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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dic. 2005
The left part and the Auslander-Reiten components of an artin algebra
Ibrahim Assem, Juan Ángel Cappa, María Inés Platzeck and Sonia Trepode
Dedicated to the memory of Ángel Rafael Larotonda
This paper was completed during a visit of the first author to the Universidad Nacional del Sur in Bahía Blanca (Argentina). He would like to thank María Inés Platzeck and María Julia Redondo, as well as all members of the argentinian group, for their invitation and warm hospitality. He also acknowledges partial support from NSERC of Canada. The other three authors gratefully acknowledge partial support from Universidad Nacional del Sur and CONICET of Argentina, and the fourth from ANPCyT of Argentina. The second author has a fellowship from CONICET, and the third and the fourth are researchers from CONICET.
Abstract. The left part 
of the module category of an artin algebra 
consists of all indecomposables whose predecessors have projective dimension at most one. In this paper, we study the Auslander-Reiten components of 
(and of its left support 
) which intersect 
and also the class 
of the indecomposable Ext-injectives in the addditive subcategory add
generated by 
.
Key words and phrases. artin algebras, Auslander-Reiten quivers, sections, left and right supported algebras
2000 Mathematics Subject Classification. 16G70, 16G20, 16E10
INTRODUCTION
Let 
be an artin algebra and mod
denote the category of finitely generated right 
modules. The class 
, called the left part of mod
, is the full subcategory of mod
having as objects all indecomposable modules whose predecessors have projective dimension at most one. This class, introduced in [15], was heavily investigated and applied (see, for instance, the survey [4]).
Our objective in this paper is to study the Auslander-Reiten components of an artin algebra which intersect the left part. Some information on these components was already obtained in [2, 3]. Here we are interested in the components which intersect the class 
of the indecomposable Ext-injectives in the full additive subcategory add 
having as objects the direct sums of modules in 
. We start by proving the following theorem.
THEOREM (A). Let 
be an artin algebra, and 
be a component of the Auslander-Reiten quiver of 
. If 
, then:
(a) Each 
-orbit of 
intersects 
exactly once.
(b) The number of 
-orbits of 
equals the number of modules in 
.
(c) 
contains no module lying on a cycle between modules in 
.
If, on the other hand, 
, then either 
or else 
.
We recall that, by [3] (3.3), the class 
contains only finitely many non-isomorphic modules (hence only finitely many Auslander-Reiten components intersect 
).
As a consequence, we give a complete description of the Auslander-Reiten components lying entirely inside the left part.
We then try to describe the intersection of 
with a component 
of the Auslander-Reiten quiver 
. We find that, in general, 
is not a section in 
(in the sense of [20, 23]) but is very nearly one. This leads us to our second theorem, for which we recall that a component 
of 
is called generalised standard if 
for all 
, see [23].
THEOREM (B). Let 
be an artin algebra and 
be a component of 
(mod
) such that all projectives in 
belong to 
. If 
, then:
(a) 
is a section in 
.
(b) 
is generalised standard.
(c) 
is a tilted algebra having 
as a connecting component and 
as a complete slice.
In particular, such a component 
has only finitely many 
-orbits.
The situation is better if we look instead at the intersection of 
with the Auslander-Reiten components of the left support 
of 
. We recall from [3, 24] that the left support 
of 
is the endomorphism algebra of the direct sum of the indecomposable projective 
-modules lying in 
. It is shown in [3, 24] that every connected component of 
is a quasi-tilted algebra (in the sense of [15]). We prove the following theorem.
THEOREM (C). Let 
be an artin algebra and 
be a component of the Auslander-Reiten quiver of the left support 
of 
. If 
, then:
(a) 
is a section in 
.
(b) 
is directed, and generalised standard.
(c) 
is a tilted algebra having 
as a connecting component and 
as a complete slice.
We then apply our results to the study of left supported algebras. We recall from [3] that an artin algebra 
is left supported provided add
is contravariantly finite in mod
. Several classes of algebras are left supported, such as all representation-finite algebras, and all laura algebras which are not quasi-tilted (see [3, 4]). It is shown in [1] that an artin algebra 
is left supported if and only if 
consists of all the predecessors of the modules in 
. We give here a proof of this fact which, in contrast to the homological nature of the proof in [1], uses our theorem and the full power of the Auslander-Reiten theory of quasi-tilted algebras. Our proof also yields a new characterisation: an algebra 
is left supported if and only if every projective 
-module which belongs to 
is a predecessor of 
. We end the paper with a short proof of the theorem of D. Smith [25] (3.8) which characterises the left supported quasi-tilted algebras.
Clearly, the dual statements about the right part of the module category, also hold true. Here, we only concern ourselves with the left part, leaving the primal-dual translation to the reader.
We now describe the contents of the paper. After a brief preliminary section 1, the sections 
and 
are respectively devoted to the proofs of our theorems (A), (B) and (C). In our final section 
, we consider the applications to left supported algebras.
1.1. Notation. For a basic and connected artin algebra 
let mod
denote its category of finitely generated right modules and ind
a full subcategory consisting of exactly one representative from each isomorphism class of indecomposable modules. We sometimes consider 
as a category, with objects a complete set 
of primitive orthogonal idempotents, and where 
is the set of morphisms from 
to 
. An algebra 
is a full subcategory of 
if there is an idempotent 
, which is a sum of some of the distinguished idempotents 
, such that 
. It is convex in 
if, for any sequence 
of objects of 
such that 
(with 
and 
, 
objects of 
, all 
are in 
.
Given a full subcategory 
of mod
, we write 
to indicate that 
is an object in 
, and we denote by add
the full subcategory with objects the direct sums of summands of modules in 
. Given a module 
, we let pd
stand for its projective dimension. We also denote by 
the Auslander-Reiten quiver of 
and by 
, 
the Auslander-Reiten translations. For further notions or facts needed on mod
, we refer to [7, 22].
1.2. Paths. Let 
be an artin algebra and 
ind
. A path 
is a sequence
are non-zero morphisms and the 
lie in ind
. We call 
a predecessor of 
and 
a successor of 
. A path from 
to 
involving at least one non-isomorphism is a cycle. An indecomposable module 
lying on no cycle is called directed . A path 
is called sectional if each 
is irreducible and 
for all 
. A refinement of 
is a path 
with 
for all 
. A full subcategory 
of ind
is convex if, for any path 
with 
, 
, all the 
lie in 
. 2. EXT-INJECTIVES IN THE LEFT PART.
2.1. Let 
be an artin algebra. The left part 
of mod
is the full subcategory of ind
defined by
An indecomposable module 
is called Ext-projective (or Ext-injective) in add
if Ext
(or Ext
, respectively), see [9]. While the Ext-projectives in add
are the projective modules lying in 
(see [3] (3.1)), the Ext-injectives are more interesting. Before stating their characterisations we recall that, by [9] (3.7), 
is Ext-injective in add
if and only if 
.
LEMMA [5] (3.2), [3] (3.1). Let 
.
(a) The following are equivalent:
such that Hom
. 
and a path 
. 
and a sectional path 
.
(b) The following conditions are equivalent for 
which does not satisfy conditions (a):
such that Hom
. 
and a path 
. 
and a sectional path 
.
Letting 
(or 
) denote the set of all 
verifying (a) (or (b), respectively), and setting 
, then 
is Ext-injective in add
if and only if 
. 
2.2. The following lemma will also be useful.
LEMMA [3] (3.2) (3.4). (a) Any path of irreducible morphisms in 
is sectional.
(b) Let 
and 
with 
. Then this path can be refined to a sectional path and 
. In particular, 
is convex in ind
. 
2.3. The following immediate corollary will be useful in the proof of our theorem (A).
COROLLARY All modules in 
are directed.
Proof. Assume 
is a cycle in ind
, with 
. By (2.2) above, such a cycle can be refined to a sectional cycle with all indecomposables lying in 
. Now compose two copies of this cycle to form a larger cycle in 
of irreducible morphisms. By (2.2), this cycle is also sectional, in contradiction to [11, 12] . 
2.4. THEOREM (A). Let 
be an artin algebra, and 
be a component of the Auslander-Reiten quiver of 
. If 
, then:
(a) Each 
-orbit of 
intersects 
exactly once.
(b) The number of 
-orbits of 
equals the number of modules in 
.
(c) 
contains no module lying on a cycle between modules in 
.
If, on the other hand, 
, then either 
or else 
.
Proof. Assume first that 
, that is, the component 
contains an Ext-injective in add
.
(a) If 
contains an injective module, then the statement follows from [3] (3.5). We may thus assume that 
contains no injective. But then 
, and therefore 
. Thus, by (2.1), there exist an indecomposable projective 
in 
such that 
a module 
and a sectional path 
. Now let 
Since 
contains no injective, there exists 
such that 
is a successor of 
Hence 
Since 
itself lies in 
there exists 
such that 
but 
so that 
is Ext-injective in add 
This shows that every 
orbit of 
intersects 
at least once.
Furthermore, it intersects it only once: if 
and 
(with 
) both belong to 
then, by (2.2), all the modules on the path
. In particular, 
and this contradicts the Ext-injectivity of 
. This completes the proof of (a). (b) It follows from (a) that the number of 
-orbits in 
does not exceed the cardinality of 
(note that by [3] (3.3), the cardinality of 
is finite and does not exceed the rank of the Grothendieck group 
of 
). Since clearly, any element of 
belongs to exactly one 
-orbit in 
, this establishes (b).
(c) Let 
be a cycle with 
and all 
in 
Clearly, all 
belong to 
By (2.2) and (2.3), none of the 
belongs to 
and none of the 
factors through an injective module. Indeed, if 
factors through the injective 
, then some indecomposable summand of 
would belong to 
and thus 
would lie in 
, contradicting (2.3). Then the cycle 
induces a cycle 
, and every module in this cycle belongs to 
We can iterate this procedure and deduce that, for any 
, the module 
lies on a cycle in 
. However, as shown in (a), there exists 
such that 
does not belong to 
, and this contradiction proves (c).
Now assume that the component 
contains no Ext-injective, that is, 
If 
contains both a module in 
and a module which is not in 
, then there exists an irreducible morphism 
with 
and 
Since 
then 
But this is a contradiction, because 
and Hom
This shows that either 
or 
as required. 
We observe that part (c) of the theorem was already proven in [3] (1.5) under the additional hypothesis that 
contains an injective module.
2.5. COROLLARY [3] (1.6). Let 
be a representation-finite artin algebra. Then 
is directed. 
2.6. We have a good description of the Auslander-Reiten components which completely lie in 
. We need to recall a definition. The endomorphism algebra 
of the direct sum of all the projective modules lying in 
is called the left support of 
, see [3, 24]. Clearly, 
is (isomorphic to) a full convex subcategory of 
, closed under successors, and any 
-module lying in 
has a natural 
-module structure. It is shown in [3] (2.3), [24] (3.1) that 
is a product of connected quasi-tilted algebras, and that 
. The following corollary generalises [3] (5.5).
COROLLARY. Let 
be a representation-infinite not hereditary artin algebra, and 
be a component of 
(mod
lying entirely in 
. Then 
is one of the following: a postprojective component, a regular component (directed, stable tube or of type 
), a semiregular tube without injectives, or a ray extension of 
.
Proof. Indeed, the component 
lies entirely in mod
and thus is a component of 
mod
. Since 
, then 
is a component of 
mod
lying in the left part 
. The statement then follows from the well-known description of the Auslander-Reiten components of quasi-tilted algebras, as in [13, 18]. 
3. EXT-INJECTIVES AS SECTIONS IN 
(mod
).
3.1. We recall the following notion from [20, 23]. Let 
be an artin algebra and 
be a component of 
(mod
). A full connected subquiver 
of 
is called a section if it satisfies the following conditions:
(
) 
contains no oriented cycle.
(
) 
intersects each 
-orbit of 
exactly once.
(
) 
is convex in 
.
(
) If 
is an arrow in 
with 
, then 
or 
.
(
) If 
is an arrow in 
with 
, then 
or 
.
As we show next, the intersection of 
with a component of 
(mod
) satisfies several of these conditions (but generally not all).
PROPOSITION. Assume 
is a component of 
(mod
) which intersects 
. Then 
satisfies (
), (
), (
) above, and the following conditions
(
) 
intersects each 
-orbit of 
at most once.
(
) If 
is an arrow in 
with 
and 
non-projective, then 
or 
.
Proof. (
) follows from Theorem (A) (c).
(
) follows from Theorem (A) (a).
(
) follows from (2.2).
(
) If 
, then 
and (2.2) imply 
. Otherwise, since 
is non-projective, there exists an arrow 
. Since 
, then 
. Since 
, we get 
.
(
) If 
is injective then, since it lies in 
(because it precedes 
), it belongs to 
. So assume it is not and consider the arrow 
. If 
then, again, 
while, if 
, then 
and (2.2) imply 
. 
3.2. EXAMPLE. Let 
be a field and 
be the radical square zero 
-algebra given by the quiver
Here, 
is representation finite and 
consists of the two indecomposable projectives 
and 
corresponding to the points 
and 
, respectively. Clearly, 
is not a section in 
(mod
) : indeed, there is an arrow 
with 
and, moreover, 
does not intersect each 
-orbit of 
(mod
).
3.3. We are now in a position to prove our second main theorem.
THEOREM (B). Let 
be an artin algebra and 
be a component of 
(mod
) such that all projectives in 
belong to 
. If 
, then:
(a) 
is a section in 
.
(b) 
is generalised standard.
(c) 
is a tilted algebra having 
as a connecting component and 
as a complete slice.
Proof.
(a) We start by observing that, if 
is an arrow in 
, with 
and 
projective then, by hypothesis, 
. Thus, (2.2) implies 
. This shows that (
) is satisfied. In view of the lemma, it suffices to show that 
cuts each 
orbit of 
.
We claim that if 
and 
lie in two neighbouring orbits, then 
intersects the 
-orbit of 
. This claim and induction imply the statement. We assume that 
does not intersect the orbit of 
and try to reach a contradiction. There exist 
and an arrow 
or 
, with 
in the 
-orbit of 
, where we may suppose, without loss of generality, that 
is minimal.
Suppose first that 
. If there exists an arrow 
then there exists an arrow 
, a contradiction to the minimality of 
. If, on the other hand, there exists an arrow 
, then there is a path in 
of the form 
. Since 
then 
. Hence 
. In particular, 
is not projective, so there exists an arrow 
, contrary to the minimality of 
.
Suppose now that 
. If there exists an arrow 
, then there exists an arrow 
, a contradiction to the minimality of 
. If, on the other hand, there exists an arrow 
, then there is a path in 
of the form 
. Hence 
. In particular, 
is not injective (otherwise, 
, a contradiction). Hence there exists an arrow 
, contrary to the minimality of 
.
We have thus shown that necessarily 
, that is, there is an arrow 
or 
. If 
then, by (
), 
or 
, in any case a contradiction. If 
, then (3.1) yields 
or 
, again a contradiction in any case. This completes the proof of (a).
(b) By [23], Theorem 2, it suffices to show that for any 
, we have Hom
. But 
implies pd
. Therefore the Ext-injectivity of 
in add
implies that
(c) This follows directly from [20] (2.2). 
3.4. EXAMPLE. Let 
be a field and 
be the radical square zero algebra given by the quiver
Let 
be the component containing the injective 
corresponding to the point 
. Clearly, 
, so that 
. On the other hand, the only projective lying in 
is 
, and it belongs to 
. Thus, the hypotheses of the theorem apply here. Note that 
is equal to the left support 
of 
, that is, the full convex subcategory with objects 
.
4. EXT-INJECTIVES AND THE LEFT SUPPORT
4.1. In this section we study the intersection of 
with the components of the Auslander-Reiten quiver of the left support 
of the artin algebra 
.
We observe first that if 
is an 
-module and 
then 
. In particular, 
is not projective in mod
. Indeed, since mod
is closed under extensions in mod
, then the inclusion 
implies that the almost split sequence in mod
ending at 
is entirely contained in mod
(See also [7], p. 187). Similarly, if 
, then 
, and 
is not an injective 
-module.
LEMMA. If an indecomposable injective 
-module 
is a predecessor of 
, then 
.
Proof. This is clear if 
is an indecomposable injective 
-module. So assume it is not. Since 
precedes 
, then 
. By the above observation we obtain that 
, because 
is 
-injective. This proves that 
, as desired. 
4.2. The following is an easy consequence of (3.1) and the results in [3].
LEMMA. Let 
. Then 
is a convex partial tilting 
-module . In particular, 
.
Proof. Indeed, since 
(see [3], (2.1)), 
implies pd
. Since Ext
and 
is a full convex subcategory of 
, we also have Ext
. Finally, the convexity of 
in ind
follows from its convexity in ind
(see (2.2)). 
4.3. THEOREM C. Let 
be an artin algebra and 
be a component of the Auslander-Reiten quiver of the left support 
of 
. If 
, then:
(a) 
is a section in 
.
(b) 
is directed, and generalised standard.
(c) 
is a tilted algebra having 
as a connecting component and 
as a complete slice.
Proof. (a) In order to show that 
is a section in 
, we just have to check the conditions of the definition in (3.1). Clearly, (
) follows from (2.3) and the observation that any cycle in ind
induces one in ind
. Also, (
) follows from (4.2). We start by proving (
) and (
).
(
) Assume 
is an arrow in 
, with 
. If 
, then (2.2) implies 
. Assume 
. Then, in particular, 
is not a projective 
-module. Since 
is an 
-module, it is not a projective 
-module either, so there is an irreducible morphism 
in mod
. Then 
precedes 
and therefore lies in 
. Thus, as we observed in (4.1), 
. Since 
, we conclude that 
, as required.
(
) Assume 
is an arrow in 
, with 
. If 
, then 
and, again by the observation in 4.1, we know that 
is not an injective 
-module. Hence 
. Since there is an arrow 
, we conclude that 
, as required.
There remains to prove (
), that is, that 
intersects each orbit of 
exactly once. We use the same technique as in the proof of Theorem (B). Clearly, the situation is different and so the arguments vary slightly.
We start by proving that 
intersects each orbit of 
at least once. We claim that if 
and 
lie in two neighbouring orbits, then 
intersects the 
-orbit of 
. This claim and induction imply the statement. We assume that 
does not intersect the orbit of 
and try to reach a contradiction. There exist 
and an arrow 
or 
, with 
in the 
-orbit of 
, where we may suppose, without loss of generality, that 
is minimal.
Suppose first that 
. If there exists an arrow 
then there exists an arrow 
, a contradiction to the minimality of 
. If, on the other hand, there exists an arrow 
, then there is a path in 
of the form 
. Now, 
implies 
. By [7] p. 186, there exists an epimorphism 
. Hence 
and so 
. In particular, 
is not a projective 
-module, so there exists an arrow 
, contrary to the minimality of 
.
Suppose now that 
. If there exists an arrow 
, then there exists an arrow 
, a contradiction to the minimality of 
. If, on the other hand, there exists an arrow 
, then there is a path in 
of the form 
, hence 
is a predecessor of 
. Since 
, by hypothesis, then we know by (4.1) that 
is not injective in mod
. Hence there exists an arrow 
, contrary to the minimality of 
.
This shows that necessarily 
, that is, there is an arrow 
or 
. If 
then (
) yields 
or 
, in any case a contradiction. If 
, then (
) yields 
or 
, again a contradiction in any case.
We proved that 
intersects each 
-orbit of 
. Suppose now that 
and 
with 
. Then the epimorphism 
yields a path 
, so that 
. This is a contradiction because 
. Thus (a) is proven.
(b) Since, by [13], directed components of quasi-tilted algebras are postprojective, preinjective or connecting, thus always generalised standard (see [20, 23]), it suffices to show that 
is directed. If this is not the case then, by [18] (4.3), 
is a stable tube, of type 
or obtained from one of these by finitely many ray or coray insertions.
We first notice that by (2.3), any 
is directed in ind
, hence in ind
. In particular, 
is neither a stable tube, nor of type 
. Therefore 
is obtained from one of these by ray or coray insertions.
Assume first that 
is an inserted tube or component of type 
, and let 
. We claim that 
. Indeed, if this is not the case, then there exists an injective 
-module 
such that Hom
, by (2.1). However, 
implies that 
is an 
-module, so that 
is an injective 
-module. But this is impossible because no injective 
-module precedes an inserted tube or component of type 
. This establishes our claim. Thus, there exists an indecomposable projective module 
such that Hom
, by (2.1). On the other hand, 
, therefore there exist a non-directed projective 
and a path 
in 
. This is clear if 
is an inserted tube, and follows from [10, 17] if 
is an inserted component of type 
. Hence there exists a path 
in ind
. Since 
, then 
. However, 
, hence 
is a projective 
-module lying in 
, a contradiction.
Assume next that 
is a co-inserted tube or component of type 
, and let 
. Then, among the predecessors of 
lies a cycle 
, with all 
. Since all 
precede 
and, by hypothesis, 
, then this cycle lies in 
. This contradicts Theorem (A) (c) (also [3] (1.5) (b)).
(c) This follows directly from [20] (2.2). 
4.4. EXAMPLE. It is important to underline that, while the components of 
which cut 
are directed, and even generalised standard, the same does not hold for the components of 
. Indeed, let 
be a field and 
be given by the quiver
bound by 
, 
and 
. Letting, as usual, 
and 
denote respectively the indecomposable projective and the simple modules corresponding to the point 
, we have an almost split sequence 
.
Moreover, rad
, where 
lies in a regular component of type 
in the Auslander-Reiten quiver of the wild hereditary algebra 
which is the full subcategory of 
with objects the points 
, 
and 
. Now, the projective 
is also injective and lies in 
(because its unique proper predecessor is 
), hence in 
. Therefore, the component of 
containing it is neither directed, nor generalised standard.
4.5. LEMMA. Let 
be a component of 
.
(a) If 
is a non-connecting postprojective component, then 
(b) If 
is a non-connecting preinjective component, then 
(c) If 
intersects 
, then 
is connecting.
(d) If a connected component 
of 
is not tilted, then 
Proof. (a) Assume that 
is a non-connecting postprojective component of
such that 
Let 
be the (unique) connected component of 
such that 
consists of 
-modules. We claim that 
does not contain every indecomposable projective 
module. Indeed, if this is not the case, then the number of 
orbits in 
coincides with rk 
. By Theorem (C) (a), 
intersects each 
orbit of 
exactly once. Hence 
has rk 
elements. From this and (4.2) we deduce that 
is a convex tilting 
-module. By [6], (2.1), 
is a complete slice in mod
. But this is a contradiction, because 
was assumed to be non-connecting. This establishes our claim.
Now, let 
be an indecomposable projective 
module. Since 
is a connected algebra, there exists a walk of projective 
modules 
with 
Thus there exists 
such that 
and 
Since 
does not receive morphisms from other components of 
then Hom
By [21] (2.1) there exists, for each 
, a path
with 
irreducible. Since 
is as large as we want, and 
intersects each 
orbit of 
we may choose 
so that 
is a proper successor of 
On the other hand, 
is a projective 
module, hence a projective 
module lying in 
Thus 
Since 
is a successor of 
by (2.2), 
a contradiction which proves (a).
(b) Assume that 
is a non-connecting preinjective component of 
such that 
Using the same reasoning as in (a), there exist 
which is a proper predecessor of 
and an indecomposable injective 
module 
such that Hom
Since 
precedes 
then, by (4.1), 
The convexity of 
yields the contradiction 
This establishes (b).
(c) It is shown in [3, 24] that every connected component of 
is quasi-tilted. By Theorem (C), 
intersects only directed components of 
Furthermore, directed components of quasi-tilted algebras are necessarily postprojective, preinjective or connecting. Now the result follows from (a) and (b).
(d) Is a consequence of (c). 
4.6. PROPOSITION Let 
be a connected component of the left support 
such that 
Then 
is a tilted algebra and 
is a complete slice in 
.
Proof. Let 
be a component of 
such that 
. By (c) of the previous lemma, we know that 
is a connecting component. Since, on the other hand, 
intersects each 
-orbit of 
exactly once (by Theorem (C) (a)), we have 
. But by (4.2), 
. Hence 
and the direct sum of the modules in 
is a convex tilting 
-module. The result then follows from [6](2.1). 
Observe that if 
is a component of 
such that 
, then it follows from the proof of (4.6) that 
is a complete slice in mod
. Therefore, by [23], 
4.7. EXAMPLE. It is possible to have 
, and 
, while 
. Indeed, let 
be given by the quiver
bound by 
, 
, 
,, 
, 
, 
. Let 
denote the (tilted) full subcategory of 
having as objects 
, 
and 
, and 
denote the (tubular) full subcategory of 
having as objects 
, 
, 
, 
, 
, 
, 
, 
. Then 
, 
(it consists of the indecomposable modules 
, 
and 
) while 
(because 
is a tubular algebra).
5.1. An artin algebra 
is left supported if add
is contravariantly finite in mod
, in the sense of [8]. It is shown in [3] (5.1) that an artin algebra 
is left supported if and only if each connected component of 
is tilted and the restriction of 
to this component is a complete slice. Several other characterisations of left supported algebras are given in [1, 3]. In particular, it is shown in [1] that 
is left supported if and only if 
, where Pred
denotes the full subcategory of ind
having as objects all the 
such that there exists 
and a path 
. Our objective in this section is to give another proof of this theorem, using the results above. Our proof also yields a new characterisation of left supported algebras.
THEOREM. Let 
be an artin algebra. Then the following conditions are equivalent:
(a) 
is left supported.
(b) 
Pred 
(c) Every projective 
module which belongs to 
is a predecessor of 
Proof. (a) implies (b). Assume that 
is left supported. By [3](4.2), 
is cogenerated by the direct sum of the modules in 
. In particular, 
Pred 
. Since the reverse inclusion is obvious, this completes the proof of (a) implies (b).
Clearly (b) implies (c). To prove that (c) implies (a) we assume that every projective 
module which belongs to 
is a predecessor of 
Let 
be a connected component of 
and 
be an indecomposable projective 
-module. Since 
, there exist 
and a path 
in 
, hence in mod
. Therefore, mod
. By (4.6), 
is a tilted algebra and 
is a complete slice in 
. Hence 
is left supported. 
5.2. We end this paper with a short proof of a result by D. Smith.
THEOREM.([25] (3.8)) Let 
be a quasi-tilted algebra. Then 
is left supported if and only if 
is tilted having a complete slice containing an injective module.
Proof. Since 
is quasi-tilted, then 
. Assume that 
is left supported. Then 
. By (5.1), 
is tilted and 
is a complete slice in 
(mod
). Furthermore, since 
is quasi-tilted, then all projective 
-modules lie in 
, so that 
and 
. Thus 
must contain an injective module.
Conversely, if 
has a complete slice containing an injective, then there exists a complete slice 
having all its sources injective. By (2.1), 
. Since 
, it follows from [3] (3.3) that 
is left supported. 
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Ibrahim Assem
Département de Mathématiques,
Faculté des Sciences,
Université de Sherbrooke,
Sherbrooke, Québec,
Canada, J1K 2R1
ibrahim.assem@usherbrooke.ca
Juan Ángel Cappa
Instituto de Matemática,
Universidad Nacional del Sur,
8000 Bahía Blanca, Argentina
jacappa@yahoo.com.ar
María Inés Platzeck
Instituto de Matemática,
Universidad Nacional del Sur,
8000 Bahía Blanca, Argentina
platzeck@uns.edu.ar
Sonia Trepode
Departamento de Matemática,
Facultad de Ciencias Exactas y Naturales,
Funes 3350,
Universidad Nacional de Mar del Plata,
7600 Mar del Plata, Argentina
strepode@mdp.edu.ar
Recibido: 26 de enero de 2006
Aceptado: 7 de agosto de 2006
