U.U.D.M. Project Report 2020:44
Global dimension of (higher) Nakayama algebras
Sandra Berg
Examensarbete i matematik, 30 hp Handledare: Julian Külshammer Examinator: Denis Gaidashev Augusti 2020
Department of Mathematics Uppsala University
Global dimension of (higher) Nakayama 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 d An introduced by Jasso and Külshammer in [5] following Iyama’s higher Auslander-Reiten theory. In particular, we restrict the algebras and consider ∼ ` d A = kQ/I and look at the special form of higher Nakayama algebras A` , 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 d of the form A` . Sandra Berg - Global dimension of (higher) Nakayama algebras
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.
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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 d 5.1 Higher Nakayama algebras of type An ...... 41 d 5.2 Nakayama algebras of type A` ...... 45 5.2.1 Projective dimension ...... 51 5.2.2 Global dimension ...... 55
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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.
Example 2.2. For any k-algebra A we can construct the upper triangular matrix algebra
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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.
op Definition 2.5. For every k-algebra A we can define the opposite algebra A 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
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(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 as AA 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 = mi1 a1 + ··· + mis as 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 AA is 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.
Using that M rad A = rad M the top of M is a right A/ rad A-module.
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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.
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Definition 2.17. A sequence as the following,
hn−1 hn hn+1 · · · → Mn−1 −−−→ Mn −→ Mn+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 −→u M −→r N → 0 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. ∼ Lm ∼ Ln 2. If M = i=1 Mi = j=1 Nj, 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
dn+3 dn+2 dn+1 dn dn−1 d2 d1 d0 C• : ... −−−→ Cn+2 −−−→ Cn+1 −−−→ Cn −→ Cn−1 −−−→ ... −→ C1 −→ C0 −→ 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
0 1 2 n−2 n−1 n n+1 n+2 C• : 0 −→d C0 −→d C1 −→d ... −−−→d Cn−1 −−−→d Cn −→d Cn+1 −−−→d Cn+2 −−−→d ...
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.
f C B 0
g g0 P
For definition of commutative diagram, see Section 2.5. In a similar way, we will define the dual notion of injective modules.
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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, f 0 C B g g0 I Definition 2.22. We define a projective resolution of a right A-module M to be a complex hm h1 P• : · · · → Pm −−→ Pm−1 → · · · → P1 −→ P0 → 0 of projective A-module such that the complex together with an epimorphism h0 : P0 → M makes the sequence
hm h1 h0 · · · → Pm −−→ Pm−1 → · · · → P1 −→ P0 −→ 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
hm h1 h0 · · · → Pm −−→ Pm−1 → · · · P1 −→ P0 −→ M → 0 in mod A is called a minimal projective resolution of M if for every i ≥ 1 the h0 homomorphism hi : Pi → Im hi and P0 −→ M are projective covers. 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
1 m+1 I• : 0 → I0 −→d I1 → · · · → Im −−−→d Im+1 → · · · of injective A-modules which together with a monomorphism d0 : M → I0 of right A-modules which makes the following sequence exact.
0 1 m+1 0 → M −→d I0 −→d I1 → · · · → Im −−−→d Im+1 → · · ·
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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 1 m+1 0 → M −→d I0 −→d I1 → · · · → Im −−−→d Im+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.
op Theorem 2.29. Let A be a finite dimensional k-algebra and D : mod A → mod A the standard duality as defined above, then the following holds. 1. The sequence 0 → L −→u N −→r M → 0 in mod A is a short exact sequence if and D(r) D(u) only if the induced sequence 0 → D(M) −−−→ D(N) −−−→ D(L) → 0 is exact in 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.
3. A module S ∈ mod A is simple if and only if D(S) ∈ mod Aop is simple.
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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 AA we 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 AA into 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. ∼ ∼ Note that eiA = ejA is equivalent to S(i) = S(j).
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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,
α1 α2 α` a = a0 −→ a1 −→ a2 → ... −→ 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
β 1 2 α
Example 2.37. A quiver with a cycle
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α 0 1 δ β 2 3 γ
Example 2.38. An acyclic quiver α 0 1 β
γ 2
3
Example 2.39. A quiver with a loop α
1
Example 2.40. β γ 0 −→α 1 −→ 2 −→ 3 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
β 0 −→α 1 −→ 2 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:
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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
α
1 where α is the only arrow and using this we can construct a basis {α0, α, α2, α3,... } 0 k ` k+` where α = ε1 is the trivial path and multiplication is defined as α ◦ α = α for k k k k, ` ≥ 0 and ε1 ◦ α = α = α ◦ 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
op Proposition 2.44. For a path algebra kQ the opposite algebra (kQ) 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
β 0 −→α 1 −→ 2.
op op The opposite algebra A is then isomorphic to kQ , i.e. the path algebra over the opposite quiver, βˆ 0 ←−αˆ 1 ←− 2.
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.
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Example 2.47. Let Q be the quiver
β 0 −→α 1 −→ 2.
A representation M of Q is then given by ! 1 1 0 0 0 3 2 k −−−−−−−−→ k −−−→ k .
Another representation M 0 of Q is given by
1 1 k −→ k −→ k.
Example 2.48. Consider the quiver α 0 1 δ β 2 3. γ
One representation of this quiver is then
! 1 0 2 k k ! 1 0 1 0 1 2 k k . 0 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.
i In the same way, we define the ideal RQ 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 2 Q is acyclic, any I ⊆ RQ is admissible.
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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 m m on the fact that I by definition is bounded by R (m ≥ 2) and thus kQ/R 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, m X ρ = λiwi i=1 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
β γ 0 −→α 1 −→ 2 −→ 3, bounded by the admissible ideal generated by γβα = 0. Now, some representations are as follows, ! 1 0 1 2 0 k −→ k −−−→ k −→ 0 or 1 0 0 2 1 0 −→ k −−−−−→ k −→ k. In particular, because the algebra is bounded by the admissible ideal, the representations
16 Sandra Berg - Global dimension of (higher) Nakayama algebras can no longer be as follows ! 1 1 0 0 0 3 2 1 2 k −−−−−−−−→ k −−−→ k −→ k or ! 1 0 1 0 1 2 k −→ k −−−→ k −−−−−→ 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 2 with the vectors in a basis of the k-vector space eb(rad A/ rad A)ea. Example 2.55. Let A be the matrix algebra of the form k 0 0 A = 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,
1 0 0 0 0 0 0 0 0 e1 = 0 0 0 , e2 = 0 1 0 , e3 = 0 0 0 . 0 0 0 0 0 0 0 0 1
0 0 0 2 It can be shown that the radical of A equals rad A = k 0 0 and rad A = 0. From k 0 0 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 can conclude that the ordinary quiver of A will be the following
17 Sandra Berg - Global dimension of (higher) Nakayama algebras
1 α β
2 3.
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 −→ repk(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,
β γ 0 −→α 1 −→ 2 −→ 3
18 Sandra Berg - Global dimension of (higher) Nakayama algebras bounded by the admissible ideal γβα = 0. The simple modules are then,
0 0 0 S(0) = k −→ 0 −→ 0 −→ 0
0 0 0 S(1) = 0 −→ k −→ 0 −→ 0 0 0 0 S(2) = 0 −→ 0 −→ k −→ 0 0 0 0 S(3) = 0 −→ 0 −→ 0 −→ k 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 β 1 2. α
The A-modules
0 S(1) = k 0 0
0 S(2) = 0 k 0 and 1 Sλ = k k 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.
19 Sandra Berg - Global dimension of (higher) Nakayama algebras
0 0 0 0 2. Let rad(P (a)) = P (a)b, φβ , then P (a)b = P (a)b when a 6= b. P (a)a is the k-vector space spanned by the set of all w¯ = w + I, where w is a non-stationary 0 path with both source and target at a. φβ = φβ for any arrow β with target b 6= a. 0 0 For an arrow α that has a as target we get φα = φα|P (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
β γ 0 −→α 1 −→ 2 −→ 3, 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 ∼ P (3) = 0 → k → k → k.
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Example 2.64. Let Q be the quiver
β δ 1 2 3 α γ bound by δβ = 0 = γα. The indecomposable injective modules of the bound quiver algebra of Q are given by
0 0 ∼ I(1) = k 0 0 0 0
! 1 0 0 ∼ 2 I(2) = k k 0 ! 0 0 1
! ! 0 1 1 0 0 0
∼ 2 2 I(3) = k k k. ! ! 0 0 0 1 0 1
The indecomposable projective modules are given by
0 0 ∼ P (3) = 0 0 k 0 0
0 1 0 ∼ 2 P (2) = 0 k k 0 0 1
21 Sandra Berg - Global dimension of (higher) Nakayama algebras
! 0 0 1 0 1 0
∼ 2 2 P (1) = k k k . ! 0 1 0 1 0 0
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
β 1 2. α
Let M be the representation of Q as follows,
1 k 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: 0 k 0. 0
For the module e2A we instead have the span of e2, α, and β. The corresponding repre- sentation is the following:
hβ, αi he2i
22 Sandra Berg - Global dimension of (higher) Nakayama algebras
which is isomorphic to the projective module P2: 1 0
2 k k. 0 1
p0 For our projective resolution to be minimal, we need P0 −→ M to be a projective cover in the short exact sequence p1 p0 P1 −→ 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:
1 k k λ
a b 1 0
2 k k. 0 1
For this diagram to commute we need to choose appropriate morphisms a and b. Without loss of generality we choose b : → to be the identity. The linear map a : 2 → be a k k k k ! 1 2 × 1 matrix that should satisfy b 1 0 = 1 ◦ a and b 0 1 = λ ◦ a, hence b = . λ Next, we identify the kernel of a, which we set to the identity, to be {0} and the kernel ! 1 of b = to be isomorphic to . We can now extend our diagram further and recognize λ k the lowest module to be isomorphic to P (1)
23 Sandra Berg - Global dimension of (higher) Nakayama algebras
1 k k λ
a b 1 0
2 k k 0 1 c d
0 k 0. 0
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
β γ 0 −→α 1 −→ 2 −→ 3 and I is generated by γβα = 0. We will calculate a minimal projective resolution of the simple modules of A. These are
∼ 0 0 0 S(0) = k −→ 0 −→ 0 −→ 0 ∼ 0 0 0 S(1) = 0 −→ k −→ 0 −→ 0 ∼ 0 0 0 S(2) = 0 −→ 0 −→ k −→ 0 ∼ 0 0 0 S(3) = 0 −→ 0 −→ 0 −→ k. 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 ∼ P (3) = 0 → k → k → k.
24 Sandra Berg - Global dimension of (higher) Nakayama algebras
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 n ExtA(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,
f A B i g C D h 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
Ker(dn) Hn(C•) = Im(dn+1)
25 Sandra Berg - Global dimension of (higher) Nakayama algebras
and n • Ker(dn) H (C ) = 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•
hm h1 h0 0 → Pm −−→ Pm−1 → ...P1 −→ P0 −→ 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). n Now, we can define ExtA(M,N): Definition 2.73. Let A be a k-algebra. We define, for n ≥ 0, the n-th extension bifunctor as n n ExtA(M,N) = H (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(h1,N) HomA(h2,N) HomA(P•,N) : 0 → HomA(P0,N) −−−−−−−−→ HomA(P1,N) −−−−−−−−→ ...
HomA(hm+1,N) → HomA(Pm,N) −−−−−−−−−−→ HomA(Pm+1,N) → ... n Next to get ExtA(M,N) we compute the n-th cohomology k-vector space, i.e.
n Ker(HomA(hn+1,N)) H (HomA(P•,N)) = Im(HomA(hn,N).)
n Remark. The definition of ExtA(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
β γ 0 −→α 1 −→ 2 −→ 3
26 Sandra Berg - Global dimension of (higher) Nakayama algebras
k and I is generated by γβα = 0. We will calculate ExtA(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.
k k We know that ExtA(M,N) = H (Hom(P•,N)), where P• is the projective resolution of k M and H = Kerdk/Imdk−1. We begin by applying the contravariant HomA(−,S(i)) to the projective modules and their homomorphisms and hence getting
d−1 d0 d1 0 −−→ HomA(P (1),S(i)) −→ HomA(P (0),S(i)) −→ 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
d−1 d0 d1 0 −−→ k −→ 0 −→ 0. Hence the homology equals, 0 ∼ H = Kerd0/Imd−1 = k and 1 ∼ H = Kerd1/Imd0 = 0. Now let i = 0 and we get
d−1 d0 d1 0 −−→ 0 −→ k −→ 0 and the homology
0 ∼ H = Kerd0/Imd−1 = 0 and 1 ∼ H = Kerd1/Imd0 = k. From this we conclude that 0 ∼ ExtA(S(1),S(1)) = k 1 ∼ ExtA(S(1),S(1)) = 0 0 ∼ ExtA(S(1),S(0)) = 0 1 ∼ ExtA(S(1),S(0)) = k. Theorem 2.75. [1, Theorem A.4.5 (a) on page 428] 0 ∼ For M,N ∈ Mod A, there exists a functorial isomorphism ExtA(M,N) = HomA(M,N).
27 Sandra Berg - Global dimension of (higher) Nakayama algebras
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
2 irrA(M,N) = rad(M,N)/ rad (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
28 Sandra Berg - Global dimension of (higher) Nakayama algebras 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 β 0 −→α 1 −→ 2
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 (2) =∼ I(0)
P (1) I(1)
P (0) =∼ S(0) S(1) I(2) =∼ S(2)
∼ Example 3.6. Let A = kQ/I where Q is the quiver
β γ 0 −→α 1 −→ 2 −→ 3 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 ∼ P (3) = 0 → k → k → k
29 Sandra Berg - Global dimension of (higher) Nakayama algebras
∼ 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
2 ∼ P (2)/ rad P (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)/ rad2 P (2) I(2)
P (0) S(1) S(2) I(3)
∼ Example 3.7. Let A = kQ/I for the following quiver 2 3
1
4 5
3 where I = RQ. Then the Auslander-Reiten quiver will be as follows,
30 Sandra Berg - Global dimension of (higher) Nakayama algebras
P (5) P (1) P (2) P (3) P (4) P (5)
P (4)/ rad2 P (4) P (5)/ rad2 P (5) P (1)/ rad2 P (1) P (2)/ rad2 P (2) P (3)/ rad2 P (3) P (4)/ rad2 P (4)
S(4) S(5) S(1) S(2) S(3) 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/ soci M)) where p : M → M/ soci M 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 ⊂ soc2 M ⊂ soc3 M ⊂ · · · ⊂ socm M = M.
Since M is finite dimensional, it has a finite composition length, i.e. there exists a least positive m such that socm M = 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 ⊃ rad2 M ⊃ rad3 M ⊃ · · · ⊃ radm M = 0.
Because M has a finite composition series, there is also a least positive integer m such that radm M = 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.
31 Sandra Berg - Global dimension of (higher) Nakayama algebras
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 ⊃ rad2 M ⊃ · · · ⊃ 0 is a composition series.
4. The socle series 0 ⊂ soc M ⊂ soc2 M ⊂ · · · ⊂ 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/ rad2 P 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 2 is right serial if and only if, for every vertex a ∈ (QA)0, the A-module rad P (a)/ rad P (a) = 2 ea(rad A/ rad A) 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 2 that ea(rad A/ rad A)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:
32 Sandra Berg - Global dimension of (higher) Nakayama algebras
1.
• • ... • 0 1 n
2.
0
n 1
n − 1 2
i
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 i ∼ note that every proper arrow ideal R , 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 1 ≤ t ≤ ``(P ) such that M =∼ P/ radt P. In particular, A is representation-finite.
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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
1 1 0 → · · · → k −→ ... −→ k → 0 · · · → 0 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. (M(0, k) if k < ` P (k) = M(k − ` + 1, k) if k ≥ `
2. (M(k, k + ` − 1) if k ≤ n − ` I(k) = 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 1 1 0 0 0 k −→ ... −→ k −→ 0 −→ ... −→ 0 0 k n−1 which is precisely the interval module M(0, k). Analogously when k ≥ `, P (k) is isomorphic to
0 0 0 1 1 0 0 0 0 −→ ... −→ 0 −→ k −→ ... −→ k −→ 0 −→ ... −→ 0 0 k−`+1 k n−1 which is precisely the interval module M(k − ` + 1, k).
34 Sandra Berg - Global dimension of (higher) Nakayama algebras
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(n − 2, n − 1)
M(0, 0) M(1, 1) M(2, 2) ... 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 → M 0 → M → M 00 → 0 be a short exact sequence consisting of A-modules, with pd(M) ≤ n and pd(M 00) ≤ n + 1, then for the projective dimension of M 0 we get pd(M 0) ≤ n. Proof. We construct a long exact sequence of 0 → M 0 → M → M 00 → 0 with the extension functor, Ext, and the Hom-functor using Theorem 2.75. Applying an alternative k characterisation of projective dimension pd via vanishing of the functors ExtA(M, −) [3, i Proposition XII.8.3] and using that if pd(M) < n, we get that ExtA(M,N) = 0 for all N n+1 and all i ≥ n + 1 and ExtA (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
Ω0(L) Ω−1(L) Ω−k(L) Ω−r(L)
35 Sandra Berg - Global dimension of (higher) Nakayama algebras 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,
(M(0, x − 1) if y < ` ΩM(x, 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 0 ≤ r < `, then the projective dimension is given by
36 Sandra Berg - Global dimension of (higher) Nakayama algebras
(2q + 1 if `` < ` − r pd M(x, y) = 2q + 2 if `` ≥ ` − r.
3. Let n − 1 = q0` + r0, 0 ≤ r0 ≤ ` − 1. Then the projective dimension is
j n−1 k l n−1 m 0 0 ` + ` if r = 0 or j ≤ r pd M(n − j, n − 1) = j k l m n−1 n−1 ` + ` − 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 x − 1 − (y − ` + 1) + 1 = ` − y + x − 1 and ` − r0. We want the length to be larger
37 Sandra Berg - Global dimension of (higher) Nakayama algebras
than, or equal to ` − r0 so we can apply the formula. This is indeed the case since
` − r0 = ` − r − y + x − 1 ≤ ` − y + x − 1.
Then pd ΩM(x, y) = pd M(y − ` + 1, ` − y + x − 1) = 2q0 + 2. Now using ? we get
pd M(x, y) = pd M(y − ` + 1, ` − y + x − 1) + 1 = 2q0 + 2 + 1 = 2(q − 1) + 3 = 2q + 1
as wanted since y − x + 1 < ` − r.
Next we still let y ≥ ` but now assume y − x + 1 = `` ≥ ` − r. Here we want the projective dimension to equal 2q + 2. We go about this in a similar way as the previous case, and begin to calculate;
(y − ` + 1) − 1 = x − 1 + y − x − ` + 1 = q` + r + y − x − ` + 1.
Since ` − r ≤ y − x + 1 < ` we get 0 ≤ r + y − x − ` + 1 < r ≤ ` − 1. So x − 1 + y − x − ` + 1 = q` + r + y − x − ` + 1, where we let q be as defined but r0 = r + y − x − ` + 1. Again, we want to use the syzygy of M(x, y) and hence compare its length, ` − y + x − 1 to ` − r0 = 2` − r − y + x − 1. Note that ` > r and this motivates the following, 2` − r − y + x − 1 > ` − y + x − 1. Hence pd ΩM(x, y) = pd M(y − ` + 1, x − 1) = 2q + 1 and using again ? gives us
pd M(x, y) = pd M(y − ` + 1, x − 1) + 1 = 2q + 1 + 1 = 2q + 2
as wanted and this completes the proof of part (2) of the proposition.
3. Note that the length of the module M(n − j, n − 1) is n − 1 − (n − j) + 1 = j. We will prove (3) by using (2), hence we again write n − j − 1 = q` + r, equivalently n − 1 = q` + j + r. First, we assume j < ` − r, hence q0 = q and r0 = j + r < `. This corresponds to the first part in (2), so we want to confirm that the projective dimension is 2q + 1. Now r0 6= 0 but j ≤ j + r = r0, so we will use the following
38 Sandra Berg - Global dimension of (higher) Nakayama algebras
n − 1 n − 1 q` + j + r q` + j + r + = + ` ` ` ` j + r j + r = 2q + + ` ` 0 as wanted. Secondly, we assume j ≥ ` − r, so j + r ≥ `. As before we calculate, n − 1 = q` + j + r = (q + 1)` + (r + j − `) = q0` + r0. where q0 = q + 1 and r0 = r + j − `. We note that if j ≤ r0 then ` ≤ r, which is a contradiction, thus we conclude that j > r0. The last case needs to be divided into two different cases, depending on whether r0 = 0 or not. If r0 = 0 we get n − 1 n − 1 q0` q0` + = + = 2q0 = 2(q + 1) = 2q + 2 ` ` ` ` as wanted. If r0 6= 0 we get n − 1 n − 1 q0` + r0 q0` + r0 + − 1 = + − 1 ` ` ` ` r + j − ` r + j − ` = 2q0 + + − 1 ` ` ` as wanted. This concludes the proof. 4.3.1 Global dimension Proposition 4.18. Let A be an acyclic Nakayama algebra. Then n − 1 n − 1 gl. dim A = + ` ` j n−1 k l n−1 m Proof. From 4.17 (3), we know that gl. dim A ≤ ` + ` . The modules M(n − j, n − 1) are exactly the non-projective injective modules since I(n − j) = M(n − j, n − 1) for n − j > n − ` and the modules can’t be projective as well since if the length of the module would be j = ` then n − j = n − `, not n − j < n − `, which is the condition for a module to be both injective and projective in Proposition 4.13. Now we know 39 Sandra Berg - Global dimension of (higher) Nakayama algebras by Theorem 4.15 that these modules have the highest projective dimension and hence j n−1 k l n−1 m ` + ` is equal to the global dimension of A. Some examples of how these results are used are presented below. ∼ ` Example 4.19. Let A = kQ/I , where n = 5 and ` = 2. Then QA equals, β γ 0 −→α 1 −→ 2 −→ 3 −→θ 4 and the admissible ideal is generated by θγ = γβ = βα = 0. The Auslander-Reiten quiver of A is M(0, 1) M(1, 2) M(2, 3) M(3, 4) M(0, 0) M(1, 1) M(2, 2) M(3, 3) M(4, 4). We calculate the projective dimension of the modules M(0, 1) and M(3, 3) using the formulas above, Consider M(0, 1), since x = 0 we can clearly see that the module is projective, hence the projective dimension equals pd M(0, 1) = 0. For the module M(3, 3) we get x − 1 = 3 − 1 = 2 ⇒ q = 1, r = 0 and hence the projective dimension equals pd M(3, 3) = 2 · 1 + 1 = 3. The global dimension of A can be found among the non-projective injective modules, hence we only consider the projective dimension of M(4, 4), 5 − 1 5 − 1 gl. dim A = pd M(4, 4) = + = 2 + 2 = 4. 2 2 ∼ ` Example 4.20. Let A = kQ/I , where n = 6 and ` = 4. The quiver equals, β γ ξ 0 −→α 1 −→ 2 −→ 3 −→θ 4 −→ 5 and the admissible ideal is generated by ξθγβ = θγβα = 0. The Auslander-Reiten quiver of A equals, 40 Sandra Berg - Global dimension of (higher) Nakayama algebras M(0, 3) M(1, 4) M(2, 5) M(0, 2) M(1, 3) M(2, 4) M(3, 5) M(0, 1) M(1, 2) M(2, 3) M(3, 4) M(4, 5) M(0, 0) M(1, 1) M(2, 2) M(3, 3) M(4, 4) M(5, 5). We calculate the global dimension of A, 6 − 1 6 − 1 gl. dim A = + = 1 + 2 = 3. 4 4 5 Higher Nakayama algebras In this section we introduce higher Nakayama algebras, which as the name suggests are higher analogues of the classical Nakayama algebras presented above. Here we present the main results of this paper. In this section we will make use of the notation of Jasso and Külshammer as we present many results from their paper [5]. d 5.1 Higher Nakayama algebras of type An We need some notations and definitions to aid the introduction of higher Nakayama algebras. Notation. Let P = (P, ≤) be a poset and d a positive integer. We provide P with the Cartesian product, P d = P × · · · × P | {z } d times and the product order as follows: given tuples x = (x1, . . . , xd) and y = (y1, . . . , yd) in d P , we define x y if and only if for each i ∈ {1, . . . , d} the relation xi ≤ yi is satisfied. The new poset with these properties is denoted Pd := (P d, ). Definition 5.1. Let P = (P, ≤) be a poset and d a positive integer. 1. If for the d tuples x, y ∈ P d we have that x1 ≤ y1 ≤ x2 ≤ y2 ≤ · · · ≤ xd ≤ yd we say that x interlaces y. This relation is denoted x y. From this definition we see that if x interlaces with itself, x is an ordered sequence of length d. Consequently x is an ordered sequence of length d if and only if x1 ≤ x2 ≤ · · · ≤ xd. 41 Sandra Berg - Global dimension of (higher) Nakayama algebras We denote the set of ordered sequences of length d in P by osd(P). 2. We define the poset algebra P as isomorphic to a bounded path algebra kQ/I. The quiver Q consists of the vertices of the poset Q0 = P and the arrows Q1 = {x → y | x < y and 6 ∃z; x < z < y | x, z, y ∈ P }. For every sub-quiver of this quiver below there is such a relation of the form βα − δγ, z α 1 β x y. γ δ z2 3. We define the d-cone of P to be the idempotent quotient d d cone(P ) := P /(Pd\osd(P)). From the definition of interlacing and ordered sequences of length d in a poset we can make the following observation. Proposition 5.2. [5, Proposition 1.10] Let P = (P, ≤) be a poset and d a positive d integer. Then for every pair of tuples x, y ∈ P there is an interlacing x y if and only if x y and [x, y] ⊂ osd(P). In particular, ( k if x y cone(Pd)(x, y) =∼ 0 otherwise. d We will now use this theory to introduce the higher Nakayama algebras. Our An in the form of a quiver was defined by Iyama in Definition 6.5 and Theorem 6.7 in [4]. We will see that our definition using the definition of the d-cone of a poset will agree with Iyama’s d d d construction. The ordinary quiver Q of An has as vertices the set osn := os (An) of ordered sequences of length d in A. This can also be described as tuples λ = (λ1, . . . , λd) of integers which satisfy 0 ≤ λ1 ≤ · · · ≤ λd ≤ n − 1. d d Now for the arrows in the quiver Q of An we consider the following, for each λ ∈ osn and each i ∈ {1, . . . , d} such that λ + ei is also an ordered sequence of length d, the arrows will be of the form ai(λ): λ → λ + ei where ei is a standard basis vector of the form (0,..., 0, 1, 0,..., 0) where 1 is at position i. d ∼ Using this notation we can establish An = kQ/I where I is the admissible ideal of kQ 42 Sandra Berg - Global dimension of (higher) Nakayama algebras generated by the relations aj(λ + ei)ai(λ) − ai(λ + ej)aj(λ) d for each λ ∈ osn for each i, j ∈ {1, . . . , d} such that i 6= j. As previously, we say that ai(λ) = 0 whenever λ or λ + ei are not vertices of Q, thus some of these relations 1 will be zero relations. As an example, An is just the path algebra over the quiver 0 → 1 → 2 → · · · → n − 1. More specific examples of these quivers are seen below. 1 Example 5.3. Here we see the quiver of A4 0 → 1 → 2 → 3 2 Example 5.4. The quiver of A4 03 02 13 01 12 23 00 11 22 33 3 Example 5.5. The quiver of A4 003 013 023 033 002 012 022 113 123 133 001 011 112 122 223 233 000 111 222 333 3 The quiver of A4 can also be presented in this way. 43 Sandra Berg - Global dimension of (higher) Nakayama algebras 033 023 133 013 123 233 003 113 223 333 022 012 122 002 112 222 011 001 111 000 Remark. For those that are interested, the last is the way Iyama presents the quivers and may give us some insight of how they are tied together. d Remark. Note that the quiver of An has the higher Auslander-Reiten quiver corresponding d+1 to An . The higher Auslander-Reiten quiver, which is part of higher Auslander-Reiten theory, is described in Theorem 6.12 in Iyama’s paper [4]. This tells us that in the 1 examples presented above, the higher Auslander-Reiten quiver of A4 is the same as the 2 quiver of A4, but the nodes would change from xy to M(x, y) to represent the isomorphism classes of indecomposable modules instead. d+1 Definition 5.6. Let λ ∈ osn , then the interval module denoted M(λ) equals M(λ) := M ((λ1, . . . , λd), (λ2, . . . , λd+1)) . d+1 Lemma 5.7. [5, Lemma 2.9] Let λ ∈ osn . Then the interval module M(λ) has Loewy length ``(λ) = λd+1 − λ1 + 1 d Proposition 5.8. [5, Proposition 2.5] Let λ ∈ osn. Then the following statements hold. d 1. The projective An-module at the vertex λ is precisely P (λ) = M(0, λ1, . . . , λd). 44 Sandra Berg - Global dimension of (higher) Nakayama algebras d 2. The injective An-module at the vertex λ is precisely I(λ) = M(λ1, . . . , λd, n − 1). Proposition 5.9. [5, Proposition 2.7] d+1 1. Let λ ∈ osn be such that λ1 6= 0 (from Proposition 5.8 we know this implies M(λ) is not projective) and 0 → P d → P d−1 → · · · → P 0 → M(λ) → 0 a minimal projective resolution of M(λ). Then 0 ∼ P = M(0, λ2, . . . , λd+1) and for each i ∈ {1, . . . , d} there is an isomorphism i ∼ P = M(0, λ1 − 1, . . . , λi − 1, λi+2, . . . , λd+1). d+1 2. Let λ ∈ osn be such that λd+1 6= n − 1 (from Proposition 5.8 we know this implies M(λ) is not injective) and 0 → M(λ) → I0 → · · · → Id−1 → Id → 0 a minimal injective coresolution of M(λ). Then 0 ∼ I = M(λ1, . . . , λd, n − 1) and for each i ∈ {1, . . . , d} there is an isomorphism i ∼ P = M(λ1, . . . , λd−i, λd−i+2 + 1, . . . , λd+1 + 1, n − 1). d 5.2 Nakayama algebras of type A` ∼ In the classical case the Nakayama algebras A = kQ/I are admissible quotients of path d algebras. Now as we define the higher Nakayama algebras A` we notice that they are idempotent quotient of the higher Auslander algebras as defined in Iyama’s paper [4]. We begin with introducing Kupisch series of type An Definition 5.10. Let ` = (`0, `1, . . . , `n−1) be an n-tuple of positive integers where `0 = 1 and for all i 6= 0 the inequalities holds 2 ≤ `i ≤ `i−1 + 1. Then ` is a Kupisch series of type An. Note that this definition of the Kupisch series gives us that ` will be of the form that it can not increase by more than 1 by each step, while it can decrease by more than 1, but 45 Sandra Berg - Global dimension of (higher) Nakayama algebras cannot go lower than 2 after the first position. Some examples are presented below. Note that the first one is the minimal Kupisch series, while the next is maximal in KS(An), which is the set of Kupisch series endowed with the structure of a poset. Example 5.11. ` = (1, 2, 2, 2,..., 2) Example 5.12. ` = (1, 2, 3, 4, . . . , n) Example 5.13. ` = (1, 2, 3, 4, 5, 2, 2, 3) Example 5.14. ` = (1, 2, 3, 4, 2, 3, 4) Example 5.15. ` = (1, 2, 3, 4, 5, 5, 5, 5) We will now define the restriction of Kupisch series on the ordered sequences, higher Nakayama algebra and its modules. Definition 5.16. Let ` be a Kupisch series of type An. d+1 1. The `-restriction of osn is the subset d+1 n d+1 o os` := λ ∈ osn | ``(λ) ≤ `λd+1 . 2. The (d+1)-Nakayama algebra with Kupisch series ` is the finite dimensional algebra (d+1) (d+1) d+1 d+1 A` := An / osn \ os` . d d 3. The An-module M` is by definition d M n d+1o M` := M(λ) | λ ∈ os` . d+1 d+1 Remark. The condition for λ ∈ osn to also belong to os` for some Kupisch series of type An only imposes conditions on (λ1, λd+1). Theorem 5.17. [5, Theorem 2.18 (i)] Let ` be a Kupisch series then for each i ∈ d {0, 1, . . . , n − 1} the indecomposable projective A` -module at the vertex (i, . . . , i) has length `i. The previous theorem tells us that there exists an equivalence between the lengths of the d indecomposable projective modules in A` and the Kupisch series `. This makes it clear how their quivers are constructed, some examples are given below. 2 3 Example 5.18. Consider the higher Nakayama algebras A` (top) and A` (bottom) for the Kupisch series ` = (1, 2, 2, 3) and their quivers are as follows, 46 Sandra Berg - Global dimension of (higher) Nakayama algebras 13 01 12 23 00 11 22 33 113 123 133 001 011 112 122 223 233 000 111 222 333 2 3 Example 5.19. Consider the higher Nakayama algebras A` (top) and A` (bottom) for the Kupisch series ` = (1, 2, 3, 2) and their quivers are as follows, 02 01 12 23 00 11 22 33 002 012 022 001 011 112 122 223 233 000 111 222 333 d We will from now on consider the higher Nakayama algebras of type A` with a special case of `. This special case equals, ` = (1, 2, . . . , `, . . . , `) where ` will be the maximal length of an indecomposable projective module. Some examples of quivers of this form are presented below. 2 3 Example 5.20. The quivers of A` and A` (presented in two ways), for ` = (1, 2, 3, 3, 3), 02 13 24 01 12 23 34 00 11 22 33 44 47 Sandra Berg - Global dimension of (higher) Nakayama algebras 002 012 022 113 123 133 224 234 244 001 011 112 122 223 233 334 344 000 111 222 333 444 244 234 344 224 334 444 133 123 233 113 223 333 022 012 122 002 112 222 011 001 111 000 2 3 Example 5.21. The quiver of A` and A` for ` = (1, 2, 3, 4, 4, 4) 03 14 25 02 13 24 35 01 12 23 34 45 00 11 22 33 44 55. 48 Sandra Berg - Global dimension of (higher) Nakayama algebras 255 245 355 235 345 455 225 335 445 555 144 134 244 124 234 344 114 224 334 444 033 023 133 013 123 233 003 113 223 333 022 012 122 002 112 222 011 001 111 000 d The higher Nakayama algebras of type A` where the Kupisch series ` is of the form ` = (1, 2, . . . , `, . . . , `) are the ones for which we will find a formula for the global dimension, similar to the one in Vaso’s paper [6]. Consequently we will again find the formulas for the injective and projective modules and the syzygies of the modules, but for the higher dimensional case. d Proposition 5.22. For each vertex λ ∈ osn and ` as in the special case of Kupisch series introduced above, the following statements hold. 49 Sandra Berg - Global dimension of (higher) Nakayama algebras d 1. The projective A` -module at the vertex λ is precisely (M(0, λ , . . . , λ ) if λ < ` P (λ) = 1 d d M(λd − ` + 1, λ1, . . . , λd) if λd ≥ `. d 2. The injective A` -module at the vertex λ is precisely (M(λ , . . . , λ , λ + ` − 1) if λ ≤ n − ` I(λ) = 1 d 1 1 M(λ1, . . . , λd, n − 1) if λ1 > n − `. 3. M(λ) is both injective and projective if and only if the Loewy length ``(M(λ)) = λd+1 − λ1 + 1 = ` and 1 ≤ λ1 ≤ n − `. Proof. Follows from Lemma 2.61 and Lemma 2.62 which gives us the modules in terms d of paths in the ordinary quiver of A` . This can also be proven by describing a basis of d the A` projective (and injective) modules using Proposition 5.2. This proposition shows d the existence of at most one non-zero path between any two vertices in A` and that the existence is determined by the interlacing relation defined above. Note that the case when d = 1 coincides with Proposition 4.13. d+1 d Proposition 5.23. Let M(λ) 6= 0 for λ ∈ os` be a non-projective A` -module. Then the syzygy for the non-projective modules are as follows, (M(0, λ − 1, . . . , λ − 1) if λ < ` ΩdM(λ) = 1 d d+1 M(λd+1 − ` + 1, λ1 − 1, . . . , λd − 1) if λd+1 ≥ ` Proof. Similar to the classical one dimensional case we will prove this by construction. We will prove the second case of the proposition, the first one is proved in a similar manner. We begin by noting that since M(λ) is non-projective we have that λd+1 − ` + 1 < λ1. Let 0 → ΩdM(λ) → P d−1 → · · · → P 0 → M(λ) → 0 be a part of a minimal projective resolution of M(λ). Calculations using Proposition 5.22 give us that 0 ∼ P = M(λd+1 − ` + 1, λ2, . . . , λd+1), and for the rest of the projective modules, for each i ∈ {1, . . . , d − 1} we have i ∼ P = M(λd+1 − ` + 1, λ1 − 1, . . . , λi − 1, λi+2, . . . , λd+1), and finally d ∼ Ω M(λ) = M(λd+1 − ` + 1, λ1 − 1, . . . , λd − 1). d+1 We can also see that (λd+1 − ` + 1, λ1 − 1, . . . , λd − 1) ∈ os` by the definition of our special case ` and that λd+1 − ` + 1 < λ1 − 1. 50 Sandra Berg - Global dimension of (higher) Nakayama algebras 2 Example 5.24. Let the higher Nakayama algebra be the following A` , where ` = (1, 2, 3, 3, 3, 3), the ordinary quiver looks as follows 02 13 24 35 01 12 23 34 45 00 11 22 33 44 55. 2 We will present the projective resolution of the module I(44) = M(445) in A` . Calculations using Proposition 5.22 give us: M(001) M(002) M(222) M(123) M(133) M(335) M(345) M(445) =∼ =∼ =∼ =∼ =∼ =∼ =∼ =∼ 0 P (01) P (02) P (22) P (23) P (33) P (35) P (45) I(44) 0. Using Proposition 5.23 instead we get Ω2(M(445)) = M(333) Ω2·2(M(445)) = M(122) Ω2·3(M(445)) = M(001) Thus Ω6(I(44)) is projective and we can confirm from both calculations that the projective dimension of M(445) equals 6. 5.2.1 Projective dimension We will now present the main results of the thesis. d d+1 Theorem 5.25. Let A` , where ` = (1, 2, . . . , `, . . . , `) and for λ ∈ osn we have the d+1 modules M(λ) ∈ mod A` 1. Let M(λ) 6= 0 and assume that λ1 = 0 or that the Loewy length of the module, `` = λd+1 − λ1 + 1, equals `, then pd M(λ) = 0. 2. Let M(λ) 6= 0 and assume that λ1 > 0 or that the Loewy length of the module, `` = λd+1 − λ1 + 1, is less than `. Consider the following sets of modules with the same projective dimension, pd(M(λ)) S = λ ∈ osd+1 = i i ` d 51 Sandra Berg - Global dimension of (higher) Nakayama algebras Then we have, when λ1 < ` + d n d+1 o S1 = λ ∈ os` |λd+1 < ` n d+1 o S2 = λ ∈ os` |λd+1 ≥ `, and λd < ` + 1 n d+1 o S3 = λ ∈ os` |λd ≥ ` + 1, and λd−1 < ` + 2 ... n d+1 o Sd+1 = λ ∈ os` |λ2 ≥ ` + d − 1, and λ1 < ` + d For any λ1 we get n d+1 o Sk = Sq(d+1)+r = λ ∈ os` |λ − q(` + d − 1, . . . , ` + d − 1) ∈ Sr d+1 To determine this we consider some M(λ), where λ ∈ os` and we have some q, such that λ1 − q(` + d − 1) < ` + d. Then λ − q(` + d − 1, . . . , ` + d − 1) ∈ Sr for some r ∈ {1, 2, . . . , d + 1} and we get pd M(λ) = d(q(d + 1) + r) Proof. 1. It is clear from Proposition 5.22 that the interval modules where λ1 = 0 or `` = ` are projective modules. Hence we know that the projective dimension equals to 0. 2. Now we calculate the projective dimension using Ωd as in Proposition 5.23, the projective dimension equals how many times we need to take the dth syzygy before it equals a projective module, so if Ωp·d(M(λ)) is a projective module, pd M(λ) = d·p. We begin when λ1 < ` + d and prove the statement from the first (d + 1) possible dimensions (divided by d). pd(M(λ)) We start by assuming λd+1 < ` for M(λ) and confirm that d = 1. We use Proposition 5.23 and since we assumed λd+1 < ` it is the first case of the proposition and we get d Ω (M(λ)) = M(0, λ1 − 1, . . . , λd − 1) pd(M(λ)) which is clearly projective as it starts with 0 and consequently d = 1 as wanted. pd(M(λ)) Next we assume λd+1 ≥ ` and λd < `+1 for M(λ) to confirm that d = 2. We calculate the dth syzygy and now since λd+1 ≥ ` it is the second case of Proposition 5.23 and we get d Ω (M(λ)) = M(λd+1 − ` + 1, λ1 − 1, . . . , λd − 1). 52 Sandra Berg - Global dimension of (higher) Nakayama algebras Since λd < ` + 1 ⇔ λd − 1 < ` the first case of the proposition is applied. 2d Ω (M(λ)) = (0, λd+1 − `, . . . , λd−1 − 2) pd(M(λ)) This module is projective and we can confirm that d = 2. Assuming now for M(λ) that λd ≥ `+1 and λd−1 < `+2 we calculate the projective dimension, again using the dth syzygy. The module Ωd(M(λ)) looks the same, but now λd − 1 ≥ ` and we apply the second case of the proposition for dth syzygies and for Ω2d we get the following, 2d Ω (M(λ)) = (λd − `, λd+1 − `, . . . , λd−1 − 2). We assumed λd−1 < ` + 2 and hence λd−1 − 2 < `, again we apply the first case of Proposition 5.23 and get for our next dth syzygy, 3d Ω (M(λ)) = (0, λd−1 − ` − 1, . . . , λd−2 − 3) pd(M(λ)) which is projective. Thus d = 3 when λd ≥ ` + 1 and λd−1 < ` + 2. The criteria continue in the same way and the projective dimension can be computed in a similar way. pd(M(λ)) Lastly we want to confirm d = d + 1 when λ2 ≥ ` + d − 1 but we still have that λ1 < ` + d. The first (d − 1) times we apply the second case of Proposition 5.23 until λ2 is at the last coordinate and get the following, (d−1)·d Ω (M(λ)) = (λ3 − ` − d + 3, . . . , λ1 − d + 1, λ2 − d + 1). Since λ2 ≥ ` + d − 1 by assumption, we have that λ2 − d + 1 ≥ ` and again apply the second case of the dth syzygy and get d·d Ω (M(λ)) = (λ2 − ` − d + 2, λ3 − ` − d + 2, . . . , λ1 − d) Now λ1 − d < ` and the next d-th syzygy becomes (d+1)·d Ω (M(λ)) = (0, λ2 − ` − d + 1, . . . , λd+1 − d + 1) pd(M(λ)) which is clearly a projective module and we can conclude that d = d + 1 when λ2 ≥ ` + d − 1. Furthermore we show that if λ1 ≥ ` + d the projective dimension divided by d, will at least be d + 2. Note that if λ1 is assumed to be larger than or equal to ` + d, then 53 Sandra Berg - Global dimension of (higher) Nakayama algebras all of λ2, . . . , λd+1 ≥ ` + d, hence we apply the second case of Proposition 5.23. d Ω (M(λ)) = M(λd+1 − ` + 1, λ1 − 1, λ2 − 1, . . . , λd − 1) Here λd − 1 ≥ ` by assumption, so we apply second case again. 2d Ω (M(λ)) = M(λd − `, λd+1 − `, λ1 − 2, . . . , λd−1 − 2) ... (this is repeated d + 1 times until our λ1 is in last position) d(d+1) d Ω = Ω M(λ2 − ` + 1 − d, . . . , λ1 − d) = M(λ1 − d − ` + 1, . . . , λd+1 − ` − d) The last syzygy, which is isomorphic to Ωd(d+1)(M(λ)) is still not a projective module since we assumed λ1 ≥ ` + d, whence λ1 − d − ` + 1 ≥ 1. Thus the projective pd M(λ) dimension d ≤ d + 1 if and only if λ1 < ` + d. d As a last part of the proof we will show that Ω of a module in Sk lies in Sk−1 by induction. To motivate that all other Sk after p(d + 1) end up in one of S1,S2,...,Sd+1. For the base case, we let λ ∈ S2, meaning λd+1 ≥ ` and λd < ` + 1 and then d Ω (M(λ)) = M(λd+1 − ` + 1, λ1 − 1, . . . , λd − 1). By assumption λd < ` + 1 ⇔ d λd−1 < ` and thus Ω (M(λ)) lies in S1. For the induction step we have two different cases for Sk = Sq(d+1)+r, one case when r = 1 and another when r ∈ {2, 3, . . . , d+1}. In the case when r = 1 we are between two bundles of (d + 1) different projective dimensions, such as in Sd+1 and Sd+2. We begin considering when r = 2 and r = 3 up to r = d + 1. When r = 2 we have M(λ1, . . . , λd+1) where λd+1 − q(` + d − 1) ≥ ` and λd − q(` + d − 1) < ` + 1 satisfies the conditions for r = 2. So for Ωd(M(λ)) = M(λd+1−`−d+1, λ1−1, . . . λd−1) we have the last coordinate λd−1−q(`+d−1) < ` which satisfies the conditions for (λd+1 − ` − d + 1, λ1 − 1, . . . λd − 1) − q(` + d − 1, . . . , ` + d − 1) ∈ S1 d and Ω (M(λ)) lies in Sk−1 as wanted. If r = 3 for Sk, then M(λ1, . . . , λd+1) where λd − q(` + d − 1) ≥ ` + 1 and d λd−1 − q(` + d − 1) < ` + 2, since λ − q(` + 1) ∈ S3. Then the module Ω (M(λ)) has λd − 1 − q(` + 1) ≥ ` and λd−1 − 1 − q(` + 1) < ` + 1, which are the conditions d for r = 2 and Ω (M(λ)) lies in Sk−1 as wanted. This goes on in a similar manner until r = d + 1 for Sk = Sq(d+1)+r, then the module M(λ1, . . . , λd+1) satisfies the conditions λ2 − q(` + d − 1) ≥ ` + d − 1 and d λ1−q(`+d−1) < `+d. Considering Ω (M(λ)) = M(λd+1−`−d+1, λ1−1, . . . λd−1) we have that the second coordinate becomes λ1 −1 and hence λ1 −1−q(`+d−1) < ` + d − 1 and λ2 − 1 − q(` + d − 1) ≥ ` + d − 2 which are the conditions for when d r = d and we can conclude that Ω (M(λ)) lies in Sk−1. For the case when r = 1 for Sk = Sq(d+1)+r we have M(λ1, . . . , λd+1), where 54 Sandra Berg - Global dimension of (higher) Nakayama algebras λd+1 − q(` + d − 1) < ` and since λ is an ordered sequence all λ1 − q(` + d − 1), λ2 − q(` + 1), . . . , λd − q(` + d − 1) are also less than `. Then the d-th syzygy becomes d Ω (M(λ)) = M(λd+1 − ` − d + 1, λ1 − 1, . . . λd − 1) and we want to prove that (λd+1 −`+1−(q−1)(`+d−1), λ1 −1−(q−1)(`+d−1), . . . , λd −1−(q−1)(`+d−1)) lies in Sd+1, since Sk−1 is in the previous group of (d + 1) sets. Hence we want λ1 − 1 − (q − 1)(` + d − 1) ≥ ` + d − 1, which is equivalent to (λ1 − 1) − q(` + d − 1) ≥ 0 ⇔ λ1 − q(` + d − 1) ≥ 1. This holds since λ − q(` + d − 1, . . . , ` + d − 1) ∈ S1 and λ1 ≥ 1. Furthermore we want λd+1 − ` − d + 1 − (q − 1)(` + d − 1) < ` + d, which can be written as λd+1 − (` + d − 1) − (q − 1)(` + d − 1) < ` + d ⇔ λd+1 − q(` + d − 1) < ` + d this holds by assumption, and we are done. 2 Example 5.26. Consider A` for ` = (1, 2, 3, 3, 3, 3) and its ordinary quiver. 02 13 24 35 01 12 23 34 45 00 11 22 33 44 55 We will use the previous proposition to calculate the projective dimension of the modules M(112),M(334),M(455) and M(555). Note that ` = 3 and d = 2. pd M(112) For M(112) we have that λ1 < ` + d = 3 + 2 and λ3 < ` = 3 hence 2 = 1. Looking at M(334) we see that we still have λ1 < 5 but now λ3 > 3 and λ2 < ` + 1 = 4, pd M(334) hence we end up in S2 and 2 = 2. pd M(455) Next, the module M(455) still has λ1 < 5 but λ2 ≥ 4 hence 2 = 3. Lastly for the module M(555), we have λ1 = 5 hence we look for what q that satisfies 5 − q(3 + 2 − 1) = 5 − 4q < 5, which is q = 1. Then (5, 5, 5) − 1 · (4, 4, 4) = (1, 1, 1) which pd M(555) lies in S1, thus q = 1 and r = 1 and 2 = 3 · 1 + 1 = 4. 5.2.2 Global dimension We know that among the non-projective injective modules we have the modules with the maximal projective dimension. From Proposition 5.22 we get that the non-projective 55 Sandra Berg - Global dimension of (higher) Nakayama algebras injective modules are of the form M(n − j, . . . , n − 1), where j < ` is the Loewy length of the module and the integers between n − j and n − 1 are arbitrary. We will prove that the interval module M(n − 1, . . . , n − 1) has the maximal projective dimension in order to find the global dimension. d Proposition 5.27. For a bounded higher Nakayama algebra A` , where d+1 ` = (1, 2, . . . , `, . . . , `), we let M(λ1, . . . , λd+1) ∈ mod A` be a module of maximal projective dimension. Then M(n − 1, . . . , n − 1) has the same projective dimension. Proof. Using Proposition 5.25 we let q be such that λ1 − q(` + d − 1) < ` + d. Then λ − q(` + d − 1, . . . , ` + d − 1) ∈ Sr for some r ∈ {1, 2, . . . , d + 1} and M(λ) ∈ Sq(d+1)+r. We will prove that the projective dimension of M(λ1, . . . , λd+1) is the same as pd M(n − 1, . . . , n − 1) in two steps, first we will show that q is the same for both of them and second that r is the same. 1. To show that q is the same we begin by assuming (n − 1) − q(` + d − 1) < 0 and λ1 − q(` + d − 1) ≥ 0. Then n − 1 < λ1 which is a contradiction to the definition of (d+1) osn . Now instead we assume (n − 1) − q(` + d − 1) ≥ ` + d and then there exists a q0 > q 0 such that (n − 1) − q (` + d − 1) < ` + d. Then M(n − 1, . . . , n − 1) ∈ S(d+1)q0+r0 0 for some r ∈ {1, . . . , d + 1} and by assumption M(λ1, . . . , λd+1) ∈ S(d+1)q+r, but then (d + 1)q0 + r0 ≥ (d + 1)q0 + 1 > (d + 1)(q + 1) which is also a contradiction. Thus we have shown that q must be the same for both modules. 2. Now we will prove that r is the same. We assume the contrary, that (n − 1, . . . , n − 0 0 1)−q(`+d−1, . . . , `+d−1) ∈ Sr0 for some r < r, then (n−1)−q(`+d−1) ≥ `+r −2 and (n − 1) − q(` + d − 1) < ` + r0 − 1. We know for some k we have λk − q(` + d − 1) ≥ ` + r − 2 and λk − q(` + d − 1) < ` + r − 1 0 which implies (n − 1) − q(` + d − 1) < ` + r − 1 ≤ ` + r − 2 ≤ λk − q(` + d − 1) d+1 0 which is a contradiction to the definition of osn . The reason ` + r − 1 ≤ ` + r − 2 holds is since we assumed r0 < r. If r0 > r we immediately get a contradiction since then M(λ) would no longer have the maximal projective dimension. Thus we have shown that r is the same and that M(n − 1, . . . , n − 1) has the maximal projective dimension. d Theorem 5.28. Let A` be a bounded higher Nakayama algebra, where the Kupisch series d is ` = (1, 2, . . . , `, . . . , `). Then the global dimension of A` is as follows. Consider n − 1 q = ` + d − 1 56 Sandra Berg - Global dimension of (higher) Nakayama algebras and (1 if n − 1 − q(` + d − 1) < ` r = n − 1 − q(` + d − 1) − ` + 2 otherwise. Then using q and r we get the global dimension (d · (d + 1)q if ` + d − 1| n − 1 gl. dim Ad = ` d · ((d + 1)q + r) otherwise. Proof. From Proposition 5.27 we know that the module M(n − 1, . . . , n − 1) will have the maximal projective dimension and thus we calculate the global dimension by calculating the projective dimension of M(n − 1, . . . , n − 1) using Proposition 5.25. We divide the proof into two cases, first when n − 1 < ` + d and we get q = 0 and hence r = n − ` + 1 and consequently the global dimension equals d · r. The second case is when n − 1 ≥ ` + d. 1. When n − 1 < ` + d we get from Proposition 5.25 that the projective dimension of M(n − 1, . . . , n − 1) will be d, 2d, 3d, . . . , or (d + 1)d. d d a) If n − 1 < ` then the higher Nakayama algebra A` is isomorphic to An, which has global dimension d by results of Iyama. We can see that other values than d as global dimension are not possible since we note that the Kupisch series is an n-tuple with entries (1, 2, 3, . . . , `, . . . , `), so ` cannot be bigger than n. d Hence gl. dim A` = d as wanted. b) If n − 1 = ` ⇔ n = ` + 1 then since λd+1 = n − 1 and λd = n − 1 we get (n − 1, . . . , n − 1) ∈ S2 and pd M(n − 1, . . . , n − 1) = 2d which coincides with d gl. dim A` = d(n − ` + 1) = d(` + 1 − ` + 1) = 2d as wanted. c) If n−1 = `+1 ⇔ n = `+2 then (n−1, . . . , n−1) ∈ S3 and pd M(n−1, . . . , n− d 1) = 3d. This coincides with gl. dim A` = d(n − ` + 1) = d(` + 2 − ` + 1) = 3d as wanted. d) This proceeds in a similar manner until the last case when n − 1 = ` + d − 1 ⇔ n = ` + d and from this we get that (n − 1, . . . , n − 1) ∈ Sd+1 and the d gl. dim A` = d(n − ` + 1) = d(` + d − ` + 1) = (d + 1)d as wanted. 2. We proceed to prove the second part of the proposition by calculating pd M(n − 1, . . . , n − 1). When n − 1 ≤ ` + d we need to apply the second part of Proposition 5.25 and need a q such that n − 1 − q(` + d − 1) < ` + d, by the Euclidean algorithm for division, this q will be n − 1 q = . ` + d − 1 We get from Proposition 5.25 that d · r = pd M(n − 1 − q(` + d − 1), . . . , n − 1 − q(` + d − 1)) and use this to calculate r. 57 Sandra Berg - Global dimension of (higher) Nakayama algebras By definition of q we know that n − 1 − q(` + d − 1) < ` + d and r ∈ {1, 2, . . . , d + 1} and hence r is of the following form: If n − 1 − q(` + d − 1) < ` then r = 1 If n − 1 − q(` + d − 1) = ` then r = 2 If n − 1 − q(` + d − 1) = ` + 1 then r = 3 If n − 1 − q(` + d − 1) = ` + 2 then r = 4 ... If n − 1 − q(` + d − 1) = ` + d − 1 then r = d + 1. This is equivalent to (1 if n − 1 − q(` + d − 1) < ` r = n − 1 − q(` + d − 1) − ` + 2 otherwise. j n−1 k n−1 When ` + d − 1 divides n − 1 we get that q = `+d−1 = `+d−1 . Then d · r = pd M(n − 1 − q(` + d − 1), . . . , n − 1 − q(` + d − 1)) = pd M(0,..., 0) = 0 and thus we set our r = 0. From this we conclude that the two cases for the global dimension become, (d · (d + 1)q if ` + d − 1| n − 1 gl. dim Ad = ` d · ((d + 1)q + r) otherwise. 58 Sandra Berg - Global dimension of (higher) Nakayama algebras References [1] I. Assem, D. Simson, A. Skowronski Elements of the Representation Theory of Associative Algebras, London Mathematical Society, (2006) [2] M. Auslander, I. Reiten, S. Smalø Representation theory of Artin algebras Vol. 36. Cambridge University Press (1997) [3] P. Grillet Abstract algebra Vol. 242. Springer Science & Business Media, (2007) [4] O. Iyama Cluster Tilting For Higher Auslander Algebras Advances in Mathematics 226.1 (2011) [5] G. Jasso, J. Külshammer Higher Nakayama Algebras I: Construction, Advances in Mathematics 351, (2019) [6] L. Vaso n-Cluster tilting subcategories of representation-directed algebras, J. Pure Appl. 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