Examensarbete i matematik, 30 hp
Handledare: Julian Külshammer
Examinator: Denis Gaidashev
Augusti 2020
Global dimension of (higher) Nakayama algebras
algebras
Sandra Berg
August 30, 2020
Abstract
In this thesis we look at bounded Nakayama algebras A ∼= kQ/I where
Q is an acyclic quiver of the form of a line and I an admissible ideal of
kQ. Furthermore we consider their higher dimensional analogues of the form
Adn introduced by Jasso and Külshammer in [5] following Iyama’s higher Auslander-Reiten theory. In particular, we restrict the algebras and consider
A ∼= kQ/I` and look at the special form of higher Nakayama algebras Ad `, for the Kupisch series ` = (1, 2, . . . ) as bounded version of the algebra. We reprove Vaso’s formula for projective dimension and global dimension 5.2 in [6] to the classical Nakayama algebras and from this develop a formula for the projective dimension and the global dimension of higher Nakayama algebras of the form Ad`.
Acknowledgements
I would like to thank my supervisor Julian Külshammer for all his support, patience, meticulous notes for improvement and entrusting me with this interesting topic. Thank you Laertis Vaso for the conversations that helped me understand theory that was happening behind your formulas. I am also grateful for the support from Anton Gregefalk, he always encourages me and helped me write a program in Python for drawing my quivers.
Contents
1 Introduction 4
2 Preliminaries 4
2.1 Algebras and modules . . . 4
2.2 Quivers and path algebras . . . 12
2.3 Representations of quivers . . . 14
2.4 Projective and injective modules . . . 18
2.5 Homological algebra . . . 25
3 Auslander-Reiten Theory 28 3.1 The Auslander-Reiten quiver of an algebra . . . 28
4 Nakayama algebras 31 4.1 Nakayama algebras . . . 31
4.2 Bounded Nakayama algebras . . . 33
4.3 Projective and injective modules . . . 34
4.3.1 Global dimension . . . 39
5 Higher Nakayama algebras 41 5.1 Higher Nakayama algebras of type Adn . . . 41
5.2 Nakayama algebras of type Ad` . . . 45
5.2.1 Projective dimension . . . 51
1 Introduction
This thesis will explore the theory of Nakayama algebras and their higher analogues introduced by Jasso and Külshammer in their paper Higher Nakayama Algebras I:
Con-struction [5]. The higher Nakayama algebras are part of higher Auslander-Reiten theory
by Iyama. We will also cover the formulas for projective and global dimension of the classical Nakayama algebras by Vaso in [6], and thereafter expand this to the setting of higher Nakayama algebras. The main results, which compute the global dimension of certain higher Nakayama algebras, and the projective dimension of interval modules for them, are presented in Section 5 with explicit combinatorial formulas. The formula for projective dimension of interval modules of higher Nakayama algebras is found in Theorem 5.25, and in Theorem 5.28 we present the formula for global dimension of certain higher Nakayama algebras.
Throughout this paper we will let A be a finite dimensional associative algebra over an algebraically closed field, k. Furthermore we denote the category of right A-modules as Mod A, and the finitely generated right A-modules, as mod A, see the definitions in Section 2.1.
2 Preliminaries
In order to make this paper as self-contained as possible, we present the basics of algebras, modules, their relation to quivers and homological algebra. For those who are familiar with this theory, this part is optional. The notation will largely follow Elements of the
Representation Theory of Associative Algebras by Assem, Simson and Skowronski [1],
unless stated otherwise.
2.1 Algebras and modules
We begin with introducing the basic terminology and theory of algebras and modules; with ideals, radical, socle, idempotents and projective and injective modules. Some category theory, such as functors, will be used freely, if the reader is not familiar with this they are referred to Section 2.5 or literature such as Grillet’s Abstract Algebra [3].
Definition 2.1. Let k be a field. An algebra over k is a pair (A, ◦), where A is a
k-vector space and ◦ is a bilinear map
◦ : A × A → A.
To ease the notation, we write a ◦ b = ab.
Tn(k), for an n × n matrix with coefficients in A. A k-basis of the algebra equals k k . . . k 0 k . . . k .. . . .. ... 0 0 . . . k .
The algebra is said to be associative if the bilinear map is associative. The k-algebra A is said to be finite dimensional if the dimension dimkA of the k-vector space A is finite. Throughout this paper we will let A be a finite dimensional associative algebra over an algebraically closed field k.
Definition 2.3. A k-vector subspace B of a k-algebra A is a k-subalgebra of A if B contains the identity element of A and it is closed under multiplication, i.e. a ◦ b ∈ B for all a, b ∈ B.
Definition 2.4. A right ideal I of A is a k-vector subspace of a k-algebra A such that the following is fulfilled; xa ∈ I for all x ∈ I and a ∈ A. A left ideal is defined analogously. If an ideal is both a left ideal and a right ideal, it is said to be a two-sided ideal.
A two-sided ideal of an algebra is usually just called an ideal of A.
Definition 2.5. For every k-algebra A we can define the opposite algebra Aop of A. This is the k-algebra which has the same underlying set and vector space structure as A,
but the bilinear map in Aop is now defined as a ◦ b = ba.
Definition 2.6. The (Jacobson) radical rad A of a k-algebra A is the intersection of all maximal right ideals in A.
Furthermore, rad A is the intersection of all maximal left ideals in A. Especially, the Jacobson radical rad A is a two-sided ideal.
Example 2.7. The radical rad A of the upper triangular matrix algebra A = Tn(k)
consists of all matrices in A where all the diagonal entries equal zero. From this, it is clear that (rad A)n= 0.
Definition 2.8. Let A be a k-algebra. A right A-module, MA is a k-vector space M
together with a binary operation · : M × A → M, (m, a) 7→ m · a which satisfies, for all x, y ∈ M, a, b ∈ A and λ ∈ k:
(a) (x + y)a = xa + ya (b) x(a + b) = xa + xb (c) x(ab) = (xa)b (d) x1 = x
(e) (xλ)a = x(aλ) = (xa)λ
We define AM , a left A-module, analogously and from Definition 2.5 it is clear that
right Aop-modules can be identified with left modules. If we want to consider the algebra
A itself via the normal multiplication, as a left or right A-module, we denote it asAA or
AA. Similar to the dimension of an algebra, we define the dimension of a module to be dimkM and M to be finite dimensional if dimkM < ∞.
We say that an A-module M is generated by some elements {mi} ⊆ M, i ∈ I if any element m in M can be written as m = mi1a1+ · · · + misas for some elements a1, . . . , as
in A and ij ∈ I and j ∈ {1, . . . , s}. If the module is generated by a finite subset of M , then M is said to be finitely generated.
Definition 2.9. Let M, N be right A-modules and h : M → N a k-linear map, then h is said to be an A-module homomorphism if h(ma) = h(m)a for all m ∈ M and a ∈ A. If the homomorphism is injective it is said to be a monomorphism, while a surjective one is said to be an epimorphism. A bijective A-module homomorphism is called an isomorphism. An A-module homomorphism h : M → M is said to be an
endomorphism of M .
Example 2.10. Let k[t] be the algebra of all polynomials in one variable t and coefficients in k. All modules in Mod k[t] can be viewed as a pair (V, h) where V is the underlying
k-vector space and h : V → V is the k-linear endomorphism v 7→ vt.
Definition 2.11. A k-subspace N of MA is an A-submodule if N is closed under the
action of A, i.e. n · a ∈ N for all n ∈ N and all a ∈ A.
If a (non-zero) module has no other submodules than itself or zero module, we call it a
simple module.
Definition 2.12. The socle of a module M is the submodule of M generated by all simple submodules of M . We denote this soc M .
Remark. Note that for a module M , soc (soc M ) = soc(M ).
Definition 2.13. The (Jacobson) radical rad M of a right A-module is the intersection of all the maximal submodules of M .
From the definition of the radical of an algebra, we can see that the radical of the right
A-module AAis the same as the radical of A itself.
Another property of the radical is that for M ∈ mod A, we have M rad A = rad M . A proof of this can be found in [1, Prop I.3.7 p.15]. We can define the top of M , as follows
topM = M/ rad M.
Proposition 2.14. Let A be a finite dimensional k-algebra and M a module in mod A. Then there exists a chain of submodules of M ,
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mm= M
such that Mj+1/Mj is simple for j ∈ 0, 1, . . . , m − 1. This chain is called a composition series of M and the simple modules of the form Mj+1/Mj are called the composition factors of M .
Theorem 2.15. (Jordan-Hölder theorem)
If A is a finite dimensional algebra, M, N ∈ mod A and Mj and Ni are submodules of
M and N , respectively, and we have the composition series
0 = M0 ⊂ M1⊂ M2⊂ · · · ⊂ Mm = M 0 = N0⊂ N1⊂ N2⊂ · · · ⊂ Nn= M
then m = n and there exists a permutation σ such that Mj+1/Mj ∼= Nσ(j+1)/Nσ(j).
Here, we mostly care about the m, which we call the length of the module. If a (non-zero) module M has no direct sum decomposition, we say that M is an indecomposable module, i.e. if we write M ∼= N ⊕ L, then one of L and M must be zero and the other is hence M .
We will now define standard dualities and as previously mentioned we will use some prerequisites of homological algebra.
Definition 2.16. Let A be a finite dimensional k-algebra. We define the functor
D : mod A → mod Aop
by first assigning to every right module M in mod A the dual k-vector space D(M ) =
Homk(M, k), which is endowed with a left A-module structure given by (aξ)(m) = ξ(ma)
for ξ ∈ Homk(M, k), a ∈ A and m ∈ M . Then we also assign to every A-module
homomorphism h : M → N the dual k-homomorphism
D(h) = Homk(h, k) : D(N ) → D(M )
ξ 7→ ξh
of left A-modules. This shows that D is a duality of categories (see Definition 2.67), called the standard k-duality.
The next definition will be useful to us throughout the paper and will also use some homological algebra.
Definition 2.17. A sequence as the following, · · · → Mn−1 hn−1 −−−→ Mn−→ Mhn n+1 hn+1 −−−→ Mn+2 → . . .
where Mi ∈ Mod A and they are connected by A-homomorphisms, is called an exact
sequence if Ker hn= Im hn−1 for any n.
In particular, we call an exact sequence short exact sequence if the exact sequence is of the following form,
0 → L−→ Mu −→ N → 0r
where u is a monomorphism and r is an epimorphism. Since the sequence is exact, we know that Ker r = Im u.
Theorem 2.18. (Krull-Remak-Schmidt theorem or Unique decomposition theorem) Let A be a finite dimensional k-algebra.
1. Every module M in mod A has a decomposition M ∼= M1 ⊕ · · · ⊕ Mm, where
M1, . . . , Mm are indecomposable modules.
2. If M ∼=Lm
i=1Mi ∼=Lnj=1Nj, where Mi and Nj are indecomposable modules, then
m = n and there exists a permutation σ of k = {1, . . . , n} such that Mi ∼= Nσ(k)
for each k.
Definition 2.19. A chain complex in the category Mod A is a sequence C• : . . . dn+3 −−−→ Cn+2 dn+2 −−−→ Cn+1 dn+1 −−−→ Cn dn −→ Cn−1 dn−1 −−−→ . . . d2 −→ C1 d1 −→ C0 d0 −→ 0
of right A-modules connected by A-homomorphisms such that dn+1◦ dn= 0, ∀n ≥ 0.
In a similar manner we define a cochain complex as a sequence
C•: 0−→ Cd0 0 d−→ C1 1 d−→ . . .2 −−−→ Cdn−2 n−1 d−−−→ Cn−1 n d−→ Cn n+1 d−−−→ Cn+1 n+2 d−−−→ . . .n+2
Definition 2.20. A right A-module P is said to be projective if for any epimorphism f : C → B and any homomorphism g : P → B, there exists a homomorphism g0 : P → C
such that f ◦ g0 = g. We can illustrate this with the following commutative diagram.
C B 0
P f
g g0
For definition of commutative diagram, see Section 2.5. In a similar way, we will define the dual notion of injective modules.
Definition 2.21. A left A-module I is said to be injective if for any monomorphism f : C → B and any homomorphism g : C → I, there exists a homomorphism g0 : B → I
where g0◦ f = g. We can illustrate this with the following commutative diagram,
0 C B
I f g
g0
Definition 2.22. We define a projective resolution of a right A-module M to be a complex
P• : · · · → Pm hm
−−→ Pm−1 → · · · → P1 −→ Ph1
0 → 0
of projective A-module such that the complex together with an epimorphism h0: P0→ M
makes the sequence
· · · → Pm hm −−→ Pm−1 → · · · → P1 h1 −→ P0 h0 −→ M → 0 exact.
Definition 2.23. 1. An A-submodule N of M is superfluous if for every submodule L of M , N + L = M implies L = M.
2. An A-epimorphism h : M → N in mod A is minimal if Ker h is superfluous in M.
It can be shown that for an arbitrary finite dimensional right A-module M , there exists a projective resolution of M in mod A. The epimorphism h0 : P0 → M is called a
projective cover of M if h0 is a minimal epimorphism. Intuitively this implies that P0
covers M in an optimal way, no submodule of P0 would suffice.
Definition 2.24. An exact sequence
· · · → Pm−−→ Phm m−1 → · · · P1 −→ Ph1 0 −→ M → 0h0
in mod A is called a minimal projective resolution of M if for every i ≥ 1 the homomorphism hi : Pi→ Im hi and P0 −→ M are projective covers.h0
We proceed to define the dual notion of an injective resolution.
Definition 2.25. An injective resolution of M is defined to be a complex I• : 0 → I0 d−→ I1 1 → · · · → Im d−−−→ Im+1 m+1 → · · ·
of injective A-modules which together with a monomorphism d0 : M → I0 of right
A-modules which makes the following sequence exact.
Definition 2.26. 1. An A-submodule N of M is essential if for every submodule H of M , H ∩ N = {0} implies H = {0}.
2. An A-module monomorphism u : L → M in mod A is minimal if Imu is essential in M.
In a similar way as above d0: M → I0 in mod A is called an injective envelope of M if d0 is a minimal monomorphism.
Definition 2.27. An exact sequence
0 → M d
0
−→ I0 d−→ I1 1 → · · · → Im d−−−→ Im+1 m+1 → · · ·
in mod A is called a minimal injective resolution of M if Imdm → Im is an injective
envelope for all m ≥ 1 and d0 : M → I0 is an injective envelope.
Definition 2.28. Let M ∈ mod A. The syzygy of M , Ω(M ), is the kernel of a projective cover of M . The cosyzygy, Ω−1(M ) is the cokernel of an injective envelope.
The syzygy is not unique and thus only defined up to isomorphism. We can use the previous theory to show the form of every minimal projective resolution will be the following,
. . . P (Ω2(M )) P (Ω(M )) P (M ) M 0
Ω2(M ) Ω(M )
and every minimal injective resolution will have the following form, 0 L I(L) I Ω−1(L)
I Ω−2(L) . . .
Ω−1(L) Ω−2(L)
Here P (M ) denotes the projective cover of M and I(L) the injective envelope.
Theorem 2.29. Let A be a finite dimensional k-algebra and D : mod A → mod Aop the standard duality as defined above, then the following holds.
1. The sequence 0 → L −→ Nu −→ M → 0 in mod A is a short exact sequence if andr
only if the induced sequence 0 → D(M )−−−→ D(N )D(r) −−−→ D(L) → 0 is exact inD(u) mod Aop.
2. A module P in mod A is projective if and only if the module D(P ) is injective in
mod Aop. And vice versa, a module I ∈ mod A is injective if and only if D(I) is projective in mod Aop.
4. An epimorphism h : P → M in mod A is a projective cover if and only if D(h) : D(M ) → D(P ) is an injective envelope in mod Aop. And vice versa for an injective envelope in mod A.
This theorem above shows us that the theory of projective and injective modules are dual for finite dimensional modules.
The Wedderburn-Artin theorem below show us that a semisimple algebra that is finite dimensional over a field k is isomorphic to a finite product of matrix algebras.
Theorem 2.30. (Wedderburn-Artin theorem) For any k-algebra, A, the following condi-tions are equivalent:
(a) AA is semisimple.
(b) Every right A-module is semisimple (c) rad A = 0
(d) There exist positive integers, m1, . . . , ms and a k-algebra isomorphism
A ∼= Mm1(k) × · · · × Mms(k)
Definition 2.31. We call an element e ∈ A idempotent if e2 = e. Idempotent elements e1 and e2 are orthogonal if e1e2= e2e1 = 0
Also we say that an idempotent e is primitive if it cannot be written as a sum of orthogonal idempotent elements, e = e1+ e2. In AAwe have the trivial idempotents 1 and 0, which are clearly orthogonal. For any idempotent e ∈ A, eA is a submodule of AA, furthermore eA is an indecomposable module if and only if e is a primitive idempotent. Now let {e1, . . . , en} be a set of primitive idempotents that are pairwise orthogonal and 1 = e1+ . . . en. Then {e1, . . . , en} is called a complete set of primitive orthogonal
idempotents. From this we get a decomposition of AAinto indecomposable A-modules,
AA = e1A ⊕ · · · ⊕ enA. An algebra A is said to be connected if 1 and 0 are the only idempotents of A, meaning it is not isomorphic a direct product of two algebras.
Theorem 2.32. Let AA= e1A ⊕ · · · ⊕ enA be a decomposition of A into indecomposable
submodules.
1. Every simple right A-module is isomorphic to one of the modules S(1) = top(e1A), S(2) = top(e2A), . . . , S(n) = top(enA).
2. Every indecomposable projective right A-module is isomorphic to one of the modules P (1) = e1A, P (2) = e2A, . . . , P (n) = enA.
3. Every indecomposable injective right A-module is isomorphic to one of the modules I(1) = D(Ae1), I(2) = D(Ae2) . . . , I(n) = D(Aen),
where D(Aei) is an injective envelope of the simple module S(i).
Proof. The statements follow from the Unique decomposition theorem 2.18.
Definition 2.33. We call an algebra A with a complete set of primitive orthogonal idempotents {e1, . . . , en} a basic algebra if eiA 6∼= ejA for all i 6= j
Remark. This definition is independent of the choice of complete set of primitive orthogonal
idempotents by the Unique decomposition theorem 2.18.
2.2 Quivers and path algebras
This section will introduce quivers and the algebraic structures we can construct from them, meaning the structures of the path algebras.
Definition 2.34. A quiver is a quadruple Q = (Q0, Q1, s, t), where Q0 are the vertices
and Q1 are the arrows. The function s(α) : Q1→ Q0 gives the source of an arrow α, and the function t(α) : Q1 → Q0 its target.
Definition 2.35. Let Q = (Q0, Q1, s, t) be a quiver and let a and b be vertices of the
quiver. A path of length ` ≥ 1 with source a and target b is a sequence α`, . . . , α1),
where αi∈ Q1 for all 1 ≥ i ≥ ` and we have s(α1) = a, t(αi) = s(αi+1) and t(α`) = b.
This path will look as follows,
a = a0 −→ aα1 1 −→ aα2 2 → . . .
α`
−→ a` = b
The paths of length ` = 0 are called the trivial paths, and are denoted εi, for the vertex
i. Intuitively these paths correspond to staying at the same vertex. A few examples of
their construction are presented below.
If a path of length ` ≥ 1 has the same source and target, we call this a cycle. A cycle of length 1 is called a loop. A quiver with no cycles is said to be acyclic.
Example 2.36. The Kronecker quiver
1 2
β
α Example 2.37. A quiver with a cycle
0 1 2 3 α β γ δ
Example 2.38. An acyclic quiver
0 α β γ 1 2 3
Example 2.39. A quiver with a loop
1
α
Example 2.40.
0−→ 1α −→ 2β −→ 3γ
Definition 2.41. For a given quiver Q we define the path algebra kQ over it. We define a vector space with a basis that consists of all the different paths in the quiver Q. The operation is now (if possible) the concatenation of paths. That is, if α and β are
paths in Q the concatenation of them α ◦ β exists if the vertex t(β) coincides with the vertex s(α). Otherwise α ◦ β equals 0. The concatenation is defined on a basis of paths and then extended bilinearly.
Remark. Note that kQ is an associative algebra. We will prove this, starting by considering
the paths α, β, γ. Then the concatenation of them are α ◦ (β ◦ γ) and (α ◦ β) ◦ γ, meaning in both cases taking α last, β in the middle and γ first. In case the conditions t(β) = s(α) and t(γ) = s(β) are satisfied, it is clear that the concatenation is associative. Otherwise, if the conditions are not satisfied we get the zero element by definition. Since the concatenation is defined on a basis of paths we can extend it bilinearly and kQ is an associative algebra.
The dimension of the path algebra is finite if and only if the underlying quiver is finite and acyclic.
Example 2.42. Consider the quiver
0−→ 1α −→ 2β
whose path algebra has a basis of paths {ε0, ε1, ε2, α, β, βα} and the operation is
concate-nation of paths, i.e. β ◦ α = βα but α ◦ β = 0. The full multiplication of basis elements table equals:
left|right ε0 ε1 ε2 α β βα ε0 ε0 0 0 0 0 0 ε1 0 ε1 0 α 0 0 ε2 0 0 ε2 0 β βα α α 0 0 0 0 0 β 0 β 0 βα 0 0 βα βα 0 0 0 0 0
Example 2.43. Consider the following cyclic quiver
1
α
where α is the only arrow and using this we can construct a basis {α0, α, α2, α3, . . . } where α0 = ε1 is the trivial path and multiplication is defined as αk◦ α` = αk+` for
k, ` ≥ 0 and ε1◦ αk = αk = αk◦ ε1, for all k ≥ 0.
Remark. Note that this path algebra is isomorphic to the polynomial algebra k[t] in one
variable t. The k-linear map which proves this is the following,
ε1 7→ 1 and α 7→ t
Proposition 2.44. For a path algebra kQ the opposite algebra (kQ)op is isomorphic to the path algebra over the opposite quiver Qop. In this quiver the arrows have opposite direction.
Example 2.45. Consider the path algebra A = kQ, where Q is the quiver
0−→ 1α −→ 2.β
The opposite algebra Aop is then isomorphic to kQop, i.e. the path algebra over the
opposite quiver,
0←− 1αˆ ←− 2.βˆ
2.3 Representations of quivers
In this section the two previous sections will be combined, we will explore modules, radical and idempotents of the path algebra A ∼= kQ. We will also define admissible quotients A ∼= kQ/I of the path algebra .
Definition 2.46. A (k-linear) representation M of a finite quiver Q is denoted as M = (Ma, ϕα). The representation is defined by a k-vector space Ma associated to each
vertex a ∈ Q0 and a k-linear map ϕα: Ma→ Mb associated to each arrow α : a → b
The representation is said to be finite dimensional if each vector space Ma is finite dimensional.
Example 2.47. Let Q be the quiver
0−→ 1α −→ 2.β
A representation M of Q is then given by
k3 1 0 0 −−−−−−−−→ k 1 0 ! −−−→ k2. Another representation M0 of Q is given by
k−→ k1 −→ k.1
Example 2.48. Consider the quiver
0 1 2 3. α β γ δ
One representation of this quiver is then
k k2 k k2. 1 0 ! 1 0 0 1 ! 0 1 1
Definition 2.49. The arrow ideal RQ of the path algebra kQ over a finite quiver Q is
generated by all the arrows of Q.
In the same way, we define the ideal RiQ of kQ, which is generated, as a k-vector space, by the set of all paths of length ≥ i.
An ideal I is said to be admissible if there exists m ≥ 2 such that
Rm⊆ I ⊆ R2.
It follows directly from the definition that an admissible ideal is a two-sided ideal that does not contain any arrows of Q but contains all paths of length at least m. If the quiver
The admissible ideal is used to construct a bound quiver, and generate the bound
path algebra A ∼= kQ/I. Why this is interesting to us is motivated by the following proposition.
Proposition 2.50. Let kQ be a path algebra of a finite quiver Q and I an admissible ideal of the path algebra, then the bounded path algebra kQ/I is finite dimensional. Proof. Can be found on page 56 in Assem, Simson and Skowronski’s Elements of the Representation Theory of Associative Algebras [1]. The main idea of the proof is based
on the fact that I by definition is bounded by Rm (m ≥ 2) and thus kQ/Rm has finitely many equivalence classes of paths, hence the same applies to kQ/I.
What this proposition means is that when the path algebra is bounded by an admissible ideal, we no longer require Q to be acyclic for the algebra to be finite dimensional. We will now introduce relations in a quiver, since it is convenient to generate the admissible ideals by them.
Definition 2.51. Let Q be a quiver and ρ an element of kQ such that, ρ =
m
X
i=1
λiwi
where λi are scalars (not all zero) from k and wi are pairwise distinct paths in Q of length at least 2 such that, if i 6= j, then the target (source respectively) of wi coincides
with the target (source respectively) of wj. Then ρ is a relation in Q.
If we have that m = 1, the previous relation is called a zero relation. We will present examples below where the relations generate an admissible ideal.
Example 2.52. Consider the path algebra over the quiver
0−→ 1α −→ 2β −→ 3,γ
bounded by the admissible ideal generated by γβα = 0. Now, some representations are as follows, k−→ k1 1 0 ! −−−→ k2 0−→ 0 or 0−→ k0 2 1 0 −−−−−→ k−→ k.1
can no longer be as follows k3 1 0 0 −−−−−−−−→ k 1 0 ! −−−→ k2 1−→ k2 or k−→ k1 1 0 ! −−−→ k2 1 0 −−−−−→ k.
Lemma 2.53. The radical of a bound path algebra, rad(kQ/I) equals RQ/I, where I
is an admissible ideal of kQ and RQ is the arrow ideal of kQ. Furthermore, kQ/I is a
basic algebra.
Seeing as we defined a bound path algebra A ∼= kQ/I from a bounded quiver, we want to also find the quiver for a given basic connected finite dimensional algebra A.
Definition 2.54. Let A be a basic connected finite dimensional k-algebra and e1, e2, . . . , en
be a complete set of primitive orthogonal idempotents of A. The underlying quiver, called the ordinary quiver of A, denoted QA, is defined as follows,
1. The vertices of QA denoted 1, 2, . . . , n have a bijective correspondence with the
idempotents of A, i.e. e1, e2. . . , en.
2. Given two vertices a, b of QA, the arrows α : a → b have a bijective correspondence
with the vectors in a basis of the k-vector space eb(rad A/ rad2A)ea.
Example 2.55. Let A be the matrix algebra of the form
A = k 0 0 k k 0 k 0 k .
It is clear that one complete set of primitive orthogonal idempotents of A will correspond to the vertices of QA, contains the three matrix idempotents,
e1 = 1 0 0 0 0 0 0 0 0 , e2 = 0 0 0 0 1 0 0 0 0 , e3= 0 0 0 0 0 0 0 0 1 .
It can be shown that the radical of A equals rad A =
0 0 0 k 0 0 k 0 0 and rad 2A = 0. From
straightforward calculations we get that e3(rad A)e1 and e2(rad A)e1 are one dimensional, while the remaining spaces are zero. These will correspond to the arrows in QA and we
1
2 3.
α β
For a path algebra the trivial paths make a set of primitive orthogonal idempotent elements (clearly εi◦ εi= εi). For a bounded path algebra kQ/I, the set {ea= εa+ I | a ∈ Qa}
is a complete set of primitive orthogonal idempotents.
Theorem 2.56. [1, Theorem II.3.7](Gabriel’s structure theorem) Every basic finite dimensional algebra A is isomorphic to kQA/I for some admissible ideal I ⊂ kQA. In the theorem below we will justify the introduction of the concepts of modules and representations. We want to study the category mod A, where A is a finite dimensional algebra. Above we showed that there exists a finite quiver QA and an admissible ideal I of kQA such that A ∼= kQA/I. We will now show that the category of k-linear representations of QA bound by I, denoted Repk(Q, I) is equivalent to Mod A.
Theorem 2.57. Let A be a bound path algebra as defined above, meaning A = kQ/I. There exists a k-linear equivalence of categories
F : Mod A−'→ Repk(Q, I)
that can be restricted to an equivalence between the categories of the finite dimensional modules and finite dimensional representations, F : mod A−→ rep' k(Q, I)
Remark. As a special case, this equivalence of categories holds for any finite and acyclic
quiver Q. This is clear since Q is acyclic, hence the algebra kQ is finite dimensional and the rest follows from previous theorem, when letting I = 0.
Since representations and modules are equivalent we will only denote them as modules from now on.
2.4 Projective and injective modules
This subsection will cover the simple, projective and injective modules of the bound path algebra A ∼= kQ/I.
Definition 2.58. Let A ∼= kQ/I be a bound path algebra and a ∈ Q0. The simple
module S(a) of A (up to isomorphism) is the representation corresponding to a ∈ Q0.
The vertex a in the representation corresponds to a vector space k, all other vector spaces on the vertices and all linear maps associated with the arrows in Q equal 0. Hence, S(a) is a one dimensional module.
Example 2.59. Consider the path algebra over the quiver,
bounded by the admissible ideal γβα = 0. The simple modules are then, S(0) = k−→ 00 −→ 00 −→ 00
S(1) = 0−→ k0 −→ 00 −→ 00
S(2) = 0−→ 00 −→ k0 −→ 00
S(3) = 0−→ 00 −→ 00 −→ k0
Remark. This is a specialization of Theorem 2.32 as a definition of simple modules.
Furthermore, it is not always true that all simple modules are the one dimensional ones. It is however true for basic algebras over algebraically closed fields. An unbound path algebra with a cycle has infinitely many simple modules of finite dimension, see the example below,
Example 2.60. Let A = kQ be the (unbounded) path algebra over the quiver
1 2. β α The A-modules S(1) = k 0 0 0 S(2) = 0 k 0 0 and Sλ= k k 1 λ
with λ ∈ k, are all simple modules. Clearly Sλ is not one dimensional.
Lemma 2.61. [1, Lemma III.2.4] Let A ∼= kQ/I, where, as before, Q is a quiver, I an
admissible ideal of kQ and P (a) = eaA for a vertex a ∈ Q0.
1. If P (a) = (P (a)b, φβ), then P (a)b is the k-vector space with the spanning set of all
the ¯w = w + I, where w is a path from b to a. For an arrow β : b → c, the k-linear map φβ : P (a)c→ P (a)b is given by the right multiplication by ¯β = β + I.
2. Let rad(P (a)) = P0(a)b, φ0β
, then P0(a)b = P (a)b when a 6= b. P0(a)a is the k-vector space spanned by the set of all ¯w = w + I, where w is a non-stationary
path with both source and target at a. φ0β = φβ for any arrow β with target b 6= a. For an arrow α that has a as target we get φ0α = φα|P0(a)
a
We say that P (a) is the indecomposable projective A-module associated to the vertex
a ∈ Q0.
Intuitively, these projective modules for a vertex a of the bounded path algebra, will correspond to a representation where all the vertices that we can reach with a path, i.e.
a is the target of the path, will be represented with a k-vector space, but we have to
take the relation of I into account. From this and Theorem 2.32 we can also see that
S(a) ∼= top(P (a)).
Lemma 2.62. [1, Lemma III.2.6] Let A ∼= kQ/I, where Q is a quiver, I an admissible
ideal of kQ and I(a) = D(Aea) for a vertex a ∈ Q0.
1. Given a ∈ Q0, the simple module S(a) is isomorphic to soc(I(a)).
2. If I(a) = (I(a)b, φβ), then I(a)b is the dual of the k-vector space spanned by ¯
w = w + I, where w is a path from a to b. For an arrow β : b → c, the k-linear map φβ : I(a)c→ I(a)b is given by the dual of the left multiplication by ¯β = β + I. The module I(a) is the indecomposable injective A-module associated to the vertex
a ∈ Q0. Here the intuition of I(a) is the dual object, it will correspond to a representation
where we represent the vertices that we can reach with a path where a is the source, and we take the admissible ideal into account.
Example 2.63. Again, consider the path algebra A over the quiver
0−→ 1α −→ 2β −→ 3,γ
bounded by the admissible ideal generated by γβα = 0. The injective modules of A are the following,
I(0) ∼= k → k → k → 0
I(1) ∼= 0 → k → k → k
I(2) ∼= 0 → 0 → k → k
I(3) ∼= 0 → 0 → 0 → k.
The projective modules of A are the following,
P (0) ∼= k → 0 → 0 → 0
P (1) ∼= k → k → 0 → 0
P (2) ∼= k → k → k → 0
Example 2.64. Let Q be the quiver 1 2 β α δ γ 3
bound by δβ = 0 = γα. The indecomposable injective modules of the bound quiver algebra of Q are given by I(1) ∼= k 0 0 0 0 0 0 I(2) ∼= k2 k 1 0 ! 0 1 ! 0 0 0 I(3) ∼= k2 k2 0 1 0 0 ! 0 0 1 0 ! 1 0 ! 0 1 ! k.
The indecomposable projective modules are given by
P (3) ∼= 0 0 0 0 0 0 k P (2) ∼= 0 k 1 0 0 1 0 0 k2
P (1) ∼= k k2 0 0 1 0 ! 0 1 0 0 ! 1 0 0 1 k2.
The next examples will calculate the projective resolutions for modules over path algebras.
Example 2.65. Let A be the path algebra of the Kronecker quiver Q
1 2.
β
α Let M be the representation of Q as follows,
k 1
k.
λ
We will calculate a minimal projective resolution of M . We know that the indecomposable projective modules are of the form e1A and e2A where e1 and e2 correspond to the trivial
paths of Q, which form a complete set of primitive orthogonal idempotents.
Let us consider the module e1A, which is the span of all paths that end at vertex 1. In this
case this is only the trivial path e1, i.e. e1A = spanhe1i. Since modules are equivalent to
representations, we map e1A to the representation below:
he1i 0
which is isomorphic to the projective module P1:
k 0
0. 0
For the module e2A we instead have the span of e2, α, and β. The corresponding
repre-sentation is the following:
which is isomorphic to the projective module P2: k2 1 0 k. 0 1
For our projective resolution to be minimal, we need P0 −→ M to be a projective cover inp0
the short exact sequence
P1
p1
−→ P0 p0
−→ M → 0.
To find the minimal projective resolution of M , we try to find a surjection P (2) → M . Consider the diagram, where M is the upper row and P (2) the lower:
k k k2 k. 1 λ 1 0 0 1 a b
For this diagram to commute we need to choose appropriate morphisms a and b. Without loss of generality we choose b : k → k to be the identity. The linear map a : k2 → k be a 2 × 1 matrix that should satisfy b1 0= 1 ◦ a and b0 1= λ ◦ a, hence b = 1
λ
!
. Next, we identify the kernel of a, which we set to the identity, to be {0} and the kernel
of b = 1
λ
!
to be isomorphic to k. We can now extend our diagram further and recognize the lowest module to be isomorphic to P (1)
k k k2 k k 0. 1 λ 1 0 0 1 0 0 a b c d
Now to determine c and d, we need to make sure ac = 0 = bd. Therefore we have c = 0 and choose d =λ −1. The projective resolution ends here since P (1) ∼= ΩP (2).
We can now conclude that our minimal projective resolution is
0 → P (1) → P (2) → M → 0.
Example 2.66. Let A ∼= kQ/I where the quiver Q is 0−→ 1α −→ 2β −→ 3γ
and I is generated by γβα = 0. We will calculate a minimal projective resolution of the simple modules of A. These are
S(0) ∼= k−→ 00 −→ 00 −→ 00
S(1) ∼= 0−→ k0 −→ 00 −→ 00
S(2) ∼= 0−→ 00 −→ k0 −→ 00
S(3) ∼= 0−→ 00 −→ 00 −→ k.0
As shown in example 2.63 the indecomposable projective modules are P (0) ∼= k → 0 → 0 → 0
P (1) ∼= k → k → 0 → 0
P (2) ∼= k → k → k → 0
Some projective resolutions are, 0 P (0) P (2) P (3) S(3) 0 Ω2S(3) ΩS(3) 0 P (1) P (2) S(2) 0 ΩS(2) ∼= P (1) 0 P (0) P (1) S(1) 0 ΩS(1) ∼= P (0).
Since we note that S(0) ∼= P (0) we have a trivial projective resolution of S(0) as the
following,
0 P (0) S(0) 0.
2.5 Homological algebra
Most of what is covered in this section builds up to defining the n-th extension bifunctor ExtnA(M, N ) which will be a useful tool in important proofs later. Most of the theory is fundamental to the main findings of the paper, i.e. the computation of projective and global dimension for (higher) Nakayama algebras.
Definition 2.67. Let D : C → D be a contravariant functor which is an equivalence of categories. Then D is called a duality.
Definition 2.68. A diagram in the category C is commutative if the composition of morphisms along any two paths with the same source and target are equal. For example, consider the following diagram,
A B C D f g h i it is commutative if g ◦ f = h ◦ i
Definition 2.69. For a chain complex C•, respectively a cochain complex C•, we define
their n-th homology and cohomology, an A-module constructed as follows Hn(C•) =Ker(dn)Im(d
and Hn(C•) =Ker(dn)
Im(dn−1).
Definition 2.70. The projective dimension of an A-module M , pd(M ) = m is the length of a minimal projective resolution P•
0 → Pm −−→ Phm m−1 → . . . P1
h1
−→ P0 h0
−→ M → 0.
If there is no finite projective resolution of the module M the projective dimension is said to be infinite, pd M = ∞.
Remark. The projective dimension is independent of the choice of projective resolution. Definition 2.71. Let A be a finite dimensional k-algebra. Then we define the global
dimension gl. dim A of an algebra A as the supremum projective dimension of all the
modules of A.
Definition 2.72. For any A-modules M, N , we can define the contravariant functor,
Hom-functor,
HomA(−, N ) : Mod A → Mod k
M 7→ HomA(M, N ). Now, we can define ExtnA(M, N ) :
Definition 2.73. Let A be a k-algebra. We define, for n ≥ 0, the n-th extension
bifunctor as
ExtnA(M, N ) = Hn(HomA(P•, N )),
where P• is a projective resolution of M , and the contravariant Hom functor and n-th
homology are as defined above. That means, given two modules M, N ∈ Mod A, we first choose a projective resolution P• of M and apply the contravariant Hom-functor to it,
thus constructing the cochain complex consisting of k-vector spaces,
HomA(P•, N ) : 0 → HomA(P0, N ) HomA(h1,N ) −−−−−−−−→ HomA(P1, N ) HomA(h2,N ) −−−−−−−−→ . . . → HomA(Pm, N ) HomA(hm+1,N ) −−−−−−−−−−→ HomA(Pm+1, N ) → . . .
Next to get ExtnA(M, N ) we compute the n-th cohomology k-vector space, i.e. Hn(HomA(P•, N )) =Ker(HomA(hn+1,N ))Im(Hom
A(hn,N ).)
Remark. The definition of ExtnA(M, N ) is independent of the choice of projective resolution up to isomorphism.
Example 2.74. Let A ∼= kQ/I where the quiver Q is 0−→ 1α −→ 2β −→ 3γ
and I is generated by γβα = 0. We will calculate ExtkA(S(i), S(1)) for i = 1, 2 and k ≥ 0.
A projective resolution, as calculated in example 2.66, equals
0 → P (0) → P (1) → S(1) → 0.
We know that ExtkA(M, N ) = Hk(Hom(P•, N )), where P• is the projective resolution of
M and Hk= Kerdk/Imdk−1. We begin by applying the contravariant HomA(−, S(i)) to
the projective modules and their homomorphisms and hence getting
0−−→ Homd−1 A(P (1), S(i))−→ Homd0
A(P (0), S(i)) d1
−→ 0.
From this we can see that k = 0, 1 are relevant since we only have the differentials d−1, d0, d1. We let i = 1, which gives us the sequence of modules
0−−→ kd−1 −→ 0d0 d1
−→ 0.
Hence the homology equals,
H0 = Kerd0/Imd−1∼= k
and
H1= Kerd1/Imd0∼= 0.
Now let i = 0 and we get
0−−→ 0d−1 d0
−→ k d1
−→ 0
and the homology
H0= Kerd0/Imd−1 ∼= 0
and
H1 = Kerd1/Imd0 ∼= k.
From this we conclude that
Ext0A(S(1), S(1)) ∼= k Ext1A(S(1), S(1)) ∼= 0 Ext0A(S(1), S(0)) ∼= 0 Ext1A(S(1), S(0)) ∼= k.
Theorem 2.75. [1, Theorem A.4.5 (a) on page 428]
3 Auslander-Reiten Theory
3.1 The Auslander-Reiten quiver of an algebra
This section of the paper will define the Auslander-Reiten quiver of an algebra. This is a useful way for us to illustrate information of the algebra and its modules, in the familiar form of a quiver. It should become clear that the vertices will represent the isomorphism classes of modules while the arrows represent certain homomorphisms in this quiver. Some preliminaries will be excluded from this theory, as we want to focus on the use and construction of the quiver. If the reader is interested in understanding more about the theory behind it she is referred to [1] and Auslander, Reiten and Smalø’s [2].
Definition 3.1. Let h : M → N and u : L → M be homomorphisms of right A-modules. 1. An A-homomorphism s : N → M is called a section of h if h ◦ s = 1N.
2. An A-homomorphism r : M → L is called a retraction of u if r ◦ u = 1L.
3. An A-homomorphism h : M → N is called a section (or a retraction) if h admits a retraction (or a section respectively).
Remark. If s is a section of h, then h is surjective and s is injective.
Definition 3.2. A homomorphism f : X → Y in mod A is said to be irreducible if f is neither a section nor a retraction and if f = g ◦ h, either g is a retraction or h is a section.
Definition 3.3. Let M, N ∈ mod A be indecomposable modules. The k-vector space of irreducible morphisms is defined as
irrA(M, N ) = rad(M, N )/ rad2(M, N ),
which represents the irreducible morphisms from M to N .
Definition 3.4. Let A be a basic finite dimensional k-algebra. The Auslander-Reiten
quiver of A, denoted Γ(mod A) of mod A is defined as follows,
1. The vertices of Γ(mod A) are the isomorphism classes [M ] of indecomposable modules M ∈ mod A.
2. For the arrows we begin with considering vertices [L] and [N ] in Γ(mod A), these vertices correspond to modules L, N ∈ mod A, which are indecomposable. The arrows [L] → [N ] in Γ(mod A) are bijectively corresponding to the vectors of a basis of irrA, the k-vector space of irreducible morphisms from L to N .
Remark. We will later only use this theory for the Nakayama algebras, thus the theory in
the examples presented below cannot be generalised to other algebras. For the curious reader, the Auslander-Reiten quivers presented below are limited since they will have at most one arrow between each vertex, i.e. the vector space of irreducible morphisms
are always one dimensional. This is not always the case. The reason for this is that Nakayama algebras are representation-finite. (See Proposition 4.12.) More on this in the next section.
Example 3.5. A ∼= kQ
0−→ 1α −→ 2β
The trivial paths corresponding to each vertex of the quiver, {e0, e1, e2} make up a complete
set of primitive orthogonal idempotents. Using this and Theorem 2.32 we get the simple, projective and injective modules,
P (0) ∼= k → 0 → 0 ∼= S(0) P (1) ∼= k → k → 0 P (2) ∼= k → k → k I(0) ∼= k → k → k ∼= P (2) I(1) ∼= 0 → k → k I(2) ∼= 0 → 0 → k ∼= S(2) S(1) ∼= 0 → k → 0
Putting this into the Auslander-Reiten quiver we get:
P (0) ∼= S(0) P (1) P (2) ∼= I(0) S(1) I(1) I(2) ∼= S(2)
Example 3.6. Let A ∼= kQ/I where Q is the quiver 0−→ 1α −→ 2β −→ 3γ
and I generated by the relation γβα = 0. The trivial paths corresponding to each vertex of the quiver, {e0, e1, e2, e3} make up a complete set of primitive orthogonal idempotents.
Using this and Theorem 2.32 we get the simple, projective and injective modules, P (0) ∼= k → 0 → 0 → 0 ∼= S(0)
P (1) ∼= k → k → 0 → 0
P (2) ∼= k → k → k → 0
I(0) ∼= k → k → k → 0 I(1) ∼= 0 → k → k → k I(2) ∼= 0 → 0 → k → k I(3) ∼= 0 → 0 → 0 → k ∼= S(3) S(1) ∼= 0 → k → 0 → 0 S(2) ∼= 0 → 0 → k → 0
The last indecomposable module, will be
P (2)/ rad2P (2) ∼= 0 → k → k → 0.
(Why this is a indecomposable module in A is motivated by a theorem later in the thesis, see 4.12.)
Putting this into the Auslander-Reiten quiver we get: P (2) ∼= I(0) P (3) ∼= I(1)
P (1) P (2)/ rad2P (2) I(2)
P (0) S(1) S(2) I(3)
Example 3.7. Let A ∼= kQ/I for the following quiver
1 2
3
4
5
P (4)/ rad2P (4) P (5) S(4) P (5)/ rad2P (5) P (1) S(5) P (1)/ rad2P (1) P (2) S(1) P (2)/ rad2P (2) P (3) S(2) P (3)/ rad2P (3) P (4) S(3) P (4)/ rad2P (4) P (5) S(4)
where the dashed lines indicate where the quiver repeats itself.
4 Nakayama algebras
4.1 Nakayama algebras
This section introduces the representation theory of Nakayama algebras, or sometimes called generalised uniserial algebras. Nakayama algebras are characterised by the fact that all their indecomposable modules are uniserial, meaning they have unique composition series. We begin with defining series of radical and socle. To do so, we need to define higher iterations of the socle.
Definition 4.1. Let M ∈ Mod A, then soci+1(M ) = p−1(soc(M/ sociM )) where p : M → M/ sociM is the canonical epimorphism.
Using this we can now define the socle series, or as we will call it ascending Loewy series.
Definition 4.2. For an A-module M we define the ascending Loewy series
0 ⊂ soc M ⊂ soc2M ⊂ soc3M ⊂ · · · ⊂ socmM = M.
Since M is finite dimensional, it has a finite composition length, i.e. there exists a least positive m such that socmM = M , this is called the length of the ascending Loewy series and is denoted s`(M ) = m. Now we define the dual notion, the radical series. Definition 4.3. For an A-module M we define the descending Loewy series
M ⊃ rad M ⊃ rad2M ⊃ rad3M ⊃ · · · ⊃ radmM = 0.
Because M has a finite composition series, there is also a least positive integer m such that radmM = 0, which is the length of the descending Loewy series, denoted r`(M ) = m. For all finite dimensional modules these lengths of the series coincide, i.e. s`(M ) = r`(M ) for all M , a proof of this can be found on page 162 in [1, Proposition
V.1.3]. This common value is defined as the Loewy length, denoted ``(M ). Naturally we want to know for which modules M we have that `(M ) = ``(M ). This leads us to the following definition.
Definition 4.4. A module M ∈ mod A is called uniserial if it has a unique composition series.
From this we can see that if M is uniserial, so is every submodule and every quotient of
M . Moreover a uniserial module M has a simple top and a simple socle and hence must
be indecomposable.
Lemma 4.5. The following conditions are equivalent for an A-module MA;
1. M is uniserial 2. `(M ) = ``(M )
3. The radical series M ⊃ rad M ⊃ rad2M ⊃ · · · ⊃ 0 is a composition series. 4. The socle series 0 ⊂ soc M ⊂ soc2M ⊂ · · · ⊂ M is a composition series.
Now we want to describe those algebras where all indecomposable projective module are uniserial.
Definition 4.6. An algebra A is said to be right serial if all indecomposable projective right A-modules are uniserial.
A left serial algebra is defined analogously.
Lemma 4.7. [1, Lemma V.2.5 p.165] An algebra A is right serial if and only if for every indecomposable projective right module P the module rad P/ rad2P is simple or zero. Theorem 4.8. A basic k-algebra A is right serial if and only if, in its ordinary quiver QA, for every vertex a, there exists at most one arrow of target a.
Proof. (Follows the proof of theorem V.2.6 p.166 in [1].) From Lemma 4.7 we know that A
is right serial if and only if, for every vertex a ∈ (QA)0, the A-module rad P (a)/ rad2P (a) = ea(rad A/ rad2A) is simple or zero. In other words, the module is at most one dimensional as a k-vector space, which occurs if and only if there is at most one point b ∈ (QA)0 such
that ea(rad A/ rad2A)eb 6= 0, then that k-vector space is at most one dimensional. By definition of the ordinary quiver QA, this occurs if and only if there is at most one point
b ∈ (QA)0 such that there is an arrow a → b. Furthermore, then there is at most one
such arrow.
Definition 4.9. We call a finite dimensional algebra A a Nakayama algebra if all indecomposable projective and indecomposable injective modules are uniserial.
This definition is equivalent to saying that A is a Nakayama algebra if the algebra is both left and right serial. Note that if A is a Nakayama algebra, so is Aop.
Theorem 4.10. A basic and connected algebra A is a Nakayama algebra if and only if its ordinary quiver QA is one of the following quivers:
1. • • . . . • 0 1 n 2. 0 1 2 i n − 1 n
Proof. This follows directly from Theorem 4.8, since A is a Nakayama algebra if and
only if every vertex of QA is the source of at most one arrow and the target of at most one arrow.
We will from now on only consider the acyclic quiver of Nakayama algebras as the corresponding path algebra is a finite dimensional algebra.
4.2 Bounded Nakayama algebras
Again we can consider the bounded path algebra, if A ∼= kQ is a finite dimensional Nakayama algebra, so is A ∼= kQ/I. This is Lemma V.3.3 in [1]. Theorem 4.10 gives us a condition on the ordinary quiver QA while the admissible ideal I is arbitrary. We note that every proper arrow ideal Ri, where i ≥ 2 of a Nakayama algebra A ∼= kQ, is admissible if Q is acyclic. Later we will also use the notation A ∼= kQ/I`, where ` indicates the minimal length of the paths in the admissible arrow ideal I.
Definition 4.11. A finite dimensional k-algebra is defined to be representation-finite if the number of isomorphism classes of indecomposable finite dimensional right A-modules is finite.
Theorem 4.12. Let A be a basic Nakayama algebra and let M be an indecomposable A-module. There exists an indecomposable projective A-module P and an integer t, with
We will now change the setting somewhat, but still consider the bounded Nakayama algebras and present some formulas introduced by Vaso in [6].
From now on, we will use that every indecomposable A-module is isomorphic to an interval module M (x, y) of the form
0 → · · · → k−→ . . .1 −→ k → 0 · · · → 01
where the leftmost k is at position x and the rightmost k is at position y. We will use the convention that M (x, y) = 0 if the coordinates (x, y) do not define a module. Clearly the dimension of the module will equal the length of the interval module, meaning dimkM (x, y) = y − x + 1. Since we are still considering Nakayama algebras, we can also
note that the length of the module also equals its Loewy length, i.e. `` (M (x, y)) = y−x+1.
4.3 Projective and injective modules
Since our main results are for the global dimension of Nakayama algebras, we need to consider the injective and projective modules of the bounded path algebras, in the interval module setting.
Proposition 4.13. Let A ∼= kQ/I` be a bounded acyclic Nakayama algebra for a quiver Q and the admissible ideal I`⊂ Q and let M (x, y) 6= 0 be an interval module of A. Then
the projective and injective modules for the k-th vertex are respectively, 1. P (k) = ( M (0, k) if k < ` M (k − ` + 1, k) if k ≥ ` 2. I(k) = ( M (k, k + ` − 1) if k ≤ n − ` M (k, n − 1) if k > n − `
3. M (x, y) is both injective and projective if and only if y − x + 1 = ` and 1 ≤ x < n − `. Proof. 3. follows directly from 1. and 2. We prove this for 1., the projective modules,
since 2., the injective modules are proved similarly. Both proofs follow from Lemma 2.61 and Lemma 2.62. For k < ` the projective module P (k) as an interval module is isomorphic to k 0 1 − → . . .−→ k1 k 0 − → 0−→ . . .0 −→ 00 n−1
which is precisely the interval module M (0, k). Analogously when k ≥ `, P (k) is isomorphic to 0 0 0 − → . . .−→ 00 −→0 k k−`+1 1 − → . . .−→ k1 k 0 − → 0−→ . . .0 −→ 00 n−1 which is precisely the interval module M (k − ` + 1, k).
Using this proposition, and the theory in the previous section we can compute the Auslander-Reiten quiver of An∼= kQ/I`. The resulting quiver is presented below.
M (0, ` − 1) . . . M (n − `, n − 1) . . . . . . . . . M (0, 1) M (1, 2) . . . . . . M (0, 0) M (1, 1) M (2, 2) . . . . . . M (n − 2, n − 1) M (n − 2, n − 2) M (n − 1, n − 1)
Remark. Note that the projective and injective modules are placed along the slopes of
the trapezoid, while the both injective and projective modules are at the top.
Lemma 4.14. Let 0 → M0 → M → M00 → 0 be a short exact sequence consisting of
A-modules, with pd(M ) ≤ n and pd(M00) ≤ n + 1, then for the projective dimension of
M0 we get pd(M0) ≤ n.
Proof. We construct a long exact sequence of 0 → M0 → M → M00 → 0 with the
extension functor, Ext, and the Hom-functor using Theorem 2.75. Applying an alternative characterisation of projective dimension pd via vanishing of the functors ExtkA(M, −) [3, Proposition XII.8.3] and using that if pd(M ) < n, we get that ExtiA(M, N ) = 0 for all N and all i ≥ n + 1 and Extn+1A (M, N ) = 0 for all N .
Theorem 4.15. Let A be an algebra such that gl. dim A < ∞, then there exists a non-projective injective module M ∈ mod A, such that
pd M = gl. dim A.
Proof. Since global dimension of a finite dimensional algebra is defined as the maximal
projective dimension, it is clear that the maximum of the projective dimensions of the injec-tive modules is smaller than or equal to the global dimension, i.e. max{pd(I)} ≤ gl. dim A. We know that the global dimension is finite, hence it suffices to show that if all injective modules have projective dimension less than some arbitrary positive integer d = pd M , i.e the injective module with maximal projective dimension, then all modules have projective dimension less than or equal to d.
We will use that for every algebra with finite global dimension every module L ∈ mod A has a finite injective coresolution. Let the following be an arbitrary finite injective coresolution for L
0 L I0 I1 . . . Ik−1 Ik . . . Ir−1 Ir 0
for some r and k. Now we will prove by induction over this coresolution that pd(L) ≤ d. We begin by considering Ω−r(L). Note that Ω−r(L) ∼= Ir by definition of finite injective dimension, then pd Ω−r(L) = pd Ir≤ d since we defined d to be the highest projective dimension of the injective modules, and this proves our basis.
Now assume for some k that pd Ω−(k+1)(L) ≤ d. For the induction step, we want to show that pd Ω−k(L) ≤ d. This part of the injective coresolution can also be written as a sequence of short exact sequences,
0 → Ω−k(L) → Ik→ Ω−(k+1)(L) → 0.
where we also note that pd Ik ≤ d by how we defined d. We can now apply Lemma 4.14 and conclude that pd Ω−k(L) ≤ d as wanted. Since L ∼= Ω0(L) it holds by induction that pd L ≤ d. This concludes the proof.
The previous theorem will be useful when we calculate the global dimension of Nakayama algebras, since global dimension is defined as the maximal projective dimension. From this theorem, we know to consider only the non-projective injective modules when finding the global dimension of an algebra.
Proposition 4.16. Let M (x, y) 6= 0 be a non-projective module. Then the syzygy for the non-projective modules are as follows,
ΩM (x, y) =
(
M (0, x − 1) if y < `
M (y − ` + 1, x − 1) if y ≥ `
Proof. We consider the short exact sequences constructed from the projective resolution
(see Definition 2.24) and projective modules (see Definition 4.13); 0 → M (0, y − (y − x + 1)) → M (0, y) → M (x, y) → 0
0 → M (y − ` + 1, y − (y − x + 1)) → M (y − ` + 1, y) → M (x, y) → 0
Hence M (0, y−(y−x+1)) = M (0, x−1) and M (y−`+1, y−(y−x+1)) = M (y−`+1, x−1) are the syzygies of the non-projective modules.
Proposition 4.17. (Proposition 5.2 in Vaso’s [6]) Let A ∼= kQ/IQ` be a bounded
Nakayama algebra.
1. Let M (x, y) 6= 0 and assume that x = 0 or that the length of the module `` = `, then pd M (x, y) = 0.
2. Let M (x, y) 6= 0 and assume x > 0 and `` < `. Now we let x − 1 = q` + r with
pd M (x, y) =
(
2q + 1 if `` < ` − r
2q + 2 if `` ≥ ` − r.
3. Let n − 1 = q0` + r0, 0 ≤ r0≤ ` − 1. Then the projective dimension is
pd M (n − j, n − 1) = jn−1 ` k +ln−1` m if r0= 0 or j ≤ r0 j n−1 ` k +ln−1` m− 1 otherwise. Proof.
1. This follows directly from Proposition 4.13, since then the module is projective.
2. Throughout the proof we will use the property pd M (x, y) = pd ΩM (x, y) + 1, we will denote this ?.
We begin by letting y ≤ ` and note by construction of interval modules that x ≤ y. Then x − 1 < x ≤ y ≤ ` and hence we have for our rewriting of x − 1, that q = 0 and are left with x − 1 = r. By Proposition 4.16 ΩM (x, y) = M (0, x − 1), this is projective by Proposition 4.13 since the rightmost position is equal to zero and then pd ΩM (x, y) = 0. Therefore, using ?, pd M (x, y) = pd ΩM (x, y) + 1 = 0 + 1 = 0(·2) + 1 as wanted since (y − x + 1) = `` ≤ ` − x + 1 = ` − r ⇔ y ≤ `.
Now we continue by letting y ≥ ` instead and we prove by induction on y. The base case was proved in the previous case. For the induction assumption we assume the statement holds for ` ≤ y ≤ k, for some k ∈ N. Now let M (x, y) be such that
y = k. Then by Proposition 4.16 ΩM (x, y) = M (y − ` + 1, x − 1), in particular our
formula holds for this module as well, by our induction assumption.
Let x − 1 = q` + r and first assume the length of the module M (x, y) to be
y − x + 1 = `` < ` − r. Now consider,
(y − ` + 1) − 1 = x − 1 + y − x − ` + 1 = q` + r − x + y − ` + 1 = (q − 1)` + r − x + y + 1
where r−x+y+1 < ` since we assumed y−x+1 < `−r. So x−1+y−x−`+1 = q0`+r0
where q0 = q − 1 and r0 = r + y − x + 1. We want to apply the induction hypothesis to ΩM (x, y), and hence want to compare the length of ΩM (x, y) which equals
