REPRESENTATIONS AND QUIVERS FOR RING THEORISTS

JUN-ICHI MIYACHI

Contents

1. Modules and Representations 1 2. Quivers and Path Algebras 5 3. Quivers with Relations 9 4. Base Extensions and Representations 13 5. Examples related to Tachikawa’s Conjecture 15 5.1. The Case of Algebras 15 5.2. The Case of Rings 18 6. Appendix 20 References 23

1. Modules and Representations

Throughout this note, k is a field, and we deal with associative k-algebras. A k-algebra A is a k-vector space with a k-bilinear map μ : A × A → A satisfying

\{ \begin{matrix}1_{A} ∈ A \\ μ(1_{A}, a) & = a (^{∀}a ∈ A) \\ μ(a, 1_{A}) & = a (^{∀}a ∈ A) \\ μ ∘ (μ × 1) & = μ ∘ (1 × μ) \end{matrix} (1.1)

A × A × A ⟶^{μ×1} A × A
1×μ↓  ↓^{μ}
A × A ⟶^{μ} A

In this note, for a k-algebra A, we fix a complete set {e_{i}|1 ≤ i ≤ n} of orthogonal primitive idempotents of A. Then we have

A = ⨁_{1≤i,j≤n} e_{i}Ae_{j}

as a k-vector space and a family of k-bilinear maps

μ_{ijk} : e_{i}Ae_{j} × e_{j}Ae_{k} → e_{i}Ae_{k}

Date: December 2000.

This is a seminar note of which I gave a lecture at Yamaguchi University in December 2000.

such that

\{ \begin{matrix}e_{i} ∈ e_{i}Ae_{i} & (^{∀}i) \\ μ_{iij}(e_{i}, a_{ij}) & = a_{ij} (^{∀}a_{ij} ∈ e_{i}Ae_{j}) \\ μ_{ijj}(a_{ij}, e_{j}) & = a_{ij} (^{∀}a_{ij} ∈ e_{i}Ae_{j}) \\ μ_{ikl} ∘ (μ_{ijk} × 1) & = μ_{ijl} ∘ (1 × μ_{jkl}) \end{matrix} (1.2)

e_{i}Ae_{j} × e_{j}Ae_{k} × e_{k}Ae_{l} ⟶^{μ_{ijk}×1} e_{i}Ae_{k} × e_{k}Ae_{l}
1×μ_{jkl}↓  ↓^{μ_{ikl}}
e_{i}Ae_{j} × e_{j}Ae_{l}  ⟶^{μ_{ijl}}  e_{i}Ae_{l}

Conversely, a system (e_{i}Ae_{j} (1 ≤ i, j ≤ n); μ_{ijk} (1 ≤ i, j, k ≤ n)) of k-vector spaces satisfying the equation 1.2 defines a k-algebra A = ⨁_{1≤i,j≤n} e_{i}Ae_{j} (in this case we define the other multiplications to be 0).

A (left) A-module M is a k-vector space with a k-bilinear map ϕ^{M} : A×M → M satisfying

\{ \begin{matrix}ϕ^{M}(1_{A}, m) & = m (^{∀}m ∈ M) \\ ϕ^{M} ∘ (1 × ϕ^{M}) & = ϕ^{M} ∘ (μ × 1) \end{matrix} (1.3)

A × A × M ⟶^{μ×1} A × M
1×ϕ^{M} ↓  ↓_{ϕ^{M}}
A × M ⟶^{ϕ^{M}} M

As an equivalent notion, a representation M of A is a k-vector space with a k-algebra map ψ : A → End_{k}(M), where End_{k}(M) is the k-vector space of k-linear endomaps of M.

For a complete set {e_{i}|1 ≤ i ≤ n} of orthogonal primitive idempotents of A, we have

M = ⨁_{1≤i≤n} e_{i}M

as a k-vector space and a family of k-bilinear maps

ϕ_{ji}^{M} : e_{j}Ae_{i} × e_{i}M → e_{j}M

such that

\{ \begin{matrix}ϕ_{ii}^{M}(e_{i}, m_{i}) & = m_{i} (^{∀}m_{i} ∈ e_{i}M) \\ ϕ_{kj}^{M} ∘ (1 × ϕ_{ji}^{M}) & = ϕ_{ki}^{M} ∘ (μ_{kji} × 1) \end{matrix} (1.4)

e_{k}Ae_{j} × e_{j}Ae_{i} × e_{i}M ⟶^{μ_{kji}×1} e_{k}Ae_{i} × e_{i}M
1×ϕ_{ji}^{M} ↓  ↓^{ϕ_{ki}^{M}}
e_{k}Ae_{j} × e_{j}M  ⟶^{ϕ_{kj}^{M}}  e_{k}M

As an equivalent notion, a family (e_{i}M)_{1≤i≤n} of k-vector spaces with k-linear maps ψ_{ij}^{M} : e_{i}Ae_{j} → Hom_{k}(e_{j}M, e_{i}M) such that

\{ \begin{matrix}ψ_{ii}^{M}(e_{i}) & = 1_{e_{i}M} \\ ψ_{ik}^{M}(a_{ij}b_{jk}) & = ψ_{ij}^{M}(a_{ij}) ∘ ψ_{jk}^{M}(b_{jk}) \\ (^{∀}a_{ij} ∈ e_{i}Ae_{j},^{∀}b_{jk} ∈ e_{j}Ae_{k}) \end{matrix} (1.5)

\xymatrix{e_{k}M \ar[d]_{ψ_{jk}^{M}(b_{jk})} \ar[dr]^{ψ_{ik}^{M} (a_{ij}b_{jk})} \\ e_{j}M \ar[r]_{ψ_{ij}^{M} (a_{ij})} & e_{i}M}

where Hom_{k}(e_{j}M, e_{i}M) is the k-vector space of k-linear maps from e_{j}M to e_{i}M. Example 1.6. For a left A-module Ae_{r}, a family (e_{i}Ae_{r})_{1≤i≤n} with k-linear maps ψ_{ij}^{Ae_{r}} : e_{i}Ae_{j} → Hom_{k}(e_{j}Ae_{r}, e_{i}Ae_{r}) defined by ψ_{ij}^{Ae_{r}}(a_{ij}) = μ_{ijr}(a_{ij}, −).

e_{j}Ae_{r} ⟶^{μ_{ijr}(a_{ij},−)} e_{i}Ae_{r} (a_{ij} ∈ e_{i}Ae_{j})

For representations M, N, an A-homomorphism f : M → N is a k-linear map satisfying f ∘ ψ^{M}(a) = ϕ^{N}(a) ∘ f (^{∀}a ∈ A) (1.7)

M ⟶^{ψ^{M} (a)} M
f↓ ↓^{f}
N ⟶_{ψ^{N} (a)} N

Then we have a family (f_{i} : e_{i}M → e_{i}N)_{1≤i≤n} of k-linear maps satisfying f_{i} ∘ ψ_{ij}^{M}(a_{ij}) = ϕ_{ij}^{N}(a_{ij}) ∘ f_{j} (^{∀}a_{ij} ∈ e_{i}Ae_{j}) (1.8)

e_{j}M ⟶^{ψ_{ij}^{M} (a_{ij})} e_{i}M
f_{j} ↓ ↓^{f_{i}}
e_{j}N ⟶_{ϕ_{ij}^{N}(a_{ij})} e_{i}N

Conversely, it is easy to see that a system (e_{i}M (1 ≤ i ≤ n); ψ_{ij}^{M} (1 ≤ i, j ≤ n)) of k-vector spaces defines a left A-module M = ⨁_{1≤i≤n} e_{i}M (in this case we define the other actions to be 0), and that a family (f_{i})_{1≤i≤n} of k-linear maps defines an A-homomorphism from M to N. Example 1.9. For idempotents e_{r}, e_{s} of A, an A-homomorphism μ(−, b_{sr}) : Ae_{s} → Ae_{r} is obtained by a family of k-linear maps μ_{isr}(−, b_{sr}) : e_{i}Ae_{s} → e_{i}Ae_{r} (1 ≤ i ≤ n).

e_{j}Ae_{s} ⟶^{μ_{ijs}(a_{ij},−)} e_{i}Ae_{s}
μ_{jsr}(−,b_{sr})↓  ↓^{μ_{isr}(−,b_{sr})}
e_{j}Ae_{r} ⟶^{μ_{ijr}(a_{ij},−)} e_{i}Ae_{r}

Theorem 1.10. Let Rep A be the category consisting of M = (M(i) (1 ≤ i ≤ n); ψ_{ij}^{M}(1 ≤ i, j ≤ n)) satisfying

ψ_{ii}^{M}(e_{i}) = 1_{M(i)}
ψ_{ik}^{M}(a_{ij}b_{jk}) = ψ_{ij}^{M}(a_{ij}) ∘ ψ_{jk}^{M}(b_{jk})
(^{∀}a_{ij} ∈ e_{i}Ae_{j},^{∀}b_{jk} ∈ e_{j}Ae_{k})
\xymatrix{M(k) \ar[d]_{ψ_{jk}^{M}(b_{jk})} \ar[dr]^{ψ_{ik}^{M} (a_{ij}b_{jk})} \\ M(j) \ar[r]_{ψ_{ij}^{M} (a_{ij})} & e_{i}M(i)}

as objects, and of (f_{i} : M(i) → N(i))_{1≤i≤n} satisfying

f_{i} ∘ ψ_{ij}^{M}(a_{ij}) = ϕ_{ij}^{N}(a_{ij}) ∘ f_{j}

M(j) ⟶^{ψ_{ij}^{M} (a_{ij})} M(i)
f_{j} ↓  ↓^{f_{i}}
N(j) ⟶_{ϕ_{ij}^{N}(a_{ij})} N(i)

for M, N as morphisms. Then Rep A is equivalent to the category Mod A of left A-modules.

For A-modules M, N, we denote by Hom_{A}(M, N) the set of A-homomorphisms from M to N.

Lemma 1.11. For a left A-module M, we have

Hom_{A}(Ae_{i}, M) ≅ e_{i}M

as e_{i}Ae_{i}-modules.

Proof. Let θ : Hom_{A}(Ae_{i}, M) → e_{i}M be the map defined by (f) = f(e_{i}) for f ∈ Hom_{A}(Ae_{i}, M), and η : e_{i}M → Hom_{A}(Ae_{i}, M) the map defined by η(m_{i})(ae_{i}) = am_{i} for m_{i} ∈ e_{i}M and ae_{i} ∈ Ae_{i}. Then θ, η are A-homomorphisms and θη = 1, ηθ = 1.

Corollary 1.12. Let J be the Jacobson radical of A. Assume that A is a basic artinian k-algebra, that is, Ae_{i} ≅ A̸e_{j} for i ≠ j. Then we have

Hom_{A}(Ae_{i}, Ae_{j}/Je_{j}) ≅ \begin{cases}e_{i}Ae_{i}/e_{i}Je_{i} if i = j \\ O if i ≠ j \end{cases}

Proposition 1.13. Assume that A is a finite dimensional k-algebra satisfying A/J ≅ k × ⋯ × k (i.e., e_{i}Ae_{i}/e_{i}Je_{i} ≅ k for any 1 ≤ i ≤ n). For a left A-module M, we have

dim_{k} e_{i}M = the appearance number of simple type Ae_{i}/Je_{i}

in a composition series of M.

Proof. Let

O = M_{−1} ⊂ M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{r} = M

be a composition series. Then we have an exact sequence

O → Hom_{A}(Ae_{i}, M_{t−1}) → Hom_{A}(Ae_{i}, M_{t}) → Hom_{A}(Ae_{i}, M_{t}/M_{t−1}) → O

for 1 ≤ t ≤ r. Therefore we have

dim_{k} e_{i}M = ∑_{0≤t≤r} dim_{k} Hom_{A}(Ae_{i}, M_{t}/M_{t−1}).

By Corollary 1.12, we get the statement.

Example 1.14. In the case of A/J ≅ k × ⋯ × k, we may assume that A = (⊕_{i=1}^{n}ke_{i})⊕J. A simple left A-module Ae_{r}/Je_{r} is described by (M(i); ψ_{ij}^{M}) ∈ Rep A as follows.

M(i) = \begin{cases}k if i = r \\ O if i ≠ r \end{cases}

ψ_{ij}^{M}(a_{ij}) = \begin{cases}λ if (i, j) = (r, r), a_{ij} = λe_{r} \\ 0 if (i, j) = (r, r), a_{ij} ∈ e_{r}Je_{r} \\ 0 otherwise \end{cases}

2. Quivers and Path Algebras

Definition 2.1. A quiver Q⃗ = (Q_{0}, Q_{1}) is an oriented graph, where Q_{0} is a set of vertices and Q_{1} is a set of arrows between vertices. We use h : Q_{1} → Q_{0}, t : Q_{1} → Q_{0} the maps defined by h(α) = j, t(α) = i when α : i → j is arrow from the vertex i to the vertex j. A quiver Q⃗ = (Q_{0}, Q_{1}) is called a finite quiver if #Q_{0}, #Q_{1} < ∞.

A path w = (i|α_{r}, …, α_{1}|j) from the vertex j to the vertex i in the quiver Q⃗ is a sequence of ordered arrows α_{1}, …, α_{r} such that j = t(α_{1}), h(α_{i}) = t(α_{i+1}) (1 ≤ i ≤ r − 1), h(α_{r}) = i. In this case, j (resp., i) is called the tail t(w) (resp., the head h(w)) of w, and r is called the length of a path w. For every vertex i, the path e_{i} = (i| |i) of length 0 is called the empty path. A non-empty path w is called an oriented cycle if h(w) = t(w).

Definition 2.2. Let Q_{0} = {1, …, n} and Q_{1} a set. For any i, j ∈ Q_{0}, e_{i}|Q⃗|e_{j} is the set of paths w in Q⃗ with t(w) = j, h(w) = i. For any i, j, k ∈ Q_{0} with e_{i}|Q⃗|e_{j} ≠ ϕ, e_{j}|Q⃗|e_{k} ≠ ϕ, we define a composition map μ_{ijk} : e_{i}|Q⃗|e_{j} × e_{j}|Q⃗|e_{k} → e_{i}|Q⃗|e_{k} by setting

μ_{ijk}((i|α_{s}, …, α_{r+1}|j), (j|α_{r}, …, α_{1}|k)) = (i|α_{s}, …, α_{1}|k).

Then for any i, j, k, l ∈ Q_{0} with e_{i}|Q⃗|e_{j} ≠ ϕ, e_{j}|Q⃗|e_{k} ≠ ϕ, e_{k}|Q⃗|e_{l} ≠ ϕ, we have

e_{i}|Q⃗|e_{j} × e_{j}|Q⃗|e_{k} × e_{k}|Q⃗|e_{l} ⟶^{μ_{ijk}×1} e_{i}|Q⃗|e_{k} × e_{k}|Q⃗|e_{l}
1×μ_{jkl}↓  ↓^{μ_{ikl}}
e_{i}|Q⃗|e_{j} × e_{j}|Q⃗|e_{l}  ⟶^{μ_{ijl}}  e_{i}|Q⃗|e_{l}

We denote by e_{i}kQ⃗e_{j} the k-vector space with the paths from the vertex j to i as a basis if e_{i}|Q⃗|e_{j} ≠ ϕ, and e_{i}kQ⃗e_{j} = O if e_{i}|Q⃗|e_{j} = ϕ. For any i, j, k ∈ Q_{0}, we define a k-bilinear map μ_{ijk} : e_{i}kQ⃗e_{j} × e_{j}kQ⃗e_{k} → e_{i}kQ⃗e_{k} by setting

μ_{ijk}(λ_{v}v, λ_{w}w) = λ_{v}λ_{w}vw

with λ_{v}, λ_{w} ∈ k. For any i, j, k, l ∈ Q_{0}, we have

e_{i}kQ⃗e_{j} × e_{j}kQ⃗e_{k} × e_{k}kQ⃗e_{l} ⟶^{μ_{ijk}×1} e_{i}kQ⃗e_{k} × e_{k}kQ⃗e_{l}
1×μ_{jkl}↓  ↓^{μ_{ikl}}
e_{i}kQ⃗e_{j} × e_{j}kQ⃗e_{l}  ⟶^{μ_{ijl}}  e_{i}kQ⃗e_{l}

Then, by 1.2, kQ⃗ = ⨁_{1≤i,j≤n} e_{i}kQ⃗e_{j} becomes an associative k-algebra. This algebra is called the path algebra of Q⃗ over k.

We often simply write α_{r}, …, α_{1} for (i|α_{r}, …, α_{1}|j).

Proposition 2.3. For a finite quiver Q⃗, kQ⃗ is a finite dimensional k-algebra if and only if Q⃗ has no oriented cycle.

Example 2.4. For a quiver

\xymatrix{Q⃗ : 1 \ar[r]^{α} & 2 \ar[r]^{β} & 3}

we have

\begin{matrix}e_{1}kQ⃗e_{1} =< e_{1} >_{k} & e_{2}kQ⃗e_{1} =< α >_{k} & e_{3}kQ⃗e_{1} =< βα >_{k} \\ e_{1}kQ⃗e_{2} = O & e_{2}kQ⃗e_{2} =< e_{2} >_{k} & e_{3}kQ⃗e_{2} =< β >_{k} \\ e_{1}kQ⃗e_{3} = O & e_{2}kQ⃗e_{3} = O & e_{3}kQ⃗e_{3} =< e_{3} >_{k} \end{matrix}

Then we have

kQ⃗ ≅ [\begin{matrix}k & 0 & 0 \\ k & k & 0 \\ k & k & k \end{matrix}]

Example 2.5. For a quiver

\xymatrix{Q⃗ : 1 \ar[r]^{α} \ar[r]_{β} & 2 \ar[r]^{γ} & 3}

we have

\begin{matrix}e_{1}kQ⃗e_{1} =< e_{1} >_{k} & e_{2}kQ⃗e_{1} =< α, β >_{k} & e_{3}kQ⃗e_{1} =< γα, γβ >_{k} \\ e_{1}kQ⃗e_{2} = O & e_{2}kQ⃗e_{2} =< e_{2} >_{k} & e_{3}kQ⃗e_{2} =< γ >_{k} \\ e_{1}kQ⃗e_{3} = O & e_{2}kQ⃗e_{3} = O & e_{3}kQ⃗e_{3} =< e_{3} >_{k} \end{matrix}

Then we have

kQ⃗ ≅ [\begin{matrix}k & O \\ _{A}M_{k} & A \end{matrix}] , A = [\begin{matrix}k & 0 \\ k & k \end{matrix}] , _{A}M = [\begin{matrix}k \\ k \end{matrix}] ⊕ [\begin{matrix}k \\ k \end{matrix}]

Example 2.6. For a quiver

[figure: Q⃗ : 1; α; 2; β]

we have

\begin{matrix}e_{1}kQ⃗e_{1} =< e_{1} >_{k} & e_{2}kQ⃗e_{1} =< α, β^{n}α : n ∈ ℕ >_{k} \\ e_{1}kQ⃗e_{2} = O & e_{2}kQ⃗e_{2} =< e_{2}, β^{n} : n ∈ ℕ >_{k} \end{matrix}

Then we have

kQ⃗ ≅ [\begin{matrix}k & 0 \\ k[x] & k[x] \end{matrix}]

Lemma 2.7. Let A be a ring, O → X → Y → Z → O an exact sequence of left A-modules. Then we have

pdim_{A} Y ≤ max{pdim_{A} X, pdim_{A} Z}

Proposition 2.8. For a finite dimensional k-algebra A, the following are equivalent.

1. lgldim A ≤ n.

2. pdim_{A} A/J ≤ n. In particular, the following are equivalent.

1. A is hereditary.

2. J is projective.

3. lgldim A ≤ 1.

4. pdim_{A} A/J ≤ 1.

Proposition 2.9. Let Q⃗ be a finite quiver without oriented cycles. Then kQ⃗ is hereditary, and kQ⃗/J_{kQ⃗} ≅ k × ⋯ × k, where J_{kQ⃗} is the Jacobson radical of kQ⃗.

Proof. Let Q_{0} = 1, …, n, then 1 = e_{1} + … + e_{n}. Let J_{+} be the vector space spanned by paths of length ≥ 1, then there exists t ≥ 0 such that J_{+}^{t+1} = 0. Therefore J_{+} ⊂ J_{kQ⃗}. It is easy to see that kQ⃗/J_{+} ≅ ke_{1} × ⋯ × ke_{n} as rings. Thus we have J_{+} = J_{kQ⃗}. For i ∈ Q_{0}, since Q⃗ is finite, we may assume that the set of arrows α with t(α) = i is {α_{1}, …, α_{r}}. Then we have

J_{+}e_{i} = ⊕_{i=1}^{r}kQ⃗α_{i}.

Since μ(−, α_{j}) : kQ⃗e_{h(α_{i})} → kQ⃗ is an isomorphism, J_{kQ⃗}e_{i} is a projective left kQ⃗- module, and hence J_{kQ⃗} is a projective left kQ⃗-module.

Definition 2.10. Given a quiver Q⃗ = (Q_{0}, Q_{1}), a representation M = (M(i); ψ^{M}) of Q⃗ over a field k is a family (M(i))_{i∈Q_{0}} of k-vector spaces together with a family (ψ^{M}(α) : M(j) → M(i))_{j⟶^{α}i∈Q_{1}} of k-linear maps. A representation M = (M(i); ψ^{M}) is called a finite dimensional representation if M(i) is a finite dimensional k-vector space for every i ∈ Q_{0}.

For (M(i); ψ^{M}), (N(i); ψ^{N}), a morphism f : (M(i); ψ^{M}) → (N(i); ψ^{N}) is a family (f_{i} : M(i) → N(i))_{i∈Q_{0}} of k-linear maps satisfying that we have a commutative diagram

M(j) ⟶^{ψ^{M} (α)} M(i)
f_{j} ↓  ↓^{f_{i}}
N(j) ⟶^{ψ^{N} (α)} N(i)

for any j ⟶^{α} i ∈ Q_{1}.

We denote by Rep_{k} Q⃗ (resp., rep_{k} Q⃗) the category of representations (resp., finite dimensional representations) of Q⃗ over k.

Theorem 2.11. For a finite quiver Q⃗, Rep_{k} Q⃗ is equivalent to Rep kQ⃗, and hence it is equivalent to Mod kQ⃗. Moreover, rep_{k} Q⃗ is equivalent to the category mod_{fd} kQ⃗ of finite dimensional left kQ⃗-modules.

Sketch of The Proof. For any idempotents e_{i}, e_{j} of kQ⃗, all elements of e_{i}kQ⃗e_{j} are k-linear combinations of paths from j to i. Then it is easy.

Proposition 2.12. For any collection {(M_{λ}; ψ^{M_{λ}})}_{λ∈Λ} of representations of Q⃗ over k, (⨁_{λ∈Λ} M_{λ}; ⨁_{λ∈Λ} ψ^{M_{λ}}) (resp., (∏_{λ∈Λ} M_{λ}; ∏_{λ∈Λ} ψ^{M_{λ}})) is the direct sum (resp., the direct product) of {(M_{λ}; ψ^{M_{λ}})}_{λ∈Λ}.

Example 2.13. For a quiver

\xymatrix{Q⃗ : 1 \ar[r]^{α} & 2 \ar[r]^{β} & 3}

kQ⃗ =< e_{1}, e_{2}, e_{3}, α, β, βα >_{k}. A representation M of Q⃗ over k is the following

M(1) ⟶^{ψ^{M} (α)} M(2) ⟶^{ψ^{M} (β)} M(3)

Then we define M = M(1) ⊕ M(2) ⊕ M(3) to be a left A-module as follows. For m = (m_{1}, m_{2}, m_{3}) ∈ M and a = λ_{1}e_{1} + λ_{2}e_{2} + λ_{3}e_{3} + λ_{α}α + λ_{β}β + λ_{βα}βα ∈ kQ⃗, we define

am = (λ_{1}m_{1}, λ_{2}m_{2} + λ_{α}ψ^{M}(α)(m_{1}), λ_{3}m_{3} + λ_{β}ψ^{M}(β)(m_{2}) + λ_{βα}ψ^{M}(β)ψ^{M}(α)(m_{1}))

By the standard technique of linear algebra, all indecomposable representations are up to isomorphisms the following

\begin{matrix}M_{1} : O → O → k & M_{2} : O → k → k & M_{3} : k → k → k \\ M_{4} : O → k → O & M_{5} : k → k → O & M_{6} : k → O → O \end{matrix}

It is easy to see that we have composition series of M_{3} and M_{5}

\begin{matrix}M_{1} : & O & ⟶ & O & ⟶ & k \\ ↓ & ↓ & & ↓ & & ↓ \\ M_{2} : & O & ⟶ & k & ⟶ & k \\ ↓ & ↓ & & ↓ & & ↓ \\ M_{3} : & k & ⟶ & k & ⟶ & k \end{matrix} \begin{matrix}M_{4} : & O & ⟶ & k & ⟶ & O \\ ↓ & ↓ & & ↓ & & ↓ \\ M_{5} : & k & ⟶ & k & ⟶ & O \end{matrix}

Then M_{1} ≅ kQ⃗e_{3}/Je_{3}, M_{2}/M_{1} ≅ kQ⃗e_{2}/Je_{2}, M_{3}/M_{2} ≅ kQ⃗e_{1}/Je_{1}, and M_{4} ≅ kQ⃗e_{2}/Je_{2}, M_{5}/M_{4} ≅ kQ⃗e_{1}/Je_{1}, where J is the Jacobson radical of kQ⃗. Moreover, kQ⃗e_{1} ≅ M_{3}, Je_{3} ≅ JM_{3} ≅ kQ⃗e_{2} ≅ M_{2} and Je_{2} ≅ JM_{2} ≅ kQ⃗e_{3} ≅ M_{1}.

We often write modules by using composition series

M_{1} : 3 M_{2} : \begin{matrix}2 \\ |β \\ 3 \end{matrix} M_{3} : \begin{matrix}1 \\ |α \\ 2 \\ |β \\ 3 \end{matrix} M_{4} : 2 M_{5} : \begin{matrix}1 \\ |α \\ 2 \end{matrix} M_{6} : 1

3. Quivers with Relations

Definition 3.1. A relation σ on a quiver Q⃗ over a field k is a k-linear combinations σ = ∑_{t=1}^{r} λ_{t}w_{t}, where w are paths from j to i, λ_{t} ∈ k. A pair (Q⃗, ρ) is called a quiver with relations over k if ρ = {σ_{1}, …, σ_{s}} where σ_{i} is a relation for every i. We denote k(Q⃗, ρ) = kQ⃗/ < ρ >, where < ρ > is the two-sided ideal of kQ⃗ generated by relations of ρ. We denote by J_{+} the two-sided ideal of kQ⃗ generated by arrows.

Proposition 3.2. Let (Q⃗, ρ) be a finite quiver with relations over k. If there is t such that J_{+}^{t} ⊂< ρ >⊂ J_{+}^{2}, then \overline{J}_{+} = rad(k(Q⃗, ρ)), where \overline{J}_{+} is the image of J_{+} in k(Q⃗, ρ).

Proof. Let A = k(Q⃗, ρ) and J = rad(k(Q⃗, ρ)). Since \overline{J}_{+}^{t} = O, we have \overline{J}_{+} ⊂ J. It is clearly that A/\overline{J}_{+} ≅ kQ⃗/J_{+} is semi-simple. Then (J + \overline{J}_{+})/\overline{J}_{+} = O, and hence \overline{J}_{+} ⊂ J.

Definition 3.3. For a quiver with relations (Q⃗, ρ) over k, Rep_{k}(Q⃗, ρ) (resp., rep_{k}(Q⃗, ρ)) is the full subcategory of Rep_{k} Q⃗ (resp., rep_{k} Q⃗) consisting objects M = (M(i); ψ^{M}) with ψ^{M}(σ) = 0 for any relation σ of ρ. Here ψ^{M}(w) = ψ^{M}(α_{r}) … ψ^{M}(α_{1}) for w = α_{r} … α_{1}, and ψ^{M}(σ) = Σ_{t}λ_{t}ψ^{M}(w_{t}) for σ = Σ_{t}λ_{t}w_{t}.

Theorem 3.4. For a finite quiver with relations (Q⃗, ρ) over k, Rep_{k}(Q⃗, ρ) (resp., rep_{k}(Q⃗, ρ)) is equivalent to Mod k(Q⃗, ρ) (resp., mod_{fd} k(Q⃗, ρ)).

Sketch. According to Theorem 2.11 and the explanations before the theorem, ψ^{M}(σ) = 0 means that σM = O when we consider M = ⨁_{i∈Q_{0}} M(i) as a left kQ⃗-module.

Definition 3.5. For a quiver Q⃗, the opposite quiver Q⃗^{op} is the quiver with all arrows reversed. For a quiver with relations (Q⃗, ρ) over k, (Q⃗^{op}, ρ^{op}) is similarly defined. Then k(Q⃗, ρ)^{op} = k(Q⃗^{op}, ρ^{op}).

Let D = Hom_{k}(−, k). For a representation M = (M(i); ψ^{M}) ∈ Rep_{k}(Q⃗, ρ), D M = (D M(i); ψ^{D M}), where ψ^{D M}(α) = D ψ^{M}(α). Then D M is a representation of (Q⃗^{op}, ρ^{op}) over k.

Proposition 3.6. For a quiver with relations (Q⃗, ρ) over k, D induces a duality between rep_{k}(Q⃗, ρ) and rep_{k}(Q⃗^{op}, ρ^{op}).

Remark 3.7. For a k-algebra A, idempotents e_{i}, e_{j} and a_{ij} ∈ e_{i}Ae_{j}, we have a left A-homomorphism μ(−, a_{ij}) : Ae_{i} → Ae_{j}. Then we have a commutative diagram in Mod A^{op}

Hom_{A}(Ae_{j}, A) ⟶^{Hom_{A}(μ(−,a_{ij}),A)} Hom_{A}(Ae_{i}, A)
≀↓  ↓^{≀}
e_{j}A  ⟶_{μ(a_{ij},−)}  e_{i}A

In Rep_{k} Q⃗, we have also the same result.

Example 3.8. For a quiver

\xymatrix{Q⃗ : 1 \ar[r]^{α} & 2 \ar[r]^{β} & 3}

with a relation ρ = βα. Then kQ⃗ =< e_{1}, e_{2}, e_{3}, α, β, βα >_{k} and the ideal < ρ >=< βα >_{k}. Therefore k(Q⃗, ρ) =< \overline{e}_{1}, \overline{e}_{2}, \overline{e}_{3}, \overline{α}, \overline{β} >_{k}. Let A = k(Q⃗, ρ), then we have

\begin{matrix}\overline{e}_{1}A\overline{e}_{1} =< \overline{e}_{1} >_{k} & \overline{e}_{2}A\overline{e}_{1} =< \overline{α} >_{k} & \overline{e}_{3}A\overline{e}_{1} = O \\ \overline{e}_{1}A\overline{e}_{2} = O & \overline{e}_{2}A\overline{e}_{2} =< \overline{e}_{2} >_{k} & \overline{e}_{3}A\overline{e}_{2} =< \overline{β} >_{k} \\ \overline{e}_{1}A\overline{e}_{3} = O & \overline{e}_{2}A\overline{e}_{3} = O & \overline{e}_{3}A\overline{e}_{3} =< \overline{e}_{3} >_{k} \end{matrix}

Since this algebra is a factor of the path algebra in Example 2.13, all indecomposable representations are up to isomorphisms the following

\begin{matrix}M_{1} : O → O → k & M_{2} : O → k → k \\ M_{4} : O → k → O & M_{5} : k → k → O & M_{6} : k → O → O \end{matrix}

The opposite quiver of with relations (Q⃗^{op}, ρ^{op}) is

\xymatrix{Q⃗^{op} : 1 & 2 \ar[l]_{α^{op}} & 3 \ar[l]_{β^{op}}}

with ρ^{op} = α^{op}β^{op}. Therefore we have

\begin{matrix}Ae_{3} = Ae_{3}/Je_{3} ≅ M_{1} & Ae_{2} ≅ D(e_{3}A) ≅ M_{2} \\ Ae_{2}/Je_{2} ≅ M_{4} & Ae_{1} ≅ D(e_{2}A) ≅ M_{5} & D(e_{1}A) ≅ Ae_{1}/Je_{1} ≅ M_{6} \end{matrix}

M_{1} : 3 M_{2} : \begin{matrix}2 \\ |β \\ 3 \end{matrix} M_{4} : 2 M_{5} : \begin{matrix}1 \\ |α \\ 2 \end{matrix} M_{6} : 1

Since projective resolutions of Ae_{1}/Je_{1}, Ae_{2}/Je_{2}, Ae_{3}/Je_{3} are

O ⟶ Ae_{3} ⟶ Ae_{2} ⟶ Ae_{1} ⟶ Ae_{1}/Je_{1} ⟶ O
O ⟶ 3 ⟶ \begin{matrix}2 \\ |β \\ 3 \end{matrix} ⟶ \begin{matrix}1 \\ |α \\ 2 \end{matrix} ⟶  1  ⟶ O
O ⟶ Ae_{3} ⟶ Ae_{2} ⟶ Ae_{2}/Je_{2} ⟶ O
O ⟶ 3 ⟶ \begin{matrix}2 \\ |β \\ 3 \end{matrix} ⟶  2  ⟶ O
O ⟶ Ae_{3}  Ae_{3} ⟶ O
O ⟶ 3  3 ⟶ O

by Proposition 2.8, lgldim k(Q⃗, ρ) = 2. Moreover, an injective resolution of _{A}A is

\xymatrix{O \ar[r] & _{A}A \ar[r] & D(e_{2}A) ⊕ D(e_{3}A)^{2} \ar[r] & D(e_{2}A) \ar[r] & D(e_{1}A) \ar[r] & O}

Since pdim_{A} D(e_{2}A) = pdim_{A} D(e_{3}A) = 0 and pdim_{A} D(e_{1}A) = 2, A is an Auslander regular k-algebra.

REPRESENTATIONS AND QUIVERS FOR RING THEORISTS 11

Example 3.9. For a quiver

\xymatrix{Q⃗ : 1 \ar[r]^{α} & 2 \ar[l]^{β}}

with a relation ρ = {βα}. Then

kQ⃗ =< e_{1}, e_{2}, α, β, (βα)^{h}, (αβ)^{l}α(βα)^{m}, β(αβ)^{n} : h, l, m, n ∈ ℕ >_{k} and the ideal

< ρ >=< (βα)^{h}, (αβ)^{l+1}, α(βα)^{m}, β(αβ)^{n} : h, l, m, n ∈ ℕ >_{k} .

Therefore k(Q⃗, ρ) =< \overline{e}_{1}, \overline{e}_{2}, \overline{e}_{3}, \overline{α}, \overline{β}, \overline{α}\overline{β} >_{k}. Let A = k(Q⃗, ρ), then we have

\begin{matrix}\overline{e}_{1}A\overline{e}_{1} =< \overline{e}_{1} >_{k} & \overline{e}_{2}A\overline{e}_{1} =< \overline{α} >_{k} \\ \overline{e}_{1}A\overline{e}_{2} =< \overline{β} >_{k} & \overline{e}_{2}A\overline{e}_{2} =< \overline{e}_{2}, \overline{αβ} >_{k} \end{matrix}

The opposite quiver of with relations (Q⃗^{op}, ρ^{op}) is

\xymatrix{Q⃗^{op} : 1 \ar[r]_{β^{op}} & 2 \ar[l]_{α^{op}}}

with a relation ρ^{op} = {α^{op}β^{op}}. Hence we have

Ae_{1} : \begin{matrix}1 \\ |α \\ 2 \end{matrix} : \xymatrix{k \ar[r]^{1} & k \ar[l]^{0}}  Ae_{2} ≅ D(e_{2}A) : \begin{matrix}2 \\ |β \\ 1 \\ |α \\ 2 \end{matrix} : [figure: k; ^{[ \begin{matrix}0 \\ 1 \end{matrix} ]}; k^{2}; _{[_{1 0} ]}]

D(e_{1}A) : \begin{matrix}2 \\ |β \\ 1 \end{matrix} : \xymatrix{k \ar[r]^{0} & k \ar[l]^{1}}

Ae_{1}/Je_{1} : 1 : \xymatrix{k \ar[r]^{0} & O \ar[l]^{0}} Ae_{2}/Je_{2} : 2 : \xymatrix{O \ar[r]^{0} & k \ar[l]^{0}}

Since projective resolutions of Ae_{1}/Je_{1}, Ae_{2}/Je_{2} are

O ⟶ Ae_{1} ⟶ Ae_{2} ⟶ Ae_{1} ⟶ Ae_{1}/Je_{1} ⟶ O
O ⟶ \begin{matrix}1 \\ |α \\ 2 \end{matrix} ⟶ \begin{matrix}2 \\ |β \\ 1 \\ |α \\ 2 \end{matrix} ⟶ \begin{matrix}1 \\ |α \\ 2 \end{matrix} ⟶ 1 ⟶ O
O ⟶ Ae_{1} ⟶ Ae_{2} ⟶ Ae_{2}/Je_{2} ⟶ O
O ⟶ \begin{matrix}1 \\ |α \\ 2 \end{matrix} ⟶ \begin{matrix}2 \\ |β \\ 1 \\ |α \\ 2 \end{matrix} ⟶ 2 ⟶ O

by Proposition 2.8, lgldim A = 2. A projective resolution of D(e_{1}A) is

\begin{matrix}O & ⟶ & Ae_{1} & ⟶ & Ae_{2} & ⟶ & Ae_{2} & ⟶ & D(e_{1}A) & ⟶ & O \\ O & ⟶ & \begin{matrix}1 \\ |α \\ 2 \end{matrix} & ⟶ & \begin{matrix}2 \\ |β \\ 1 \\ |α \\ 2 \end{matrix} & ⟶ & \begin{matrix}2 \\ |β \\ 1 \\ |α \\ 2 \end{matrix} & ⟶ & \begin{matrix}2 \\ |β \\ 1 \end{matrix} & ⟶ & O \end{matrix}

Moreover, an injective resolution of _{A}A is

\xymatrix{O \ar[r] & _{A}A \ar[r] & D(e_{2}A)^{2} \ar[r] & D(e_{2}A) \ar[r] & D(e_{1}A) \ar[r] & O}

Since pdim_{A} D(e_{2}A) = 0 and pdim_{A} D(e_{1}A) = 2, A is an Auslander regular kalgebra.

Definition 3.10. Let Λ be a ring, and V a Λ-bimodule. We denote by V^{⊗n} =

\overbrace{V ⊗_{Λ} ⋯ ⊗_{Λ} V}^{n times}. Then the tensor ring T(Λ, V) is Λ ⊕ (⨁_{n≥1} V^{⊗n}) as an abelian group, and its multiplication is induced by the canonical Λ-bilinear maps V^{⊗m}⊗_{Λ}V^{⊗n} → V^{⊗m+n} for m, n ≥ 0.

Lemma 3.11. Let Λ be a ring, V a Λ-bimodule and A a Λ-algebra. For a Λbimodule homomorphism f : V → A, there exists a unique Λ-algebra homomorphism f̃: T(Λ, V) → A such that f̃|_{V} = f.

Sketch of The Proof. Let ϕ : Λ → A be a ring homomorphism. A map f̃ : T(Λ, V) → A is defined by

f̃(a_{0} + ∑_{i≥1} ∑_{j} v_{i1j} ⊗ ⋯ ⊗ v_{iij}) = ϕ(a_{0}) + ∑_{i≥1} ∑_{j} f(v_{i1j}) … f(v_{iij})

for a_{0}+∑_{i≥1} ∑_{j} v_{i1j} ⊗⋯⊗v_{iij} ∈ T(Λ, V). Then this satisfies the desired property.

Definition 3.12. For a k-algebra Λ = ∏_{i=1}^{n}k and Λ-bimodule V, the quiver Q⃗_{T(Λ,V)} of T(Λ, V) consists of Q_{T(Λ,V)0} = {1, …, n}, and of the number of arrows from the vertex i to j which is dim_{k} e_{j}V e_{i}, where e_{i}, e_{j} correspond to i, j.

For a finite dimensional k-algebra A with A/J_{A} ≅ ∏_{i=1}^{n}k, the quiver Q⃗_{A} is the quiver Q⃗_{T(A/J_{A},J_{A}/J_{A}^{2})}.

Proposition 3.13. For a k-algebra Λ = ∏_{i=1}^{n}k and Λ-bimodule V, there is a kalgebra isomorphism ϕ : T(Λ, V) → kQ⃗_{T(Λ,V)}.

Proof. Since kQ⃗ = (⊕_{i=1}^{n}λ_{i}e_{i}) ⊕ J_{+}, we identify the idempotents of A/J with them of kQ⃗. For 1 ≤ i, j ≤ n, we take a k-basis {v_{ijk}|1 ≤ k ≤ n_{ij}} of e_{i}V e_{j}, and denote by α_{v_{ijk}} the arrow in kQ⃗_{T(Λ,V)} corresponding to v_{ijk}. A map ϕ : T(Λ, V) → A is defined by

ϕ(∑_{i=1}^{n} λ_{i}e_{i} + ∑_{i≥1,j} λ_{ij}u_{i1j} ⊗ ⋯ ⊗ u_{iij}) = ∑_{i=1}^{n} λ_{i}e_{i} + ∑_{i≥1,j} λ_{ij}α_{u_{i1j}} … α_{u_{iij}}

for ∑_{i=1}^{n} λ_{i}e_{i} + ∑_{i≥1,j} λ_{ij}u_{i1j} ⊗ ⋯ ⊗ u_{iij} ∈ T(Λ, V), where u_{ijk} are elements of the above basis. It is easy to see that dim_{k} e_{i}(⨁_{n≥1} V^{⊗n})e_{j} = e_{i}kQ⃗e_{j}. Hence ϕ is bijective.

Theorem 3.14. Let A be a finite dimensional k-algebra with A/J_{A} ≅ ∏_{i=1}^{n}k. Then the following hold.

1. There is a surjective ring homomorphism ϕ : T(A/J_{A}, J_{A}/J_{A}^{2}) → A such that ∐_{i≥rl(A)}(J_{A}/J_{A}^{2})^{i} ⊂ Ker ϕ ⊂ (J_{A}/J_{A}^{2})^{2}, where rl(A) is the Loewy length of A (i.e. rl A = min{t|J_{A}^{t+1} = 0}).

2. A ≅ k(Q⃗, ρ) with J_{A}^{r} ⊂< ρ >⊂ J_{A}^{2} for some r, where Q⃗ = Q⃗_{A}.

Proof. 1. By the assumption, we may assume that a split injective k-algebra homomorphism ϕ_{0} : A/J → A, A/J = ⊕_{i=1}^{n}ke_{i} and A = A/J ⊕J with J = J_{A} the Jacobson radical of A. For any e_{i}, e_{j}, we choose elements r_{ij1}, …, r_{ijn_{ij}} of e_{i}Je_{j} such that \{\overline{r}_{ij1}, …, \overline{r}_{ijn_{ij}}\} is a k-basis of e_{i}(J/J^{2})e_{j}. Let ϕ_{1} : J/J^{2} → A be an A/J-bimodule homomorphism defined by ϕ_{1}(\overline{r}_{ijk}) = r_{ijk}, then by Lemma 3.11, there exists an A/J-algebra homomorphism ϕ : T(A/J, J/J^{2}) → A such that ϕ̃|_{A/J⊕J/J^{2}} = ϕ_{0}⊕ϕ_{1} is injective. Therefore ∐_{i≥rl(A)}(J_{A}/J_{A}^{2})^{i} ⊂ Ker ϕ ⊂ (J_{A}/J_{A}^{2})^{2}, because of J^{t+1} = 0 for t = rl(A). If rl(A) = 1, then ϕ is clearly bijective. In order to prove that ϕ is surjective, it suffices to show that for any m ≥ 1 and any x ∈ J^{m}, there exists y ∈ (ϕ(J/J^{2}))^{m} such that x − y ∈ J^{m+1}. In the case of m = 1, it is trivial. In the case of m ≥ 2, for x ∈ J^{m} we have x = ∑_{i} v_{i}w_{i}, where v_{i} ∈ J and w_{i} ∈ J^{m−1}. Then there are y_{i} ∈ ϕ(J/J^{2}) and z_{i} ∈ (ϕ(J/J^{2}))^{m−1} such that v_{i} − y_{i} ∈ J^{2} and w_{i} − z_{i} ∈ J^{m}. Since v_{i} ∈ J and z_{i} ∈ J^{m−1}, v_{i}w_{i} − y_{i}z_{i} = v_{i}(w_{i} − z_{i}) + (v_{i} − y_{i})z_{i} ∈ J^{m+1} and hence x − ∑_{i} y_{i}z_{i} ∈ J^{m+1}.

2. According to Proposition 3.13, we have a surjective k-algebra homomorphism ϕ : kQ⃗ → A, where Q⃗ = Q⃗_{A}. Let t = rl(A) + 1, then ϕ induces a surjective k-algebra homomorphism ψ : kQ⃗/J_{+}^{t} → A. Since kQ⃗/J_{+}^{t} is a finite dimensional k-algebra, Ker ψ is a finitely generated ideal. Hence Ker ϕ is a finitely generated ideal < σ_{1}, …, σ_{s} > of kQ⃗, because J_{+}^{t} is a finitely generated ideal of kQ⃗. Since σ_{h} = ∑_{ij} e_{i}σ_{h}e_{j}, there is a set ρ of relations such that Ker ϕ =< ρ >.

Lemma 3.15. Let A be a hereditary finite dimensional k-algebra, I a two-sided ideal of A with I ⊂ J_{A}^{2}. Then A/I is not hereditary.

Proof. Consider the exact sequence in Mod A/I

O → I/IJ_{A} → J_{A}/IJ_{A} ⟶^{π} J_{A}/I → O.

By Nakayama’s Lemma, I/IJ_{A} ≠ O. Since J_{A} is A-projective, J_{A}/IJ_{A} is A/Iprojective. I ⊂ J_{A}^{2} implies I/IJ_{A} ⊂ J_{A}^{2}/IJ_{A} = J_{A/I}(J_{A}/IJ_{A}), If J_{A}/I is A/Iprojective, then there is η : J_{A}/I → J_{A}/IJ_{A} such that πη = 1_{J_{A}/I}, and then J_{A/I}(J_{A}/IJ_{A}) ⊕ Im η = J_{A}/IJ_{A}. By Nakayama’s Lemma, Im η = J_{A}/IJ_{A} and I/IJ_{A} = O. This is a contradiction. Hence J_{A}/I is not A/I-projective. By Proposition 2.8, we get the statement.

Proposition 3.16. Let A be a finite dimensional k-algebra with A/J_{A} ≅ k×⋯×k. Then the following are equivalent.

1. A is hereditary.

2. A ≅ kQ⃗_{A}.

Proof. 1 ⇒ 2. Let f : Ae_{i} → Ae_{j} be a non-zero A-homomorphism for primitive idempotents i, j. If f is not an isomorphism, then f is a monomorphism, because Im f is projective. Then there is no path Ae_{i_{1}} → ⋯ → Ae_{i_{n}} = Ae_{i_{1}} of non-zero A-homomorphisms which are not isomorphisms. Hence Q⃗ has no oriented cycle, kQ⃗ is a finite dimensional k- algebra. By Lemma 3.15, A ≅ kQ⃗_{A}.

2 ⇒ 1. By Proposition 2.9, it is trivial.

4. Base Extensions and Representations

Let k be a field and R a k-algebra. For a quiver with relations (Q⃗, ρ) over a field k, let e_{1}, …, e_{n} be the set of idempotents corresponding to vertices in Q⃗, A = k(Q⃗, ρ) and A^{R} = R⊗_{k}k(Q⃗, ρ). Then we can consider that A^{R} = ⨁_{path w} R\overline{w} and r\overline{w} = \overline{w}r for any r ∈ R and any path w in Q⃗.

A left A^{R}-module M is a left A-module, and it is a direct sum ⨁_{i=1}^{n}e_{i}M as an R-module. For any α ∈ Q_{1}, we have

α(rm) = (αr)m
= (rα)m
= r(αm)

with r ∈ R, m ∈ M. Then ψ^{M}(α) : e_{j}M → e_{i}M is a left R-linear map, and we get a system (e_{i}M; ψ^{M}) of left R-modules satisfying

1. e_{i}M is a left R-module for any i.

2. ψ^{M}(α) is a left R-linear map for any α ∈ Q_{1}.

3. ψ^{M}(σ) = 0 for any relation σ ∈ ρ.

For a left A^{R}-homomorphism f : M → N, we get left R-linear maps e_{i}f = f_{i} : e_{i}M → e_{i}N (1 ≤ i ≤ n) such that

f_{i} ∘ ψ^{M}(α) = ψ^{N}(α) ∘ f_{j} (4.1)

for any α ∈ Q_{1}.

e_{j}M ⟶^{ψ^{M} (α)} e_{i}M
f_{j} ↓  ↓^{f_{i}}
e_{j}N ⟶_{ψ^{N} (α)} e_{i}N

Theorem 4.2. Let A = k(Q⃗, ρ), and let Rep_{R/k}(Q⃗, ρ) be the category consisting of M = (M(i) (1 ≤ i ≤ n); ψ^{M}(α)(α ∈ Q_{1})) satisfying

1. M(i) is a left R-module for any i.

2. ψ^{M}(α) is a left R-linear map for any α ∈ Q_{1}.

3. ψ^{M}(σ) = 0 for any relation σ ∈ ρ.

as objects, and of (f_{i} : M(i) → N(i))_{1≤i≤n} satisfying

f_{i} ∘ ψ^{M}(α) = ϕ^{N}(α) ∘ f_{j}

M(j) ⟶^{ψ^{M} (α)} M(i)
f_{j} ↓  ↓^{f_{i}}
N(j) ⟶_{ϕ^{N} (α)} N(i)

for M, N as morphisms. Then Rep_{R/k}(Q⃗, ρ) is equivalent to the category Mod A^{R} of left A^{R}-modules.

Sketch of The Proof. By the above, we can construct a functor from Mod A^{R} to Rep_{R/k}(Q⃗, ρ). Conversely, given M = (M(i); ψ^{M}) ∈ Rep_{R/k}(Q⃗, ρ), let M = ⨁_{i=1}^{n}M(i). For any r ∈ R, any arrow α : i → j and m ∈ M(i), we define the left A^{R}-action

(rα)m = rψ^{M}(α)(m)

Then for any r, s ∈ R, any arrow α : i → j, β : j → l and m ∈ M(i), we have

(sβ)((rα)m) = (sβ)(rψ^{M}(α)(m))
= (sψ^{M}(β))(rψ^{M}(α)(m))
= s(rψ^{M}(β)(ψ^{M}(α)(m)))
= sr(ψ^{M}(β)ψ^{M}(α))(m))
= (srβα)(m)

Therefore M becomes a left A^{R}-module. For a family (f_{i} : M(i) → N(i))_{1≤i≤n} of morphisms , let f = ⊕_{i=1}^{n}f_{i}. For any r ∈ R, any arrow α : i → j and m ∈ M(i), we have

f_{j}(rαm) = f_{j}(rψ^{M}(α)(m))
= r(f_{j} ∘ ψ^{M}(α))(m)
= r(ψ^{N}(α) ∘ f_{i})(m)
= (rα)f_{i}(m)

Hence f becomes a left A^{R}-homomorphism. It is easy to see that this construction defines a functor from Rep_{R/k}(Q⃗, ρ) to Mod A^{R}, and it is an equivalence.

5. Examples related to Tachikawa’s Conjecture

Conjecture 5.1 (Nakayama’s Conjecture). Let A be a finite dimensional algebra over a field k, and

O → _{A}A → I^{0} → I^{1} → …

an injective resolution of a left A-module _{A}A. If all I^{i} are projective, then A is self-injective.

Tachikawa showed that the above conjecture is equivalent to the pair of the following two conjectures.

Conjecture 5.2 (Tachikawa’s Conjectures). Let A be a finite dimensional algebra over a field k, M a finitely generated left A-module.

1. If A is self-injective and Ext_{A}^{i}(M, M) = O for all i ≥ 1, then M is projective.

2. If Ext_{A}^{i}(D A, A) = O for all i ≥ 1, then A is self-injective.

R. Schultz showed that 1 of Conjecture 5.2 is not true in the case of A being an artinian ring [Sc]. I introduce his examples here.

5.1. The Case of Algebras. For a quiver

[figure: Q⃗ :; 1; x; y]

with relations ρ = {yx − δxy, x^{2}, y^{2}} where δ ∈ k^{×}. Then

kQ⃗ = the free k-algebra k < x, y >

and the ideal

\begin{matrix}< ρ >= & k < x, y > (yx − δxy)k < x, y > + \\ k < x, y > x^{2}k < x, y > +k < x, y > y^{2}k < x, y > \end{matrix}

Therefore k(Q⃗, ρ) =< 1, α, β, αβ >_{k} is a local k-algebra, where α = \overline{x}, β = \overline{y}. The multiplication of k(Q⃗, ρ) is

\begin{matrix}(a1 + b_{1}α + b_{2}β + cαβ)(a′1 + b′_{1}α + b′_{2}β + c′αβ) \\ = aa′1 + (ab′_{1} + a′b_{1})α + (ab′_{2} + a′b_{1})β + (ac′ + a′c + b_{1}b′_{2} + δb_{2}b′_{1})αβ \end{matrix}

with a, b_{1}, b_{2}, c, a′, b′_{1}, b′_{2}, c′ ∈ k. Then we have

[figure: _{A}A :; k1; 1; α; β; 1; kα; δ; β; kβ; α; 1; kαβ; k^{4}; [ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]; [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & δ & 0 & 0 \end{matrix} ]]

Since it is easy to see that A has the simple socle, A is self-injective. Indeed, D A =< D 1, D α, D β, D(αβ) >_{k}

[figure: _{A} D A :; k D(αβ); δ; α; β; 1; k D β; 1; β; k D α; α; 1; k D 1; k^{4}; [ \begin{matrix}0 & 0 & 0 & 0 \\ δ & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]; [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]]

(We calculate the action as follows. (α D(αβ))(β) = D(αβ)(βα) = D(αβ)(δαβ) = δ implies α D(αβ) = δ D β). Then every isomorphism from _{A}A to _{A} D A is the

form [ \begin{matrix}a & 0 & 0 & 0 \\ b & δa & 0 & 0 \\ c & 0 & a & 0 \\ d & c & b & a \end{matrix} ] with a ∈ k^{×}.

On the other hand, the opposite quiver with relations (Q⃗^{op}, ρ^{op}) is

[figure: Q⃗^{op} :; 1; x^{op}; y^{op}]

with relations ρ^{op} = {x^{op}y^{op} − δy^{op}x^{op}, (x^{op})^{2}, (y^{op})^{2}}. A_{A}, D A_{A} are the following

[figure: A_{A} :; k1; 1; α; β; 1; kα; 1; β; kβ; α; δ; kαβ; k^{4}; [ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & δ & 0 \end{matrix} ]; [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]; D A_{A} :; k D(αβ); 1; α; β; δ; k D β; 1; β; k D α; α; 1; k D 1; k^{4}; [ \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]; [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ δ & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]]

(Here, −− means the right action). Then every isomorphism from A_{A} to D A_{A} is the form [ \begin{matrix}a & 0 & 0 & 0 \\ b & a & 0 & 0 \\ c & 0 & δa & 0 \\ d & c & b & a \end{matrix} ] with a ∈ k^{×}. If δ = 1, then A ≅ D A as A-bimodules and A is a symmetric k-algebra. Otherwise, A ≅ D̸ A as A-bimodules and A is not a symmetric k-algebra. For n, let M_{n} = A(α + (−δ)^{n}β)

[figure: k^{2}; [ \begin{matrix}0 & 0 \\ (−δ)^{n} & 0 \end{matrix} ]; [ \begin{matrix}0 & 0 \\ δ & 0 \end{matrix} ]]

Then we have an exact sequence

\begin{matrix}O & ⟶ & A(α + (−δ)^{n−1}β) & ⟶ & A & ⟶ & A(α + (−δ)^{n}β) & ⟶ & O \\ O & ⟶ & M_{n−1} & ⟶ & A & ⟶ & M_{n} & ⟶ & O \end{matrix}

for each n ∈ ℤ, and

\begin{aligned}Hom_{A}(M_{n}, A) & = \{[ \begin{matrix}0 & 0 \\ a & 0 \\ a(−δ)^{n} & 0 \\ b & a \end{matrix} ] |a, b ∈ k\} \\ Hom_{A}(M_{m}, M_{n}) & = \{[ \begin{matrix}a & 0 \\ b & a \end{matrix}] |(−δ)^{m}a = (−δ)^{n}a, a, b ∈ k\} \end{aligned} (5.3)

And we have an exact sequence

O  ⟶ Hom_{A}(M_{0}, M_{i}) ⟶ Hom_{A}(M_{0}, A) ⟶
Hom_{A}(M_{0}, M_{i+1}) ⟶ Ext_{A}^{1}(M_{0}, M_{i}) ⟶  O

for i ≥ 1. If −δ is not a root of 1, then by the equation 5.3 we have

dim_{k} Ext_{A}^{1}(M_{0}, M_{0}) = dim_{k} Hom_{A}(M_{0}, M_{1}) − dim_{k} Hom_{A}(M_{0}, A)+
dim_{k} Hom_{A}(M_{0}, M_{0})
= 1 − 2 + 2 = 1
dim_{k} Ext_{A}^{i}(M_{0}, M_{0}) = dim_{k} Ext_{A}^{1}(M_{0}, M_{i−1})
= dim_{k} Hom_{A}(M_{0}, M_{i+1}) − dim_{k} Hom_{A}(M_{0}, A)+
dim_{k} Hom_{A}(M_{0}, M_{i−1})
= 1 − 2 + 1 = 0
for i ≥ 2

(5.4)

Proposition 5.5. Assume that −δ is not a root of 1. Let M = A(α + β), then we have Ext_{A}^{i}(M, M) = O for all i ≥ 2.

Proposition 5.6. Assume that −δ is not a root of 1. Let M = A(α + β), and ⋯ → A → A → M → O a minimal projective resolution, then all syzygy Amodules Ω^{n}M have k-dimension 2, and they are non-isomorphic each other.

5.2. The Case of Rings. Let A = k(Q⃗, ρ) be a finite dimensional k-algebra given in §5.1. Let K be a skew field which is a k-algebra, and B = A^{K}. Then Hom_{K}(_{K}−,_{K}K) and Hom_{K}(−_{K}, K_{K}) induce a duality between rep_{K/k}(Q⃗, ρ) and rep_{K/k}(Q⃗^{op}, ρ^{op}). Hence B is a local self-injective artinian ring. According to Theorem 4.2, Mod B is is equivalent to Rep_{R/k}(Q⃗, ρ). For a representation M = (M, ψ^{M}), ψ^{M}(α) is a left K-linear map for any arrow α. Then ψ^{M} is represented by the set of the right multiplications of matrices of K, and their matrix compositions are the opposite compositions of maps (i.e. we take row vectors as elements of K-vector spaces in this subsection). Therefore by taking the transpose of matrices in _{A}A of §5.1, we have a representation _{B}B in Rep_{R/k}(Q⃗, ρ)

[figure: _{B}B :; K1; 1; α; β; 1; Kα; δ; β; Kβ; α; 1; Kαβ; K^{4}; [ \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix} ]; [ \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & δ \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ]]

For λ ∈ K^{×}, let M_{λ} = B(α + λβ), then M is represented by

\xymatrix{[ \begin{matrix}0 & λ \\ 0 & 0 \end{matrix} ] \ar[r] & K^{2} & [ \begin{matrix}0 & δ \\ 0 & 0 \end{matrix} ] \ar[l]}

Lemma 5.7. The following hold.

1. Hom_{B}(M_{λ}, M_{μ}) = \{[ \begin{matrix}a & b \\ 0 & a \end{matrix}] |λa = aμ, a, b ∈ K\}

2. Hom_{B}(M_{λ}, B) = \{[ \begin{matrix}0 & a & λa & b \\ 0 & 0 & 0 & a \end{matrix}] |a, b ∈ K\}

Lemma 5.8. For n ∈ ℤ, λ ∈ K^{×} and δ ∈ k^{×}, we have an exact sequence

O → M_{λ(−δ)^{n}} ⟶^{η_{n}} B ⟶^{θ_{n+1}} M_{λ(−δ)^{n+1}} → O

where η_{n} = [ \begin{matrix}0 & 1 & λ(−δ)^{n} & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ], and θ_{n+1} = [ \begin{matrix}1 & 0 \\ 0 & λ(−δ)^{n+1} \\ 0 & δ \\ 0 & 0 \end{matrix} ].

Proposition 5.9. If δ ∈ k^{×} and λ ∈ K^{×} satisfy (i) λ and λ(−δ)^{n} are not conjugate in K^{×} for n ≥ 1, (ii) for any n ≥ 0 and any b ∈ K, there exists a ∈ K such that λa − aλ(−δ)^{n} = b,

then Ext_{B}^{i}(M_{λ}, M_{λ}) = 0 for any i ≥ 1, and End_{B}(M_{λ}) is neither left artinian nor right artinian.

Proof. By Lemma 5.8, for n ≥ 0, we have an exact sequence

O → M_{λ(−δ)^{n}} ⟶^{η_{n}} B ⟶^{θ_{n+1}} M_{λ(−δ)^{n+1}} → O.

Then in order to prove the first part, it suffices to show that

O → Hom_{B}(M_{λ}, M_{λ(−δ)^{n}}) ⟶^{Hom_{B}(M_{λ},η_{n})} Hom_{B}(M_{λ}, B)

⟶^{Hom_{B}(M_{λ},θ_{n+1})} Hom_{B}(M_{λ}, M_{λ(−δ)^{n+1}}) → O.

is an exact sequence for n ≥ 0. By Lemma 5.7 1 and assumption 1 , we have

\begin{aligned}Hom_{B}(M_{λ}, M_{λ(−δ)^{n+1}}) & = \{[ \begin{matrix}a & b \\ 0 & a \end{matrix}] |λa = aλ(−δ)^{n+1}, a, b ∈ K\} \\ & = \{[ \begin{matrix}0 & b \\ 0 & 0 \end{matrix} ] |b ∈ K\} \end{aligned}

According to Lemma 5.7 2, we have

Im Hom_{B}(M_{λ}, θ_{n+1}) = \{[ \begin{matrix}0 & λaδ+aλ(−δ)^{n+1} \\ 0 & 0 \end{matrix} ] |a ∈ K\}

By assumption 2, there exists a ∈ K such that λa − aλ(−δ)^{n} = bδ^{−1}. For the second part, by Lemma 5.7 1, we have

End_{B}(M_{λ}) = \{[ \begin{matrix}a & b \\ 0 & a \end{matrix}] |λa = aλ, a, b ∈ K\}

Let ∂_{λ} : K → K be a map defined by ∂_{λ}(a) = λa − aλ for a ∈ K. Then ∂_{λ} is an additive group homomorphism and F = Ker ∂_{λ} is a skew subfield. For any s ∈ F, a ∈ K, we have

∂_{λ}(sa) = λsa − saλ
= sλa − saλ
= s∂_{λ}(a)

Therefore K is a left F -vector space and ∂_{λ} is a left F -linear map. Similarly K is a right F -vector space and ∂_{λ} is a right F -linear map. We have dim_{F} K = dim K_{F} = ∞, because O → F → K ⟶^{∂_{λ}} K → O is exact. It is easy to see End_{B}(M_{λ}) ≅ F ⋉K (this is a trivial extension of F by K).

Proposition 5.10. There are a skew field K, its commutative subfield k, λ ∈ K^{×} and δ ∈ k^{×} such that K is a k-algebra and that they satisfy the conditions (i) and (ii) of Proposition 5.9.

Proof. According to [Co1] or [Co2] Section 8, there are a skew field L and λ ∈ L such that the inner derivation ∂_{λ} : L → L is surjective. Let K be the skew field L{X} of formal Laurant polynomials, and δ = −X. For 0 ≠ f = ∑_{i} ν_{i}X^{i} ∈ K, we denote by deg_{min} f = min{i|ν_{i} ≠ 0}. Then deg_{min} f^{−1} = − deg_{min} f. Therefore λ and λX^{n} are not conjugate for n ≥ 1, because deg_{min} λ ≠ deg_{min} λX^{n}. Let ∂_{λ,n} : K → K be the map defined by ∂_{λ,n}(a) = λa − aλX^{n}. Let g = ∑_{i} ν_{i}X^{i} ∈ K. In the case n = 0, there is μ_{i} ∈ L such that λμ_{i} − μ_{i}λ = ν_{i}. Let f = ∑_{i} μ_{i}X^{i}, then ∂_{λ,0}(f) = g. In the case n ≥ 1, f = ∑_{i=1}^{∞} λ^{−i}gλ^{i−1}X^{n(i−1)}. Hence we have

\begin{aligned}λf − fλX^{n} & = ∑_{i=1}^{∞} λ^{−i+1}gλ^{i−1}X^{n(i−1)} − ∑_{i=1}^{∞} λ^{−i}gλ^{i}X^{ni} \\ & = g. \end{aligned}

We take k = the center Z(K) of K. Then k satisfies the desired property, because of X ∈ Z(K).

6. Appendix

In this section, we recall some properties of homological algebra without proofs. The reader see e.g. [Ro] for details.

Definition 6.1 (Category). We define a category C by the following data: 1. A class Ob C of elements called objects of C.

2. For a ordered pair (X, Y) of objects a set Hom_{C}(X, Y) of morphisms is given such that Hom_{C}(X, Y)∩Hom_{C}(X′, Y′) = ϕ for (X, Y) ≠ (X′, Y′) (an element f of Hom_{C}(X, Y) is called a morphism, and denote by f : X → Y).

3. For each triple (X, Y, Z) of objects of C a map

θ(X, Y, Z) : Hom_{C}(X, Y) × Hom_{C}(Y, Z) → Hom_{C}(X, Z)

(θ is called the composition map) is given.

4. The composition map θ is associative.

5. For each object X of C, there is a morphism 1_{X} : X → X such that for any g : Y → X, h : X → Z we have 1_{X}g = g, h1_{X} = h.

Definition 6.2 (Complex). A diagram X^{▪} : … → X^{i−1} ⟶^{d^{i−1}} X^{i} ⟶^{d^{i}} X^{i+1} → … is called a (cochain) complex if d^{i+1}d^{i} = 0 for all i, that is, Im d^{i−1} ⊂ Ker d^{i} for all i. A complex X^{▪} is called exact if Im d^{i−1} = Ker d^{i} for all i. Sometimes, we call an exact sequence for an exact complex. For a complex X^{▪}, H^{n}(X^{▪}) = Ker d^{n}/ Im d^{n} is called the n-th cohomology.

Lemma 6.3. Let O → V_{0} → V_{1} → … → V_{n} → O be an exact sequence of k-vector spaces. Then we have

dim_{k} V_{0} = Σ_{i=1}^{n}(−1)^{i} dim_{k} V_{i}.

Definition 6.4. For f : X → Y in Mod A, Hom_{A}(X, Y) = the set of left Alinear maps from X to Y. For M ∈ Mod A, we have the following additive group homomorphisms

Hom_{A}(M, X) ⟶^{Hom_{A}(M,f)} Hom_{A}(M, Y)(g ↦ f ∘ g)

Hom_{A}(Y, M) ⟶^{Hom_{A}(M,f)} Hom_{A}(X, M)(h ↦ h ∘ f).

Definition 6.5 (Projective, Injective Module). A left A-module M is called Aprojective if for any surjective A-linear map X → Y we have a surjective additive

group homomorphism Hom_{A}(M, X) ⟶^{Hom_{A}(M,f)} Hom_{A}(M, Y). Similarly, a left Amodule M is called A-injective if for any injective A-linear map X → Y we have a

surjective additive group homomorphism Hom_{A}(Y, M) ⟶^{Hom_{A}(M,f)} Hom_{A}(X, M).

Proposition 6.6. A left A-module A is A-projective. In the case of A being a finite dimensional k-algebra, D A is a injective left A-module.

Proposition 6.7. For a left A-module M, the following hold.

1. M is A-projective if and only if any surjective A-linear map f : X → M splits (i.e. there exists g : M → X such that gf = 1_{M}).

2. M is A-injective if and only if any injective A-linear map f : M → Y splits (i.e. there exists g : Y → M such that fg = 1_{M}).

Proposition 6.8. For a left A-module M, the following hold.

1. There exists a set I and f : A^{(I)} → M such that f is surjective.

2. There exists a injective A-module E and g : M → E such that g is injective.

Definition 6.9 (Projective, Injective Resolution). For a left A-module M, according to Proposition 6.8, we have a surjective A-linear map ϵ_{0} : P_{0} → M with P_{0} being A-projective. For Ker ϵ_{0}, we have a surjective A-linear map ϵ_{1} : P_{1} → Ker ϵ_{0} with P_{1} being A-projective. Therefore we have an exact complex

… → P_{n} → … → P_{1} → P_{0} → M → O,

with P_{i} being A-projective The complex P_{▪} : … → P_{n} → … → P_{1} → P_{0} is called projective resolution of M.

Similarly, we have an exact complex

O → M → I^{0} → I^{1} → … → I^{n} → …,

with I^{i} being A-injective The complex I^{▪} : I^{0} → I^{1} → … → I^{n} → … is called injective resolution of M.

When we have a projective resolution

O → P_{n} → … → P_{1} → P_{0} → M → O,

we say that the projective dimension of M is at most n, denote by pdim_{A}M ≤ n. Similarly, when we have an injective resolution

O → M → I^{0} → I^{1} → … → I^{n} → O,

we say that the injective dimension of M is at most n, denote by idim_{A}M ≤ n.

The left global dimension lgldim A of A is the supremum of pdim M of left Amodules M.

Theorem 6.10 (Higher Extension Groups). The following hold.

1. Let … → P_{n} → … → P_{1} → P_{0} → X → O be a projective resolution of a left A-module X. Then for any Y ∈ Mod A and any n ≥ 0, H^{n} Hom_{A}(P_{▪}, Y) is determined independent of choice of projective resolutions.

2. Let O → Y → I^{0} → I^{1} → … → I^{n} → … be an injective resolution of a left A-module Y. Then for any M ∈ Mod A and any n ≥ 0, H^{n} Hom_{A}(X, I^{▪}) is determined independent of choice of injective resolutions.

3. For X, Y ∈ Mod A, we have H^{n} Hom_{A}(P_{X▪}, Y) ≅ H^{n} Hom_{A}(X, I_{Y}^{▪}) for n ≥ 0, where P_{X▪} (resp., I_{Y}^{▪}) is a projective (resp., an injective) resolution of X (resp., Y).

The additive group H^{n} Hom_{A}(P_{X▪}, Y) ≅ H^{n} Hom_{A}(X, I_{Y}^{▪}) is called the n-th Extension group Ext_{A}^{n}(X, Y).

Proposition 6.11. The following hold.

1. If P is A-projective, then Ext_{A}^{n}(P, Y) = 0 for n ≥ 1.

2. If I is A-injective, then Ext_{A}^{n}(X, I) = 0 for n ≥ 1.

3. For an exact sequence O → X → Y → Z → O in Mod A, we have long exact sequences

O → Hom_{A}(M, X) → Hom_{A}(M, Y) → Hom_{A}(M, Z) →
Ext_{A}^{1}(M, X) →  Ext_{A}^{1}(M, X) → Ext_{A}^{1}(M, X) →
Ext_{A}^{2}(M, X) → …,

and

O → Hom_{A}(Z, M) → Hom_{A}(Y, M) → Hom_{A}(X, M) →
Ext_{A}^{1}(Z, M) →  Ext_{A}^{1}(Y, M) → Ext_{A}^{1}(X, M) →
Ext_{A}^{2}(Z, M) → ….

Lemma 6.12 (Nakayama’s Lemma). Let A be a ring with unity, J the Jacobson radical of A, and M a finitely generated left A-module. For a left A-submodule N of M, if JM + N = M, then N = M.

Definition 6.13 (Minimal Projective resolution). Let M be a finitely generated left A-module. A projective resolution of M

… → P_{n} → … ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} → M → O

is called a minimal projective resolution provided that Im d_{i} ⊂ JP_{i−1} for all i ≥ 1. This resolution does not exists in general. In the case of A being left artinian, a minimal projective resolution exists for any finitely generated left A-module.

Definition 6.14 (Indecomposable Module). A left A-module M is called indecomposable provided that if M = X ⊕ Y, then X or Y = O.

Definition 6.15. Let A and B be k-algebras. The tensor product A ⊗_{k} B is the k-algebra defined by

\begin{aligned}(a ⊗ b)(a′ ⊗ b′) & = aa′ ⊗ bb′ \\ 1_{A⊗B} & = 1_{A} ⊗ 1_{B}. \end{aligned}

Then we have

\begin{aligned}(1_{A} ⊗ b)(a ⊗ 1_{B}) & = a ⊗ b \\ & = (a ⊗ 1_{B})(1_{A} ⊗ b). \end{aligned}

Definition 6.16 (The Skew Field of Formal Laurant Polynomials). For a skew field L, let

L{X} = {Σ_{i=n}^{∞}a_{i}X^{i}|n ∈ ℤ, a_{i} ∈ L}.

We define the multiplication of Σ_{i=m}^{∞}a_{i}X^{i}, Σ_{j=n}^{∞}b_{j}X^{j} ∈ L{X} by

(Σ_{i=m}^{∞}a_{i}X^{i})(Σ_{j=n}^{∞}b_{j}X^{j}) = Σ_{k=m+n}^{∞}(Σ_{i+j=k}a_{i}b_{j})X^{k},

and define

deg_{min}(Σ_{i=m}^{∞}a_{i}X^{i}) = m

if a_{m} ≠ 0. Then we have

deg_{min}(fg) = deg_{min}(f) + deg_{min}(g)

for non-zero polynomials f, g ∈ L{X}. It is easy to see that L{X} is a skew field.

References

[ARS] M. Auslander, I. Reiten and Sverre O. Smalø, “Representation Theory of Artin Algebras,” Cambridge Studies in Advanced Math. 36, Cambridge, UK 1997 (corrected paperback edition).

[Co1] P. M. Cohn, The range of derivations on a skew field and the equation ax − xa = c, J. Indian Math. Soc. 37 (1973) 61-69.

[Co2] P. M. Cohn, “Skew Fields,” Encyclopedia of Math. and Its Appl. 57, Cambridge, UK 1995.

[Rl] C.M. Ringel, “Tame Algebras and Integral Quadratic Forms,” Lecture Notes in Math. 1099, Springer-Verlag, Berlin, 1984.

[Ro] J. J. Rotman, “An introduction to homological algebra,” Pure and Applied Mathematics 85, Academic Press, 1979.

[Sc] R. Schultz, A non-projective module without self-extensions, preprint.

J. Miyachi: Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184-8501, Japan

   E-mail address: miyachi@u-gakugei.ac.jp