Iterated Extensions and Uniserial Length Categories

In this paper, we study length categories using iterated extensions. We consider the problem of classifying all indecomposable objects in a length category, and the problem of characterizing those length categories that are uniserial. We solve the last problem, and obtain a necessary and sufficient criterion for uniseriality under weak assumptions. This criterion turns out to be known by Amdal and Ringdal already in 1968; we give a new proof that is both elementary and constructive. The first problem is the most fundamental one, and its general solution is"the main and perhaps hopeless purpose of representation theory"according to Gabriel. We solve the problem in the case when the length category is uniserial, using our constructive methods. As an application, we classify all graded holonomic $D$-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when $D$ is the first Weyl algebra. Finally, we show that the iterated extensions are completely determined by the noncommutative deformations of its simple factors. This tells us precisely what we can learn about a length category by studying its species; it gives the tangent space of the noncommutative deformation functor, or the infinitesimal deformations, but not the obstructions for lifting these deformations.


Introduction
Let S = {S α : α ∈ I} be a family of non-zero, pairwise non-isomorphic objects in an Abelian k-category A, where k is a field. We consider the minimal full subcategory A(S) ⊆ A that contains S and is closed under extensions. The family S is called a family of orthogonal points if End(S α ) is a division algebra and Hom(S α , S β ) = 0 for all α, β ∈ I with α = β. In this case, A(S) ⊆ A is a length category with S as its simple objects.
We use the category Ext(S) of iterated extensions of S to study the length category A(S). An iterated extension of S is a couple (X, C) where X is an object in A, and C is a cofiltration X = C n fn −→ C n−1 → · · · → C 2 f2 − → C 1 f1 − → C 0 = 0 where f i : C i → C i−1 is surjective and K i = ker(f i ) ∼ = S α(i) with α(i) ∈ I for 1 ≤ i ≤ n. Hence the assignment (X, C) → X defines a forgetful functor Ext(S) → A(S). When we work with the category Ext(S) of iterated extensions, the order vector α = (α(1), . . . , α(n)) ∈ I n is an invariant, in addition to the usual invariants in the length category A(S) such as the length n, the simple factors {K 1 , . . . , K n }, and their multiplicities.
An important special case is when A = Mod A is the category of modules over an associative k-algebra A, and S is a subset of the simple A-modules. If S is the family of all simple modules, then A(S) is the category of all modules of finite length. There are also many other interesting applications, for example when A is the category of graded modules over a graded k-algebra, or the category of coherent sheaves over a k-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 such an embedding.
We say that A(S) is a uniserial length category if any indecomposable object in A(S) has a unique composition series, and that at point S in S is k-rational if End(S) = k. When S is a family of k-rational orthogonal points, we show that A(S) is a uniserial length category if and only if the family S satisfies the condition β∈I dim k Ext 1 A (S α , S β ) ≤ 1 for all α ∈ I (UC) α∈I dim k Ext 1 A (S α , S β ) ≤ 1 for all β ∈ I It turns out that the condition (UC) and the characterization of uniserial length categories was known already in the 60's; see Section 8.3 in Gabriel [7]. As far as we know, it first appeared in Amdal, Ringdal [1], where it is stated without proof. We give an elementary and constructive proof of the result that A(S) is uniserial if and only if (UC) is satisfied, using the category Ext(S) of iterated extensions. In fact, after showing that the condition is necessary, we explicitly construct all indecomposable objects in A(S) when (UC) holds, and prove that these objects are uniserial.
Theorem. Let S = {S α : α ∈ I} be a family of orthogonal k-rational points in an Abelian k-category A. If S satisfies (UC), then the indecomposable objects in A(S) of length n are given by {X(α) : α ∈ J }, up to isomorphism in A(S), where the subset J ⊆ I n consists of the vectors α such that the following conditions hold: (i) Ext 1 A (S α(i−1) , S α(i) ) = 0 for 2 ≤ i ≤ n (ii) If σ i ∈ Ext 1 A (S α(i−1) , S α(i) ) is non-zero for 2 ≤ i ≤ n, then the matric Massey product σ 2 , σ 3 , . . . , σ n is defined and contains zero. Moreover, the indecomposable objects X(α) are uniserial, and can be constructed from the family S and their extensions.
As an application, we show that the category grHol D of graded holonomic D-modules is uniserial when D = Diff(A) is the ring of differential operators on a monomial curve A defined over the field k = C of complex numbers. Moreover, we classify all indecomposable objects in grHol D . We build upon the results in Eriksen [4], where we studied this category. We obtain the most explicit result in the case when A = k[t] and D = A 1 (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.
In the last section, we prove that for a swarm S of orthogonal points in an Abelian k-category A, the iterated extensions of the family S are completely determined by the noncommutative deformations of its simple factors. Hence the length category A(S) is also determined by these deformations. If the noncommutative deformations are unobstructed, then they are determined by the species of A(S). This is the case for modules over a hereditary ring, such as the ring D of differential operators on a monomial curve over the complex numbers. In general, we need both the species of A(S), which defines the noncommutative deformations on the tangent level, and the obstructions for lifting these deformations, to determine the iterated extensions in Ext(S).

Iterated extensions
Let k be a field, let A be an Abelian k-category, and let S = {S α : α ∈ I} be a fixed family of non-zero, pairwise non-isomorphic objects in A. In this section, we define the category Ext(S) of iterated extensions of the family S, equipped with a forgetful functor Ext(S) → A(S) into the minimal full subcategory A(S) ⊆ A that contains S and is closed under extensions, and study its properties.
An object of Ext(S) is a couple (X, C), where X is an object of the category A and C is a cofiltration of X in A of the form The integer n ≥ 0 is called the length, the objects K 1 , . . . , K n are called the factors, and the vector α = (α 1 , . . . , α n ) is called the order vector of the iterated extension (X, C).
Let (X, C) and (X ′ , C ′ ) be a pair of objects in Ext(S) of lengths n, n By convention, C i = X for all i > n and C ′ i = X ′ for all i > n ′ . The category Ext(S) has a dual category defined by filtrations. An object of this category is a couple (X, F ), where X is an object of A and F is a filtration of X in A of the form Given an object (X, F ) in the dual category, the corresponding object in Ext(S) is (X, C), where the cofiltration C is defined by C i = X/F i for 0 ≤ i ≤ n, with the natural surjections f i : C i → C i−1 . Conversely, if the object (X, C) in Ext(S) is given, then the corresponding filtration of X is given by F i = ker(X → C i ) for 0 ≤ i ≤ n, where X → C i 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.
We recall that a short exact sequence 0 → Y → Z → X → 0 in A is called an extension of X by Y , and that Ext 1 A (X, Y ) denotes the set of all extensions of X by Y , modulo equivalence. The set Ext 1 A (X, Y ) has a natural End A (Y )-End A (X) bimodule structure, inherited from the bimodule structure on Hom A (X, Y ).
As the name suggests, the category Ext(S) can be characterized in terms of extensions. In fact, for any object (X, C) in Ext(S) of length n and for any integer i with 2 ≤ i ≤ n, the cofiltration C induces a commutative diagram where the rows are exact and Z = f −1 i (K i−1 ). We define ξ i ∈ Ext 1 A (C i−1 , K i ) and τ i ∈ Ext 1 A (K i−1 , K i ) to be the extensions corresponding to the upper and lower row. By construction, ξ i → τ i under the map Ext 1 In particular, C 2 is an extension of C 1 = K 1 by K 2 , C 3 is an extension of C 2 by K 3 , and in general, C i+1 is an extension of C i by K i+1 for 1 ≤ i ≤ n − 1. It follows that X = C n is obtained from the factors {K 1 , . . . , K n } ⊆ S by an iterated use of extensions, and this justifies the name iterated extensions.
Let us consider the natural forgetful functor Ext(S) → A given by (X, C) → X, and the full subcategory A(S) ⊆ A defined in the following way: An object X in A belongs to A(S) if there exists a cofiltration C of X such that (X, C) is an object of Ext(S). The following lemma proves that A(S) ⊆ A is the minimal full subcategory that contains S and is closed under extensions: Lemma 1.1. Let (X ′ , C ′ ), (X ′′ , C ′′ ) be iterated extensions of the family S. If X is an extension of X ′ by X ′′ in A, then there is a cofiltration C of X such that (X, C) is an iterated extension of the family S. In particular, the full subcategory A(S) ⊆ A is closed under extensions.
Proof. Let us assume that (X ′ , C ′ ) and (X ′′ , C ′′ ) are iterated extensions of the family S of lengths n ′ , n ′′ . Since X is an extension of X ′′ by X ′ , we can construct a cofiltration of X of length n = n ′ + n ′′ in the following way: Let f : X ′ → X and g : X → X ′′ be the maps given by the extension 0 → X ′ → X → X ′′ → 0, let F ′ be the filtration of X ′ dual to the cofiltration C ′ , and let F ′′ be the filtration of X ′′ dual to the cofiltration C ′′ . We define for n ′′ ≤ i ≤ n. Let C be the cofiltration of X dual to the filtration F . Then it follows by construction that (X, C) is an iterated extension of the family S of length n.
We recall that A(S) ⊆ A is called an exact Abelian subcategory if the inclusion functor A(S) → A is an exact functor. It is well-known that this is the case if and only if A(S) is closed in A under kernels, cokernels and finite direct sums. It is clear that A(S) is closed under finite direct sums since it closed under extensions. But in general, it is not closed under kernels and cokernels. (i) End A (S α ) is a division algebra for all α ∈ I (ii) Mor A (S α , S β ) = 0 for all α, β ∈ I with α = β If this is the case, then S is the set of simple objects in A(S), up to isomorphism.
Proof. This follows from Theorem 1.2 in Ringel [10], and the comments preceding it. Let us use the notation from Ringel [10], and say that an object X in A is a point if End A (X) is a division ring, and that two points X, Y in A are orthogonal if Mor A (X, Y ) = 0 and Mor A (Y, X) = 0. Moreover, we shall write k(X) = End A (X) for the division algebra over k associated with a point X, and say that X is a k-rational point if k(X) = k.

Length categories
A length category is an Abelian category such that any of its objects has finite length, and such that the isomorphism classes of objects form a set. We recall some well-known facts about length categories; see for instance Gabriel [7]: (1) The Jordan-Hölder Theorem: Any object X in a length category has a composition series; that is, it has a filtration The length n and the simple factors K 1 , . . . , K n in a composition series are unique, up to a permutation of the simple factors.
(2) The Krull-Schmidt Theorem: Any object X in a length category is a finite direct sum X = X 1 ⊕ X 2 ⊕ · · · ⊕ X r of indecomposable objects. The indecomposable direct summands X 1 , . . . , X r are unique, up to a permutation.
(3) Mitchell's Embedding Theorem: A length category is exact equivalent to an exact subcategory of Mod A for an associative ring A. Let S be a family of orthogonal points in an Abelian k-category A. It follows from Proposition 1.2 that A(S) is a length category, with S as its simple objects. In fact, any length category which is an Abelian k-category is of this type.
Our goal is to classify and explicitly construct the indecomposable objects in the length category A(S). Even though this is a quite hopeless task in general, we prove a classification result in the special case of uniserial length categories in Section 3 and 4. Our philosophy is to start with the family S, and use iterated extensions in Ext(S) to build larger indecomposable modules.
The species of the length category A(S) consists of the family {k(S α ) : α ∈ I} of division algebras of its simple objects, and the family If S is a family of orthogonal k-rational points, the species of A(S) can be represented by a quiver Λ, with I as nodes, and with dim k Ext 1 A (S α , S β ) arrows from node α to node β for all α, β ∈ I. The quiver Λ (and more generally, the species) of the length category A(S) contains a lot of information about A(S) and its indecomposable objects.
In fact, we shall show in Section 6 that the iterated extensions in Ext(S) are completely determined by noncommutative deformations of its simple factors. In the unobstructed case, these deformations are determined by the species of A(S).

Uniserial length categories
Let S be a family of orthogonal k-rational points in an Abelian k-category A, and let A(S) be the corresponding length category. We denote by Λ the quiver of the species of A(S).
We say that an object X in A(S) is uniserial if its lattice of subobjects is a chain. If this is the case, then this chain is the unique decomposition series of X. It follows that X is uniserial if and only if any two cofiltrations of X are isomorphic. Any uniserial object in A(S) is indecomposable, but the opposite implication does not hold in general. We say that A(S) is a uniserial category if every indecomposable object in A(S) is uniserial. Lemma 3.1. Let X be an object in A(S), and consider the following conditions: In particular, all conditions are equivalent if and only if A(S) is a uniserial category.
for i = 1, 2 and this contradicts (2). The last part follows directly from the definition.
The implication (3) ⇒ (1) in Lemma 3.1 clearly holds if X has length n = 2, since an indecomposable object of length 2 is a non-split extension of two objects in S. But already for n = 3, it is easy to find examples where this implication fails: Proof. Notice that there exist non-split extensions U, V of S by T such that U and V are not isomorphic in A(S). In fact, if U, V are non-split extensions of S by T , then any isomorphism u : U → V satisfies u(T ) ⊆ T since T is the unique minimal subobject of U, V in A(S). Therefore, u induces automorphisms on T , and on S, which are given by multiplication in k * since S, T are k-rational points. This means that not all non-split extensions are isomorphic in A(S); otherwise, we would have that dim k Ext 1 We see that X has length n = 3, that U, V ⊆ X are subobjects in A(S) of length n = 2, and that T is the unique minimal subobject of X in A(S). In fact, if T ′ is another minimal subobject of X, then T ′ is not contained in U, V since they are uniserial. Hence U ⊕ T ′ = X = V ⊕ T ′ , and this implies that U is isomorphic to V , which is a contradiction. Since X has a unique minimal subobject in A(S), it is indecomposable, and it is non-uniserial since are different composition series of X. Proof. Let ξ 1 , ξ 2 = 0 be non-split extensions of U by S, and of U by T , given by short exact sequences to be the pullback of g 1 and g 2 . Then the short exact sequence . It is clear that S, T are minimal subobjects of X. Moreover, X is clearly indecomposable; otherwise, it would have S or T as a direct summand, and this is not possible since ξ 1 , ξ 2 = 0. Proof. Let ξ 1 , ξ 2 = 0 be non-split extensions of S by U , and of T by U , given by short exact sequences Define X be the push-out of f 1 and f 2 . Then the induced short exact sequence . Clearly, there are natural injections E 1 → X and E 2 → X, which gives the composition series Hence, X is not uniserial. Moreover, U is the unique minimal subobject of X since ξ 1 , ξ 2 = 0, and it follows that X is indecomposable.
Proposition 3.5. Let S = {S α : α ∈ I} be a family of orthogonal k-rational points in an Abelian k-category A, and let Λ be the quiver of the species of the length category A(S). If A(S) is uniserial, then following conditions hold: (i) For any α ∈ I, there is at most one arrow in Λ with source α.
(ii) For any β ∈ I, there is at most one arrow in Λ with target β.
Proof. It follows from Lemma 3.2, Lemma 3.3 and Lemma 3.4 that the length category A(S) is not uniserial if the quiver Λ contains a subquiver of one of the forms with β = β ′ in the middle quiver and α = α ′ in the right quiver.
The two conditions in Proposition 3.5 are equivalent to the following condition, which we call the uniseriality criterion and refer to as (UC): We claim that the length category A(S) is uniserial if and only if the condition (UC) holds. It turns out that this was known already in the 60's; see Section 8.3 in Gabriel [7]. As far as we know, this result first appeared in Amdal, Ringdal [1], where it is stated without proof. We shall give an elementary and constructive proof of this characterization in the next section. The proof is constructive in the sense that we classify and explicitly construct all indecomposable objects in A(S) when (UC) holds, and show that these indecomposable objects are uniserial.

Construction of indecomposable objects
Let S be a family of orthogonal k-rational points in an Abelian k-category A, and let A(S) be the corresponding length category. We denote by Λ the quiver of the species of A(S), and assume that the condition (UC) holds. In this situation, we shall classify and explicitly construct all indecomposable objects in A(S).
We consider the full subcategory Ext(S, n, * ) ⊆ Ext(S) of iterated extensions (X, C) of length n such that ξ 2 , . . . , ξ n = 0. Any indecomposable object X in A(S) of length n has a cofiltration C such that (X, C) is an iterated extension in Ext(S) with ξ n = 0. The idea is that many indecomposable objects, though not necessarily all, have a cofiltration C such that (X, C) is in Ext(S, n, * ), and we start by classifying these indecomposable objects.
induced by the inclusion K i−1 ⊆ C i−1 is an isomorphism for all integers i such that 2 ≤ i ≤ n and for all simple objects K ∈ S. In particular, τ i = 0 for 2 ≤ i ≤ n.
Proof. We show the result by induction on n. Since C 1 = K 1 by definition, the result is clearly true for n = 2. So let n ≥ 3, and assume that the result holds for all integers less than n and all simple objects K ∈ S. In particular, this implies that is an isomorphism. Since ξ n−1 → τ n−1 under this map when K = K n−1 , it follows that τ n−1 = 0, and in particular, that Ext 1 A (K n−2 , K n−1 ) = k · τ n−1 ∼ = k by (UC). Hence we also have Ext 1 A (C n−2 , K) = k · ξ n−1 ∼ = k. Let us consider the long exact sequence of the functor Hom A (−, K) applied to the extension ξ n−1 , given by For all simple objets K ∈ S, we claim that Hom A (K n−1 , K) ∼ = Ext 1 A (K n−2 , K) and that Hom A (K n−1 , K) → Ext 1 A (C n−2 , K) is an isomorphism: If K = K n−1 , we have that End A (K n−1 ) ∼ = k and that Ext 1 A (K n−2 , K n−1 ) = k · τ n−1 ∼ = k. This proves the claim, since Ext 1 A (C n−2 , K n−1 ) = k · ξ n−1 ∼ = k by the comments above and the identity on K n−1 maps to ξ n−1 by construction. If K is not isomorphic to K n−1 , we have that Hom A (K n−1 , K) = 0 and that Ext 1 A (K n−2 , K) = 0 by orthogonality and (UC). This proves the claim, since Ext 1 A (C n−2 , K) ∼ = Ext 1 A (K n−2 , K) = 0 by the induction hypothesis. We conclude that for all simple objects K ∈ S, the k-linear map is injective, and it is enough to show that it is an isomorphism to conclude the proof. If K = K n , then ξ n → τ n under this map, and by injectivity, it follows that τ n = 0 since ξ n = 0. Therefore, Ext 1 A (K n−1 , K n ) = k · τ n ∼ = k by (UC), and the map is an isomorphism. If K is not isomorphic to K n , then it follows from (UC) that Ext 1 A (K n−1 , K) = 0, and the map is an isomorphism also in this case.
We consider the map v n : Ext(S, n, * ) → I n , which maps an iterated extension (X, C) to its order vector α, and say that a vector α ∈ I n is admissible if α ∈ im(v n ). For any iterated extension (X, C) in Ext(S, n, * ), it follows from Lemma 4.1 that τ i = 0 for 2 ≤ i ≤ n, and that This means that if α is admissible, and (X, C), (X ′ , C ′ ) are two iterated extensions in Ext(S, n, c) with order vector α, then X ∼ = X ′ in A(S). In fact, we have that ξ ′ i = c i ξ i for 2 ≤ i ≤ n with c i ∈ k * , and it is well-known that the extensions of C i−1 by K i in the same k * -orbit of Ext 1 A (C i−1 , K i ) are isomorphic in A(S). Let α ∈ I n be an admissible vector. Then there is an iterated extension (X, C) in Ext(S, n, * ) with order vector α, and it follows from the comments above that X is unique, up to isomorphism in A(S). We shall write X(α) for this object in A(S) when α is admissible. It turns out that X(α) is an indecomposable and uniserial object in A(S): Proposition 4.2. Let α ∈ I n be an admissible vector. If S satisfies (UC), then X(α) is indecomposable and uniserial.
Proof. We claim that if (X, C) is an iterated extension in Ext(S, n, * ), then there is a unique minimal subobject of X. The claim clearly holds if n = 2, and we shall prove the claim by induction on n. We therefore assume that n ≥ 3, and that the claim holds for all iterated extensions of length less than n. To prove that it holds for iterated extensions of length n, it is enough to prove to φ(K) = K n for any injective homomorphism φ : K → X. We consider the commutative diagram given by the cofiltration C, where the horizontal rows represent τ n and ξ n . Assume that φ(K) = K n , which implies that φ(K) ∩ K n = 0 since K n is simple, and consider the induced morphism f n • φ : K → C n−1 . We claim that this morphism is injective. In fact, we have that ker(f n • φ) = φ −1 (K n ) = 0. By the induction hypothesis, this means that f n • φ(K) = K n−1 . This implies that φ(K) ⊆ f −1 n (K n−1 ), which is a contradiction since τ n = 0 by Lemma 4.1, and it follows that φ(K) = K n . We have therefore proven the induction step, which means that X is indecomposable with a unique minimal subobject in A(S). Finally, X is uniserial since C i has a unique minimal submodule in A(S) for 2 ≤ i ≤ n. In fact, we can see this by applying the argument above to the iterated extension (C i , C ′ ) in Ext(S, i, * ), where C ′ is the cofiltration The next step in the classification, is to characterize the vectors α ∈ I n that are admissible. If α is admissible, then by definition there is an iterated extension (X, C) in Ext(S, n, * ) with order vector α, and it follows from Lemma 4.1 that This means that α corresponds to a path of length n − 1 in the quiver Λ, with an arrow from node α(i − 1) to node α(i) for 2 ≤ i ≤ n. Conversely, if α ∈ I n is a vector corresponding to a path of length n − 1 in the quiver Λ, such that The vector α is admissible if there is an iterated extensions (X, C) of length n in Ext(S) with τ i = σ i for 2 ≤ i ≤ n, where τ i is the extension induced by the cofiltration C. This is clearly the case when n = 2, since σ 2 = 0 is a non-split extension In fact, we may choose X = E and the cofiltration C of length n = 2 given by E → K 1 → 0, which has τ 2 = σ 2 . However, if n ≥ 3, there are obstructions for the existence of such an iterated extension: Proposition 4.3. Let α ∈ I n be a vector corresponding to a path of length n − 1 in the quiver Λ, and choose a non-zero extension σ i ∈ Ext 1 A (K i−1 , K i ) for 2 ≤ i ≤ n. If S satisfies (UC), then α is admissible if and only if the matric Massey product σ 2 , σ 3 , . . . , σ n is defined and contains zero.
Proof. Since A(S) is exact equivalent to an exact subcategory of a category of modules over an associative k-algebra, we may assume that A is such a module category without loss of generality. In this case, the result follows from Proposition 4 in Eriksen, Siqveland [6], and the preceding construction.
We remark that the use of an exact embedding of A into a module category is a choice of convenience in the proof of Proposition 4.3, and not essential. Matric Massey products are tied to noncommutative deformations and may be computed directly in many Abelian k-categories; see Eriksen, Laudal, Siqveland [5].
We say that A(S) is a hereditary length category if Ext 2 A (S, T ) = 0 for any objects S, T ∈ S. If this is the case, then the obstruction in Proposition 4.3 vanishes. This is clear from the construction of matric Massey products.
We claim that any indecomposable object X in A(S) has the form X(α) for an admissible vectors α ∈ I n , and this would complete the classification. To prove the claim, we must show that any indecomposable object X in A(S) has a cofiltration C such that (X, C) ∈ Ext(S, n, * ): Proof. The result clearly holds if n = 2, and we shall prove the claim by induction on n. We therefore assume that n ≥ 3, and that the claim holds for all iterated extensions of length less than n. For an iterated extension (X, C) of length n in Ext(S), where X is indecomposable, there is a non-split, short exact sequence 0 → K n → X → C n−1 → 0, and C n−1 has a direct decomposition C n−1 = Y 1 ⊕ · · · ⊕ Y q such that Y j is an indecomposable object in A(S) of length n j for 1 ≤ j ≤ q. If q = 1, then ξ i = 0 for 2 ≤ i ≤ n − 1 by the induction hypothesis, and ξ n = 0 since X in indecomposable. Hence we have proved the induction step if q = 1. Next, we suppose that q > 1, and show that this leads to a contradiction: Choose a cofiltration of Y j given by we have ξ n = (ξ n,1 , . . . , ξ n,q ), and we claim that ξ n,j = 0 for 1 ≤ j ≤ q. In fact, if ξ n,j = 0 for some j, then the short exact sequence 0 → K n → f −1 n (Y j ) → Y j → 0 splits, hence there is a section of X → Y j making Y j a direct summand of X. This is a contradiction, and therefore we must have ξ n,j = 0 for 1 ≤ j ≤ q. Since Y j is indecomposable of length n j < n for 1 ≤ j ≤ q, it follows from the induction hypothesis that the extensions with induced surjective morphism f n : X j → Y j = C j,nj . Then (X j , C j * ) is an iterated extension in Ext(S, n j , * ), given by Therefore, it follows from Lemma 4.1 that Ext 1 A (N j , K n ) → Ext 1 A (K j,nj , K n ) is an isomorphism and that the image τ n,j of ξ n,j is non-zero for 1 ≤ j ≤ n. Hence, we must have that α(1, n 1 ) = α(2, n 2 ) = · · · = α(q, n q ) is the unique node in Λ with an arrow to node α(n). In a similar manner, it follows that is an isomorphism and therefore that τ j,nj −i+1 = 0 for 1 ≤ j ≤ q and 1 ≤ i ≤ min(n 1 , . . . , n q ) − 1. Hence we must have that Without loss of generality, we may assume that n 1 ≤ n 2 ≤ · · · ≤ n q . From the argument above, it follows that Y 1 ⊆ Y 2 ⊆ · · · ⊆ Y q . For any injective morphism Y 1 → Y 1 ⊕ Y 2 that has the form y 1 → (y 1 , Cy 1 ) with C ∈ k, there is an induced injective map i : Y 1 → C n−1 = N 1 ⊕ · · · ⊕ N q , and we may consider the commutative diagram where the first row is the extension ξ n = (ξ n,1 , . . . , ξ n,q ). We claim that we may choose C such that second row is a split extension. In fact, this follows from the fact that there are c, c i ∈ k * such that τ n,1 = c · τ n,2 and for 1 ≤ i ≤ n 1 − 1. It follows that Y 1 is a direct summand in X, and this contradicts the assumption q > 1.
Theorem 4.5. Let S = {S α : α ∈ I} be a family of orthogonal k-rational points in an Abelian k-category A. If S satisfies (UC), then the indecomposable objects in A(S) of length n are given by {X(α) : α ∈ J }, up to isomorphism in A(S), where the subset J ⊆ I n consists of the vectors α such that the following conditions hold: is non-zero for 2 ≤ i ≤ n, then the matric Massey product σ 2 , σ 3 , . . . , σ n is defined and contains zero. Moreover, the indecomposable objects X(α) are uniserial, and can be constructed from the family S and their extensions.
Proof. This follows from the results in this section. The explicit construction of X(α) is obtained by iteratively constructing C i as an extension of C i−1 by K i for 2 ≤ i ≤ n. Proof. This follows from Proposition 3.5 and Theorem 4.5.
There is a more general form of Corollary 4.6, where the points in S are not assumed to be krational; see Gabriel [7]. We have chosen to work with k-rational points out of convenience, and also because all points are k-rational in the applications we have in mind. However, it would be possible to prove the general form of Theorem 4.5 and Corollary 4.6 using the methods of this paper.

Graded holonomic D-modules on monomial curves
Let Γ ⊆ N 0 be a numerical semigroup, generated by positive integers a 1 , . . . , a r without common factors, and let A = k[Γ] ∼ = k[t a1 , . . . , t ar ] be its semigroup algebra over the field k = C of complex numbers. We call A a monomial curve since X = Spec(A) is the affine monomial curve X = {(t a1 , t a2 , . . . , t ar ) : t ∈ k} ⊆ A r k . We studied the positively graded algebra D of differential operators on the monomial curve A = k[Γ] in Eriksen [3], and the category grHol D of graded holonomic left D-modules in Eriksen [4]. We recall that any D-module M satisfies the Bernstein inequality d(M ) ≥ 1, that M is holonomic if d(M ) = 1, and that this condition holds if and only if M has finite length; see Proposition 4 and Proposition 5 in Eriksen [4]. This implies that grHol D is a length category, and its simple objects are given by where J * = {α ∈ C : 0 ≤ Re(α) < 1, α = 0}; see Theorem 10 in Eriksen [4]. Moreover, the graded extensions of the simple objects are given by : α ∈ J * ∪ {0, ∞}, w ∈ Z} is the family of simple objects in grHol D , and it is a family of orthogonal k-rational points that satisfies (UC). In particular, the category grHol D of graded holonomic D-modules is a uniserial category.
Proof. Since k = C is algebraically closed, it follows from the main theorem in Quillen [9] that End D (M α [w]) = k for all α ∈ J * ∪ {0, ∞}, w ∈ Z. Moreover, the comments above show that S is the family of simple objects in grHol D , and therefore a family or orthogonal k-rational points, which satisfies (UC).
It is, in principle, possible to construct all indecomposable objects in grHol D using the constructive proof of Theorem 4.5. As an illustration, we shall classify the indecomposable objects in the case A = 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 when A = k[t], the ring D of differential operators on A is the first Weyl algebra  where n ≥ 1, α ∈ J * , β ∈ {0, ∞}, and w(β, n) is the alternating word on n letters in t and ∂, ending with ∂ if β = 0, and in t if β = ∞.
Proof. Let us write I = J * ∪ {0, ∞}, such that S = {M α [w] : (α, w) ∈ I × Z} is the family of simple objects in grHol D . It follows from the computation of the graded extensios above that for any length n ≥ 1 and any (α, w) ∈ I × Z, there is a unique path The corresponding vector is admissible since D = A 1 (k) is a hereditary graded ring; see for instance Coutinho [2]. Note that if α ∈ J * , then α(i) = α and w i = w for 1 ≤ i ≤ n, and if α ∈ {0, ∞}, then we have for α ∈ J * and n ≥ 2, are non-split.

Iterated extensions and noncommutative deformations
Let S = {S α : α ∈ I} be a family of orthogonal points in an Abelian k-category A, and let A(S) be the corresponding length category. In this section, we consider the noncommutative deformations of finite subfamilies of S; see Laudal [8] and also Eriksen, Laudal, Siqveland [5], and show that they determine the iterated extensions in Ext(S). This will shed light on some of the results for uniserial length categories in this paper, and in particular Propostion 4.3. It will also provide a useful tool for future study of length categories that are more complicated than the uniserial ones.
Let (X, C) be an iterated extension in Ext(S) of length n with order vector α. We define the extension type of (X, C) to be the ordered quiver Γ with nodes {α(1), α(2), . . . , α(n)} and edges γ i−1,i from node α(i − 1) to node α(i) for 2 ≤ i ≤ n. The quiver is ordered in the sense that there is a total order γ 12 < γ 23 < · · · < γ n−1,n on the edges in Γ. Clearly, the extension type Γ is uniquely defined by the order vector α, and isomorphic iterated extensions have the same extension type. We denote by E(S, Γ) the set of isomorphism classes of iterated extensions of the family S with extension type Γ.
To fix notation, we shall give the set J = {α(1), α(2), . . . , α(n)} ⊆ I of nodes in Γ a total order, and write S(Γ) = {S α : α ∈ J} = {X 1 , . . . , X r } for the associated subfamily of S, considered as an ordered set with X 1 < X 2 < · · · < X r . Note that for 1 ≤ l ≤ r, we have that X l = S α(i) for at least one value of i with 1 ≤ i ≤ n, and that r ≤ n, with r < n if there are repeated factors.
The path algebra k[Γ] of the ordered quiver Γ is the k-algebra with base consisting of paths γ i−1,i · γ i,i+1 · · · · · γ j−1,j of length j − i + 1 for 2 ≤ i ≤ j ≤ n. The product of two paths γ · γ ′ is given by juxtaposition when the last arrow γ j−1,j in the first path γ is the predecessor of the first arrow γ j,j+1 in the second path γ ′ in the total ordering, and otherwise the product γ · γ ′ = 0. We consider e i as a path of length 0 for 1 ≤ i ≤ r. For example, an iterated extension of length three with α(1) < α(2) < α(3) has r = n = 3, and its extension type Γ has path algebra We recall that the category a r of Artinian r-pointed algebras consists of Artinian k-algebras R with r simple modules fitting into a diagram k r → R → k r , where the composition is the identity. It is clear that if Γ is the extension type of an iterated extension of r objects in S of length n, then the path algebra k[Γ] is an algebra in a r (n), where a r (n) is the full subcategory of a r consisting of algebras R such that I(R) n = 0, with I(R) = ker(R → k r ). Noncommutative deformation functors are defined on the category a r , and noncommutative deformations are parameterized by r-pointed Artinian algebras; see Chapter 3 of Eriksen, Laudal, Siqveland [5] for details. Let Γ be an extension type, and write S(Γ) = {X 1 , . . . , X r } for the corresponding ordered subfamily of S. We consider the noncommutative deformation functor Def S(Γ) : a r → Sets of the finite family S(Γ) in the Abelian category A. We shall assume, without loss of generality, that A is the category of right modules over an associative k-algebra A in the rest of this section. This is a choice of convenience, as noncommutative deformations can be computed directly in many other Abelian k-categories; see Eriksen, Laudal, Siqveland [5]. Proof. Let us write S(Γ) = {X 1 , . . . , X r }, and let α be the order vector corresponding to the extension type Γ. There is a unique s with 1 ≤ s ≤ r such that S α(1) = X s . Any noncommutative deformation X Γ ∈ Def S(Γ) (k[Γ]) has the form X Γ = (k[Γ] ij ⊗ k X j ) as a left k[Γ]-module by flatness, with a right multiplication of A. Let X Γ (s) = e s · X Γ ⊆ X Γ , which is closed under right multiplication with A. A path in e s · k[Γ] is called leading if it has the form γ 12 γ 23 · γ i−1,i and non-leading otherwise. By convention, we consider the path e s as leading, and define where the sum is taken over all non-leading paths γ in e s · k[Γ]. Notice that X N L Γ (s) ⊆ X Γ (s) is closed under right multiplication by A. We define X = X Γ (s)/X N L Γ (s), which has an induced right A-module structure. As a k-linear space, we have that X ∼ = ⊕ 1≤i≤n (γ 12 γ 23 . . . γ i−1,i ) ⊗ k S α(i) with S α(i) = X l for some l with 1 ≤ l ≤ r, and we claim that there is a cofiltration C of X such that (X, C) is an iterated extension of S with extension type Γ. In fact, we may choose the cofiltration C dual to the filtration F given by (γ 12 γ 23 . . . γ i−1,i ) ⊗ k S α(i) for 0 ≤ j ≤ n, where F j ⊆ X is closed under right multiplication with A. Conversely, if (X, C) is a iterated extension of S with extension type Γ, then it follows from the construction in Section 3 of Eriksen, Siqveland [6] that X ∼ = K n ⊕ K n−1 ⊕ · · · ⊕ K 2 ⊕ K 1 as a k-linear vector space, with K i ∼ = S α(i) and with right multiplication of A given by (m n , . . . , m 2 , m 1 )a = (m n · a + n−1 i=1 ψ in a (m i ), . . . , m 2 · a + ψ 12 a (m 1 ), m 1 · a) for m i ∈ K i , a ∈ A. Let I l = {i : α(i) = l} for 1 ≤ l ≤ r. Then the right multiplication of A on X Γ = (k[Γ] ij ⊗ k X j ) given by (e l ⊗ m l ) · a = e l ⊗ (m l · a) + i∈I l i+1≤j≤n (γ i,i+1 γ i+1,i+2 . . . γ j−1,j ) ⊗ ψ ij a (m) for 1 ≤ l ≤ r, a ∈ A, m l ∈ X l defines a noncommutative deformation X Γ ∈ Def S(Γ) (k[Γ]).
We say that S(Γ) is a swarm if dim k Ext 1 A (X i , X j ) if 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 deformation functor Def S(Γ) has a miniversal object (H, X H ) by general results; see Eriksen, Laudal, Siqveland [5], where H is the pro-representing hull in the pro-category a r , and X H ∈ Def