) > l(wi) for all i2Z.

We set Oimap :¼Oim

p \Oap and Oimsp :¼Oim

p \Osp.

The next theorem follows from Theorem 3 in Bondarenko and Drozd(1977, 1982) and Theorem 1.

Theorem 2. The Y-matrices B(T(d)), where

d 2 Oima qOim

s f0; 1gqOimap Ind0 k½xqOim

sp M;

constitute an exhaustive list of pairwise non-isomorphic non-zero indecom-posable Y-matrices in ImF.

For a given generalized walk u¼ u1� � �un in GSt and m2Z, we definea sequence t((u,m)) :¼w1� � �wnþ1, where wi2 j Cj, by putting

� wi :¼o(ui�1)ui�1½muði � 1Þ þm0þif 1 < i < nþ 1 and ui�1, ui2P�1.

� wi :¼o(u�1i�1)½muði � 1Þ þm0�

if 1 < i < nþ 1 and u�1i�1,u

�1i 2P�1.

� wi :¼o(u�1i�1)½muði � 1Þ þm1�

if 1 < i < nþ 1 and u�1i�1, ui2P�1.


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� wi :¼o(ui�1)ui�1½muði � 1Þ þm1þif 1 < i < nþ 1 and ui�1,u

�1i 2P�1.

� wi :¼ (o(u1))�½m1�if i¼ 1, u12P�1 and o(u1)[1]2Y�.

� wi :¼ (o(u�11 )u�1

1 )�½m0�if i¼ 1,u�1

1 2P�1 and o(u�11 )u�1

1 [1]2Y�.� wi :¼o(u1)½m0�

if i¼ 1, u12P�1 and o(u1)[1]2Y(1).� wi :¼o(u�1

1 )u�11 ½m0þ

if i¼ 1, u�11 2P�1 and o(u�1

1 )u�11 [1]2Y(1).

� wi :¼o(un)un½muðnÞ þm0þif i¼ nþ 1, un2P�1 and o(un)un[1]2Y�.

� wi :¼o(u�1n )½muðnÞ þm1�

if i¼ nþ 1, u�1n 2P�1 and o(u�1

n )[1]2Y�.� wi :¼o(un)un½muðnÞ þm1þ

if i¼ nþ 1, un2P�1 and o(un)un[1]2Y(1).

� wi :¼o(u�1n )½muðnÞ þm1�

if i¼ nþ 1,u�1n 2P�1 and o(u�1

n )[1]2Y(1).

And, for a given closed generalized walk u¼ u1� � �un in GBaand m2Z, we define a Z-sequence tc((u,m)) :¼ (vi)i2Z, where vi2 jCj,by putting vlnþi :¼wi for any l2N and any 0 < i� n, where wi are asabove.

Lemma 7. Let u¼ u1� � �un2 GSt. Then

1. t((u, m)) is a C(A)-word for any m2Z.2. If u 2 GBa, then tc((u, m)) is a periodic C(A)-word for any m2Z.3. t((u�1,m))¼ (t((u,mþ mu(n))))� and tc((v

�1,m))¼(tc((v,mþ mv(n))))� for any u2GSt, v2GBa and any m2Z;

4. tc((u[i ],m))¼ t((u, mþ mu(i)))[i ] for any u2GBa and any m2Z.5. If u2GSt is symmetric (resp., asymmetric), then t((u, m)) is sym-

metric (resp., asymmetric) for any m2Z.6. If u2GBa is symmetric (resp., asymmetric), then tc((u, m)) is sym-

metric (resp., asymmetric) for any m2Z.

Proof. 1. Let t((u,m))¼w1� � �wnþ1. We are going to show that wi�jwiþ1

for any 1� i� n. We distinguish the following cases.

(a) i¼ 1, u12P�1 and o(u1)[1]2Y�.Then w1¼ (o(u1))�½m1�, w1

� ¼o(u1)½m1� and w2¼o(u1)u1½mþ 1sþ forsome s2f0,1g. Therefore w1

� 2F(m(u1),m) and w22E(m(u1),m), hence w1�jw2.


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(b) i¼ 1, u�11 2P�1 and o(u�1

1 )u�11 [1]2Y�.

Then w1¼ (o(u�11 )u�1

1 )�½m0�, w1� ¼o(u�1

1 )u�11 ½m0þ and w2¼o(u�1

1 )½m� 1s� for some s2f0, 1g. Therefore w1

� 2Eðmðu�11Þ;m�1Þ and w22

Fðmðu�11Þ;m�1Þ, hence w1

�jw2.

(c) i¼ 1, u12P�1 and o(u1)[1]2Y(1).Then w1¼o(u1)½m0�, w1

� ¼o(u1)½m1� and w2¼o(u1)u1½mþ 1sþ forsome s2f0, 1g. Therefore w1

� 2F(m(u1),m) and w22E(m(u1),m), hence w1�jw2.

(d) i¼ 1, u�11 2P�1 and o(u�1

1 )u�11 [1]2Y(1).

Then w1¼o(u�11 )u�1

1 ½m0þ, w1� ¼o(u�1

1 )u�11 ½m1þ and w2¼o(u�1

1 )½m� 1s� for some s2f0, 1g. Therefore w1

� 2Eðmðu�11Þ;m�1Þ and w22

Fðmðu�11Þ;m�1Þ, hence w1

�jw2.

(e) 1 < i� n and ui�1, ui2P�1.Then wi¼o(ui�1)ui�1½muði � 1Þ þm0þ, wi

� ¼ (o(ui�1)ui�1)�½muði � 1Þ

þm0� and wiþ1¼o(ui)ui½muðiÞ þmsþ for some s2f0,1g. Since ui�1ui 2GSt, we have m((o(ui�1)ui�1)

�)¼m(o(ui)ui)¼m(ui) and therefore wi� 2

F(m(ui),mu(i� 1)þm) and wiþ12E(m(ui),mu(i� 1)þm), hence wi�jwiþ1.

(f) 1 < i� n and ui�1, u�1i 2P�1.

Then wi¼o(ui�1)ui�1½muði � 1Þ þm1þ, wi� ¼ (o(ui�1)ui�1)

�½muði � 1Þþm1þ and wiþ1¼o(u�1

i )½muðiÞ þms� for some s2f0, 1g. Sinceui�1ui 2 GSt, we have m((o(ui�1)ui�1)

�)¼m(o(u�1i ))¼m(u�1

i ) and there-fore wi

� 2Eðmðu�1iÞ;muðiÞþmÞ and wiþ12Fðmðu�1

iÞ;muðiÞþmÞ, hence wi

�jwiþ1.

(g) 1 < i� n and u�1i�1, u

�1i 2P�1.

This, in a sense, is the dual of case (e).

(h) 1 < i� n and u�1i�1, ui2P�1.

This, in a sense, is the dual of case (f).

2. Since u; u½1 2 GSt, this statement follows from the statement 1.

3. Let t((u,m))¼w1� � �wnþ1, (t((u,m)))� ¼wnþ1�� w1

� ¼ x1� � �xnþ1

and t((u�1,mþ mu(n)))¼ y1� � �ynþ1. It is necessary to prove that xi¼ yifor all i. We distinguish the following cases.

(a) un2P�1,o(un)un[1]2Y�.Then y1¼ (o(un)un)�½mþ muðnÞ0� and x1¼ (wnþ1)

� ¼ (o(un)un½muðnÞþm0þ)� ¼ (o(un)un)�½mþ muðnÞ0�, hence x1¼ y1.

(b) u�1n 2P�1,o(u�1

n )[1]2Y�.Then y1¼ (o(u�1

n ))�½mþ muðnÞ1� and x1¼ (wnþ1)� ¼ (o(u�1

n )½muðnÞþm1�)� ¼ (o(u�1

n ))�½mþ muðnÞ1�, hence x1¼ y1.


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(c) un2P�1,o(un)un[1]2Y(1).Then y1¼o(un)un½mþ muðnÞ0þ and x1¼ (wnþ1)

� ¼ (o(un)un½muðnÞ þm1þ)� ¼o(un)un½mþ muðnÞ0þ, hence x1¼ y1.

(d) u�1n 2P�1,o(u�1

n )[1]2Y(1).Then y1¼o(u�1

n )½mþ muðnÞ0� and x1¼ (wnþ1)� ¼ (o(u�1

n )½muðnÞþm1�)� ¼o(u�1

n )½mþ muðnÞ0�, hence x1¼ y1.

(e) u12P�1,o(u1)[1]2Y�.Then ynþ1¼o(u1)½mþ muðnÞ þ mu�1ðnÞ1� and xnþ1¼ (w1)

� ¼ ((o(u1))�

½m1�)� ¼o(u1)½m1�, hence xnþ1¼ ynþ1 by Lemma 6.

(f) u�11 2P�1,o(u�1

1 )u�11 [1]2Y�.

Then ynþ1¼o(u�11 )u�1

1 ½mþ muðnÞ þ mu�1ðnÞ0þ and x1¼ (w1)� ¼ ((o(u�1

1 )u�11 )�½m0�)� ¼o(u�1

n )u�11 ½m0þ, hence xnþ1¼ ynþ1 by Lemma 6.

(g) u12P�1,o(u1)[1]2Y(1).Then ynþ1¼o(u1)½mþ muðnÞ þ mu�1ðnÞ1� and xnþ1¼ (w1)

� ¼ (o(u1)½m0�)� ¼o(u1)½m1�, hence xnþ1¼ ynþ1 by Lemma 6.

(h) u�11 2P�1,o(u�1

1 )u�11 [1]2Y(1).

Then ynþ1¼o(u�11 )u�1

1 ½mþ muðnÞ þ mu�1ðnÞ1þ and xnþ1¼ (w1)�

¼ (o(u�11 )u�1

1 ½m0þ)� ¼o(u�11 )u�1

1 ½m1þ, hence xnþ1¼ ynþ1 by Lemma 6.

(i) u�1n�iþ2, u

�1n�iþ1 2P�1, where 1 < i < nþ 1.

Then we have yi¼o(u�1n�iþ2)u

�1n�iþ2½muðnÞ þmþ mu�1ði � 1Þ0þ and

xi¼ (wn�iþ2)� ¼ (o(u�1

n�iþ1)½muðn� i þ 1Þ þm0�)� ¼ o(u�1n�iþ2)u

�1n�iþ2

½muðn� i þ 1Þ þm0þ), hence xi¼ yi by Lemma 6.

(j) u�1n�iþ2, un�iþ12P�1, where 1 < i < nþ 1.

Then we have yi¼o(u�1n�iþ2)u

�1n�iþ2½muðnÞ þmþ mu�1ði � 1Þ1þ and

xi¼ (wn�iþ2)� ¼ (o(un�iþ1)un�iþ1½muðn� i þ 1Þ þm1þ)� ¼ o(u�1

n�iþ2)u�1n�iþ2½muðn� i þ 1Þ þm1þ), hence xi¼ yi by Lemma 6.

(k) un�iþ2, un�iþ12P�1, where 1 < i < nþ 1.This, in a sense, is the dual of case (i).

(l) un�iþ2, u�1n�iþ1 2P�1, where 1 < i < nþ 1.

This, in a sense, is the dual of case (j).

4. Evident.

5. The statement follows from 3.

6. The statement follows from 4. &


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Corollary 4. The map t (resp., tc) is a map from GSt Z (resp.,GBa Z) to O (resp., Op).

We denote by GStnta (resp., GStnts ) the set of all nontrivial asymmetric(resp., symmetric) generalized strings.

Lemma 8.

1. The maps t : GSt Z ! Oim and tc : GBa Z ! Oimp are bi-

jections.2. We can choose GSt and Oa such that t(GStnta Z)¼Oim

a .3. We can choose GSt and Os such that t(GStnts Z)¼Oim

s .4. We can choose GBa and Oap such that tc(GBaa Z)¼Oim

ap.

5. We can choose GBa and Osp such that tc(GBas Z)¼Oimsp .

Proof. 1. We define maps t0 : Oim ! GSt Z and t0c : Oimp ! GBa Z

by the following rule.Let w¼w1� � �wnþ12Oim, where n� 1. Suppose first that wiþ12E,

where 1� i� n. Since wi�jwiþ1 and l(w�

i ) < l(wiþ1), we have wi� ¼ xi½misi�

and wiþ1¼ xiyi½mi þ 1tiþ for some xi2P, yi2P�1, mi2Z and si,ti2f0, 1g. Then we set ui :¼ yi.

Suppose finally that wiþ12F. Since wi�jwiþ1 and l(w�

i ) > l(wiþ1), wehave wi

� ¼ xiyi½misiþ and wiþ1¼ xi½mi � 1ti� for some xi2P, yi2P�1,mi2Z and si, ti2f0,1g. Then we set ui :¼ y�1

i .We will show that u ¼ u1 � � � un 2 GSt, that is that uiuiþ1 2 GSt for

1� i < n. As for this, we distinguish four cases.

(a) wiþ1, wiþ22E, where 1� i < n.Thenui, uiþ12P�1. Sincewiþ1¼ xiui½mi þ 1tiþ,wiþ1

� ¼ xiþ1½miþ1siþ1

� andwiþ2¼ xiþ1uiþ1½miþ1 þ 1tiþ1

þ (see above), we have (xiui)� ¼ xiþ1 and hence

e(ui)¼ s(uiþ1). If e(ui) 62Sp, then xiui 6¼xiþ1 and therefore uiuiþ12 J becoseof xiþ1uiþ1 62 J and A is skewed-gentle. Hence uiuiþ1 2 GSt.

(b) wiþ12E, wiþ22F, where 1� i < n.Then ui, u�1

iþ1 2P�1. Since wiþ1¼ xiui½mi þ 1tiþ, wiþ1� ¼ xiþ1

u�1iþ1½miþ1siþ1

þ and wiþ2¼ xiþ1½miþ1 � 1tiþ1

� (see above), we have (xiui)� ¼

xiþ1u�1iþ1 and hence e(ui)¼ e(u�1

iþ1)¼ s(uiþ1). If e(ui) 62Sp, then xiui 6¼xiþ1u

�1iþ1 and therefore uiuiþ12St. Hence uiuiþ1 2 GSt.

(c) wiþ1, wiþ22F, where 1� i < n.This, in a sense, is the dual of case (a).

(d) wiþ12F, wiþ22E, where 1� i < n.This, in a sense, is the dual of case (b).


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Then we set t0(w) :¼ (u, m1).Let v¼ (vi)i2Z be a periodic C(A)-word from Oim

p and nþ 1 be theperiod of it. Then w¼ v1� � �vnþ12Oim and we set tc0(v) :¼ t0(w). Sincet0ðwwÞ 2 GSt, we have t0cðvÞ 2 GBa.

It is easy to see that tt0, t0t, tct0c and t0ctc are identity maps.The others statements follow from the statement 1 and lemma 7. &

5.4 The Main Theorem

For a walk w¼w1� � �wn let us write m(w) for the minimum of the mw(i),i¼ 0, . . . , n and let us introduce also the following additional notations.

� Qc¼fa2Q1 j 9 a1, . . . , am2Q1 such that s(aiþ1)¼ e(ai),s(a1)¼ e(am), a1¼ a, aiaiþ1, ama12 Ig.

� GStc ¼ fw 2 GSt j l(w) > 0 and 9 a2Qc such that aw 2 GSt, andm(w)¼ 0g.

� GStc ¼ fw 2 GSt j l(w)¼ n > 0and 9 a2Qc such thatwa

�1 2 GSt,and m(w)¼ mw(n)g.

� GStc ¼ fw ¼ w1 � � �wn 2 GStc j if w12Qc then w2 � � �wn 62 GStcg:� GStc ¼ fw ¼ w1 � � �wn 2 GSt

c j ifw�1n 2Qc thenw1 � � �wn�1 62 GSt

cg.� GStcc ¼GStc\GStc.

Given w¼w1� � �wn2P�1(Q, J), we set

�ww :¼ a; if 9 a 2 Q1 such that aw1 2 I ;0; otherwise:

�

Lemma 9. Let A be a skewed-gentle algebra and let w2P�1(Q, J). Thenwe have

1. If s(w)2Sp, thenKer( p(w) p(gw))T¼A(w�,�w�g)þA(gw�, � gw�g);2. If s(w) 62Sp, then Ker p(w)¼Ker p(wg)¼Aw�þA(gw�);3. b(P�

w)�¼P�

w for any w¼w1w2 such that w�11 and w2 are in

P�1(Q, J ).

Proof. 1 is obvious and, for 2, let us observe that, since Aþ is gentle,we have Ker p(w�1

1 )\Ker p(w2)¼ Ker p(w�11 g)\Ker p(w2g)¼ 0. &

As consequence we obtain the following

Lemma 10. Let A be a skewed-gentle algebra. Then we have

1. For every w2GBaa and f2 Ind0 k[x], b(P�w;f )

�¼P�w;f .

2. For every w2GBas and M2M,b(P�w;M)�¼P�

w;M .


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3. In order that P�bðP�

wÞ� 62Kb(A� pro) for some w2GSt it is necessaryand sufficient that w2GStc or w2GStc.

The complexes in Db(A) isomorphic to T i( b(P�g)�), where

g2GStc[GStc, i2Z and P�g is an asymmetric (resp., a dimidiate) genera-

lized string , will be called asymmetric (resp., dimidiate) periodic string.Now we are in a good position to state and proof our main theorem

which gives all the indecomposable objects of the derived category.

Theorem 3. Let A be a finite-dimensional skewed-gentle algebra and letus keep our foregoing notations. Then each indecomposable object ofDb(A) is a string (asymmetric or dimidiate) or a periodic string (asymmetricor dimidiate) or a band (asymmetric or dimidiate). The complexes T i(P�

d)and T i(b(P�

g)�), where

d 2 GStaqGSts f0; 1gqGBaa Ind0 k½xqGBas M; ð3Þ

g 2 GStcqðGStcnGStccÞ and i 2 Z; ð4Þ

(T being the translation functor) constitute an exhaustive list of pairwisenon-isomorphic indecomposable objects of Db(A).

Proof. A straightforward calculation, which we omit, shows that forany nontrivial d satisfying (3) and any i2Z we have F(Ti(P�

d))ffiB(T(t(d,i))). Therefore it follows from Lemma 5, Theorem 2 and Lemma8 that, for d satisfying (3), the complexes Ti(P�

d) constitute an exhaustivelist of pairwise non-isomorphic indecomposable objects of Kb(A-pro).

We end up our proof with the following observation. It follows fromLemma 10 that fb(M�)�jM� 2Ver p(A) and P�

bðM�Þ� 62Kb(A-pro)g ¼fTi(b(P�

g)�) j g2GStcq(GStc nGStcc) and i2Zg. &

As consequence we obtain the following

Corollary 5. Let A be a finite-dimensional skewed-gentle algebra. Then

(i) A is derived tame.(ii) A is derived discrete if and only if GBa¼;.(iii) A is derived finite if and only if jGStj < 1.

Remark 4. If A has finite global dimension, the statement (i) of thecorollary follows from Geiss and de la Pena (1999).


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ACKNOWLEDGMENTS

The research was done during visits of the first author to theUniversityof Sao Paulo and University of Natal. The financial support of FAPESPand CNPq and the hospitality offered by these universities are gratefullyacknowledged.

The first author (V.B.) was supported by FAPESP (Grant 98=14538-0)and CNPq (Grant 301183=00-7). The second author (E.N.M.) was sup-ported by CNPq.

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Received August 2001Revised August 2002


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indecomposables in derived categories of skewed-gentle algebras

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