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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dez. 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
![(*) M = M -f1→ M -f2→ ⋅⋅⋅ -→ M -f→t M = N 0 1 t-1 t](/img/revistas/ruma/v46n2/2a02139x.png)
![fi](/img/revistas/ruma/v46n2/2a02140x.png)
![Mi](/img/revistas/ruma/v46n2/2a02141x.png)
![A](/img/revistas/ruma/v46n2/2a02142x.png)
![M](/img/revistas/ruma/v46n2/2a02143x.png)
![N](/img/revistas/ruma/v46n2/2a02144x.png)
![N](/img/revistas/ruma/v46n2/2a02145x.png)
![M](/img/revistas/ruma/v46n2/2a02146x.png)
![M](/img/revistas/ruma/v46n2/2a02147x.png)
![M](/img/revistas/ruma/v46n2/2a02148x.png)
![M](/img/revistas/ruma/v46n2/2a02149x.png)
![(*)](/img/revistas/ruma/v46n2/2a02150x.png)
![fi](/img/revistas/ruma/v46n2/2a02151x.png)
![τAMi+1 ⁄= Mi- 1](/img/revistas/ruma/v46n2/2a02152x.png)
![i](/img/revistas/ruma/v46n2/2a02153x.png)
![(*)](/img/revistas/ruma/v46n2/2a02154x.png)
![′ f′1 ′ f′2 ′ f′s ′ M = M 0 -→ M 1 -→ ⋅⋅⋅ - → M s-1- → M s = N](/img/revistas/ruma/v46n2/2a02155x.png)
![σ : {1,⋅⋅⋅ ,t - 1} - → {1,⋅⋅⋅ ,s - 1}](/img/revistas/ruma/v46n2/2a02156x.png)
![Mi = M ′σ(i)](/img/revistas/ruma/v46n2/2a02157x.png)
![i](/img/revistas/ruma/v46n2/2a02158x.png)
![C](/img/revistas/ruma/v46n2/2a02159x.png)
![A](/img/revistas/ruma/v46n2/2a02160x.png)
![(*)](/img/revistas/ruma/v46n2/2a02161x.png)
![M](/img/revistas/ruma/v46n2/2a02162x.png)
![N ∈ C](/img/revistas/ruma/v46n2/2a02163x.png)
![Mi](/img/revistas/ruma/v46n2/2a02164x.png)
![C](/img/revistas/ruma/v46n2/2a02165x.png)
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
![LA = {M ∈ indA ∣ pdL ≤ 1 for any predecessor L of M }.](/img/revistas/ruma/v46n2/2a02170x.png)
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:
Hom.
and a path
.
and a sectional path
.
(b) The following conditions are equivalent for which does not satisfy conditions (a):
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
![Y → * → τ-A 1Y → ⋅⋅⋅ → τ-A tY](/img/revistas/ruma/v46n2/2a02266x.png)
![L A](/img/revistas/ruma/v46n2/2a02267x.png)
![τ-1Y ∈ L A A](/img/revistas/ruma/v46n2/2a02268x.png)
![Y](/img/revistas/ruma/v46n2/2a02269x.png)
(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
![2 // \\ 1 ----------3](/img/revistas/ruma/v46n2/2a02433x.png)
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
![HomA(X, τAY ) ≃ D Ext1A(Y, X) = 0.](/img/revistas/ruma/v46n2/2a02527x.png)
(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
![f1 f2 fs f Pi = M0 - → M1 - → M2 - → ⋅⋅⋅-→ Ms = L - → Pi+1](/img/revistas/ruma/v46n2/2a02839x.png)
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