中山多元環の τ 傾加群 TAKAHIDE ADACHI

Throughout this note, K is an algebraically closed field and Λ is a basic finite dimensional K-algebra. We denote by mod Λ the category of finite dimensional right Λ-modules and by τ the Auslander-Reiten translation of mod Λ. 1. Main result. Recently, as a generalization of tilting modules, the notion of τ -tilting modules was introduced in [AIR]. In this note, we give a classification of τ -tilting modules over Nakayama algebras. For simplicity, we assume some conditions for Nakayama algebras. Our main result of this note is the following. Theorem 1. Let Λ be a Nakayama algebra with n simple modules. Assume that the Loewy length of each indecomposable projective Λ-module is not less than n. Then there are bijections between (a) (b) (c) (d)

the set τ -tilt Λ of isomorphism classes of basic τ -tilting Λ-modules, the set psτ -tilt Λ of isomorphism classes of basic proper support τ -tilting Λ-modules, the set T (n) of triangulations of the n-gon with a puncture, P the set Z(n) of sequences (a1 , a2 , · · · , an ) of nonnegative integers with ni=1 ai = n.

In this note, we only give the proof of the case (a) ↔ (c).

2. τ -tilting modules. We recall the definition of τ -tilting modules. We denote by |M | the number of pairwise nonisomorphic indecomposable summands of a module M in mod Λ. Definition 2. (1) We call M in mod Λ τ -rigid if HomΛ (M, τ M ) = 0. (2) We call M in mod Λ τ -tilting if M is τ -rigid and |M | = |Λ|. (3) We call M in mod Λ support τ -tilting if there exists an idempotent e ∈ Λ such that M is a τ -tilting (Λ/hei)-module. If moreover e 6= 0, M is said to be a proper support τ -tilting Λ-module. We refer to [AIR] for the details about τ -tilting modules. In this note, we need the following property. Proposition 3. [AIR, Theorem 2.12] Let M be a τ -rigid Λ-module. Then M is a τ -tilting Λ-module if and only if M is a maximal τ -rigid Λ-module (i.e. if M ⊕ N is τ -rigid for some Λ-module N , then N ∈ add M ). We denote by τ -rigid Λ the set of isomorphism classes of indecomposable τ -rigid Λmodules. 3. Nakayama algebras. We recall some properties of Nakayama algebras. For more details, we refer to [ASS, Section V]. Let Λ be a Nakayama algebra with n simple modules. Assume that the Loewy length of each indecomposable projective Λ-module is not less than n. Namely, the quiver QΛ of 1

2

TAKAHIDE ADACHI

Λ is the following. 1

x xx xx x x x{ x αn

n

aCC CC α1 CC CC 2O

αn−1

α2



n−1

FF FF F αn−2 FFF #

···

{= {{ { {{ α {{ 3

3

Then we denote by Pi (respectively, Si ) the indecomposable projective (respectively, the simple) Λ-module corresponding to the vertex i in QΛ . We collect basic results for Nakayama algebras. We denote the Loewy length of a module M by ℓ(M ). Proposition 4. [ASS, V.3.5, V.4.1 and V.4.2] Let Λ be a Nakayama algebra and M ∈ mod Λ an indecomposable nonprojective module with ℓ(M ) = t. Then there exists i ∈ {1, 2, · · · , n} such that M ≃ Pi / radt (Pi ). In this case, we have τ M ≃ rad(Pi )/ radt+1 (Pi ) and ℓ(τ M ) = ℓ(M ). By Proposition 4, any indecomposable Λ-module M is uniquely determined by its simple top Sj and the Loewy length ℓ(M ) = t. Namely, M has a unique composition series of the form Sj =: Si1 , Si2 , · · · , Sit , where i1 , i2 , · · · , it ∈ {1, 2, · · · , n} with il+1 = il − 1 (mod n) for any l. Let Λrn be a self-injective Nakayama algebra with n simple modules and the Loewy length ℓ(Λ) = r. For example, the Auslander-Reiten quiver of Λ54 is given by the following. 3 2 1 4 3

2 1 4

1

>> >> >> >> > {= {{ { {{ {{ o HHH HHH HH $ 8 q q q q q qo q

? 3 2 1 4

2 1

o CC CC CC CC ! : vvv v v vvv Mo MM MMM M&

4 3 2 1 4

3 2 1

2

>> >> >> >> > {= {{ { {{ {{ o HHH HHH HH $ 8 q q q q q qo q

? 4 3 2 1

3 2

1 4 3 2 1

o

CC CC CC CC ! : vvv v v vvv oM MMM MMM &

4 3 2

3

>> >> >> >> > {= {{ { {{ {{ o HHH HHH HH $ 8 q q q q q q oq

? 1 4 3 2

4 3

o CC CC CC CC ! : vvv v v vvv Mo MM MMM M&

2 1 4 3 2

1 4 3

4

>> >> >> >> > {= {{ { {{ {{ o HHH HHH HH $ 8 q q q q q qo q

? 2 1 4 3

1 4

o CC CC CC CC ! : vvv v v vvv Mo MM MMM M&

3 2 1 4 3

2 1 4

1

We can easily understand the exsitence of homomorphisms between indecomposable modules. Lemma 5. Let Λ be a Nakayama algebra with n simple modules. Assume that M = Pi / radj Pi and N = Pk / radl Pk . Then the following are equivalent. (1) HomΛ (M, N ) 6= 0. (2) i ∈ {k, k − 1, · · · , k − l + 1 (mod n)} and k − l + 1 ∈ {i, i− 1, · · · , i− j + 1 (mod n)}. By Lemma 5, we give a criterion for indecomposable modules to be τ -rigid.

中山多元環の τ 傾加群

3

Corollary 6. Let Λ be a Nakayama algebra with n simple modules and M an indecomposable nonprojective Λ-module. Then M is a τ -rigid Λ-module if and only if ℓ(M ) < n. Proof. Let M = Pi / radj Pi by Proposition 4. The assertion follows from ( i ∈ {i − 1, i − 2, · · · , i − j (mod n)} 5 HomΛ (M, τ M ) 6= 0 ⇐⇒ i − j ∈ {i, i − 1, · · · , i − j + 1 (mod n)} ⇐⇒ ℓ(M ) = j ≥ n.  4. Triangulations of polygons with a puncture. We recall the definition and properties of triangulations of polygons with a puncture. Let Gn be an n-gon with a puncture. We label the points of Gn clockwise around the boundary by 1, 2, · · · , n (mod n). Definition 7. Let i, j ∈ Zn := Z/nZ. (1) An outer arc hi, ji in Gn is a path from the vertex i to j the boundary path i, i + 1, · · · , i + l = j (mod n) such that 2 ≤ l ≤ n. Then we call l the length of hi, ji. (2) A projective arc h•, ji in Gn is a path from the puncture to j. We denote by O(n) (respectively, P (n)) the set of outer (respectively, projective) arcs. We call an element of A(n) := P (n) ⊔ O(n) an admissible arc. We distinguish the following admissible arcs. Remark 8. Let i, j ∈ Zn with i 6= j. (1) hi, ji = 6 hj, ii. (2) hi, ii = 6 h•, ii.

j j−1

j j−1 i+1 i

hi, ji

i+1 i

hj, ii

i

hi, ii

i

h•, ii

Figure 1. Admissible arcs in a polygon with a puncture We give the definition of triangulations. Definition 9. Two admissible arcs in Gn are called compatible if they do not intersect in Gn . A triangulation of the polygon Gn is a maximal set of distinct pairwise compatible arcs in Gn .

Figure 2. Triangulations of G4 We have the following properties for triangulations.

4

TAKAHIDE ADACHI

Proposition 10. (1) T (n) 6= ∅. (2) Each triangulation of the polygon Gn has a projective arc and consists of n admissible arcs. 5. Proof of Theorem 1. We give a bijection between (a) and (c) in Theorem 1. First, we study a connection between indecomposable τ -rigid Λ-modules and admissible arcs in Gn . By Corollary 6, each indecomposable nonprojective τ -rigid Λ-module M is uniquely determined by its simple socle Sk and its simple top Sj . Thus we denote by Mk−2,j the module M . Moreover, we let M•,i := Pi . Proposition 11. There is a bijection τ -rigid Λ ←→ A(n) given by Mi,j 7→ hi, ji, where i ∈ Zn ⊔ {•} and j ∈ Zn . Proof. It is clear that the map Pi 7→ h•, ii is a bijection. Next, we claim that the map Mi,j 7→ hi, ji is well-defined for any nonprojective module Mi,j . Since the length of the outer arc corresponding to Mi,j is equal to ℓ(Mi,j ) + 1, the claim thus follows from Corollary 6. Moreover, we can easily check that the map is a bijection.  By Proposition 11, we have the following result. Theorem 12. The map in Proposition 11 induces a bijection between τ -tilt Λ and T (n). Proof. We remark that each τ -tilting Λ-module is a maximal τ -rigid Λ-module and has n indecomposable summands. On the other hand, each triangulation is a maximal set of compatible arcs and has n admissible arcs. Thus we only have to show the following claim. Let i, k ∈ Zn ⊔ {•} and j, l ∈ Zn . Then Mi,j ⊕ Mk,l is a τ -rigid Λ-module if and only if the admissible arcs hi, ji and hk, li are compatible. Indeed, it follows from Lemma 5. Thus the map in Proposition 11 gives a well-defined map sτ -tilt Λ → T (n). Hence the assertion follows.  Finally, we give maps (i) τ -tilt Λ → psτ -tilt Λ and (ii) τ -tilt Λ → Z(n). (i) By Proposition 10 and Theorem 12, each τ -tilting Λ-module has a nonzero projective module as a direct summand. We decompose M ∈ τ -tilt Λ as M = Mnp ⊕Mpr , where Mpr is a maximal projective direct summand of M . Thus we can show that Mnp is a proper support τ -tilting Λ-module and the map M 7→ Mnp gives a bijection between τ -tilt Λ and psτ -tilt Λ. (ii) The dimension vector of a Λ-module M is defined to be the vector dimM := (dimK M e1 , dimK M e2 , · · · , dimK M en ) where e1 , e2 , · · · , en are primitive orthogonal idempotents of Λ corresponding to the vertices 1, 2, · · · , n of QΛ . Since every indecomposable Λ-module has a simple top, we have dim top(M ) ∈ Z(n). Thus we can show that the map M 7→ dim top(M ) gives a bijection τ -tilt Λ and Z(n). As an application of Theorem 1, we have the following. Corollary 13. Let Λ := Λrn is a self-injecitve Nakayama algebra with r ≥ n. Then the
cardinality of the set of isomorphism classes of support τ -tilting Λ-modules is equal to 2n n .
Proof. Since the cardinality of Z(n) is equal to 2n−1 n−1 , the assertion follows from Theorem 1. 

中山多元環の τ 傾加群

T (4)

Z(4)

τ -tilt Λ54

5

τ -tilt Λ44

psτ -tilt Λ44

1

4

2

(1, 1, 1, 1)

1 4 3 2 1

2 1 4 3 2

3 2 1 4 3

4 3 2 1 4

1 4 3 2

2 1 4 3

3 2 1 3

(2, 1, 1, 0)

1 4 3 2 1

2 1 4 3 2

3 2 1 1 4 3

1 4 3 2

2 1 4 3

3 2 1 1 4

(2, 0, 2, 0)

1 3 4 2 3 1 1 3 2 4 1 3

(3, 1, 0, 0)

1 4 3 2 1

2 1 1 4 4 1 3 2

(2, 1, 0, 1)

1 4 3 2 1

2 1 1 4 4 4 3 2

(4, 0, 0, 0)

4 3 2 1

{0}

3

1

1 3 4 1 2 3 3 1 2 4

1 3

1 4 3 2

2 1 1 1 4 4 3

1 1 4

1 4 3 2

2 1 1 4 4 4 3

1 4 4

1 4 1 1 3 4 4 1 2 3 1

1 1 4 4 1 1 3 3 4 2

1 1 4 4 1 3

(3, 0, 1, 0)

1 4 1 3 4 1 3 2 3 1

1 1 4 4 1 3 3 3 2

1 4 1 3 3

(2, 0, 1, 1)

1 4 1 4 3 4 3 3 2 3 1

1 1 4 4 4 3 3 3 3 2

1 4 4 3 3 3

(2, 0, 0, 2)

1 4 1 4 3 4 3 4 2 3 1

1 1 4 4 4 4 3 3 3 2

1 4 4 3 4 3

(3, 0, 0, 1)

1 4 1 1 3 4 4 4 2 3 1

1 1 4 4 1 4 3 3 4 2

1 1 4 4 4 3

Table 1. Example of Theorem 1

6

TAKAHIDE ADACHI

References [AIR] T. Adachi, O. Iyama, I. Reiten, τ -tilting theory. preprint (2012), arXiv: 1210.1036. [ASS] I. Assem, D. Simson, A. Skowro´ nski, Elements of the Representation Theory of Associative Algebras. Vol. 1. London Mathematical Society Student Texts 65, Cambridge university press (2006). Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan E-mail address: [email protected]