Download TILTED ALGEBRAS OF TYPE

TILTED ALGEBRAS OF TYPE Ae n
Jessica Levesque
Abstract
We give a short proof of the classications of tilted algebras of
types A n and Aen , obtained respectively by Assem 1] and Roldan 18].
Introduction.
The class of tilted algebras, introduced by Happel and Ringel 13], plays a
prominent role in the representation theory of algebras. There exist many
classication results for tilted algebras. In his Ph.D. thesis, O. Roldan 18]
gave a classication of the tilted algebras of euclidean type Aen . This result
has been applied in several contexts for instance in 4, 6, 14]. The proof
of this classication has never been published in any journal. On the other
hand, Ringel's classication of the representation-innite tilted algebras of
euclidean type as truncated branch extensions and coextensions 17], the
classication of the iterated tilted algebras of type Aen due to Assem and
Skowronski 4] and a technique due to F. Huard 15] allow us to give a short
and conceptual proof of this result. The objective of this paper is to give this
proof.
Throughout this paper, k always denotes an algebraically closed eld and
A a nite dimensional associative and indecomposable k-algebra.
We recall that, by 4], an algebra is iterated tilted of type Aen if and only
if it is gentle and its bound quiver contains exactly one cycle satisfying the
clock condition (see (1.3)).
We prove the following result, which is a reformulation of Roldan's theorem :
Theorem. 18] Let A = kQ=I be an iterated tilted algebra of type Aen . Then
A is tilted if and only if its bound quiver (Q I ) contains no bound subquiver
of one of the following forms or their duals :
1
a)
c
1
- c - c . . .
2
with t 4, = 0, = 0.
c c . . .
b)
c
1
2
- c - c
c
t;1
t
- c - c
c
t;1
t
with t 5, = 0, = 0, 1 and 2 lie on the cycle while t ; 1 and t do not.
1 c
c)
t c
6
c
?. . .
c
c
- c
with t 4, = 0, = 0, only 1 and t do not lie on the cycle.
d)
1 c
c
6
?. . . c
t
c
c
c
with t 4, = 0, = 0, only 1 and t do not lie on the cycle.
In each of the cases a), b), c) and d) above, there are no other relations
than the specied ones, and the arrows whose orientations are not specied
may be oriented arbitrarily.
This paper consists of two sections. The rst is devoted to preliminaries
and the second to the proof of the theorem.
1 Preliminaries.
1.1. Notation. We recall that a quiver Q is dened by a set of points
Q0 and a set of arrows Q1. A relation from x 2 Q0 to y 2 Q0 is a linear
combination of paths from x to y of length at least two. A relation is called
a zero-relation if it equals a path. Let I denote the ideal of kQ generated
by a set of relations, then the pair (Q I ) is called a bound quiver. A path is
called a zero-path if it belongs to I . It is well-known that, if A is a basic and
connected nite dimensional k-algebra, then there exists a connected bound
2
quiver (QA I ) such that A = kQA=I . For a point a in the quiver of A,
we denote by P (a) the corresponding indecomposable projective A-module.
Given an A-module M , we denote by SuppM the full bound subquiver of
QA generated by the points a such that HomA(P (a) M ) 6= 0. We say that
A is triangular whenever its quiver QA has no oriented cycles.
For general properties of the category mod A of nitely generated right
A-modules, we refer the reader to 17]. For tilted and iterated tilted algebras,
we refer the reader to 2, 11, 17].
1.2. Gentle algebras. Following 4] (1.1), we call a triangular algebra A
gentle if it is isomorphic to a bound quiver algebra kQ=I , where the bound
quiver (Q I ) satises :
(G1) The number of arrows in Q with a prescribed source or target is at
most two.
(G2) For any 2 Q1, there is at most one 2 Q1 and one 2 Q1 such
that and do not belong to I .
(G3) For any 2 Q1 , there is at most one 2 Q1 and one 2 Q1 such that
and belong to I .
(G4) I is generated by a set of paths of length two.
In particular, gentle algebras are string algebras, in the sense of 8] hence indecomposable modules over a gentle algebra are given by strings and
bands. If a gentle algebra is representation-nite, then its indecomposable
modules are given by strings.
1.3. We recall the classication results for iterated tilted algebras of type
An
and Aen .
Theorem. 3] An algebra A = kQ=I is iterated tilted of type A n if and only
if it is gentle, and Q is a tree. Theorem. 4] An algebra A = kQ=I is iterated tilted of type Aen if and only
if it is gentle, and (Q I ) satises :
(G5) Q contains a unique (non-oriented) cycle C .
3
(G6) (The clock condition). On C , the number of clockwise oriented relations equals the number of counterclockwise oriented relations.
Furthermore, A is representation-nite if and only if the cycle C is bound by
at least two relations.
e Ie) the universal cover
1.4. Let A = kQ=I be a gentle algebra. Denote by (Q
of the bound quiver (Q I ) (see, for instance, 16]). It follows directly from
e Ie is a gentle algebra with Q
e a tree. Therefore any
the denition that kQ=
e Ie) is the bound quiver of an iterated tilted
nite full bound subquiver of (Q
algebra of type A n (see (1.3)). This implies that, if M is a nite dimensional
e Ie, then M is multiplicity-free and hence is
indecomposable module over kQ=
uniquely determined by its support.
2 Proof of the theorem.
2.1. De
nition. 14, 15] Let A = kQ=I be a gentle algebra. A reduced walk
w is called a double-zero in (Q I ) if w contains exactly two zero-relations
and these two zero-relations point to the same direction in w.
e Ie) of a
It follows directly from the denition that the universal cover (Q
gentle algebra kQ=I contains a double-zero if and only if (Q I ) contains a
double-zero.
2.2. Lemma. Let A = kQ=I be an iterated tilted algebra of type A n and
M be an indecomposable A-module of projective dimension larger than one.
Then (Q I ) contains a reduced walk of the form
c
- c . . . c - c - c
s
p
with s a source of SuppM , p 2= (SuppM )0 and the unique zero-relation.
Proof. First of all, note that all indecomposable A-modules are multiplicityfree in this case.
Let f : P ! M be a projective cover of M . Since pdM > 1, we have that
L = Kerf is not projective. Thus there exists an indecomposable module
L0 2 addL which is not projective.
4
Now, let P 0 be the projective cover of L0 . Since L0 6= P 0, there exists
q 2 (SuppP 0)0 n (SuppL0 )0 and an arrow : p ! q with p 2 (SuppL0)0 .
Clearly, p 2= (SuppM)0. Let v be a non-zero path in SuppP starting at a
source s of SuppM , and ending at p. Since q 2= (SuppL0 )0, we have v = 0,
and since A is gentle, there exist an arrow and a non-zero path v0 such
that v = v0 and = 0. 2.3. The following proposition has been proved by Huard and Liu 15] (3.4)
in the more general context of string algebras. We give a short proof in our
context of gentle algebras.
Proposition. Let A = kQ=I be a representation-nite gentle algebra. Then
A is tilted if and only if (Q I ) contains no double-zero.
Proof. First, we note that A is tilted if and only if it is quasi-tilted since A
is representation-nite 12].
Necessity. Suppose that (Q I ) contains a double-zero of the form
c
1
- c - c
2
c
3
...c
c
- c - c
t;2
t;1
t;2
with t 4 and where and are the zero-relations (the points and the
zero-relations are not necessarily distinct).
If t = 4, then, by 9, 10], gl.dim.A > 2, hence A is not tilted. Thus,
e Ie) be the universal cover of (Q I ). Let M
suppose that t 5 and let (Q
e Ie-module corresponding to the indecomposable
be an indecomposable kQ=
A-module N of support
3
; ; t-2
and let P be the projective cover of M .
We use the same notation for the points of SuppM and SuppN . Let s be
the source of SuppM such that there exists a path from s to t ; 2 in SuppM .
Then the kernel of the canonical morphism P (s) ! M has a non-projective
direct summand and hence pdM > 1. We deduce that pdN > 1.
Similarly, one proves that idN > 1. Thus, by 12], A is not quasi-tilted
and hence is not tilted.
5
Suciency. Suppose that A is not tilted. Then A is not quasi-tilted. If
gl.dim.A > 2, then, by 9, 10], (Q I ) contains a double-zero. If this is not
the case, then by 12], there exists an indecomposable A-module N such that
pdN > 1 and idN > 1. Denoting by M an indecomposable module over
e Ie) of (Q I ) corresponding to N , we have pdM > 1
the universal cover (Q
e Ie) is the bound quiver of an iterated tilted algebra
and idM > 1. Since (Q
of type A n (see(1.4)), it follows directly from the lemma and its dual that
e Ie) contains a double-zero, hence so does (Q I ) (see 2.1). (Q
2.4 We easily deduce the classication of the tilted algebras of type A n .
Corollary. 1] An iterated tilted algebra A = kQ=I of type A n is tilted if and
only if (Q I ) contains no double-zero. 2.5. Proof of the theorem. Necessity. Assume that (Q I ) contains
a bound subquiver of one of the forms a), b), c) or d). If A = kQ=I is
representation-nite, a straightforward analysis of all possible cases (as done
in 14]) shows that (Q I ) contains a double-zero. The result then follows
from the proposition. If now A is representation-innite, the result follows
from 17] (4.9).
Suciency. If A is representation-nite and is not tilted, then, by the
proposition, (Q I ) contains a double-zero. It is easy to see that in all cases,
(Q I ) contains a bound subquiver of the form a).
Thus, suppose that A is representation-innite and that (Q I ) does not
contain a bound subquiver of one of the forms a), b), c) or d). Since A is
iterated tilted of type Aen , it follows that (Q I ) is obtained from a hereditary
algebra of type Aen by some branch extensions and coextensions, in the sense
of 5] (2.1). Since (Q I ) contains no bound subquiver of the forms c) and
d), it cannot simultaneously contain an extension branch and a coextension
branch. Thus we may assume that (Q I ) contains no coextension branches
(the other case is similar).
If the branch of an extension is not a truncated branch (in the sense of
5] (2.3)), then there exists a vertex b such that the unique reduced walk
between b and the root of the extension contains a relation. But then (Q I )
contains a bound subquiver of one of the forms a) or b) thus each branch is
necessarily truncated. Finally, since (Q I ) contains no bound subquiver of
the form a), we see that (Q I ) is a domestic tubular extension in the sense
6
of 17] (4.9). We infer from 17] (4.9) that A is tilted. 2.6. Remark. The main result of 14] says that if a tilted algebra is gentle,
then it is of type A n or Aen . Hence our main theorem together with (2.4)
provides a complete classication of the tilted gentle algebras.
Acknowledgments. This paper is part of my M.Sc. thesis. I would like to thank
Professor I. Assem for his precious help and direction, and Professors S. Liu and F. Huard
for useful discussions. I would also like to thank the N.S.E.R.C. (Gov. of Canada) for
nancial support.
References
1] Assem, I. : Tilted algebras of type A n , Comm. Algebra 10(19), 2121-2139 (1982).
2] Assem, I. : Tilting theory - an introduction, Topics in algebra, Banach Center Publ.
26, Vol. 1 (PWN. Warsaw, 1990) 127-180.
3] Assem, I. and Happel, D. : Generalized tilted algebras of type A n , Comm. Algebra 9
(1981) 2101-2125.
4] Assem, I. and Skowronski, A. : Iterated tilted algebras of type Aen , Math. Z. 195,
(1987) 269-290 .
5] Assem, I. and Skowronski, A. : Algebras with cycle-nite derived categories, Math.
Ann. 280 (1988) 441-463
6] Assem, I. and Zhang, Y. : Endomorphism algebras of exceptional sequences over path
algebras of type Aen , Colloq. Math. Vol. 77, No. 2 (1998) 271-292.
7] Bongartz, K. and Gabriel, P. : Covering spaces in representation theory, Invent.
Math. 65 (1981/82) 331-378.
8] Butler, M.C.R. and Ringel, C.M. : Auslander-Reiten sequences with few middle terms
and applications to string algebras, Comm. Algebra 15 (1987) 145-179.
9] Gauvreau, C. : A new proof of a theorem of Green, Happel and Zacharia, Ann. Sci.
Math. Quebec 21 (1997) 83-89.
10] Green, E., Happel, D. and Zacharia, D. : Projective resolutions over artin algebras
with zero relations, Ill. J. Math., Vol. 29, No. 1 (1985) 180-190.
11] Happel, D.: Triangulated categories in the representation theory of nite dimensional
algebras, London Math. Soc. Lecture Note Series 119, Cambridge Univ. Press (1988).
12] Happel, D., Reiten, I. and Smal, S.O. : Tilting in abelian categories and quasitilted
algebras, Mem. Amer. Math. Soc., Vol. 120, No. 575 (1996).
13] Happel, D. and Ringel, C.M. : Tilted algebras, Trans. Amer. Math. Soc. 274 (1982),
No.2, 399-443.
7
14] Huard, F. : Tilted gentle algebras, Comm. Algebra, Vol. 26, No. 1 (1998)
15] Huard, F. and Liu, S. : Tilted string algebras, to appear.
16] Martinez-Villa, R. and de la Pe~na, J.A. : The universal cover of a quiver with relations, Journal of Pure and Applied Algebra 30 (1983) 277-292.
17] Ringel, C.M. : Tame algebras and integral quadratic forms, Springer Lecture Notes
1099 (1984).
18] Roldan, O. : Tilted algebras of types Aen Ben Cen and BfC n , Ph.D. thesis, U. Carleton
(1983).
Jessica Levesque
Departement de Mathematiques et Informatique
Universite de Sherbrooke
Sherbrooke, Quebec
J1K 2R1, Canada
[email protected]
8