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Expansions of abelian categories

Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications.


Introduction
In this note we discuss certain functors between abelian categories which we call expansions. Such functors arise naturally in finite dimensional representation theory and provide a tool for induction/reduction arguments. We describe the formal properties of these functors and give some applications.
Roughly speaking, an expansion is a fully faithful and exact functor B → A between abelian categories that admits an exact left adjoint and an exact right adjoint. In addition one requires the existence of simple objects S λ and S ρ in A such that S ⊥ λ = B = ⊥ S ρ , where B is viewed as a full subcategory of A and S ⊥ λ = {A ∈ A | Hom A (S λ , A) = 0 = Ext 1 A (S λ , A)}, ⊥ S ρ = {A ∈ A | Hom A (A, S ρ ) = 0 = Ext 1 A (A, S ρ )}. In fact, these simple objects are related by an almost split sequence 0 → S ρ → S → S λ → 0 in A such that S is a simple object in B. In terms of the Ext-quivers of A and B, the expansion B → A turns the vertex S into an arrow S λ → S ρ .
It is interesting to note that an expansion B → A induces a recollement of derived categories where mod ∆ denotes the category of finitely generated right modules over the associated division ring ∆ = End A (S λ ) which is isomorphic to End A (S ρ ).
Our motivation for studying expansions is the following result that characterizes the abelian categories arising as categories of coherent sheaves on weighted projective lines in the sense of Geigle and Lenzing [7].
Theorem. Let k be an arbitrary field. A k-linear abelian category A is equivalent to coh X for some weighted projective line X over k (the exceptional points being rational with residue field k) if and only if there exists a finite sequence A 0 ⊆ A 1 ⊆ . . . ⊆ A r = A of full subcategories such that A 0 is equivalent to coh P 1 k and each inclusion A l → A l+1 is a non-split expansion with associated division ring k.
This result is part of Theorem 4.2.1 and based on a technique which is known as reduction of weights [8]. There is also an explicit construction for categories of coherent sheaves which increases the weight type and is therefore called insertion of weights [11]. The reduction of weights involves perpendicular categories; these Chen was supported by the Alexander von Humboldt Stiftung and the National Natural Science Foundation of China (No. 10971206).
were introduced in the context of quiver representations by Schofield [15] but appear already in early work of Gabriel [5]. A common feature of such reduction techniques is the reduction of the rank of the Grothendieck group. Note that in most cases the Grothendieck group is free of finite rank. Further induction/reduction techniques in representation theory include one-point extensions of algebras and the shrinking of arrows; see Ringel's report on tame algebras in [13]. In a more categorical setting, the trivial extensions introduced by Fossum, Griffith and Reiten [4] should be mentioned. One-point extensions are probably the most important among these techniques, but they require the existence of enough projective objects. A category with Serre duality has no non-zero projective objects, and in that sense expansions seem to be the appropriate variant for dealing with sheaves on weighted projective lines.
This paper is organized as follows. Some preliminaries about abelian categories are collected in Section 2. The central subject of this work is treated in Section 3, including a couple of generic examples. The applications to weighted projective lines are discussed in the final Section 4.

Preliminaries
In this section we fix our notation and collect some basic facts about abelian categories. The standard reference is [5].
Let F : A → B be an additive functor between additive categories. The kernel Ker F of F is the full subcategory of A formed by all objects A such that F A = 0. The essential image Im F of F is the full subcategory of B formed by all objects B such that B is isomorphic to F A for some A in A. Observe that F is fully faithful if and only if F induces an equivalence A ∼ − → Im F ; in this case we usually identify A with Im F , and identify F with the inclusion functor Im F B.
2.1. Serre subcategories and quotient categories. A non-empty full subcategory C of an abelian category A is a Serre subcategory provided that C is closed under taking subobjects, quotients and extensions. This means that for every exact sequence 0 → A ′ → A → A ′′ → 0 in A, the object A belongs to C if and only if A ′ and A ′′ belong to C.
Given a Serre subcategory C of A, the quotient category A/C is by definition the localization of A with respect to the collection of morphisms that have their kernel and cokernel in C. The quotient category A/C is an abelian category and the quotient functor Q : A → A/C is exact with Ker Q = C.
We observe that the kernel Ker F of an exact functor F : A → B between abelian categories is a Serre subcategory of A. Given any Serre subcategory C ⊆ Ker F , the functor F induces a unique functorF : A/C → B such that F =F Q; moreover, the functorF is exact.
2.2. Perpendicular categories. Let A be an abelian category. The quotient functor A → A/C with respect to a Serre subcategory C admits an explicit description if there exists a right adjoint; this is based on the use of the perpendicular category C ⊥ .
For any class C of objects in A, its perpendicular categories are by definition the full subcategories The next result shows that this definition of a perpendicular category is appropriate in the abelian context. The lemma provides a useful criterion for an exact functor to be a quotient functor and it describes the right adjoint of a quotient functor.
. Let F : A → B be an exact functor between abelian categories and suppose that F admits a right adjoint G : B → A. Then the following are equivalent: The functor G is fully faithful.
Moreover, in that case (Ker F ) ⊥ = Im G and Ker F = ⊥ (Im G).
Next we characterize the fact that a quotient functor admits a right adjoint.
In that case the functor A → C sending A to A ′ is a right adjoint of the inclusion C → A, and the functor A → C ⊥ sending A toĀ is a left adjoint of the inclusion C ⊥ → A.

Expansions of abelian categories
In this section we introduce the concept of expansion and contraction for abelian categories. 1 Roughly speaking, an expansion is a fully faithful and exact functor B → A between abelian categories that admits an exact left adjoint and an exact right adjoint. In addition one requires the existence of simple objects S λ and S ρ in A such that S ⊥ λ = B = ⊥ S ρ , where B is viewed as a full subcategory of A. In fact, these simple objects are related by an exact sequence 0 → S ρ → S → S λ → 0 in A such that S is a simple object in B. In terms of the Ext-quivers of A and B, the expansion B → A turns the vertex S into an arrow S λ → S ρ . On the other hand, is a contraction of A in the sense that the left adjoint of B → A identifies S ρ with S, whereas the right adjoint identifies S λ with S.
In the following we use the term 'expansion' but there are interesting situations where 'contraction' yields a more appropriate point of view. So one should think of a process having two directions that are opposite to each other.
Proof. (1) and (2)   The following lemma describes for any abelian category A a bijective correspondence between localizable objects in A and left expansions B → A.
For an object X in A, we denote by add X the full subcategory consisting of all finite direct sums of copies of X and their direct summands. Proof.
(1) We identify B = Im i. Let S be an indecomposable object in ⊥ B. Then S is a simple object and Ext 1 A (S, S) = 0 since ⊥ B = add S is semisimple. For each object A in A, we use the natural exact sequence (2.2.3) This sequence induces the following isomorphisms.
Here we use the condition Ext 2 A (S, S) = 0. It follows that Hom A (S, A) and (2) The proof follows closely that of [8,Prop. 3.2]. Set B = S ⊥ and observe that this is an exact abelian subcategory since Ext 2 is surjective and letĀ be its cokernel. It is easily checked that the composite A → B →Ā is the universal morphism into B. Thus we define i λ A =Ā.
Next observe that kernel and cokernel of the adjunction morphism A → i λ A belong to C = add S for each object A in A. Moreover, C is a Serre subcategory of A since S is simple and Ext 1 A (S, S) = 0. Thus we can apply Lemma 2.2.2 and infer that the quotient functor A → A/C admits a right adjoint. In fact the right adjoint identifies A/C with C ⊥ , by Lemma 2.2.1, and therefore i λ identifies with the quotient functor. In particular, i λ is exact. We have ⊥ B = C by Lemma 2.2.1, and Hom A (S, −) induces an equivalence C ∼ − → mod End A (S). Thus the inclusion B A is a left expansion.

3.2.
Expansions of abelian categories. We are ready to introduce the central concept of this work. Let us fix some notation for an expansion i : B → A. We identify B with the essential image of i. We denote by i λ the left adjoint of i and by i ρ the right adjoint of i. We choose an indecomposable object S λ in ⊥ B and an indecomposable object S ρ in B ⊥ . Thus ⊥ B = add S λ and B ⊥ = add S ρ . Finally, setS = i λ (S ρ ).

Proposition 3.2.2. Let A be an abelian category.
(1) Given an expansion i : B → A, there exist a localizable object S λ and a colocalizable object Then the inclusion S ⊥ λ A is an expansion. Proof. Apply Lemma 3.1.4 and its dual.
If the expansion is non-split, then the exact sequences (2.2.3) for S λ and S ρ are of the form ( Now apply i ρ to the first and i λ to the second sequence in (3.2.3). Then This implies l = 1 = r and therefore i λ (S ρ ) ∼ = i ρ (S λ ).
(2) We have a sequence of isomorphisms which takes a morphism φ to i λ φ. Thus i λ induces an equivalence add S ρ ∼ − → add i λ (S ρ ).
An expansion i : B → A of abelian categories determines a division ring ∆ such that ⊥ B and B ⊥ are equivalent to mod ∆; we call ∆ the associated division ring. If the expansion does not split, then the sequences in (3.2.3) yield an essentially unique non-split extension which is called the connecting sequence of the expansion i. This sequence is almost split; see Proposition 3.7.1.
It is interesting to observe that this diagram induces a recollement of triangulated categories [1] when one passes from abelian categories to their derived categories.
Recall that a diagram of exact functors between triangulated categories  (1) F * and G * form an adjoint pair of exact functors and F * is fully faithful. Proof.
(1) Fix a pair of complexes X ∈ D b (B) and Y ∈ D b (A). The unit Id B → GF yields a morphism η X : X → G * F * (X) and we obtain a map by sending φ to (G * φ)η X . A simple application of Beilinson's lemma shows that this map is bijective. The same lemma yields that F * is fully faithful.
(2) We construct a left adjoint L : D b (A) → Ker G * as follows. For X ∈ D b (A) complete the counit F * G * (X) → X to an exact triangle F * G * (X) → X → X ′ →. The assignment X → L(X) = X ′ is functorial and yields an exact left adjoint for the inclusion Ker G * → D b (A); see for example [2, Lemma 3.3] for details. A right adjoint of a fully faithful functor is, up to an equivalence, a quotient functor, by [6, Prop. I.1.3]. Thus it remains to observe that Ker L = Im F * , which is obvious from the triangle defining L.
We point out that in general the unlabeled functor is not induced from an exact functor between the abelian categories. In fact, this functor equals the right derived functor of j ρ , and it equals the left derived functor of k λ .
Proof. The assertion is an immediate consequence of Lemma 3.3.1 and its dual. It only remains to observe that the inclusion j induces an equivalence D b (mod ∆)  (1) If S is a simple object in B and S ∼ =S, then iS is simple in A and i λ iS ∼ = S.
(2) There is an Proof.
(1) Let 0 = U ⊆ iS be a subobject. Then Hom B (i λ U, S) ∼ = Hom A (U, iS) = 0 shows that U ∈ Ker i λ . Thus i λ U = S, and therefore iS/U belongs to Ker i λ = add S λ . On the other hand, Hom A (iS, S λ ) ∼ = Hom B (S,S) = 0. Thus iS/U = 0, and it follows that iS is simple. Finally observe that i λ iA ∼ = A for every object A in B.
(3) This is a general fact: A quotient functor A → A/C takes each simple object of A not belonging to C to a simple object of A/C. Here, we take C = Ker i λ and identify i λ with the corresponding quotient functor.
If S ∼ = S ρ , then i λ S ∼ =S and therefore ii λ S is simple by (1). Thus the canonical morphism S → ii λ S is an isomorphism.
The Ext-groups of most simple objects in A can be computed from appropriate Ext-groups in B. This follows from an adjunction formula; see Lemma 3.1.2. The remaining cases are treated in the following lemma. For an abelian category A denote by A 0 the full subcategory formed by all finite length objects; it is a Serre subcategory. (1) The functor i and its adjoints i λ and i ρ send finite length objects to finite length objects. Let A be an abelian category. We call an object A torsion-free if Hom A (S, A) = 0 for each simple object S. An object A is called 1-simple if it becomes a simple object in the quotient category A/A 0 , equivalently, if for each subobject A ′ ⊆ A either A ′ or A/A ′ is of finite length, but not both. Proof. Using adjunctions, it is clear that i and i ρ preserve torsion-free objects. Now fix a torsion-free object A ∈ A and a morphism φ : and observe that A ′ = 0. The composite iS → ii λ (A) → A ′′ has a non-zero kernel, since A ′′ belongs to add S λ , and iφ maps this kernel to A. This implies φ = 0.
The statement on 1-simple objects follows from Proposition 3.4.3. Here we note that i λ and i ρ induce functors A/A 0 → B/B 0 that are quasi-inverse to the functor B/B 0 → A/A 0 induced by i.

3.5.
The Ext-quiver. The Ext-quiver of an abelian category A is a valued quiver Σ = Σ(A) which is defined as follows. The set Σ 0 of vertices is a fixed set of representatives of the isomorphism classes of simple objects in A. For a simple object S, let ∆(S) denote its endomorphism ring, which is a division ring. There is an arrow Given an expansion B → A, the Ext-quiver Σ(A) can be computed explicitly from the Ext-quiver Σ(B), and vice versa. The following statement makes this precise.  for all A, B in A, where D = Hom k (−, k) denotes the standard k-duality. Let S λ be a simple object in A with Ext 1 A (S λ , S λ ) = 0 and set S ρ = τ S λ . Then by Serre duality S ⊥ λ = ⊥ S ρ . Note that the category A is hereditary, that is, Ext 2 A (−, −) = 0. It follows that S λ is localizable and S ρ is colocalizable. By Proposition 3.2.2 the inclusion S ⊥ λ A is an expansion, and this is non-split since S λ ≇ S ρ . A specific example of an abelian category having the above properties is the category of finite dimensional nilpotent representations of a quiver Γ n of typeÃ n with cyclic orientation. In that case, S ⊥ λ is equivalent to the category of finite dimensional nilpotent representations of Γ n−1 .
Example 3.6.2. Let k be a field. Consider a finite quiver Γ having two vertices a, b that are joined by an arrow ξ : a → b which is the unique arrow starting at a and the unique arrow terminating at b. We obtain a new quiver Γ ′ by identifying a and b and removing ξ.
Let kΓ be the path algebra of Γ and let A = kΓ/I be a finite dimensional quotient algebra with respect to some admissible ideal I. Denote by A = mod A op the abelian category of finite dimensional left A-modules, which is viewed as a full subcategory of the category of k-linear representations of Γ.
Consider the full subcategory B of A formed by all modules which correspond to representations of Γ such that ξ is represented by an isomorphism. Note that B is equivalent to the category of finite dimensional left modules over a finite dimensional algebra A ′ = kΓ ′ /I ′ for some ideal I ′ .
For each vertex i of Γ, denote by P i (respectively, I i ) the projective cover (respectively, injective hull) in A of the corresponding simple module S i . Note that the arrow ξ induces morphisms P ξ : P b → P a and I ξ : Assume that the morphism P ξ is a monomorphism and that I ξ is an epimorphism. For example, this happens when the admissible ideal I is generated by paths which neither begin with nor end with ξ. Then we claim that the inclusion B A is a non-split expansion.
The assumption yields the following two exact sequences It follows that the simple module S a is a localizable object and that S b is a colocalizible object of A. Moreover, we observe from the two exact sequences above that S ⊥ a = B = ⊥ S b . Thus by Proposition 3.2.2 the inclusion functor B A is a non-split expansion.
Observe that the algebra A ′ is Morita equivalent to the universal localization [14] of A at the map P ξ , since B equals the full subcategory consisting of all A-modules X such that Hom A (P ξ , X) is invertible. Now let A = kΓ/I be an arbitrary finite dimensional algebra over k, where Γ is any finite quiver and I is an admissible ideal. Then one can show that for any non-split expansion i : B → A = mod A op , there is a unique arrow ξ of Γ satisfying the above conditions such that i identifies B with the full subcategory formed by all representations inverting ξ.

3.7.
An Auslander-Reiten formula. Given a non-split expansion B → A, the corresponding simple objects S λ and S ρ in A are related by an Auslander-Reiten formula. Proof. Recall that ⊥ B = add S λ and B ⊥ = add S ρ . Fix an object A in A and consider the corresponding exact sequence (2.2.3) The morphism A ′ → A induces the first and the third isomorphism in the sequence below, while the second isomorphism follows from the isomorphism Hom follows from the first by duality. Let ξ : 0 → S ρ → E → S λ → be a non-split extension. Using the formula for D Ext 1 A (−, S ρ ), one shows that the pullback along any morphism φ : A → S λ is a split extension, provided that φ is not a split epimorphism. Analogously, one shows via the formula for D Ext 1 A (S λ , −) that the pushout along any morphism φ : S ρ → A is a split extension, provided that φ is not a split monomorphism. Thus ξ is an almost split sequence.

Applications to coherent sheaves on weighted projective lines
In this section we present some results on weighted projective lines that are based on expansions. We begin with a brief description of weighted projective lines and their categories of coherent sheaves.
Throughout this section we fix an arbitrary field k.

4.1.
Coherent sheaves on weighted projective lines. Let P 1 k be the projective line over k, let λ = (λ 1 , . . . , λ n ) be a (possibly empty) collection of distinct rational points of P 1 k , and let p = (p 1 , . . . , p n ) be a weight sequence, that is, a sequence of positive integers. The triple X = (P 1 k , λ, p) is called a weighted projective line. Geigle and Lenzing [7] have associated to each weighted projective line a category coh X of coherent sheaves on X, which is the quotient category of the category of finitely generated L(p)-graded S(p, λ)-modules, modulo the Serre subcategory of finite length modules. Thus coh X = mod L(p) S(p, λ)/(mod L(p) S(p, λ)) 0 .
Here L(p) is the rank 1 additive group k . Geigle and Lenzing showed that coh X is a hereditary abelian category with finite dimensional Hom and Ext spaces. The free module S(p, λ) yields a structure sheaf O, and shifting the grading gives twists E( x) for any sheaf E and x ∈ L(p).
Every sheaf is the direct sum of a torsion-free sheaf and a finite length sheaf. A torsion-free sheaf has a finite filtration by line bundles, that is, sheaves of the form O( x). The finite length sheaves are easily described as follows. There are simple sheaves S x in bijection to closed points x in P 1 k λ, and S ij (1 ≤ i ≤ n, 1 ≤ j ≤ p i ) satisfying for any r ∈ Z that Hom(O(r c), S ij ) = 0 if and only if j = 1, and the only extensions between them are Here k(x) denotes the residue field at each closed point x. For each simple sheaf S and l > 0 there is a unique sheaf with length l and top S, which is uniserial, meaning that it has a unique composition series. These are all the finite length indecomposable sheaves.

4.2.
A characterization of coherent sheaves on weighted projective lines.
The following result describes in terms of expansions the abelian categories that arise as categories of coherent sheaves on weighted projective lines.
Theorem 4.2.1. A k-linear abelian category A is equivalent to coh X for some weighted projective line X over k if and only if there exists a finite sequence A 0 ⊆ A 1 ⊆ . . . ⊆ A r = A of full subcategories such that A 0 is equivalent to coh P 1 k and each inclusion A l → A l+1 is a non-split expansion with associated division ring k.
In that case each inclusion A l → A l+1 induces a recollement Proof. The first part of the assertion is covered by [3, Thm. 6.8.1] and based on work of Lenzing in [10]. A detailed proof can be found in [3]. The existence of a recollement induced by the inclusion A l → A l+1 is an immediate consequence of Proposition 3.3.2. Now fix a weighted projective line X = (P 1 k , λ, p). We provide the argument that gives the filtration A 0 ⊆ A 1 ⊆ . . . ⊆ A r = A for A = coh X; this is known as reduction of weights. If p i = 1 for all i, then A = coh P 1 k . Otherwise, choose some j such that p j > 1. Then S = S j1 is a simple object satisfying Ext 1 A (S, S) = 0. So we can apply Example 3.6.1 and obtain an expansion S ⊥ A. It follows from the arguments given in [10, Prop. 1] that S ⊥ is equivalent to coh X ′ for X ′ = (P 1 k , λ, p ′ ), where p ′ i = p i − δ ij for 1 ≤ i ≤ n. Thus we can proceed and obtain a trivial weight sequence (1, . . . , 1) after n i=1 (p i − 1) steps. Next we provide an explicit construction of an expansion coh X ′ → coh X, where X ′ = (P 1 k , λ, p ′ ) with p ′ i = p i − δ ij for 1 ≤ i ≤ n and some fixed j; see also [8, §9]. There is an injective map φ : L(p ′ ) −→ L(p) Here, l is in its normal form, that is, 0 ≤ l i < p ′ i for all i. Note that l ∈ L(p) belongs to the image of φ if and only if l j = p j − 1. Observe that φ is not a morphism of abelian groups.
Consider the following functor The functor F identifies the category mod L(p ′ ) S(p ′ , λ) with the full subcategory of mod L(p) S(p, λ) consisting of the modules N such that multiplication with x j induces an isomorphism N l → N l+ xj whenever l j = 0.
The functor F admits a left adjoint F λ which sends N to F λ N such that (F λ N ) l = N φ( l)+ xj . Similarly, there is a right adjoint F ρ sending N to F ρ N such that (F ρ N ) l = N φ( l) if l j = 0, and (F ρ N ) l = N φ( l)+ xj otherwise. Note that x j acts on (F λ N ) l as x 2 j if l j = p j − 2, and on (F ρ N ) l as x 2 j if l j = 0. We remark that F ρ N = F λ (N ( x j ))(− x j ).
All three functors are exact and preserve finite length modules. So they induce functors between coh X ′ and coh X. In particular, this yields the pursued non-split expansion coh X ′ → coh X. Note that F O( l) = O(φ( l)) for all l in L(p ′ ).
Let us remark that the kernel of F λ on the category of sheaves is of the form add S λ with S λ a simple sheaf concentrated at (x j ). Similarly, the kernel of F ρ equals add S ρ with S ρ another simple sheaf concentrated at (x j ). More precisely, there is a presentation 0 → O(− x j ) xj − → O → S λ → 0 and S ρ = S λ (− x j ). Using the notation from 4.1, we have S λ = S j1 and S ρ = S j,pj .
Remark 4.2.2. Let X = (P 1 k , λ, p). The proof shows that the length r of the filtration of A = coh X is determined by the weight sequence p. More precisely, r = n i=1 (p i − 1) and each category A l is of the form coh X ′ for some weighted projective line X ′ = (P 1 k , λ, p ′ ) such that p ′ i ≤ p i for 1 ≤ i ≤ n.

4.3.
Vector bundles on weighted projective lines. Let A = coh X be the category of coherent sheaves on a weighted projective line X and denote by vect X the full subcategory formed by all torsion-free objects. Note that the line bundles in A are precisely those objects that are torsion-free and 1-simple. The category vect X admits a Quillen exact structure such that vect X has enough projective and injective objects. Moreover, projective and injective objects coincide; they are precisely the direct sums of line bundles. Thus vect X is a Frobenius category and its stable category vect X carries a triangulated structure; see [9] for details. Theorem 4.3.1. Let X = (P 1 k , λ, p) and X ′ = (P 1 k , λ, p ′ ) be a pair of weighted projective lines such that p ′ i ≤ p i for 1 ≤ i ≤ n. Then there is a fully faithful and exact functor coh X ′ → coh X that induces the following recollement Proof. The reduction of weights described in the proof of Theorem 4.2.1 yields a fully faithful functor i : coh X ′ → coh X which is a composite of i (p i − p ′ i ) expansions. This functor has left and right adjoints, and all three functors preserve vector bundles and line bundles by Proposition 3.4.4. It follows that i restricts to an exact functor vect X ′ → vect X which admits exact left and right adjoints. For the exactness of these functors, one uses Serre duality and the fact that a sequence ξ : 0 → E ′ → E → E ′′ → 0 in vect X is exact with respect to the specified Quillen exact structure if and only if Hom A (L, −) sends ξ to an exact sequence for each line bundle L, if and only if Hom A (−, L) sends ξ to an exact sequence for each line bundle L. Thus the induced functors between vect X ′ and vect X form the left hand side of a recollement, while the right half exists for formal reasons, as explained before in Lemma 3.3.1.
Not much seems to be known about the right hand term in the recollement of vect X. This is in contrast to the recollement of D b (coh X) given in Theorem 4.2.1, where one has a very explicit description.