Iterated Extensions and Uniserial Length Categories

https://doi.org/10.1007/s10468-020-09946-0

Iterated Extensions and Uniserial Length Categories

Eivind Eriksen¹

Received: 30 October 2018 / Accepted: 8 January 2020 /

©The Author(s) 2020

Abstract

In this paper, we study length categories using iterated extensions. We fix a field k, and for any familySof orthogonalk-rational points in an Abeliank-categoryA, we consider the categoryExt(S)of iterated extensions ofSinA, equipped with the natural forgetful functorExt(S)→A(S)into the length categoryA(S). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel’s criterion, we give a complete classification of the indecomposable objects inA(S)when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the familySinA. As an application, we classify all graded holonomicD-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, whenDis the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish.

Keywords Finite length categories·Uniserial categories·Iterated extensions· Noncommutative deformations

Mathematics Subject Classification (2010) Primary 18E10; Secondary 16G99·14F10

1 Introduction

Let S = {Sα : α ∈ I} be a family of non-zero, pairwise non-isomorphic objects in an Abelian k-categoryA, wherek is a field. We consider the minimal full subcategory A(S)⊆Athat containsSand is closed under extensions. The familySis called a family of orthogonal points if End(S_α)is a division algebra and Hom(S_α, S_β)=0 for allα, β ∈I

Presented by: Michel Brion Eivind Eriksen

eivind.eriksen@bi.no

1 Department of Economics, BI Norwegian Business School, Oslo, Norway

withα=β. In this case,A(S)⊆Ais a length category withSas its simple objects. When End(Sα)=k, we callSαak-rational point.

An important special case is whenA = ModA is the category of modules over an associativek-algebraA, andSis a subset of the simpleA-modules. IfSis the family of all simple modules, thenA(S)is the category of all modules of finite length. There are also many other interesting applications, for example whenAis the category of graded modules over a gradedk-algebra, or the category of coherent sheaves over ak-scheme. Note that any length category is exact equivalent to an exact subcategory of a module category.

Nevertheless, it is often better to work directly in the Abelian category of interest than to use an embedding into a module category.

We shall use the categoryExt(S)ofiterated extensionsofSto study the length category A(^S)whenSis a family of orthogonalk-rational points. An iterated extension ofSis a couple(X, C), whereXis an object inAandCis a cofiltration

X=Cn f_n

−→C_n−1→ · · · →C2 f₂

−→C1 f₁

−→C0 =0

wherefi :Ci →C_i−1 is surjective andKi =ker(fi) ∼=Sαi withαi ∈I for 1≤i ≤n.

The assignment(X, C)→Xdefines a forgetful functorExt(S)→A(S).

The categoryExt(S)of iterated extensions has some interesting invariants, in addition to the lengthn, the simple factors{K₁, . . . , K_r}withr ≤ n, and their multiplicities. The order vectorα =(α₁, . . . , α_n)∈ Iⁿis an invariant ofExt(S). Moreover, when(X, C)is an iterated extension with order vectorα, there are induced commutative diagram of short exact sequences

0 //K_i //C_i ^fi //C_i−1 //0

0 //Ki //

OO

Z f_i //

OO

Ki−1 //

OO

0

We call the extensionsτi∈Ext¹_A(K_i−1, Ki)for 2≤i≤nthesimple extensionsof(X, C).

These simple extensions are also invariants ofExt(S).

We say thatA(S)is a uniserial length category if any indecomposable object inA(S)has a unique composition series. WhenSis a family ofk-rational orthogonal points, thenA(S) is a uniserial length category if and only ifSsatisfies the following criterion, due to Gabriel:

β∈I

dimkExt¹_A(Sα, Sβ)≤1 and

β∈I

dimkExt¹_A(Sβ, Sα)≤1 for allα∈^S The Gabriel quiver ofA(S)has nodesS= {S_α : α ∈ I}, and dim_kExt¹_A(S_α, S_β)edges from nodeαto nodeβ. Gabriel’s criterion forA(S)to be uniserial is a condition on the Gabriel quiver, and we use it to classify and explicitly construct all indecomposable objects inA(S)when the length category is uniserial.

Letn≥2, and assume that simple objectsK1, . . . , KninSand extensionsτ2, . . . , τnin Aare given, such thatτ_i ∈Ext¹_A(K_i−1, K_i)for 2≤i ≤n. Ifn= 2, then it is clear that any extension ofK₁ byK₂is an iterated extension inExt(S)with simple factorsK₁, K₂ and simple extensionτ₂. Forn ≥ 3, we give a necessary and sufficient condition for the existence of an iterated extension inExt(S)with simple factorsK1, . . . , Kn and simple extensionsτ2, . . . , τn. The condition is given in terms of matrix Massey products in the sense of May, and we show that it can be interpreted as obstructions for lifting infinitesimal noncommutative deformation. In order to do this, we describe the close link between iterated extensions and noncommutative deformations, following Laudal [10].

Theorem LetS= {S_α :α ∈I}be a family of orthogonalk-rational points in an Abelian k-categoryA. IfA(S)is a uniserial length category, then the indecomposable objects in A(S)of lengthnare given by

{X(α):α∈Iⁿis admissible}

up to isomorphism inA(S). Moreover, an order vectorα∈Iⁿis admissible if and only if it satisfies the following conditions:

1. The order vectorαcorresponds to a path of lengthn−1in the Gabriel quiverΛ.

2. Whenn≥3, the matric Massey productτ₂, τ₃, . . . , τ_n ⊆Ext²_A(S_α1, S_αn)is defined and contains zero for all non-split extensionsτ_i ∈Ext¹_A(S_αi−1, S_αi).

WhenA(S)is hereditary, the obstructions vanish, and any path of lengthn−1 in the Gabriel quiver ofA(S)gives rise to an indecomposable object of lengthninA(S). Conse- quently, there is a bijective correspondence between indecomposable objects of lengthnin A(S)and paths of lengthn−1 in the Gabriel quiver ofA(S). This result is well-known;

see for instance Chen and Krause [3].

As an application, we show that the categorygrHol_Dof graded holonomicD-modules is uniserial whenD = Diff(A)is the ring of differential operators on a monomial curve Adefined over the fieldk =Cof complex numbers. Moreover, we classify all indecom- posable objects ingrHol_D. We build upon the results in Eriksen [6], where we studied this category. We obtain the most explicit result in the case when A = k[t]andD = A1(k) is the first Weyl algebra. The classification is similar in the other cases, since all rings of differential operators on monomial curves are Morita equivalent.

Theorem LetD = A1(k)be the first Weyl algebra. Then the categorygrHol_Dof graded holonomicD-modules is uniserial, and the indecomposableD-modules ingrHol_Dare, up to graded isomorphisms and twists, given by

M(α, n)=D/D (E−α)ⁿ, M(β, n)=D/Dw(β, n)

wheren≥1,α∈J^∗= {α∈k:0≤Re(α) <1, α=0},β∈ {0,∞}, and whereE=t∂

is the Euler derivation andw(β, n)is the alternating word onnletters intand∂, ending with∂ifβ=0, and intifβ= ∞.

This result resembles Boutet de Monvel’s classical result, giving all regular holonomic D-modules over the ringDof differential operators on the ringOof germs of holomorphic function onX=Caround 0.

We also include an example of a non-hereditary uniserial length category, and compute the obstructions. The simplest example isA=k[x]/(x²)with the familyS= {S}consisting of the unique simple leftA-modules, given byS = A/(x). We show that in this case, the obstructions do not vanish, and there are no indecomposable modules of lengthn≥3.

2 Iterated Extensions and Length Categories

Let kbe a field, letAbe an Abelian k-category, and letS = {Sα : α ∈ I}be a fixed family of non-zero, pairwise non-isomorphic objects inA. In this section, we define the categoryExt(S)ofiterated extensionsof the familyS, equipped with a forgetful functor Ext(S)→A(S)into the minimal full subcategoryA(S)⊆Athat containsSand is closed under extensions, and study its properties.

An object ofExt(S)is a couple(X, C), whereXis an object of the categoryAandCis a cofiltration ofXinAof the form

X=Cn f_n

−→C_n−1→ · · · →C2 f₂

−→C1 f₁

−→C0 =0

wherefi :Ci →C_i−1 is surjective andKi =ker(fi) ∼=Sα_i withαi ∈I for 1≤i ≤n.

The integern ≥0 is called thelength, the objectsK1, . . . , Kn are called thefactors, and the vectorα=(α₁, . . . , α_n)is called theorder vectorof the iterated extension(X, C).

Let(X, C)and(X, C)be a pair of objects inExt(S)of lengthsn, n≥0. A morphism φ:(X, C)→(X, C)inExt(S)is a collection{φ_i :0≤i≤N}of morphismsφ_i:C_i → C_iinAsuch thatφ_i−1fi =f_iφifor 1≤i ≤N, whereN =max{n, n}. By convention, Ci=Xfor alli > nandC_i=Xfor alli > n.

The categoryExt(S)has a dual category defined by filtrations. An object of this category is a couple(X, F ), whereXis an object ofAandFis a filtration ofXinAof the form

0=F_n⊆F_n−1⊆ · · · ⊆F₀=X

such thatK_i = F_i−1/F_i ∼= S_αi withα_i ∈ I for 1 ≤ i ≤ n. Given an object(X, F ) in the dual category, the corresponding object inExt(S) is(X, C), where the cofiltrationC is defined byCi = X/Fi for 0 ≤ i ≤ n, with the natural surjectionsfi : Ci → C_i−1. Conversely, if an object(X, C)inExt(S)is given, then the corresponding filtration ofX is given by Fi = ker(X → Ci) for 0 ≤ i ≤ n, whereX → Ci is the composition f_i+1◦ · · · ◦f_n:C_n→C_i. It is clear from the construction that the dual objects(X, C)and (X, F )have the same length, the same factors, and the same order vector.

As the name suggests, the categoryExt(S)can be characterized in terms of extensions.

In fact, for any object(X, C)inExt(S)of lengthnand for any integeri with 2≤i ≤n, the cofiltrationCinduces a commutative diagram

0 //K_i //C_i ^fi //C_i−1 //0

0 //Ki //

OO

Z f_i //

OO

Ki−1 //

OO

0

inA, where the rows are exact andZ=f_i⁻¹(K_i−1). We defineξi ∈Ext¹_A(C_i−1, Ki)and τ_i ∈ Ext¹_A(K_i−1, K_i)to be the extensions corresponding to the upper and lower row, and callτ2, τ3, . . . , τnthesimple extensionsof(X, C). By construction,ξi →τiunder the map Ext¹_A(Ci−1, Ki)→Ext¹_A(Ki−1, Ki)induced by the inclusionKi−1 ⊆Ci−1. In particular, C2is an extension ofC1=K1byK2,C3is an extension ofC2byK3, and in general,C_i+1

is an extension ofC_ibyK_i+1for 1≤i ≤n−1. It follows thatX=C_nis obtained from the factors{K₁, . . . , K_n} ⊆^Sby an iterated use of extensions, and this justifies the name iterated extensions.

Let us consider the natural forgetful functorExt(S) →Agiven by(X, C)→ X, and the full subcategoryA(S) ⊆ Adefined in the following way: An objectXinAbelongs toA(S)if there exists a cofiltrationCofXsuch that(X, C)is an object ofExt(S). The following lemma proves thatA(S)⊆Ais the minimal full subcategory that containsSand is closed under extensions:

Lemma 1 Let(X, C), (X, C)be iterated extensions of the familyS. IfXis an extension ofXbyXinA, then there is a cofiltrationCofXsuch that(X, C)is an iterated extension of the familyS. In particular, the full subcategoryA(S)⊆Ais closed under extensions.

Proof Let us assume that(X, C)and(X, C)are iterated extensions of the familySof lengthsn, n. SinceXis an extension ofXbyX, we can construct a cofiltration ofXof lengthn=n+nin the following way: Letf :X→Xandg :X→Xbe the maps given by the extension 0 →X →X → X → 0, letFbe the filtration ofXdual to the cofiltrationC, and letFbe the filtration ofXdual to the cofiltrationC. We define F_i =g⁻¹(F_i)for 0≤i ≤n, andF_i =f (F_i−n )forn≤i ≤n. ThenF is a filtration ofX, and we have thatF_i−1/Fi ∼=ker(X→C_i−1 )/ker(X→C_i)∼=K_ifor 0≤i ≤n, and also thatFi−1/Fi ∼= K_i−n forn ≤ i ≤ n. LetC be the cofiltration ofX dual to the filtrationF. Then it follows by construction that(X, C)is an iterated extension of the familySof lengthn.

We recall thatA(S)⊆Ais called anexact Abelian subcategoryif the inclusion functor A(S)→Ais an exact functor. It is well-known that this is the case if and only ifA(S)is closed inAunder kernels, cokernels and finite direct sums. It is clear thatA(S)is closed under finite direct sums since it closed under extensions. But in general, it is not closed under kernels and cokernels.

Proposition 2 The full subcategoryA(S)⊆Ais an exact Abelian subcategory if and only if the following conditions hold:

1. End_A(S_α)is a division algebra for allα∈I 2. Mor_A(Sα, S_β)=0for allα,β∈Iwithα=β

If this is the case, thenSis the set of simple objects inA(S), up to isomorphism.

Proof This follows from Theorem 1.2 in Ringel [13], and the preceding comments.

We use the notation from Ringel [13], and say that an object X in A is a point if End_A(X)is a division ring, and that two pointsX, YinAareorthogonalif Mor_A(X, Y )= Mor_A(Y, X)=0. We callXak-rational pointif End_A(X)=k.

Alength categoryis an Abelian category such that any of its objects has finite length, and such that the isomorphism classes of objects form a set. IfSis a family of orthogonal points in an Abeliank-categoryA, then it follows from Proposition 2 thatA(S)is a length category, with Sas its simple objects. In fact, any length category which is an Abelian k-category is of this type.

3 Obstructions for Existence of Iterated Extensions

In this section, we assume thatAis the categoryModAof right modules over an associative k-algebraA, and letS = {S_α : α ∈ I} be a family of orthogonal points inMod_A. Let n ≥ 2 be an integer, and fix simpleA-modules K₁, . . . , K_n in Sand extensions τ_i ∈ Ext¹_A(K_i−1, K_i)for 2 ≤ i ≤ n. Ifn= 2, then any extension ofK₁byK₂ is an iterated extension inExt(S)withK1, K2as its simple factors andτ2as its simple extension. Forn≥ 3, there are obstructions, and we give a necessary and sufficient condition for the existence of an iterated extension inExt(S)withK1, . . . , K_nas its simple factors andτ2, . . . , τ_nas its simple extensions. These obstructions are expressed in terms ofmatric Massey products;

see May [11].

We writeK= ⊕K_i, and consider the Hochschild complex HC^•(A,End_k(K))ofAwith values in Endk(K)as a DGA (differential graded algebra) overk^r. It has a decomposition

HCⁿ(A,End_k(K))= ⊕

i,j

HCⁿ(A,Hom_k(Ki, Kj))

A 1-cochain in this DGA is ak-linear mapα :A→End_k(K), and it is a 1-cocycle if and only if it is a derivation. Multiplication of the 1-cochainsα,βin the DGA is defined by the compositionα·β= {(a, b)→β(b)◦α(a)}. It is well-known that its cohomology is given by

HHⁿ(A,End_k(K))= ⊕

i,j

HHⁿ(A,Hom_k(K_i, K_j))∼= ⊕

i,j

Extⁿ_A(K_i, K_j)

In particular, Ext¹_A(K_i, K_j)∼=Der_k(A,Hom_k(K_i, K_j))/IDer_k(A,Hom_k(K_i, K_j)), where IDerk(−,−)denotes the inner derivations.

We choose a derivationαi−1,i:A→Homk(Ki−1, Ki)that representsτi in Hochschild cohomology for 2≤ i ≤n. Thecup productτ2∪τ3 = τ2, τ3is the cohomology class ofα₁₂·α₂₃. It is also called a second order matric Massey product. If the cup-products τ₂, τ₃ = τ₃, τ₄ =0, then there are 1-cochainsα₁₃andα₂₄such that

d(α₁₃)=α₁₂·α₂₃ and d(α₂₄)=α₂₃·α₃₄

In that case,α = {α₁₂, α₂₃, α₃₄, α₁₃, α₂₄}is called adefining systemfor the third order matric Massey productτ₂, τ₃, τ₄, and the cohomology class of

α₁₄=α₁₃·α₃₄+α₁₂·α₂₄

is the corresponding value ofτ2, τ3, τ4. Notice that this cohomology class may depend on the defining system. Higher order matric Massey products are defined similarly:

Definition A defining system for the matric Massey productτ2, τ3, . . . , τnis a family α= {αij:1≤i < j ≤n, (i, j )=(1, n)}

of 1-cochainsαij :A →Hom_k(Ki, Kj)such thatα_i−1,i is a 1-cocycle that representsτi

for 2≤i≤n, and such that

d(αij)=αij, withαij =

j−1

l=i+1

αil·αlj

whenj−i >1. The matric Massey productτ2, τ3, . . . , τnis defined if it has a defining system. In that case,τ2, τ3, . . . , τnis the collection of cohomology classes represented by

α_1n=

n−1

l=2

α_1,l·α_l,n

for some defining systemα.

LetE₂ be a rightA-module that is an extension ofK₁byK₂, such that there is a short exact sequence 0→K₂ →E₂ →K₁ →0. Then it is well-known thatE₂ ∼=K₂⊕K₁ considered as a vector space overk, and that the right action ofAis given by

(m₂, m₁)·a=(m₂·a+ψ_a¹²(m₁), m₁·a)

whereψ¹²:A→Hom_k(K₁, K₂)is ak-linear map. Since the action ofAmust be associa- tive,ψ¹²must be a derivation. In fact, it is a derivation that represents the extensionτ₂.

LetE₃be a rightA-module that is an extension ofE₂ byK₃, such that there is a short exact sequence 0 → K3 → E3 → E2 → 0. ThenE3 ∼= K3 ⊕E2 ∼= K3⊕K2⊕K1

considered as a vector space overk, and the right action ofAis given by

(m3, m2, m1)·a=(m3·a+ψ_a²³(m2)+ψ_a¹³(m1), m2·a+ψ_a¹²(m1), m1·a) whereψⁱ³ : A → Homk(Ki, K3)is ak-linear map fori = 1,2. Since the action ofA must be associative,ψ²³must be a derivation (representing the extensionτ3), andψ¹³must satsify

−d(ψ¹³)=ψ₁₃, withψ₁₃=ψ¹²·ψ²³

such that the cup productτ₂∪τ₃ = 0. It follows by an inductive argument that in the cofiltration

E=En f_n

−→E_n−1 → · · · →E2 f₂

−→E1 f₁

−→E0=0

we have thatE=En∼=Kn⊕ · · · ⊕K2⊕K1considered as a vector space overk, with right action ofAgiven by

(m_n, . . . , m₂, m₁)·a=(m_n·a+

n−1

i=1

ψ_aⁱⁿ(m_i), . . . , m₂·a+ψ_a¹²(m₁), m₁·a) whereψ^ij : A →Hom_k(K_i, K_j)is a 1-cochain for 1≤ i < j ≤n. The condition that these cochains must satisfy for the action ofAto be associative, is that

−d(ψ^ij)=ψij, withψij=

j−1

l=i+1

ψ^il·ψ^lj

In other words, the familyα = {αij : 1 ≤i < j ≤n, (i, j ) = (1, n)}given byαij = (−1)^j−i+1ψ^ijis a defining system for the matric Massey product

τ2, τ3, . . . , τn

Moreover, the cohomology class of α1n is zero, sinceα1n = d(α1n). This proves the following result:

Proposition 3 Letn ≥ 3be an integer, letK1, . . . , Kn be simpleA-modules in S, and letτ2, . . . , τnbe extensions ofA-modules withτi ∈ Ext¹_A(Ki−1, Ki). There is an iterated extension inExt(S)withK₁, . . . , K_nas its simple factors and withτ₂, . . . , τ_nas its simple extensions if and only if the matric Massey productτ₂, . . . , τ_nis defined and contains zero.

4 Iterated Extensions and Noncommutative Deformations

Let S = {S_α : α ∈ I} be a family of orthogonal points in an Abeliank-category A, and letA(S) be the corresponding length category inA. In this section, we consider the noncommutative deformations of finite subfamilies ofS, and show that they determine the iterated extensions inExt(S).

Let(X, C)be an iterated extension inExt(S)with order vectorα. We define theexten- sion type of (X, C) to be the ordered quiver with nodes{α₁, α₂, . . . , α_n} and edges γ_i−1,i :α_i−1 →αi for 2≤i ≤n. This quiver is ordered in the sense that there is a total orderγ12 < γ23 < · · ·< γn−1,n on the edges in. The extension typeis determined by the order vectorα, and isomorphic iterated extensions have the same extension type. We denote byE(S, )the set of isomorphism classes of iterated extensions of the familySwith extension type.

LetS()⊆^Sbe the set of simple factors of an iterated extension with extension type. To fix notation, we shall writeS()= {X1, . . . , Xr}. This means that for 1 ≤j ≤r, we have thatXj =Sα_ifor at least one value ofi. Hencer≤n, withr < nif there are repeated factors.

We define thepath algebrak[]of the ordered quiverto be thek-algebra with base consisting of pathsγ_i−1,i ·γ_i,i+1· · · · ·γ_j−1,j of lengthj −i +1 for 2 ≤ i ≤ j ≤ n, and with the following multiplicative strcuture: The product of two pathsγ·γis given by juxtaposition when the last arrowγj−1,j in the first pathγ is the predecessor of the first arrowγj,j+1in the second pathγin the total ordering, and otherwise the productγ·γ=0.

We considere_i as a path of length 0.

For example, an iterated extension of lengthn = 3 with non-isomorphic simple fac- torsX1 = Sα1,X2 = Sα2, andX3 = Sα3 hasr = 3, and its extension typehas path algebra

k[] =

⎛

⎝k e1 k γ12 k γ12γ23

0 k e₂ k γ₂₃

0 0 k e₃

⎞

⎠∼=

⎛

⎝k k k 0 k k 0 0 k

⎞

⎠

We remark that the path algebrak[]of any extension typeis an object of the category a_r of Artinianr-pointed algebras. Recall that an Artinianr-pointed algebra is an Artinian k-algebra R with r simple modules fitting into a diagram k^r → R → k^r, where the composition is the identity.

Let us consider the noncommutative deformation functorDefS() : ^ar → ^Setsof the familyS() = {X₁, . . . , X_r}, which is defined on the categorya_r. We refer to Eriksen et al. [7] for details of noncommutative deformations. We shall study the deformations in DefS()(k[]). Without loss of generality, we may assume that the Abeliank-categoryAis the category of right modules over an associativek-algebraA. We remark that noncommu- tative deformations can be computed directly in many other Abeliank-categories as well, see for instance Eriksen [8] and Eriksen et al. [7].

Proposition 4 For any extension type, there is a bijective correspondence between the noncommutative deformations inDefS()(k[])and the setE(S, )of equivalence classes of iterated extensions of the familySwith extension type.

Proof Letα be the order vector corresponding to the extension type, and lets be the unique index such thatSα₁ =Xs. Any noncommutative deformationX ∈^DefS()(k[]) has the formX = (k[]ij ⊗kX_j)as a leftk[]-module by flatness, with a right multi- plication ofA, and we consider the rightA-submoduleX(s)=e_s·X ⊆X. A path in e_s·k[]is calledleadingif it has the formγ₁₂γ₂₃· · ·γ_i−1,iandnon-leadingotherwise. By convention, we consider the pathesas leading, and define

X^{N L} (s)= ⊕

γ γ·X⊆X(s)

where the sum is taken over all non-leading pathsγ ine_s·k[]. Notice thatX^{N L}(s) ⊆ X(s)is closed under right multiplication byA. We defineXto be the right A-module X=X(s)/X^{N L}(s). As ak-linear space, we have that

X∼= ⊕

1≤i≤n

γ12γ23. . . γ_i−1,i ⊗kSα_i

withS_αi =X_lfor somelwith 1 ≤l ≤r, and we claim that there is a cofiltrationCofX such that(X, C)is an iterated extension ofSwith extension type. In fact, we may choose the cofiltrationCdual to the filtrationFgiven by

Fj = ⊕

j+1≤i≤n

γ12γ23. . . γ_i−1,i ⊗kSα_i

for 0 ≤ j ≤ n, whereF_j ⊆ Xis closed under right multiplication withA. Conversely, assume that an iterated extensions (X, C) ofSwith extension type is given. Then it follows from Proposition 3 that the matric Massey productτ2, τ3, . . . , τnof its simple extensions is defined and contains zero. Hence, there is a defining systemα= {α_ij :1≤ i < j ≤n, (i, j )=(1, n)}for this matric Massey product such that the cohomology class ofα_1nis zero. By the construction preceding Proposition 3,Xhas the following form: As ak-linear vector space,X∼= Kn⊕K_n−1⊕ · · · ⊕K2⊕K1withKi =Sαi, and the right multiplication ofAis given by

(mn, . . . , m2, m1)·a=(mn·a+

n−1

i=1

ψ_aⁱⁿ(mi), . . . , m2·a+ψ_a¹²(m1), m1·a) form_i ∈K_i, a∈A, whereψ^ij =(−1)^j−i+1α_ij. This defines a right multiplication ofA onX=(k[]ij⊗kX_j), and therefore a noncommutative deformationXinDefS()(k[]).

In fact, the right multiplication is given by (el⊗ml)·a=el⊗(ml·a)+

i∈Il

i+1≤j≤n

γ_i,i+1γ_i+1,i+2. . . γ_j−1,j ⊗ψa^ij(m)

for 1≤l≤r, a∈A, m_l∈X_l, withI_l= {i :α_i=l}.

We say thatS()is aswarmif dimkExt¹_A(Xi, Xj)is finite for 1≤i, j ≤r. We shall assume that this is the case in the rest of this section. In this case, the noncommutative defor- mation functorDefS() has a miniversal object(H, X_H), consisting of a pro-representing hull H ofDefS() in the pro-category^ar, and a versal family X_H ∈ ^DefS()(H ); see Eriksen et al. [7] for details. We write X(S, ) = Mor(H, k[]) for the set of mor- phismsφ :H →k[]inar. There is a natural mapX(S, )→^DefS()(k[])given by φ→^DefS()(φ)(XH), and it is surjective by the versal property.

Lemma 5 The setX(S, )=Mor(H, k[])is an affine algebraic variety.

Proof Sincek[]is an algebra inar, and its radicalI (k[])satisfy I (k[])ⁿ = 0, any morphismφ:H →k[]inarcan be identified withφn:Hn→k[]sinceφ(I (H )ⁿ)=0.

To prove thatX(S, )=Mor(H_n, k[])is an affine algebraic variety, it is enough to notice thatH_nis a quotient of T¹_n, that Mor(T¹_n, k[])is isomorphic to affine spaceA^N, where

N=

i,j

dimkExt¹_A(Xi, Xj)·dimk

I (k[])/I (k[])²

ij

and that Mor(Hn, k[]) ⊆ Mor(T¹_n, k[])is a closed subset in the Zariski topology, with equations given by the obstructionsf_ij(l)ⁿ∈T¹_ndefiningH.

Corollary 6 The setE(S, )of equivalence classes of iterated extensions of the familyS with extension typeis a quotient of the affine algebraic varietyX(S, ).

5 Uniserial Length Categories

LetSbe a family of orthogonalk-rational points in an Abeliank-categoryA, and letA(S) be the corresponding length category. We say that an objectXinA(S)isuniserialif its lattice of subobjects is a chain. If this is the case, then this chain is the unique decomposi- tion series ofX. It follows thatXis uniserial if and only if any two cofiltrations ofXare isomorphic. Any uniserial object inA(S)is indecomposable, but the opposite implication does not hold in general. We say thatA(S)is auniserial categoryif every indecomposable object inA(S)is uniserial.

Theorem 7(Gabriel) LetS= {Sα :α ∈ I}be a family of orthogonalk-rational points in an Abeliank-categoryA. ThenA(S)is uniserial if and only if the following conditions hold:

β∈I

dimkExt¹_A(Sα, S_β)≤1 and

β∈I

dimkExt¹_A(S_β, Sα)≤1 for allα∈^S

Proof The result is due to Gabriel; see Gabriel [9] and Amdal, Ringdal [1]. A readable proof, due to Yu Ye, appears in Chen and Krause [3].

We denote bythe Gabriel quiver ofA(S), which has the objects inSas nodes, indexed by I, and dimkExt¹_A(Sα, S_β)arrows from nodeα to node β. Note that the condition in Theorem 7 is a condition on the Gabriel quiver. We shall assume that this condition is satisfied in the rest of this section, such thatA(S)is a uniserial length category. Under this assumption, we shall classify and explicitly construct all indecomposable objects inA(S).

Let(X, C)be an object ofExt(S) of lengthnand with order vectorα such thatXis uniserial. Then it follows from Lemma 1.7.2 and the proof of Theorem 1.7.1 in Chen and Krause [3] that eachCi in the cofiltration

X=Cn f_n

−→C_n−1→ · · · →C2 f₂

−→C1 f₁

−→C0 =0

is uniserial, and that the natural map Ext¹_A(C_i−1, K_i) → Ext¹_A(K_i−1, K_i)is an isomor- phism for 2≤i ≤n. This means in particular thatτi ∈Ext¹_A(K_i−1, Ki)=0 for 2≤i ≤n.

Hence the order vectorαcorresponds to a path

α1→α2→ · · · →αn

of lengthn−1 in the Gabriel quiver. Furthermore, if(X, C)is another object ofExt(S) with the same order vectorαsuch thatXis uniserial, thenX∼=XinA. This follows from Lemma 1.7.3 in Chen and Krause [3].

Letα ∈ Iⁿ be an order vector of lengthn. We say thatα isadmissibleif there is an iterated extension(X, C)inExt(S)with order vectorαsuch thatXis uniserial. The com- ments above shows that ifαis admissible, then it corresponds to a path of lengthn−1 in the Gabriel quiver.

It also follows from the comments above that ifα is an admissible order vector, then there is a unique uniserial objectXinA(S)with the property that it admits a cofiltrationC such that(X, C)is an iterated extension inExt(S)with order vectorα. We shall writeX(α) for this uniserial object.

Conversely, letα∈Iⁿbe an order vector corresponding to a path of lengthn−1 in the Gabriel quiver. Ifn≥3, then there are obstructions forαto be admissible. In fact, we have that Ext¹_A(Sα_i−1, Sα_i) =0 for 2 ≤i ≤n, and we may choose non-split extensions

τ_i ∈ Ext¹_A(S_αi−1, S_αi), which are unique up to scalars ink^∗. It follows from Proposition 3 that the order vectorαis admissible if and only if the matric Massey product

τ2, τ3, . . . , τ_n ⊆Ext²_A(S_α1, S_αn)

is defined an contains zero. The vanishing of these matric Massey products is clearly independent of the choice of non-split extensionsτi.

Notice that the condition of Proposition 3 can be interpreted as obstructions for lifting noncommutative deformations. In fact, let us writefor the extension type corresponding toα, and let I ⊆ k[]be the Jacobson radical of the path algebra. Then the extensions τ₂, . . . , τ_ndefines an infinitesimal noncommutative deformation inDefS()(k[]/I²), and αis admissible if and only if this infinitesimal deformation can be lifted to the path algebra k[].

Theorem 8 LetS= {S_α :α∈I}be a family of orthogonalk-rational points in an Abelian k-categoryA. IfA(S)is a uniserial length category, then the indecomposable objects in A(S)of lengthnare given by

{X(α):α∈Iⁿis admissible}

up to isomorphism inA(S). Moreover, an order vectorα∈Iⁿis admissible if and only if it satisfies the following conditions:

1. The order vectorαcorresponds to a path of lengthn−1in the Gabriel quiver. 2. Whenn≥3, the matric Massey productτ2, τ3, . . . , τn ⊆Ext²_A(Sα1, Sαn)is defined

and contains zero for all non-split extensionsτi ∈Ext¹_A(Sα_i−1, Sα_i).

We note that if αis admissible, thenX(α)can be explicitly constructed from its sim- ple factors and the simple extensionsτi ∈Ext¹_A(Sα_i−1, Sαi). The construction is given by Proposition 3.

We say thatA(S)is ahereditary length categoryif Ext²_A(S, T )=0 for any objectsS, T inS. If this is the case, then the obstruction in Proposition 3 vanishes, and it follows that there is a bijective correspondence between paths of lengthn−1 in the Gabriel quiver and indecomposable objects inA(S).

6 Graded Holonomic D-Modules on Monomial Curves

Let ⊆ N0 be a numerical semigroup, generated by positive integersa1, . . . , ar without common factors, and let A = k[] ∼= k[t^a1, . . . , t^ar]be its semigroup algebra over the field k = C of complex numbers. We callA an affine monomial curve, and have that X=Spec(A)= {(t^a1, t^a2, . . . , t^ar):t ∈k} ⊆A^rk.

We studied the positively graded algebraD of differential operators on the monomial curve A = k[]in Eriksen [5], and the categorygrHol_D of graded holonomic leftD- modules in Eriksen [6]. We recall that anyD-moduleMsatisfies the Bernstein inequality d(M) ≥1, thatMis holonomic ifd(M)= 1, and that this condition holds if and only if M has finite length; see Proposition 4 and Proposition 5 in Eriksen [6]. This implies that grHol_Dis a length category, and its simple objects are given by

{M0[w] :w∈Z} ∪ {Mα[w] :α∈J^∗, w∈Z} ∪ {M_∞[w] :w∈Z}

whereJ^∗= {α ∈C:0≤Re(α) <1, α=0}; see Theorem 10 in Eriksen [6]. Moreover, the graded extensions of the simple objects are given by

Ext¹_D(Mα[w], M_β[w])0=

⎧⎨

⎩

kξ ∼=k, (α,β)=(0,∞), (∞,0)andw=w kξ ∼=k, α=β∈J^∗andw=w

0, otherwise

for simple gradedD-modulesMα[w], M_β[w]withα,β∈J^∗∪ {0,∞}andw, w∈Z; see Proposition 12 in Eriksen [6].

Proposition 9 The familyS = {M_α[w] : α ∈ J^∗∪ {0,∞}, w ∈ Z}is the family of simple objects ingrHol_D, and it is a family of orthogonalk-rational points that satisfies the condition in Theorem 7. In particular, the categorygrHol_Dof graded holonomicD-modules is a uniserial category.

Proof Sincek=Cis algebraically closed, it follows from the main theorem in Quillen [12]

that EndD(Mα[w])=kfor allα ∈J^∗∪ {0,∞}, w ∈Z. Moreover, the comments above show thatSis the family of simple objects ingrHol_D, and therefore a family or orthogonal k-rational points, which satisfies the condition in Theorem 7.

It is, in principle, possible to construct all indecomposable objects ingrHol_Dusing the constructive proof of Theorem 8. As an illustration, we shall classify the indecomposable objects in the caseA=k[t], which is the unique smooth monomial curve. The classification would be similar in the other cases, since all rings of differential operators on monomial curves are Morita equivalent. However, the indecomposable objects would be defined by more complicated equations in the singular cases.

Note that whenA = k[t], the ringD of differential operators onA is the first Weyl algebraA1(k)=k[t]∂, with generatorstand∂ =d/dt, and relation[∂, t] = 1. Let us writeE = t∂for the Euler derivation inD. The simple objects ingrHol_D, up to graded isomorphisms and twists, are given byM₀=D/D∂,M_∞=D/DtandM_α=D/D(E−α) forα∈J^∗.

Theorem 10 Let D = A₁(k) be the first Weyl algebra. The indecomposable graded holonomicD-module, up to graded isomorphisms and twists, are given by

M(α, n)=D/D (E−α)ⁿ, M(β, n)=D/Dw(β, n)

wheren≥1,α∈J^∗,β∈ {0,∞}, andw(β, n)is the alternating word onnletters intand

∂, ending with∂ifβ=0, and intifβ= ∞.

Proof Let us writeI =J^∗∪ {0,∞}, such thatS= {M_α[w] :(α, w)∈I×Z}is the family of simple objects ingrHol_D. It follows from the computation of the graded extensios above that for any lengthn≥1 and any(α, w)∈I×Z, there is a unique path

(α, w)=(α1, w1)→(α2, w2)→ · · · →(αn, wn)

insuch that Ext¹_D(M(α_i−1)[w_i−1], M(α_i)[w_i])₀ =0 for 2≤i≤n. The corresponding vector is admissible sinceD=A₁(k)is a hereditary graded ring; see for instance Coutinho