ON THE ℤD_{∞} CATEGORY ^{1}

Michel Van den Bergh

Abstract In this paper we give a direct proof of the properties of the ℤD_{∞} category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author.

1. Introduction

Below k is a field. All categories will be k-linear. An abelian or triangulated category A is Ext-finite if for all objects A, B ∈ A one has that ⊕_{i} Ext^{i}(A, B) is finite dimensional. If A is triangulated and Ext-finite then we say that A satisfies Serre duality [1] if there exists an auto-equivalence F of A together with isomorphisms

Hom_{A}(A, B) → Hom_{A}(B, F A)^{∗}

natural in A, B (where (−)^{∗} is the k-dual). If A is abelian and Ext-finite then we say that A satisfies Serre duality if this is the case for D^{b}(A). The following result can be extracted from [5, Ch. 1].

Theorem 1.1. Assume that C is an Ext-finite hereditary category without injectives or projectives. Then the following are equivalent

(1) C has almost split sequences.

(2) C satisfies Serre duality.

(3) There is an auto-equivalence V : C → C together with natural isomorphisms

(1.1)  Hom_{C}(A, B) → Ext_{C}^{1}(B, V A)^{∗}

Furthermore the functor V coincides with the Auslander-Reiten translate τ when evaluated on objects.

In the classification of noetherian Ext-finite hereditary categories with Serre duality in [5] we considered a category C defined by the following pullback diagram

(1.2)

mod(k) ⊕ mod(k) ⟶ mod(k)
↑  ↑
C  ⟶ gr(k[x])

where the horizontal map sends (V_{1}, V_{2}) to V_{1} ⊕ V_{2} and the vertical map is localizing at x followed by restricting to degree zero. It was shown by a rather indirect argument that C is a noetherian, Ext-finite, hereditary abelian category without injectives or projectives which satisfies Serre duality. It was also shown that the AR-quiver of C has two components, one equal to ℤA_{∞} (a “wing”) and the other equal to ℤD_{∞}. For this reason C was called the “ℤD_{∞}-category”.

1The paper is in a final form and no version of it will be submitted for publication elsewhere.

The aim of this paper is to give a direct proof of the above facts. In addition we will also establish a link with one-dimensional A_{n} singularities.

2. Elementary properties

It is easy to see that C, as defined in the introduction is a noetherian abelian category. It will be convenient to consider the locally noetherian Grothendieck category C̃ associated to C. It follows for example by [3, Prop. 2.14] that D^{b}(C) and D_{C}^{b}(C̃) are equivalent. Hence the Ext-groups between objects in C may be computed in C̃.

The objects of C̃ are quadruples (M, V_{0}, V_{1}, ϕ) where M is a graded k[x]-module, V_{0}, V_{1} are k-vector spaces and ϕ is an isomorphism of k[x] modules M_{x} → (V_{0} ⊕ V_{1}) ⊗_{k} k[x, x^{−1}]. Objects in C are given by the quadruples (M, V_{0}, V_{1}, ϕ) in which M is finitely generated.

Sending (M, V_{0}, V_{1}, ϕ) to M defines an faithful exact functor C̃ → Gr(k[x]) which we call the restriction functor and which we denote by (−)_{k[x]}.

We write M(n) = (M(n), V_{0}, V_{1}, ϕ) where we have identified M(n)_{x} with M_{x} through multiplication with x^{n}. Furthermore we define σ(M) = (M, V_{1}, V_{0}, ϕ).

We define T̃ ⊂ C̃ and F̃ ⊂ C̃ respectively as the inverse images of the x-torsion and x-torsion free modules in Gr(k[x]). T and F are defined similarly, but starting from C.

By C̃_{x} we denote the full subcategory of C̃ with objects the quadruples (M, V_{0}, V_{1}, ϕ) in which x acts invertibly on M.

We denote by (−)_{x} the functor C̃ → C̃_{x} which sends (M, V_{0}, V_{1}, ϕ) to (M_{x}, V_{0}, V_{1}, ϕ). Clearly if M ∈ C̃ and N ∈ C̃_{x} then the canonical maps

(2.1)  HomC̃(M, N) → HomC̃_{x}(M, N) → HomC̃_{x}(M_{x}, N)

are isomorphisms. We list a few other obvious facts. (O1) (T̃, F̃) forms a torsion pair in C̃. That is Hom(T̃, F̃) = 0 and for any M ∈ C̃ there

exists an exact sequence (necessarily unique)

0 → T → M → F → 0

with T ∈ T̃ and F ∈ F̃. (O2) If T ∈ T̃ and M ∈ C̃ then

HomC̃(T, M) = Hom_{k[x]}(T, M)
HomC̃(M, T) = Hom_{k[x]}(M, T)

(O3) The restriction functor defines an equivalence between T̃ and Tors(k[x]) where

Tors(k[x]) denotes the x-torsion modules is Gr(k[x]). (O4) The functor C̃_{x} → Mod(k) ⊕ Mod(k) which sends (M, V_{0}, V_{1}, ϕ) to V_{0} ⊕ V_{1} is an

equivalence of categories. Combining (O4) with (2.1) yields in particular (O5) If N ∈ C̃_{x} then Hom_{C}(−, N) is exact. Hence the objects in C̃_{x} are injective in C̃. We now describe the indecomposable injectives in C̃. For n ∈ ℤ let E_{n} be the graded injective k[x]-module given by k[x, x^{−1}]/x^{n+1}k[x]. Since E_{n} ∈ Tors(k[x]) there exists by (O3) a corresponding object in T̃ which we denote by the same symbol. From (O2) it follows that Hom(−, E_{n}) is exact and hence E_{n} is injective in C̃.

To construct other injectives we note that by (O5) we know that the objects in C̃_{x} are injective in C̃. Since by (O4) C̃_{x} is equivalent to Mod(k) ⊕ Mod(k) there must be two corresponding indecomposable injectives in C̃. They are given by

E^{0} = (k[x, x^{−1}], k, 0, id_{k[x,x^{−1}]})
E^{1} = (k[x, x^{−1}], 0, k, id_{k[x,x^{−1}]})

Proposition 2.1. (1) (E_{n})_{n}, E^{0}, E^{1} forms a complete list of indecomposable injectives

in C̃.

(2) Every object in C̃ has injective dimension one (and hence C̃ and C are hereditary [5, Prop. A.3]).

(3) C is Ext-finite.

Proof. Since the listed injectives are clearly indecomposable (1) follows if we can show that any indecomposable object can be embedded in a direct sum of them [4].

We prove (1) and (2) together by showing that every object M ∈ C̃ has a resolution of length at most two whose terms consist of direct sums of the injectives given in (1). By (O1) it is clearly sufficient to prove this claim separately in the cases M ∈ T̃ and M ∈ F̃.

Assume first that M ∈ T̃. Then M has an injective resolution

(2.2)  0 → M → I_{0} → I_{1} → 0

in Tors(k[x]). By (O3) this resolution corresponds to one in C̃. Furthermore by the structure of the injectives in Gr(k[x]) the I_{i} are direct sums of the E_{n} in Gr(k[x]). Again by (O3) the same is true in C̃.

Now assume that M ∈ F̃. Consider the short exact sequence

(2.3)  0 → M → M_{x} → M_{x}/M → 0

M_{x} lies in C̃_{x} and hence by (O4) is a direct sum of copies of E^{0} and E^{1}. M_{x}/M is xdivisible and lies in T̃ and so by (O3) M_{x}/M is a direct sum of copies of E_{n}. Whence (2.3) is the kind of resolution we were looking for.

To prove (3) we note that if E, F are indecomposable injectives as in (1) then dim HomC̃(E, F) ≤ 1. Thus it suffices to show that every M ∈ C has an injective resolution consisting in every degree of a finite number of indecomposable injectives. This follows easily from the construction. □

Proposition 2.2. If F ∈ F̃ and T ∈ T̃ then Ext1 ; C̃(F, T) = 0. In particular every object in C is of the form F ⊕ T with F ∈ F and T ∈ T .

Proof. It follows from (O2) that Hom(F, −) is exact on T . Since by the proof of the previous proposition T has a C̃ injective resolution inside T̃, we are done. □

Now we describe the Ext-groups between objects in F̃.

Lemma 2.3. Assume that F = (F, V_{0}, V_{1}, ϕ), F′ = (F′, V′_{0}, V′_{1}, ϕ′) are objects in F̃. Then there exists an exact sequence of the form (2.4)

0 → HomC̃(F, F′) → Hom_{k[x]}(F, F′) → Hom_{k}(V_{0}, V′_{1}) ⊕ Hom_{k}(V_{1}, V′_{0}) → Ext1 ; C̃(F, F′) → 0

Proof. We start with the short exact sequence

0 → F′ → F′_{x} → F′_{x}/F′ → 0

which according to the proof of lemma 2.1 is an injective resolution of F′, both in C̃ and in Gr(k[x]).

Applying Hom_{C}(F, −), Hom_{Gr(k[x])}(F, −) and comparing yields a commutative diagram with exact rows and columns.

[figure: 0; 0; HomC̃(F, F′); Hom_{k}(V_{0}, V′_{0}) ⊕ Hom_{k}(V_{1}, V′_{1}); HomC̃(F, F′_{x}/F′); Ext1 ; C̃(F, F′); 0; 0; Hom_{Gr(k[x])}(F, F′); Hom_{k}(V_{0} ⊕ V_{1}, V′_{0} ⊕ V′_{1}); Hom_{Gr(k[x])}(F, F′_{x}/F′); 0; Hom(V_{0}, V′_{1}) ⊕ Hom(V_{1}, V′_{0}); 0]

(2.4) now follows from the previous diagram through an easy diagram chase. □

Proposition 2.4. C has neither injectives nor projectives.

Proof. Since T is equivalent to the x-torsion modules in gr(k[x]), it is easy to see that T does not contain any injective or projectives.

If 0 ≠ (F, V_{0}, V_{1}, ϕ) in F then by considering the faithful restriction functor to gr(k[x]) we see that Hom_{C}(F, σF (−n)) = 0 for n ≫ 0. On the other hand V_{0} or V_{1} ≠ 0. It follows from the previous lemma that Ext_{C}^{1}(F, σF (−n)) ≠ 0 for n ≫ 0. Hence F is not projective. A similar argument shows that F is not injective. □

Remark 2.5. The reason why we called this section “Elementary properties” is that the stated results hold in greater generality. For example, suitably adapted versions would be valid for the pullback of

mod(k)^{⊕m} ⟶ mod(k)
↑
gr(k[x])

for any m. By contrast, the results in the next section require m = 2.

3. Serre duality

§ Our next aim is to prove that C satisfies Serre duality. First we construct a Serre functor on F. Put V M = σ(M)(−1).

The first step in proving Serre duality is constructing a “trace map” η_{M} : Ext_{C}^{1}(M, V M) → k for M ∈ F which should corresponds to the identity map in Hom_{C}(M, M) under the isomorphism (1.1).

We now use (2.4) to construct the trace map η_{F} for F = (F, V_{0}, V_{1}, ϕ) ∈ F. In this case V F = (F (−1), V_{1}, V_{0}, ϕ) and we have an exact sequence

Hom_{k[x]}(F, V F) → Hom_{k}(V_{0}, V_{0}) ⊕ Hom_{k}(V_{1}, V_{1}) → Ext_{C}^{1}(F, V F) → 0

Lemma 3.1. The composition

(3.1)  Hom_{k[x]}(F, V F) → Hom_{k}(V_{0}, V_{0}) ⊕ Hom_{k}(V_{1}, V_{1}) ⟶^{Tr_{V_{0}} + Tr_{V_{1}}} k

is the zero map.

Proof. To see this note that Hom_{k[x]}(F, V F) = Hom(F, F (−1)) and furthermore that (3.1) can be extended to a commutative diagram.

\xymatrix{Hom_{k[x]}(F, F (−1)) \ar[r] & Hom_{k}(V_{0} ⊕ V_{1}, V_{0} ⊕ V_{1}) \ar[r] \ar[d]^{Tr} & Hom_{k}(V_{0}, V_{0}) ⊕ Hom_{k}(V_{1}, V_{1}) \ar[dl]^{Tr_{V_{0}} + Tr_{V_{1}}} \\  & k}

By choosing a basis for F as graded k[x]-module one easily sees that every element of Hom(F, F (−1)) ⊂ Hom(F, F) is nilpotent. Since nilpotent elements have zero trace it follows that the composition

Hom_{k[x]}(F, F (−1)) → Hom_{k}(V_{0} ⊕ V_{1}, V_{0} ⊕ V_{1}) ⟶^{Tr} k

is zero. This proves what we want. □

From lemma 3.1 together with (2.4) there exists a unique map η_{F} : Ext_{C}^{1}(F, V F) → k which makes the following diagram commutative.

\xymatrix{Hom_{k[x]}(F, V F) \ar[r] & Hom(V_{0}, V_{0}) ⊕ Hom(V_{1}, V_{1}) \ar[r] \ar[dr]^{Tr_{V_{0}} + Tr_{V_{1}}} & Ext_{C}^{1}(F, V F) \ar[r] \ar[d]^{η_{F}} & 0 \\  &  & k}

To continue it will be convenient to use the Yoneda multiplication on Ext_{C}^{∗}(−, −). In order to have compatibility with the notation for compositions of maps we will write the Yoneda multiplication as a pairing

Ext_{C}^{∗}(B, C) × Ext_{C}^{∗}(A, B) → Ext_{C}^{∗}(A, C)

We extend η_{F} to a map Ext^{∗}(F, V F) → k by letting it act trivially on Hom(F, V F).

Lemma 3.2. Let F, G ∈ F and assume that f ∈ Ext_{C}^{∗}(F, G) and g ∈ Ext_{C}^{∗}(G, V F). Then we have η_{F} (gf) = η_{G}(V (f)g).

Proof. We may assume that f and g are homogeneous. Furthermore the cases where f,g are both of degree 0 or of degree 1 are trivial. Hence we may assume that (deg f, deg g) = (0, 1) or (deg f, deg g) = (1, 0).

Let us consider the first possibility. We check that η_{F} (−f) = η_{G}(V (f)−) as maps Ext^{1}(G, V F) → k. This amounts to the commutativity of

(3.2)

Ext^{1}(G, V F) ⟶ Ext^{1}(G, V G)
↓  ^{η_{G}}↓
Ext^{1}(F, V F) ⟶^{η_{F}}  k

Assume that F = (F, V_{0}, V_{1}, ϕ), G = (G, W_{0}, W_{1}, θ). Then f induces maps f_{0} : V_{0} → W_{0} and f_{1} : V_{1} → W_{1}. Elementary linear algebra yields that we have a commutative diagram

Hom(W_{0}, V_{0}) ⊕ Hom(W_{1}, V_{1}) ⟶ Hom(W_{0}, W_{0}) ⊕ Hom(W_{1}, W_{1})
↓  ^{Tr_{W_{0}} + Tr_{W_{1}}} ↓
Hom(V_{0}, V_{0}) ⊕ Hom(V_{1}, V_{1}) ⟶^{Tr_{V_{0}} + Tr_{V_{1}}}  k

This diagram, together with the definition of η yields the commutativity of (3.2). Now we consider the possibility (deg f, deg g) = (1, 0). Since we trivially have

(3.3)  η_{V X} ∘ V = η_{X}

it is sufficient to prove that η_{V F} (V (g)V (f)) = η_{G}(V (f)g). Replacing (V f, g) by (g, f) this reduces to the previous case. □

We are now in a position to prove Serre duality for objects in F. We will show that the pairing

(3.4)  Hom_{C}(F, G) × Ext_{C}^{1}(G, V F) → Ext^{1}(F, V F) ⟶^{η_{F}} k : (f, g) ↦ η_{F} (gf)

is non-degenerate. By lemma 3.2 the non-degeneracy of (3.4) for all F, G is equivalent to the non-degeneracy of the pairing

(3.5)  Ext_{C}^{1}(F, G) × Hom_{C}(G, V F) → Ext^{1}(F, V F) ⟶^{η_{F}} k : (f, g) → η_{F} (gf)

for all F, G. It follows also easily from lemma 3.2 that (3.4) and (3.5) are natural in F and G.

Lemma 3.3. If we have and exact sequence

(3.6)  0 → F_{1} → F → F_{2} → 0

in F and if we have non-degeneracy of (3.4) and (3.5) for two out of the three pairs (F_{1}, G), (F, G), (F_{2}, G) then we also have it for the third one. A similar statement holds for an exact sequence

(3.7)  0 → G_{1} → G → G_{2} → 0

Proof. Assume that we have an exact sequence of the form (3.6). We claim that the following diagram with exact rows is commutative.

\xymatrix{0 \ar[r] & Hom(F_{2}, G) \ar[r] \ar[d]^{α_{2}} & Hom(F, G) \ar[r] \ar[d]^{α} & Hom(F_{1}, G) \ar[r] \ar[d]^{α_{1}} & Ext^{1}(F_{2}, G) \ar[r] \ar[d]^{β_{2}} & Ext^{1}(F, G) \ar[r] \ar[d]^{β} & Ext^{1}(F_{1}, G) \ar[r] \ar[d]^{β_{1}} & 0 \\ 0 \ar[r] & Ext^{1}(G, V F_{2})^{∗} \ar[r] & Ext^{1}(G, V F)^{∗} \ar[r] & Ext^{1}(G, V F_{1})^{∗} \ar[r] & Hom(G, V F_{2})^{∗} \ar[r] & Hom(G, V F)^{∗} \ar[r] & Hom(G, V F_{1})^{∗} \ar[r] & 0}

Here the maps labeled by α are obtained from (3.4) whereas those labeled by β are obtained from (3.5).

The commutativity of this diagram follows easily from lemma 3.2 together with the observation that all horizontal arrows are obtained by Yoneda multiplying with elements of suitable Ext-groups. For example the connecting maps are obtained from multiplying with the element of Ext^{1}(F_{2}, F_{1}) representing the exact sequence (3.6).

If we now have non-degeneracy for two out of the three pairs (F_{1}, G), (F, G), (F_{2}, G) then we also have it for the third pair because of the five-lemma.

The case where we have an exact sequence as in (3.7) is treated similarly. □

To continue we define a some canonical objects in F. Let a ∈ ℕ. Then we write

F_{0a}^{0} = (x^{−a}k[x], k, 0, id_{k[x,x^{−1}]})
F_{0a}^{1} = (x^{−a}k[x], 0, k, id_{k[x,x^{−1}]})

(the reason for this notation will become clear in Section §5).

Lemma 3.4. Every object F in F has a finite filtration 0 = F_{0} ⊂ ⋯ ⊂ F_{n} = F such that the corresponding subquotients are among the F_{0a}^{i}.

Proof. By the structure of C_{x} there must be a surjective map ϕ : F_{x} → E^{i} where i = 0 or i = 1. Hence im ϕ is a non-trivial quotient. Since it is easy to see that the subobjects of E^{i} in C are of the form F_{0a}^{i} we are done. □

Using (2.4) we can compute the Hom and Ext-groups between the F_{0a}^{i}. The results are given in the next lemma.

Lemma 3.5. One has

Hom(F_{0a}^{i}, F_{0b}^{j}) = \begin{cases}k & if i = j and a ≤ b \\ 0 & otherwise \end{cases}

Ext^{1}(F_{0a}^{i}, F_{0b}^{j}) = \begin{cases}k & if i = 1 − j and a > b \\ 0 & otherwise \end{cases}

Proof. The claim for Hom is trivial, so we concentrate on Ext.

We use (2.4). This immediately yields that Ext^{1}(F_{0a}^{i}, F_{0b}^{j}) = 0 if j ≠ 1 − i. If j = 1 − i then we have the following exact sequence.

(3.8)  Hom_{k[x]}(x^{−a}k[x], x^{−b}k[x]) → k → Ext_{C}^{1}(F_{0a}^{i}, F_{0b}^{j}) → 0

This yields that Ext_{C}^{1}(F_{0a}^{i}, F_{0b}^{j}) = k if and only if Hom_{k[x]}(x^{−a}k[x], x^{−b}k[x]) = 0, i.e. if and only if a > b. □

We are now in a position to prove the main result of this section.

Theorem 3.6. C satisfies Serre duality.

Proof. We show first that C satisfies Serre duality for objects F, G in F. We will show the non-degeneracy of (3.4) and (3.5) by induction of rk_{k[x]}(F), rk_{k[x]}(G). This reduces us to the case where F = F_{0a}^{i}, G = F_{0b}^{j}. So we need to check the non-degeneracy of

(3.9)  Hom_{C}(F_{0a}^{i}, F_{0b}^{j}) × Ext_{C}^{1}(F_{0b}^{j}, F_{0,a−1}^{i′}) → Ext_{C}^{1}(F_{0a}^{i}, F_{0,a−1}^{i′}) ⟶k
(3.10)  Ext_{C}^{1}(F_{0a}^{i}, F_{0b}^{j}) × Hom_{C}(F_{0b}^{j}, F_{0,a−1}^{i′}) → Ext_{C}^{1}(F_{0a}^{i}, F_{0,a−1}^{i′}) ⟶k

where i′ = 1 − i. We will concentrate ourselves on (3.10). (3.9) is similar. By (3.5) the only non-trivial case is given by j = 1−i and a > b. In that case all vector spaces involved are equal to k and what we want to prove follows from inspecting (3.8).

Now we show that C has almost split sequences. By Theorem 1.1 this implies that C satisfies Serre duality.

By Proposition 2.2 it is clearly sufficient to construct almost split sequences ending in indecomposable objects in F and T . First let F ∈ F be indecomposable. Since Ext_{C}^{1}(F, V F) ≅ Hom_{C}(F, F)^{∗}, Ext_{C}^{1}(F, V F) has a simple socle as (left or right) Hom_{C}(F, F) module. Let ζ be a non-zero element this socle. It is well-known, and easy to see that ζ defines the almost split sequence in F ending in F.

0 → V F → M → F → 0

But this is also an almost split sequence in C since if T ∈ T then Hom(T, F) = 0. Now let T ∈ T . Then there exists an almost split sequence in T

(3.11)  0 → T′′ → T′ → T → 0

(since T is equivalent to the category of x-torsion modules over k[x]).

We have to prove that any pullback for C → T of (3.11) with C indecomposable is split. Clearly we only have to consider the case C ∈ F. But then it follows from Proposition 2.2 that the pullback is split. This finishes the proof. □

4. Relation with one-dimensional graded type A_{n}-singularities

Let C be the hereditary category which was described in the previous section. We will now show that C can be considered as a limit of certain graded simple singularities.

If m ∈ ℕ then the graded simple A_{2m−1}-singularity of dimension one is by definition the graded subring R_{m} of k[x] ⊕ k[x] generated by u = (x, x) and v = (x^{m}, 0). It is easy to see that R ≅ k[u, v]/(u^{m}v − u^{2m}) and hence this is equivalent to the classical definition (see for example [2]). We put C_{m} = mod(R_{m}).

Let us consider k[x] as being diagonally embedded in k[x] ⊕ k[x]. That is we identify x with (x, x). Clearly we have

(R_{m})_{x} = k[x, x^{−1}] ⊕ k[x, x^{−1}]

Hence if M ∈ Mod(R_{m}) then M_{x} is canonically a sum of two k[x, x^{−1}]-modules which we denote by M_{x}^{0} and M_{x}^{1} respectively. This allows us to define the following functor.

U_{m} : C_{m} → C : M ↦ (M, (M_{x}^{0})_{0}, (M_{x}^{1})_{0}, id)

Clearly U_{m} is faithful. We have inclusions

k[x] ⊂ ⋯ ⊂ R_{m+1} ⊂ R_{m} ⊂ ⋯ R_{0} = k[x] ⊕ k[x]

Dualizing these yield restriction functors

C_{0} → ⋯ → C_{m} → C_{m+1} → ⋯ → mod(k[x])

It is clear that these restriction functors are compatible with the functors (U_{m})_{m}. Define C_{∞} as the 2-direct limit of the C_{m}. That is the objects in C_{∞} are the objects in ∐_{m} C_{m} and we put

Hom_{C_{∞}}(M, N) = inj lim Hom_{C_{m}}(M, N)

The functors (U_{m})_{m} define a functor U_{∞} : C_{∞} → C.

Proposition 4.1. The functor U_{∞} defined above is an equivalence.

Proof. From the definition it is clear that U_{∞} is faithful. So we only have to show that it is full and essentially surjective.

We will first show that U_{∞} is full. Let M, N ∈ C_{m} and let f : M → N be a homomorphism in C. So f is in fact a k[x]-linear homomorphism f : M → N such that the localization f_{x} is k[x, x^{−1}] ⊕ k[x, x^{−1}]-linear.

Let y = (x, 0). Then y^{n} ∈ R_{m} for n ≥ m and R_{n} as subring of R_{m} is generated by x and y^{n}. To prove fullness of U_{∞} it is sufficient that f is y^{n}-linear for n ≫ 0. Let T be the torsion submodule of N and consider the k[x] linear map M → N given by f^{(n)} = f(y^{n}−) − y^{n}f(−). Since after localizing at x, f is y linear, it follows that the image of f^{(n)} lies in T. Since T is rightbounded it is clear that f^{(n)} must be zero if n ≫ 0. This proves what we want.

Now we prove essential surjectivity. First let F ∈ F. Then we claim that F ⊂ F_{x} is stable under multiplication by y^{n} for n ≫ 0. First note that yF is a finitely generated k[x]-submodule of F_{x}. Hence x^{n}yF ⊂ F. Since x^{n}y = y^{n+1} this proves what we want.

Now let T ∈ T . Then as graded k[x]-module T has right bounded grading and since k[x]_{<m} = (R_{n})_{<m} for n ≥ m it follows that for n ≫ 0 we may consider T as a graded R_{n}-module. This proves what we want. □

5. Representation theory

In section we construct the AR-quiver of C. From the above discussion it follows that the components of the AR-quiver of C lie either in T or in F. Since T is equivalent to the x-torsion modules in gr(k[x]) it has a unique component which is ℤA_{∞}. So the main difficulty is represented by the component(s) in F.

We now describe the indecomposable torsion free objects in C as well as the associated Auslander-Reiten quiver (see [5]). Using Proposition 4.1 this could be easily obtained by using a graded version of the results in [2]. However for completeness we give an independent proof here.

For m > 0 denote by F_{ma} the unique indecomposable projective R_{m}-module in with grading starting in degree −a (thus F_{ma} = F_{m0}(a)). For m = 0 we let F_{00}^{0}, F_{00}^{1} be the two indecomposable R_{0} modules whose gradings starts exactly at 0. We also put F_{0a}^{i} = F_{00}^{i}(a) (as in Section §3).

Finally to simplify the notation we will write F_{ma}^{i} (i = ∅, if m ≠ 0) for U_{m}(F_{ma}^{i}).

Proposition 5.1. The indecomposable objects in F are given by F_{ma}^{i}. Furthermore the associated Auslander-Reiten quiver is given by Figure 1

Proof. By Serre duality it follows that Ext_{C}^{1}(F_{ma}^{i}, V F_{ma}^{i}) is one dimensional. Therefore its unique (up to scalar multiplication) non-zero element represents the almost split sequence ending in F_{ma}^{i}.

Let us now explicitly construct non-split extensions between F_{ma}^{i} and V F_{ma}^{i}. First note

V F_{ma} = F_{m,a−1}

where for simplicity we have written F_{0a} = F_{0a}^{0} ⊕ F_{1a}^{1}, and

V F_{0a}^{i} = F_{0a−1}^{1−i}

To construct the extension associated to F_{m,a} we note that F_{m−1,a−1} and F_{m+1,a} are naturally submodules of F_{m,a} whose sum is F_{m,a} and whose intersection is F_{m,a−1}. Hence the exact sequence

(5.1)  0 → F_{m,a−1} → F_{m−1,a−1} ⊕ F_{m+1,a} → F_{m,a} → 0

yields the sought extension.

To construct the extension associated to F_{0a}^{i} we note that F_{1a} maps surjectively to F_{0a}^{i} with kernel F_{0a−1}^{1−i}. Thus in this case the sought extension is

(5.2)  0 → F_{0,a−1}^{1−i} → F_{1a} ⊕ F_{0a}^{i} → F_{m,a} → 0

[figure: F_{41}; F_{42}; F_{43}; F_{31}; F_{32}; F_{20}; F_{21}; F_{22}; F_{10}; F_{11}; F_{0−1}^{0}; F_{00}^{1}; F_{01}^{0}; F_{0−1}^{1}; F_{00}^{0}; F_{01}^{1}]

Figure 1. The Auslander-Reiten quiver of F

It is now easy to assemble the almost split sequences given by (5.1) and (5.2) into the translation quiver given by Figure 1.

To show that Figure 1 is the entire AR-quiver of F (and not just a component) we have to show that there are no other indecomposable objects.

So assume that F is an indecomposable object in F, not occurring among the F_{ma}^{i}. By lemma 3.4 there exist a non-zero map F_{0a}^{i} → F for some i, a. Using the defining property of AR-sequences we may use this to construct a non-zero map F_{mb}^{i} for some i (possibly ∅) and m, and for b arbitrarily large.

Now note that the only non-trivial torsion free quotients of F_{mb}^{i} are F_{mb}^{i} itself and F_{0b}^{0,1} (if m ≠ 0). Since all these quotients possess a non-trivial element in degree −b it follows that Hom_{C}(F_{mb}^{i}, F) = 0 for b ≫ 0. This finishes the proof. □

References

[1] A. Bondal and M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337.

[2] E. Dieterich and A. Wiedemann, The Auslander-Reiten quiver of a simple curve singularity, Trans. Amer. Math. Soc. 294 (1986), 455–475.

[3] W. Lowen and M. Van den Bergh, Deformation theory of abelian categories, math.CT/0405226.

[4] E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528.

[5] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366 (electronic).

Department WNI,

Limburgs Universitair Centrum 3590 Diepenbeek, Belgium

  E-mail address: vdbergh@luc.ac.be