GLOBAL DIMENSION IN LEFT SERIAL ALGEBRAS

MORIO UEMATSU

Let A be a finite dimentional basic conected algebra over an algebraically closed field k, and n is the number of the non isomorphic simple right modules of A. If A is serial and gl.dim.A is finite, then gl.dim.A ≤ 2n − 2 and l(A) ≤ 2n − 1 [3].

If A is quasi-hereditary, then the global dimension gl.dim.A is less than or equal to 2n − 2, and the Loewy length l(A) of A is less than or equal to 2^{n} − 1[4]. If A is serial and gl.dim.A is finite, then gl.dim.A ≤ 2n − 2 and l(A) ≤ 2n − 1. On the other hand, Yamagata constructed the algebras of large global dimension with few simple modules [5], and Deng constructed the algebras of arbitrary finite global dimension with n ≥ 2 arbitrary [1]. So, if we consider the relationship between gl.dim.A and n, we need some conditions on algebras. In this note we show that the algebra whose quiver contains unique oriented cycle has global dimension less than or equal to 2n − 2 if it has finite global dimension. Moreover, in case of quasi-hereditary algebras, we compute the global dimension of an algebra whose quiver contains some essentital oriented cycles.

1. The algebra whose quiver contains at most one oriented cycle

First we fix some notations. Let Q_{A} be the quiver of A, {1, 2, ⋯, n} be its vertices, and {e_{1}, ⋯, e_{n}} be the set of corresponding primitive idempotents of A.

A vertex i of Q_{A} is called a sink(resp., source) vertex if all arrows which contain i have i as ending(resp., starting) point. If vertex i is a sink vertex, then Ae_{i} is a simple projective module. Dually, if vertex i is a source vertex, then e_{i}A is a simple injective module.

Lemma 1.1. Suppose that Q_{A} has a sink or source vertex, that is there is a primitive idempotent e of A with simple projective module Ae or simple injective module eA. Let B = A/AeA. Then gl.dim.A ≤ gl.dim.B + 1, l(A) ≤ l(B) + 1, and these bounds are sharp.

Proof. The second inequality is trivial. For the first, assume that Ae is simple projective. Since AeA is direct sum of Ae, _{A}AeA is projective as left A-module. Let _{A}X be an arbitary left A-module. If _{A}AeAX ≠ 0, AeAX is direct sum of Ae and projective. Hence, proj.dim._{A}X ≤ proj.dim._{A}(X/AeAX). If AeAX = 0, _{A}X is a B-module. It is enough to show that if X is a B-module, then

proj.dim._{A}X ≤ 1 + proj.dim._{B}X.

This is the same as the Statement 1 of [4].

In case of eA is simple injective, we can show by duality.

Remark 1.1. In the above Lemma, the word “simple” can be replaced by “semi simple”.

Lemma 1.2. If A is an algebra whose quiver has no oriented cycles then gl.dim.A ≤ n−1.

Proof. If there are no sink vertices, each vertex is the starting point of some arrow. Then there exists an infinite chain of arrows and this forms an oriented cycle.

So we may assume that n is a sink vertex. Set B = A/Ae_{n}A. Since B has no oriented cycles, using Lemma 1.1 we complete the proof.

Example 1. In the above lemma, the bound of global dimension is sharp. Let A be an algebra defined by the following quiver with relations {α_{i+1}α_{i} = 0(1 ≤ i ≤ n − 2)}.

1 ⟶^{α_{1}} 2 ⟶^{α_{2}} ⋯ ⟶^{α_{n−1}} n

Then the global dimension of A is n − 1.

Next theorem is due to Gustafson[3].

Theorem 1.3 (Gustafson). Let A be a serial algebra. If the global dimension of A is finite, then gl.dim.A ≤ 2n − 2, l(A) ≤ 2n − 1, and these bounds are sharp.

Serial algebras have at most one oriented cycle. So, next theorem is generalization of Theorem 1.3.

Theorem 1.4. Let A be an algebra whose quiver contains at most one oriented cycle. If the global dimension of A is finite, then gl.dim.A ≤ 2n − 2, l(A) ≤ 2n − 1, and these bounds are sharp.

Proof. Using Lemma 1.2, we may assume Q_{A} has unique oriented cycle. If Q_{A} has a sink or source vertex, take the corresponding idempotent e and make new algebra B = A/AeA. Then Q_{B} contains one oriented cycle and by the Lemma 1.1 gl.dim.A ≤ gl.dim.B + 1. We continue this process until we get the algebra U that has no simple projective nor simple injective module. Then Q_{U} is the following shape.

[figure: 1; 2; ⋯; m]

U must be a serial algebra. So the Theorem 1.3, gl.dim.U ≤ 2m − 2 and l(U) ≤ 2m − 1 where m is the number of the non isomorphic simple left modules of U. Using Lemma 1 repeatedly, we can prove the theorem.

For the sharpness of bounds, these are the same as the Gustafson’s Example[3].

If A is a left or right (not necessarily both) serial algebra and has finite global dimension, the condition of the theorem holds. So the following corollary is the direct consequence of the Theorem 1.4.

Corollary 1.5. Let A be a left or right serial algebra. If the global dimension of A is finite, then gl.dim.A ≤ 2n − 2, l(A) ≤ 2n − 1, and these bounds are sharp.

More over if one of the equality holds, A is a serial algebra.

2. Quasi-hereditary algebras with some oriented cycles

Let N be the Jacobson radical of A. An idempotent e is said to be a heredity idempotent of A if eNe = 0 and AeA is projective as a right A-module.

Let {e_{1}, e_{2}, ⋯, e_{n}} be fixed ordering of the complete set of primitive orthogonal idempotents of A. An algebra A is said to be quasi-hereditary algebra with respect to this ordering if for any 1 ≤ t ≤ n, \overline{ε_{t}} is a heredity idempotent of A/Aε_{t+1}A, where ε_{j} = e_{j}+e_{j+1}+⋯+e_{n} for 1 ≤ j ≤ n and ε_{n+1} = 0. Such a sequence called a heredity sequence.

The following lemma is due to Dlab and Ringel[4].

Lemma 2.1. Suppose that A has a heredity idempotent e. Let B = A/AeA. Then gl.dim.A ≤ gl.dim.B + 2 l(A) ≤ 2l(B) + 1.

We call an oriented cycle is essential when it is made by the distinct vertices.

Theorem 2.2. Let A be a quasi-hereditary algebra with m essential oriented cycles. Then gl.dim.A ≤ n + m − 1, l(A) ≤ 2^{m}(n − m + 1), and these bouds are sharp.

Proof. If Ae_{n} is simple projective or e_{n}A is simple injective, we use Lemma 1.1 and consider the algebra B = A/Ae_{n}A. Then B is again quasi-hereditary algebra with respect to {e_{1}, e_{2}, ⋯, e_{n−1}} which has same number of essentital oriented cycles. Otherwise, if n is not sink nor source vertex, then n is belong to some essential oriented cycle. In this case, we use Lemma 2.1 and consider the algebra B = A/Ae_{n}A. B is again quasi-hereditary algebra with respect to {e_{1}, e_{2}, ⋯, e_{n−1}} that has m − 1 or less essential oriented cycles.

We continue these processes. Second case does occur at most m times. So we conclude that

gl.dim.A ≤ 2m + (n − m) = n + m − 1,
l(A) ≤ 2^{m}(n − m + 1).

Remark 2.1. In the above theorem, we may assume m ≤ n − 1. Since otherwise these bounds exceed 2n − 2 and 2^{n} − 1 respectively.

Corollary 2.3. Let A be a left serial quasi-hereditary algebra. Then gl.dim.A ≤ n and l(A) ≤ 2n − 1, and these bounds are sharp.

Example 2. Let A be a serial algebra of second type with relation {α_{i+1}α_{i} = 0(1 ≤ i ≤ n − 2), α_{1}α_{n} = 0}. Where n ≥ 3 and α_{i} is an arrow from i to i + 1 (1 ≤ i ≤ n − 1), and α_{n} is an arrow from n to 1. Then S_{n} (the simple module corresponding to the vertex n ) is of projective dimension n and the other simple modules are of projective dimension less than n. So gl.dim.A = n. A is quasi-hereditary with heredity idempotent e_{n}.

Example 3 (Gustafson). Let A be a serial algebra of second type with relation {α_{1}α_{2} ⋯ α_{n} = 0 } Then S_{i} is of projective dimension one, for 2 ≤ i ≤ n, while S_{1} is of projective dimension 2. Hence, gl.dim.A = 2, and A is quasi-hereditary. The length of Ae_{2} is 2n − 1, and this is the maximal length among the indecomposable projective modules of A. We have l(A) = max{l(e_{i}A)} = 2n − 1.

Example 4. Let A be the algebra of

\xymatrix{1 \ar[r]^{α_{1}} & 2 \ar[l]^{β_{1}} \ar[r]^{α_{2}} & 3 \ar[l]^{β_{2}} \ar[r]^{α_{3}} & 4 \ar[l]^{β_{3}} \ar[r]^{α_{4}} & 5}

with relation {α_{4}α_{3} = α_{3}α_{2} = α_{2}α_{1} = β_{3}β_{2} = β_{2}β_{1} = 0, α_{1}β_{1} = β_{2}α_{2}, α_{2}β_{2} = β_{3}α_{3}}. Then A is a quasi-hereditary algebra with respect to {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}} and has 3 essential oriented cycles. Ae_{5} is simple projective module. This is the case of n = 5 and m = 3 in Theorem 2.2, and gl.dim.A = 7. This shows that the bound of global dimension is sharp.

Example 5. Let A be the algebra of

[figure: 1; α_{1}; 2; β_{1}; α_{2}; 5; β_{2}; 4; α_{4}; 3; α_{3}; 6; β_{3}]

with relation {α_{4}α_{3} = α_{3}α_{2} = α_{1}α_{4} = 0, β_{2}β_{1} = β_{1}β_{3} = 0, β_{1}α_{1} = α_{2}β_{3} = 0}. Then A is a quasi-hereditary algebra with respect to {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}} and has 2 essential oriented cycles. Ae_{6} is not simple. Projective dimension of S_{6} is 7. Indeed,

P_{6} ⟶ P_{5} ⟶ P_{2} ⟶ P_{1} ⟶ P_{4} ⊕ P_{6} ⟶ P_{3} ⊕ P_{5} ⟶ P_{2} ⟶ P_{6} ⟶ S_{6} ⟶ 0

is the minimal projective resolution of S_{6}. Where P_{i} and S_{i} are coressponding indecomposable projective module Ae_{i} and simple left module Ae_{i}/Ne_{i} respectively. This is maxmal among S_{i}, so gl.dim.A = 7. This is the case of n = 6 and m = 2 in Theorem 2.2, and gl.dim.A = 7 = 6 + 2 − 1 is maximal.

References

[1] Deng, B., A Construction of Algebras with Arbitrary Finite Global Dimensions, Communications in algebra 26(1998), pp. 2959

[2] Drozd, Yu. A. and Kirichenko, V. V., Finite Dimensional Algebras, Springer-Verlag, Berlin Heidelberg, 1994.

[3] Gustafson, William H., Global Dimension in Serial Rings, J. Algebra 97(1985), 14-16.

[4] Dlab,V. and Ringel,C.M., Quasi-hereditary algebras, Illinois J. Math. 33(1989),280-291.

[5] Yamagata,K., A construction of algebras with large global dimensions, J. Algebra 163(1994), 57-67.

Faculty of Management Information Science Jobu University Yamanashi University 270-1 Shin-machi Tano Gunma 370-1393 JAPAN

  E-mail address: uematsu@jobu.ac.jp