Lattice structure of torsion classes for path algebras
aa r X i v : . [ m a t h . R T ] M a r LATTICE STRUCTURE OF TORSION CLASSES FOR PATH ALGEBRAS
OSAMU IYAMA, IDUN REITEN, HUGH THOMAS, GORDANA TODOROV
Abstract.
We consider module categories of path algebras of connected acyclic quivers. It isshown in this paper that the set of functorially finite torsion classes form a lattice if and only ifthe quiver is either Dynkin quiver of type A, D, E, or the quiver has exactly two vertices. Introduction
Let Λ be a finite dimensional algebra over an algebraically closed field k , and mod Λ the categoryof finite dimensional Λ-modules. In this setup a subcategory T is a torsion class if it is closedunder factor modules, isomorphisms and extensions. The set torsΛ of torsion classes is a partiallyordered set by inclusion, and it is easy to see that it is always a lattice (see Definition 1.2). Thereis however an important subset f-torsΛ of torsΛ, where f-torsΛ denotes the set of torsion classeswhich are functorially finite in mod Λ. In this setting a torsion class is functorially finite preciselywhen it is of the form
Fac X for some X in mod Λ [AS]. The set f-torsΛ is of special interest sincethe elements are in bijection with the support τ -tilting modules (see Definition 1.6), which wereintroduced in [AIR]. This bijection also induces a structure of partially ordered set on the support τ -tilting modules. A related partial order has been studied in classical tilting theory by manyauthors (e.g. [RS, HU, AI, K]). There is also a connection with the weak order on finite Coxetergroups [M].The aim of this paper is to study the following questions. Question 0.1.
Let Λ be a finite dimensional k -algebra. (a) When is f-torsΛ a complete lattice? (b)
When is f-torsΛ a lattice?
A simple answer to Question 0.1(a) is given in terms of the τ -rigid finiteness (see Definition 1.8for details): Theorem 0.2.
Let Λ be a finite dimensional k -algebra. Then the following conditions are equiv-alent. (a) f-torsΛ forms a complete lattice. (b) f-torsΛ forms a complete join-semilattice. (c) f-torsΛ forms a complete meet-semilattice. (d) f-torsΛ = torsΛ holds (i.e. any torsion class in mod Λ is functorially finite). (e) Λ is τ -rigid finite. On the other hand, Question 0.1(b) for an arbitrary algebra Λ above does not seem to have asimple answer. Hence we are mainly concerned with f-tors( kQ ) where kQ is the path algebra of afinite connected acyclic quiver Q . Our main theorem is the following. Theorem 0.3.
Let Q be a finite connected quiver with no oriented cycles. Then the followingconditions are equivalent. All of the authors were partially supported by MSRI. The first author was supported by JSPS Grant-in-Aidfor Scientific Research 24340004, 23540045 and 22224001. The second author was supported by FRINAT grants19660 and 231000 from the Research Council of Norway. The third author was partially supported by an NSERCDiscovery Grant. The fourth author was partially supported by NSF grant DMS-1103813.2010
Mathematics Subject Classification.
Key words and phrases.
Torsion class, lattice, τ -tilting module, tilting module. (a) f-tors( kQ ) forms a lattice. (b) f-tors( kQ ) forms a join-semilattice (see Definition 1.2). (c) f-tors( kQ ) forms a meet-semilattice. (d) Q is either a Dynkin quiver or has at most 2 vertices. We remark that condition (d) is equivalent to the property that all the rigid indecomposable kQ -modules are preprojective or preinjective.We also show the following result. Theorem 0.4.
Let Λ be a concealed canonical algebra (in particular a canonical algebra) or atubular algebra. Then the following conditions are equivalent. (a) f-torsΛ forms a lattice. (b) f-torsΛ forms a join-semilattice. (c) f-torsΛ forms a meet-semilattice. (d) Λ has at most 2 simple modules up to isomorphism. The paper is organized as follows. In section 1 we give a proof of Theorem 0.2 and give animportant criterion for deciding if f-torsΛ is a lattice, together with some preliminary results. Insubsection 2.1 we show our sufficient conditions for f-tors( kQ ) to be a lattice. In subsection 2.2we show that f-tors( kQ ) is not a lattice for a path algebra kQ of an extended Dynkin quiver Q with at least 3 vertices. In subsection 2.3 we deal with a path algebra kQ of a wild quiver Q with3 vertices, and show that f-tors( kQ ) is not a lattice. In subsection 2.4 we put things together toprove Theorem 0.3. In subsection 2.5 we prove Theorem 0.4. Acknowledgements.
The authors would like to thank Otto Kerner and Claus Michael Ringelfor valuable discussions. The authors are grateful to MSRI and Oberwolfach for having had theopportunity to work together in such inspiring environments.1.
Lattice structure of torsion classes for finite dimensional algebras
General results.
Let Λ be a finite dimensional k -algebra. A full subcategory F of mod Λis a torsionfree class if it is closed under submodules, isomorphisms and extensions. We denoteby torfΛ the set of all torsionfree classes in mod
Λ, and by f-torfΛ the set of all functorially finite torsionfree classes (i.e. torsionfree classes of the form
Sub X for some X ∈ mod Λ). The followingobservation is classical.
Proposition 1.1. (a)
We have a bijection torsΛ → torfΛ T 7→ T ⊥ := { X ∈ mod Λ | Hom Λ ( T , X ) = 0 } whose inverse is given by torfΛ → torsΛ F 7→ ⊥ F := { X ∈ mod Λ | Hom Λ ( X, F ) = 0 } . (b) [S] They induce bijections between f-torsΛ and f-torfΛ . Clearly torsΛ and f-torsΛ have a structure of partially ordered sets with respect to the inclusionrelation.
Definition 1.2.
Let P be a partially ordered set and x i ( i ∈ I ) be elements in P . If there existsa unique maximal element in the subposet { y ∈ P | y ≤ x i , ∀ i ∈ I } of P , we call it a meet of x i ( i ∈ I ) and denote it by V i ∈ I x i . Dually we define a join W i ∈ I x i . We say that P is a meet-semilattice (respectively, join-semilattice ) if any finite subset of P has a meet (respectively, join).We say that P is a lattice if it is a join-semilattice and a meet-semilattice. More strongly, we saythat P is a complete lattice (respectively, complete join-semilattice , complete meet-semilattice ) ifany subset of P has a meet and a join (respectively, a join, a meet).If a map f : P → P ′ between lattices preserves a join and a meet of any finite subset (respectively,any subset), we call f a morphism of lattices (respectively, complete lattices). ATTICE STRUCTURE OF TORSION CLASSES FOR PATH ALGEBRAS 3
We have the following statement.
Proposition 1.3. (a) torsΛ and torfΛ are complete lattices, and we have an isomorphism torsΛ → (torfΛ) op , T 7→ T ⊥ of complete lattices. (b) For torsion classes T i ( i ∈ I ) in mod Λ , we have ^ i ∈ I T i = \ i ∈ I T i and _ i ∈ I T i = ⊥ ( \ i ∈ I T ⊥ i ) . (c) For torsionfree classes F j ( j ∈ J ) in mod Λ , we have ^ j ∈ J F j = \ j ∈ J F j and _ j ∈ J F j = ( \ j ∈ J ⊥ F j ) ⊥ . Proof.
It is clear that a meet of torsion classes T i ( i ∈ I ) is given by T i ∈ I T i . Dually a meet oftorsionfree classes F j ( j ∈ J ) is clearly given by T j ∈ J F j .It is also clear that the bijection in Proposition 1.1 gives an isomorphism torsΛ → (torfΛ) op ofpartially ordered sets. Hence ⊥ ( T i ∈ I T ⊥ i ) gives a join of T i ( i ∈ I ), and W j ∈ J F j = ( T j ∈ J ⊥ F j ) ⊥ gives a join of F j ( j ∈ J ). (cid:3) Proposition 1.4.
Let Λ be a finite dimensional k -algebra. Then (a) We have an isomorphism of complete lattices: torsΛ → (tors(Λ op )) op , T 7→ D ( T ⊥ ) . (b) The map in (a) induces a bijection f-torsΛ → f-tors(Λ op ) . In particular, f-torsΛ forms ameet-semilattice (respectively, complete meet-semilattice) if and only if f-tors(Λ op ) formsa join-semilattice (respectively, complete join-semilattice).Proof. (a) We have an isomorphism torfΛ → tors(Λ op ), F 7→ D ( F ) of complete lattices. Thus theassertion follows from Proposition 1.3.(b) This follows from Proposition 1.1 since F is functorially finite if and only if so is D ( F ). (cid:3) We now show that f-torsΛ being a lattice is preserved by factoring by ideals h e i , where e is anidempotent element in Λ. Proposition 1.5.
Let Λ be a finite dimensional k -algebra, and e an idempotent in Λ . (a) f-tors(Λ / h e i ) is the interval {T ∈ f-torsΛ | ⊆ T ⊆ mod (Λ / h e i ) } in f-torsΛ . (b) If f-torsΛ is a lattice (respectively, complete lattice), then f-tors(Λ / h e i ) is a lattice (respec-tively, complete lattice).Proof. (a) This is shown in [AIR, Theorem 2.7] and [AIR, Proposition 2.27].(b) This is a consequence of (a), using that an interval of a lattice (respectively, complete lattice)is again a lattice (respectively, complete lattice). (cid:3) Proof of Theorem 0.2.
We denote by τ the Auslander-Reiten translation of Λ. Definition 1.6. (a) We call M ∈ mod Λ τ -rigid if Hom Λ ( M, τ M ) = 0. We call M ∈ mod Λ τ -tilting if it is τ -rigid and | M | = | Λ | holds, where | M | is the number of non-isomorphicindecomposable direct summands of M .(b) We call M ∈ mod Λ support τ -tilting if there exists an idempotent e of Λ such that M is a τ -tilting (Λ / h e i )-module.We denote by s τ -tiltΛ the set of isomorphism classes of basic support τ -tilting Λ-modules. Thenwe have the following result. Proposition 1.7. [AIR, Theorem 2.7]
There exists a bijection s τ -tiltΛ → f-torsΛ given by M Fac M . Using the bijection in Proposition 1.7, we regard s τ -tiltΛ as a partially ordered set which isisomorphic to f-torsΛ. OSAMU IYAMA, IDUN REITEN, HUGH THOMAS, GORDANA TODOROV
Definition 1.8. [DIJ] We say that Λ is τ -rigid finite if there are only finitely many indecomposable τ -rigid Λ-modules. This is equivalent to | s τ -tiltΛ | < ∞ , and to | f-torsΛ | < ∞ .For example, any local algebra is τ -rigid finite. In fact s τ -tiltΛ = { Λ , } holds in this case. Apath algebra kQ of an acyclic quiver Q is τ -rigid finite if and only if Q is a Dynkin quiver. On theother hand, any preprojective algebra of Dynkin type is τ -rigid finite [M].We say that two non-isomorphic basic support τ -tilting Λ-modules M and N are mutations ofeach other if M = X ⊕ U , N = Y ⊕ U and X and Y are either 0 or indecomposable. Then anysupport τ -tilting Λ-module has exactly n mutations.The following results play a crucial role. Proposition 1.9.
Let Λ be a finite dimensional k -algebra. (a) [AIR, Theorem 2.35] If M and N are support τ -tilting Λ -modules such that M > N , thenthere exists a mutation L of N such that M ≥ L > N . (b) [DIJ, Proposition 3.2] Assume that Λ is not τ -rigid finite. Then there exists an infinitedescending chain of mutations Λ = M > M > M > · · · . (c) [DIJ, Theorem 3.1] Λ is τ -rigid finite if and only if every torsion class in mod A is functo-rially finite. Now we are ready to prove Theorem 0.2.(d) ⇒ (a) This is immediate from Proposition 1.3(a).(a) ⇒ (c) This is clear.(c) ⇒ (e) We assume that f-torsΛ is a complete meet-semilattice and that Λ is not τ -rigid finite.Take an infinite descending chain in Proposition 1.9(b). Since s τ -tiltΛ ≃ f-torsΛ is a complete meet-semilattice by our assumption, there exists a meet M of M i ( i ≥
0) in s τ -tiltΛ. Let N , . . . , N n beall mutations of M . Since Fac M i ) Fac M , the set I i := { ≤ k ≤ n | M i ≥ N k > M } is non-emptyby Proposition 1.9(a). Since we have a descending chain I ⊃ I ⊃ I ⊃ · · · of finite non-empty sets, their intersection I := T i ≥ I i is also non-empty. Then any k ∈ I satisfies M i ≥ N k > M for all i . This is a contradiction since M is a meet of M i ( i ≥ ⇒ (d) This follows from Proposition 1.9(c).(a) ⇔ (b) We have already shown that the conditions (a), (c), (d) and (e) are equivalent. Re-placing Λ by Λ op , we have that (a) for Λ op is equivalent to (c) for Λ op . Using Proposition 1.4(b),we have the assertion. (cid:3) A criterion for the existence of joins and meets.
In this subsection, we need the fol-lowing result, which improves Proposition 1.9(a).
Proposition 1.10. [DIJ, Theorem 3.3]
Let M be a support τ -tilting Λ -module and T a torsionclass in mod Λ . (a) If Fac M ) T , then there exists a mutation N of M satisfying Fac M ) Fac N ⊃ T . (b) If Fac M ( T , then there exists a mutation N of M satisfying Fac M ( Fac N ⊂ T . Immediately we have the following property of non-functorially finite torsion classes.
Proposition 1.11.
Let Λ be a finite dimensional k -algebra, and T a torsion class in mod Λ whichis not functorially finite. (a) For any T ′ ∈ f-torsΛ satisfying T ′ ) T , there exists T ′′ ∈ f-torsΛ satisfying T ′ ) T ′′ ⊃ T . (b) For any T ′ ∈ f-torsΛ satisfying T ′ ( T , there exists T ′′ ∈ f-torsΛ satisfying T ′ ( T ′′ ⊂ T .Proof. The statement follows immediately from Propositions 1.7 and 1.10. (cid:3)
We give a more explicit criterion for existence of a meet and a join.
Theorem 1.12.
Let Λ be a finite dimensional k -algebra. (a) A subset {T i | i ∈ I } of f-torsΛ has a meet if and only if T i ∈ I T i is functorially finite. ATTICE STRUCTURE OF TORSION CLASSES FOR PATH ALGEBRAS 5 (b)
A subset {T i | i ∈ I } of f-torsΛ has a join if and only if ⊥ ( T i ∈ I T ⊥ i ) is functorially finite.Proof. We only have to prove (a) since (b) is a dual.If T i ∈ I T i is functorially finite, then it is a meet of T i ( i ∈ I ) in f-torsΛ, by Proposition 1.3.Thus we only have to prove the ‘only if’ part.Assume that T i ( i ∈ I ) has a meet S in f-torsΛ and that T := T i ∈ I T i is not functorially finite.Since S ⊂ T i for all i ∈ I , we have S ⊂ T . Since T is not functorially finite, we have S ( T .Applying Proposition 1.11, there exists S ′ ∈ f-torsΛ such that S ( S ′ ⊂ T . Thus S ′ ⊂ T i holds for any i ∈ I . This is a contradiction since S is a meet of T i ( i ∈ I ). (cid:3) Remark 1.13.
The statements in the above theorem mean that a meet (respectively, join) inf-torsΛ has to be the same as a meet (respectively, join) in the complete lattice torsΛ.2.
Lattice structure of torsion classes for path algebras
Sufficient conditions for f-tors( kQ ) to be a lattice. Let Q be a finite connected acyclicquiver. In this section we give two sufficient conditions for f-tors( kQ ) to be a lattice. Since for anartin algebra of finite representation type any subcategory is functorially finite, the first result isa direct consequence of the fact that tors( kQ ) is a lattice. Proposition 2.1. If Q is a Dynkin diagram, then f-tors( kQ ) is a lattice. When Q is a Dynkin diagram, the lattice f-tors( kQ ) was shown in [IT, Theorem 4.3] to be aCambrian lattice in the sense of Reading [Re] .The second sufficient condition is the following. Proposition 2.2.
Assume that Q has at most two vertices. Then f-tors( kQ ) is a lattice.Proof. If Q has one vertex, then kQ ∼ = k , hence the claim is obvious. Assume then that we havetwo vertices. Then our quiver Q is 1 ( n ) −−→
2, with n ≥ n ≥ Q is Dynkin. The Auslander-Reiten quiver is then of the form: A n ) ❇❇❇❇❇ A B n ) ❇❇❇❇❇ B . . . R . . .A n ) > > ⑤⑤⑤⑤⑤ A n ) > > ⑤⑤⑤⑤⑤ B n ) > > ⑤⑤⑤⑤⑤ B n ) > > ⑤⑤⑤⑤⑤ Here R consists of tubes when n = 2, and of Z A ∞ -components when n >
2. It is known that noindecomposable rigid module lies in R . The tilting modules are given by two consecutive verticesin the preprojective or preinjective component. So for i ≥ A i − ⊕ A i ,with associated torsion class T i = Fac ( A i − ⊕ A i ) which is equal to Fac A i − when i ≥
3. For i ≥ B i ⊕ B i − with associated torsion class T ′ i = Fac ( B i ⊕ B i − ) = Fac B i .There are no other tilting modules. The additional support tilting modules are the simple modules A and B , and hence we have the additional torsion classes T = Fac A and T ′ = Fac B .We have the inclusions { } ⊂ T ⊂ T ⊃ T ⊃ · · · ⊃ T i ⊃ · · · ⊃ T ′ j ⊃ · · · ⊃ T ′ for all elements off-tors( kQ ). It is clear that if neither T nor T ′ is T , then T ∨ T ′ is the larger one and T ∧ T ′ isthe smaller one. Further, T ∨ T = T (= mod kQ ) for T 6 = T , and T ∧ T = T , T ∧ T = { } for T 6 = T . (cid:3) Tame algebras.
In this section we deal with path algebras kQ of extended Dynkin quiverswith at least 3 vertices, and show that in that case the f-tors( kQ ) do not form lattices. Proposition 2.3.
Let Q be an acyclic extended Dynkin quiver with at least 3 vertices. Then f-tors( kQ ) is neither a join-semilattice nor a meet-semilattice. OSAMU IYAMA, IDUN REITEN, HUGH THOMAS, GORDANA TODOROV
Proof.
Since kQ is extended Dynkin with at least 3 vertices, there is a tube C of rank r ≥ r quasi-simple modules S , . . . , S r in C . Since S , . . . , S r are τ -rigid, we have that T = Fac S , . . . , T r = Fac S r are in f-tors( kQ ). By Theorem 1.12 there is a join of these T i in f-tors( kQ ) if and only if ⊥ ( T i ∈ I T ⊥ i ) is functorially finite, where I = { , . . . , n } . However ⊥ ( T i ∈ I T ⊥ i ) = add ( C ∪ { preinjectives } ) which is not functorially finite, since it clearly cannot bewritten as Fac Y for any Y . Therefore there is no join in f-tors( kQ ), and hence f-tors( kQ ) is not ajoin-semilattice.Since Q op is an acyclic extended Dynkin quiver with at least 3 vertices, f-tors( kQ op ) is not ajoin-semilattice. By Proposition 1.4, f-tors( kQ ) is not a meet-semilattice. (cid:3) Wild algebras.
In this section we show that f-tors( kQ ) is not a lattice when the quiver Q is connected wild, with 3 vertices.For a finite dimensional algebra Λ and a set S of Λ-modules, we denote by Filt S the fullsubcategory of mod Λ whose objects are the Λ-modules which have a finite filtration with factorsin S . Proposition 2.4.
Let Q be an acyclic quiver, and let M and N be indecomposable rigid kQ -modules such that Hom kQ ( M, N ) = 0 = Hom kQ ( N, M ) , Ext kQ ( M, N ) = 0 and Ext kQ ( N, M ) = 0 . (a) [Ri] The category A := Filt ( M, N ) is an exact abelian subcategory of mod kQ with twosimple objects M and N . (b) End kQ ( M ) ∼ = k ∼ = End kQ ( N ) holds, and M and N are regular. (c) For any ℓ ≥ , there exists an object X ℓ in A which is uniserial of length ℓ in A .Proof. (a) This is shown in [Ri, Theorem 1.2].(b) Since M and N are rigid, we have the first assertion. Since Ext kQ ( M, N ) = 0 andExt kQ ( N, M ) = 0 hold, M and N are in a cycle. Hence they are regular.(c) The assertion is clear for ℓ = 1. Assume that we have a uniserial object X ℓ of length ℓ in A . Without loss of generality, let M be the top of X ℓ in A . Then there exists an exact sequence0 → rad A X ℓ → X ℓ → M →
0. Since Ext kQ ( N, M ) = 0, there exists a non-split exact sequence0 → M → E → N →
0. Since kQ is hereditary, we have a commutative diagram of exactsequences: 0 00 / / M / / O O E / / O O N / / / / X ℓ / / O O Y / / O O N / / A X ℓ O O rad A X ℓ O O O O . O O Clearly Y belongs to the category A . We show that Y is uniserial of length ℓ + 1 in A . It is enoughto show rad A Y = X ℓ . Otherwise rad A Y is strictly contained in X ℓ , and hence rad A Y = rad A X ℓ holds since X ℓ is uniserial. Then Y / rad A Y = E holds, a contradiction since E is not semisimplein the category A . Thus the assertion follows. (cid:3) We shall also need the following.
Lemma 2.5.
Let C be a full subcategory of mod kQ closed under extensions. Then Fac C is alsoclosed under extensions.Proof. Let 0 → X → Y → Z → mod kQ , where X and Z are in Fac C .Then we have surjections f : C → X and g : C → Z , where C and C are in C . This gives riseto the exact sequence:Ext kQ ( C , C ) → Ext kQ ( C , X ) → Ext kQ ( C , Ker f ) = 0 . ATTICE STRUCTURE OF TORSION CLASSES FOR PATH ALGEBRAS 7
Thus we get the exact commutative diagrams:0 / / X / / Y ′ / / (cid:15) (cid:15) C / / (cid:15) (cid:15) / / X / / Y / / Z / / / / C / / (cid:15) (cid:15) Y ′′ / / (cid:15) (cid:15) C / / / / X / / Y ′ / / C / / C is extension closed, then Y ′′ is in C , and we have surjections Y ′′ → Y ′ → Y , so that Y isin Fac C , as desired. (cid:3) Combining the above results, we get the following.
Proposition 2.6.
Let kQ be an acyclic quiver, and M and N be kQ -modules satisfying the as-sumptions in Proposition 2.4. Let A := Filt ( M, N ) and T := Fac A . Then: (a) The subcategory T is a torsion class which is not functorially finite in mod ( kQ ) . (b) f-tors( kQ ) is neither a join-semilattice nor a meet-semilattice.Proof. (a) It follows from Lemma 2.5 that T is a torsion class. Let T := Fac M and T := Fac N .Since M and N are rigid, the subcategories T and T are in f-tors( kQ ).Assume that T is functorially finite. Then there exists a module X in T so that T = Fac X .By the definition of T , there is a module C in A and an epimorphism C → X in mod kQ , andhence T = Fac C . Now let ℓ be the Loewy length of C in A . Since the modules M and N satisfythe conditions of Proposition 2.4, there is a uniserial object X ℓ +1 of length ℓ + 1 in A . Since X ℓ +1 ∈ Fac C , there is an epimorphism C m → X ℓ +1 in mod kQ (and hence in A ) for some m ≥ X ℓ +1 is bigger than that of C .(b) If f-tors( kQ ) is a lattice, we know from section 1 that the join of Fac M and Fac N must bethe smallest torsion class containing Fac M and Fac N , which is clearly T . But since we have seenthat this is not a functorially finite subcategory of mod kQ by (a), it follows that f-tors( kQ ) is nota join-semilattice.Since the kQ op -modules DM and DN satisfy the conditions of Proposition 2.4, we have thatf-tors( kQ op ) is not a join-semilattice. By Proposition 1.4, f-tors( kQ ) is not a meet-semilattice. (cid:3) Now we are able to show the following result, where 1 ( a ) / / a multiple arrows from1 to 2. Lemma 2.7.
Let Q = 1 ( a ) / / ( c ) ( b ) / / be a quiver with a ≥ , b ≥ and c ≥ . Then thereexist M and N satisfying the conditions in Proposition 2.4.Proof. Let Q ′ := (1 ( a ) −−→
2) be a full subquiver of Q . We regard the projective kQ ′ -modulecorresponding to the vertex 1 as a kQ -module M , and let N := τ kQ M . We show that M and N satisfy the conditions in Proposition 2.4 with p > q >
0. We have dim M = (1 , a, t . Sincethe Cartan matrix of kQ (see [ASS]) is C = h a ab + c b i and the Coxeter matrix of kQ (see [ASS])is given by Φ = − C t · C − = − h a ab + c b i h − a − c − b i = (cid:20) a + abc + c − ab + bc − a − ab − ca + bc b − − bc b − (cid:21) , we have dim N = Φ · dim M = ( a b + 2 abc + c − , ab + bc, ab + c ) t . (Step 1) Since M is a rigid kQ ′ -module and Q ′ is a full subquiver of Q , it is a rigid kQ -module.Hence N is also a rigid kQ -module since τ preserves the rigidity of kQ -modules.Since M is rigid, we have Hom kQ ( M, N ) = Hom kQ ( M, τ M ) = 0. We have Ext kQ ( M, N ) =Ext kQ ( M, τ M ) ≃ D End kQ ( M ) = 0 by Auslander-Reiten duality.It remains to show that Hom kQ ( N, M ) = 0 and Ext kQ ( N, M ) = 0. OSAMU IYAMA, IDUN REITEN, HUGH THOMAS, GORDANA TODOROV (Step 2) To prove Hom kQ ( N, M ) = 0, it is enough to show Hom kQ ( M, τ − M ) = 0. Since M doesnot have S as a composition factor, it is enough to show that soc τ − M is a direct sum of copiesof S . Since S is injective, it does not appear in soc τ − M by the indecomposability of τ − M .Assume that S appears in soc τ − M . Then we have an exact sequence 0 → S → τ − M → L →
0. Applying Hom kQ ( − , S ), we have an exact sequenceExt kQ ( τ − M, S ) → Ext kQ ( S , S ) → Ext kQ ( L, S ) = 0 . Since Ext kQ ( S , S ) = 0, we have Ext kQ ( τ − M, S ) = 0. On the other hand, we have by Auslander-Reiten duality, Ext kQ ( τ − M, S ) ≃ D Hom kQ ( S , M ) = 0 , a contradiction.(Step 3) To prove Ext kQ ( N, M ) = 0, we calculate the Euler form, see [ASS]. We have h N, M i = (dim N ) t · ( C − ) t · dim M = ( a b + 2 abc + c − , ab + bc, ab + c ) h − a − c − b i h a i = − − a ( a b − b − − abc (2 a − − c ( a − , which is easily shown to be negative by our assumption a ≥ b ≥ c ≥ (cid:3) Now we show the following main result in this section.
Proposition 2.8.
Let Q be a connected acyclic wild quiver with 3 vertices. Then: (a) There exist M and N satisfying the conditions in Proposition 2.4. (b) f-tors( kQ ) is neither a join-semilattice nor a meet-semilattice.Proof. (a) Let a, b, c be integers such that a ≥ b ≥ c ≥
0. Then Q has one of the followingforms:(i) : 1 ( a ) / / ( c ) ( b ) / / , (ii) : 1 ( b ) / / ( a ) ( c ) / / , (iii) : 1 ( c ) / / ( b ) ( a ) / / , (iv) : 1 ( b ) / / ( c ) ( a ) / / , (v) : 1 ( c ) / / ( a ) ( b ) / / , (vi) : 1 ( a ) / / ( b ) ( c ) / / . First, the case (i) was shown in Lemma 2.7. Next, the case (ii) (respectively, (iii)) follows fromthe case (i) by using the reflection functor at the vertex 1 (respectively, 3). Finally the case (iv)(respectively, (v), (vi)) follows from the case (i) (respectively, (ii), (iii)) by using the k -dual.(b) This follows from (a) and Proposition 2.6. (cid:3) Remark 2.9.
When a, b, c ≥
1, it is easy to check that the modules M = S and N := k ( a ) / / ( c ) f =(1 ,..., A A ( b ) / / k c also satisfy the conditions in Proposition 2.4 with p > q > Proof of Theorem 0.3.
We need the following preparation, which is an analog of a well-known result, see [ASS, Lemma VII.2.1].
Proposition 2.10.
Let Q be a finite connected quiver. Then one of the following holds. (a) Q is a Dynkin quiver. (b) Q has at most two vertices. (c) Q has an extended Dynkin full subquiver with at least 3 vertices. (d) Q has a connected wild full subquiver with exactly 3 vertices. ATTICE STRUCTURE OF TORSION CLASSES FOR PATH ALGEBRAS 9
Proof.
First, assume that Q has multiple arrows from i to j . If Q has exactly two vertices, then wehave the case (b). If Q has at least 3 vertices, then any connected full subquiver of Q consistingof i , j and one more vertex is wild. Thus we have the case (d).Next, assume that Q has no multiple arrows. Then it follows from [ASS, Lemma VII.2.1] thatwe have either the case (a) or (c). (cid:3) Now we are ready to prove Theorem 0.3.(d) ⇒ (a) If Q is a Dynkin quiver, then f-tors( kQ ) forms a lattice by Proposition 2.1. If Q hasexactly two vertices, then f-tors( kQ ) forms a lattice by Proposition 2.2.(a) ⇒ (b) This is clear.(b) ⇒ (d) Assume that Q does not satisfy the condition (d). Then by Proposition 2.10, Q has either an extended Dynkin full subquiver with at least 3 vertices, or a connected wild fullsubquiver with exactly 3 vertices, For the former case (respectively, latter case), f-tors( kQ ) is nota join-semilattice by Propositions 2.3 (respectively, 2.8) and 1.5(b).(c) ⇔ (d) By Proposition 1.4, the condition (c) is equivalent to that f-tors( kQ op ) forms a join-semilattice. This is equivalent to that Q op is either a Dynkin quiver or has at most two vertices,by using the equivalence (b) ⇒ (d) for the quiver Q op . This is clearly equivalent to the condition(d). (cid:3) Concealed canonical algebras and tubular algebras.
Inspired by the proof that f-torsΛis not a join-semilattice for path algebras of extended Dynkin quivers with at least 3 vertices, wehave the following.
Proposition 2.11.
Let Λ be a finite dimensional k -algebra such that the set of indecomposable Λ -modules is a disjoint union P ∪ R ∪ Q , where R is a family of stable standard orthogonal tubes, Hom Λ ( R , P ) = 0 , Hom Λ ( Q , R ) = 0 and Hom Λ ( Q , P ) = 0 . If there is a tube C in R of rank r ≥ ,then f-torsΛ is neither a join-semilattice nor a meet-semilattice.Proof. We only prove the assertion for join-semilattices since the other assertion follows by Propo-sition 1.4.Let S , . . . , S r be the indecomposable modules at the border of C . Since C is standard, then S , . . . , S r are τ -rigid, and hence Fac S i is in f-torsΛ for i = 1 , . . . , n . Let T := W ni =1 Fac S i intorsΛ. Then T is the smallest torsion class in mod Λ containing C . Since add ( C , Q ) is a torsionclass by our assumptions, we have T ⊂ add ( C , Q ). Now if T is functorially finite, then there exists M ∈ T such that T = Fac M . Since Hom Λ ( Q , C ) = 0 holds by our assumption, the maximal directsummand N of M contained in add C satisfies C ⊂ Fac N . But this is impossible since add C isequivalent to the category of finite dimensional modules over the complete path algebra k b Q of thequiver Q of type e A r − by our assumption, and hence there is no upper bound of Loewy length ofobjects. (cid:3) Now we are ready to prove Theorem 0.4. It follows from Proposition 2.11 and by the propertiesof the concealed canonical (respectively, tubular algebras) listed in [SS, page 380] (respectively,[SS, Theorem XIX.3.20]) since there exists a tube C of rank r ≥ (cid:3) References [AIR] T. Adachi, O. Iyama, I. Reiten, τ -tilting theory , Compos. Math. 150 (2014), no. 3, 415–452.[AI] T. Aihara, O. Iyama, Silting mutation in triangulated categories , J. Lond. Math. Soc. (2) 85 (2012), no. 3,633–668.[ASS] I. Assem, D. Simson, A. Skowro´nski,
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Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya. 464-8602, Japan
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