Noetherian hereditary abelian categories satisfying Serre duality
Abstract
In this paper we classify noetherian hereditary abelian categories over an algebraically closed field finite satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.
As a side result we show that when our hereditary abelian categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.
Notations and conventions
Most notations will be introduced locally. The few global ones are given below. Unless otherwise specified will be an algebraically closed field and all rings and categories in this paper will be linear.
If is a ring, then will be the category of finitely generated right Similarly if modules. is a ring, then graded will be the category of finitely generated graded right modules with degree zero morphisms, and the category of all graded right If modules. is noetherian, then following Reference3 will be the full subcategory of consisting of graded modules with right bounded grading. Also following Reference3 we put .
For an abelian category we denote by the bounded derived category of .
Introduction
One of the goals of noncommutative algebraic geometry is to obtain an understanding of abelian categories linear for a field , which have properties close to those of the category of coherent sheaves over a nonsingular proper scheme. Hence some obvious properties one may impose on , in this context are the following:
 •

is , i.e. finite
for all and for all .
 •

has homological dimension i.e. ,
for and and , is minimal with this property.
Throughout this paper will denote a field, and even though it is not always necessary we will for simplicity assume that is algebraically closed. All categories will be When we say that linear. is it will be understood that this is with respect to the field finite, .
In most of this paper we will assume that is an abelian category of homological dimension at most 1, in which case we say that finite is hereditary.
A slightly more subtle property of nonsingular proper schemes is Serre duality. Let be a nonsingular proper scheme over of dimension and let , denote the category of coherent Then the classical Serre duality theorem asserts that for modules. there are natural isomorphisms
where .
A very elegant reformulation of Serre duality was given by Bondal and Kapranov in Reference8. It says that for any there exist natural isomorphisms
Stated in this way the concept of Serre duality can be generalized to certain abelian categories.
 •

satisfies Serre duality if it has a socalled Serre functor. The latter is by definition an autoequivalence such that there are isomorphisms
which are natural in .
On the other hand hereditary abelian categories finite with the additional property of having a tilting object have been important for the representation theory of finite dimensional algebras. Recall that is a tilting object in if and if , implies that is 0. These categories are important in the study of quasitilted algebras, which by definition are the algebras of the form for a tilting object Reference16, and which contain the important classes of tilted and canonical algebras. A prominent property in the representation theory of finite dimensional algebras is having almost split sequences, and also the hereditary abelian categories with tilting object have this property finiteReference16.
In view of the above it is interesting, and useful, to investigate the relationship between Serre duality and almost split sequences. In fact, this relationship is very close in the hereditary case. The more general connections are on the level of triangulated categories, replacing almost split sequences with Auslander–Reiten triangles. In fact one of our first results in this paper is the following (see §I for more complete results):
Theorem A
 (1)
has a Serre functor if and only if has Auslander–Reiten triangles (as defined in Reference14).
 (2)
If is hereditary, then has a Serre functor if and only if has almost split sequences and there is a oneone correspondence between the indecomposable projective objects and the indecomposable injective objects such that the simple top of , is isomorphic to the socle of .
Hence hereditary abelian categories with Serre duality are of interest both for noncommutative algebraic geometry and for the representation theory of finite dimensional algebras. The main result of this paper is the classification of the noetherian ones. finite
To be able to state our result we first give a list of hereditary abelian categories satisfying Serre duality.
 (a)
If consists of the finite dimensional nilpotent representations of the quiver or of the quiver with all arrows oriented in the same direction, then it is classical that , has almost split sequences, and hence Serre duality.
 (b)
Let be a nonsingular projective connected curve over with function field and let , be a sheaf of hereditary in orders (see Reference21). Then one proves exactly as in the commutative case that satisfies Serre duality.
 (c)

Let be either or with zigzag orientation. It is shown in §III.3 that there exists a noetherian hereditary abelian category which is derived equivalent to the category of finitely presented representations of and which has no nonzero projectives or injectives. Depending on , we call this category the or the category. Since Serre duality is defined in terms of the derived category, it follows that satisfies Serre duality.
If then , is nothing but the category considered in Reference30. If and then , is a skew version of the category (see §III.3). The category and the category have also been considered by Lenzing.
 (d)
We now come to more subtle examples (see §II). Let be a connected quiver. Then for a vertex we have a corresponding projective representation and an injective representation If . is locally finite and there is no infinite path ending at any vertex, the functor may be derived to yield a fully faithful endofunctor Then . behaves like a Serre functor, except that it is not in general an autoequivalence. We call such a right Serre functor (see §I.1). Luckily given a right Serre functor there is a formal procedure to invert it so as to obtain a true Serre functor (Theorem II.1.3). This yields a hereditary abelian category which satisfies Serre duality. Under the additional hypotheses that consists of a subquiver with no path of infinite length, with rays attached to vertices of then , turns out to be noetherian (see Theorem II.4.3). Here we mean by a ray an quiver with no vertex which is a sink. An interesting feature of the noetherian categories exhibiting a new type of behavior, is that they are generated by the preprojective objects, but not necessarily by the projective objects. ,
Now we can state our main result. Recall that an abelian category is connected if it cannot be nontrivially written as a direct sum .
Theorem B
Let be a connected noetherian hereditary abelian category satisfying Serre duality. Then finite is one of the categories described in (a)–(d) above.
The cases (a), (b), (c) are those where there are no nonzero projective objects. Those in (a) are exactly the where all objects have finite length. For the having some objects of infinite length, then either all indecomposable objects of finite length have finite (case (b)) or all have infinite period (case (c)). Here the object period is defined by the almost split sequence for indecomposable in and we have , .
Under the additional assumption that has a tilting object, such a classification was given in Reference19. The only cases are the categories of finitely generated modules for a finite dimensional indecomposable hereditary algebra and the categories of coherent sheaves on a weighted projective line in the sense of Reference12. From the point of view of the above list of examples the first case corresponds to the finite quivers in (d), in which case is equivalent to where , is the path algebra of over The categories . are a special case of (b), corresponding to the case where the projective curve is (see Reference23).
Our proof of Theorem B is rather involved and covers the first four sections. The main steps are as follows:
 (1)
In the first two subsections of §II we construct the categories and we show that they are characterized by the property of having noetherian injectives and being generated by preprojectives. ,
 (2)
In §II.4 we give necessary and sufficient conditions for to be noetherian, and furthermore we prove a decomposition theorem which states that an noetherian hereditary abelian category with Serre functor can be decomposed as a direct sum of a hereditary abelian category which is generated by preprojectives and a hereditary abelian category which doesn’t have nonzero projectives or injectives. finite
 (3)
We are now reduced to the case where there are no nonzero projectives or injectives. The case where all objects have finite length is treated in §III.1.
 (4)
The case where there are no nonzero projectives or injectives and at least one object of infinite length is covered in §IV. It turns out that this case naturally falls into two subcases:
 ( )
The simple objects are In that case, using the results in periodic.Reference3, we show that is of the form for a twodimensional commutative graded ring, where is the quotient category length. Using /finiteReference1 it then follows that is of the form (b).
 ( )
The simple objects are not We show that if such periodic. exists, then it is characterized by the fact that it has either one or two of simple objects. Since the orbits and category have this property, we are done.
Our methods for constructing the new hereditary abelian categories are somewhat indirect although we believe they are interesting. After learning about our results Claus Ringel has recently found a more direct construction for these categories Reference24.
All the hypotheses for Theorem B are necessary. For example the noncommutative curves considered in Reference30 are noetherian hereditary abelian categories of Krulldimension one which in general do not satisfy Serre duality (except for the special case listed in (c)). If is the opposite category to one of the categories (b), (c), (d), then it is not noetherian, but it satisfies the other hypotheses.
Nevertheless it is tempting to ask whether a result similar to Theorem B remains valid without the noetherian hypothesis if we work up to derived equivalence. In particular, is any such category derived equivalent to a noetherian one? Under the additional assumption that has a tilting object, this has been proved by Happel in Reference15, and it has recently been shown by Ringel that this is not true in general Reference25.
In the final section we use Theorem B to draw some conclusions on the structure of certain hereditary abelian categories.
To start with we discuss the “saturation” property. This is a subtle property of certain abelian categories which was discovered by Bondal and Kapranov Reference8. Recall that a cohomological functor is of finite type if for every only a finite number of are nonzero. We have already defined what it means for to have homological dimension It will be convenient to say more generally that . has finite homological dimension if for any in there is at most a finite number of with In particular, the analogue of this definition makes sense for triangulated categories. .
 •
Let be an abelian category of finite homological dimension. Then finite is saturated if every cohomological functor of finite type is of the form (i.e. is representable).
It is easy to show that a saturated category satisfies Serre duality. It was shown in Reference8 that for a nonsingular projective scheme is saturated and that saturation also holds for categories of the form with a finite dimensional algebra. Inspired by these results we prove the following result in §V.1.
Theorem C
Assume that is a saturated connected noetherian hereditary abelian category. Then finite has one of the following forms:
 (1)
where is a connected finite dimensional hereditary algebra.
 (2)
where is a sheaf of hereditary orders(see (b) above) over a nonsingular connected projective curve .
It is easy to see that the hereditary abelian categories listed in the above theorem are of the form We refer the reader to .Reference9 (see also §V.2) where it is shown in reasonable generality that abelian categories of the form are saturated.
There are also applications to the relationship between existence of tilting objects and the Grothendieck group being finitely generated. This was one of the original motivations for this work, and is dealt with in another paper Reference23.
We would like to thank Claus Ringel for helpful comments on the presentation of this paper.
I. Serre duality and almost split sequences
It has been known for some time that there is a connection between classical Serre duality and existence of almost split sequences. There is a strong analogy between the Serre duality formula for curves and the formula for artin algebras (where on which the existence of almost split sequences is based (see ),Reference5). Actually, existence of almost split sequences in some sheaf categories for curves can be proved either by using an analogous formula for graded maximal Cohen–Macaulay modules or by using Serre duality Reference4Reference27. The notion of almost split sequences was extended to the notion of Auslander–Reiten triangles in triangulated categories Reference14, and existence of such was proved for when is a of finite global dimension algebraReference14. In this case the corresponding translate is given by an equivalence of categories. On the other hand an elegant formulation of Serre duality in the bounded derived category, together with a corresponding Serre functor, was given in Reference8. These developments provide the basis for further connections, which turn out to be most complete in the setting of triangulated categories. For abelian categories we obtain strong connections in the hereditary case. In fact, we show that when is hereditary, then has Serre duality if and only if it has almost split sequences and there is a oneone correspondence between indecomposable projective objects and indecomposable injective objects such that , modulo its unique maximal subobject is isomorphic to the socle of .
I.1. Preliminaries on Serre duality
Let be a linear additive category. A right Serre functor is an additive functor finite together with isomorphisms
for any which are natural in and A left Serre functor is a functor . together with isomorphisms
for any which are natural in and Below we state and prove a number of properties of right Serre functors. We leave the proofs of the corresponding properties for left Serre functors to the reader. .
Let be given by and let , Looking at the commutative diagram (which follows from the naturality of . in )
we find for that
Similarly by the naturality of in we obtain the following commutative diagram:
This yields for the formula
and we get the following description of the functor .
Lemma I.1.1
The following composition coincides with :
Proof.
To prove this we need to show that for and one has Thanks to the formulas .EquationI.1.3, EquationI.1.4 we obtain and also Thus we obtain indeed the correct result. .
■We have the following immediate consequence.
Corollary I.1.2
If is a right Serre functor, then is fully faithful.
Also note the following basic properties.
Lemma I.1.3
 (1)
If and are right Serre functors, then they are naturally isomorphic.
 (2)
has a right Serre functor if and only if is representable for all .
From the above discussion it follows that there is a lot of redundancy in the data In fact we have the following. .
Proposition I.1.4
In order to give it is necessary and sufficient to give the action of on objects, as well as maps linear such that the composition
yields a nondegenerate pairing for all If we are given . then , is obtained from the pairing EquationI.1.5. Furthermore the action on maps
is defined by the property that for we have for all .
Proof.
It is clear from the previous discussion that the data gives rise to with the required properties. So conversely assume that we are given and the action of on objects. We define and the action of on maps as in the statement of the proposition.
We first show that is a functor. Indeed let and assume that there are maps and Then for all . we have but also , Thus by nondegeneracy we have . .
It is easy to see that the pairing EquationI.1.5 defines an isomorphism
which is natural in and .
The proof is now complete.
■A Serre functor is by definition a right Serre functor which is essentially surjective. The following is easy to see.
Lemma I.1.5
has a Serre functor if and only it has both a right and a left Serre functor.
From this we deduce the following Reference8.
Lemma I.1.6
has a Serre functor if and only if the functors and are representable for all .
Remark I.1.7
In the sequel will always be a Krull–Schmidt category (in the sense that indecomposable objects have local endomorphism rings). In that case it is clearly sufficient to specify etc. on the full subcategory of , consisting of indecomposable objects.
If one is given a right Serre functor, then it is possible to invert it formally in such a way that the resulting additive category has a Serre functor. The next result is stated in somewhat greater generality.
Proposition I.1.8
Let be an additive category as above, and let be a fully faithful additive endofunctor. Then there exists an additive category with the following properties:
 (1)
There is a fully faithful functor .
 (2)
There is an autoequivalence together with a natural isomorphism .
 (3)
For every object there is some such that is isomorphic to with .
Furthermore a quadruple with these properties is unique (in the appropriate sense).
Proof.
Let us sketch the construction of The uniqueness will be clear. .
The objects in are formally written as with and A morphism . is formally written as with where is such that , We identify . with .
The functor is defined by and the functor is defined by It is clear that these have the required properties. .
■The following lemma provides a complement to this proposition in the case that is triangulated.
Lemma I.1.9
Assume that in addition to the usual hypotheses one has that is triangulated. Let be as in the previous proposition. Then there is a unique way to make into a triangulated category such that and are exact.
Proof.
If we require exactness of and then there is only one way to make , into a triangulated category. First we must define the shift functor by and then the triangles in must be those diagrams that are isomorphic to
where
is a triangle in (note that the exactness of is equivalent to that of ).
To show that this yields indeed a triangulated category one must check the axioms in Reference34. These all involve the existence of certain objects/maps/triangles. By applying a sufficiently high power of we can translate such problems into ones involving only objects in Then we use the triangulated structure of . and afterwards we go back to the original problem by applying a negative power of .
■In the sequel we will denote by the category which was constructed in Proposition I.1.8. Furthermore we will consider as a subcategory of through the functor Finally we will usually write . for the extended functor .
Below we will only be interested in the special case where is a right Serre functor on In that case we have the following. .
Proposition I.1.10
The canonical extension of to is a Serre functor.
Proof.
By construction is an automorphism on To prove that . is a Serre functor we have to construct suitable maps Pick . , Then we have .
We define as the composition of these maps. It follows easily from Lemma I.1.1 that the constructed map is independent of and it is clear that , has the required properties.
■We shall also need the following easily verified fact.
Lemma I.1.11
If is a direct sum of additive categories, then a (right) Serre functor on restricts to (right) Serre functors on and .
I.2. Connection between Serre duality and Auslander–Reiten triangles
In this section we prove that existence of a right Serre functor is equivalent to the existence of right Auslander–Reiten triangles, in triangulated Krull–Schmidt finite Hence the existence of a Serre functor is equivalent to the existence of Auslander–Reiten triangles. categories.
In the sequel is a finite KrullSchmidt triangulated category. Following linearReference14 a triangle in is called an Auslander–Reiten triangle if the following conditions are satisfied:
 (AR1)
and are indecomposable.
 (AR2)
.
 (AR3)
If is indecomposable, then for every nonisomorphism we have .
It is shown in Reference14 that, assuming (AR1) and (AR2), then (AR3) is equivalent to
 (AR3)
If is indecomposable, then for every nonisomorphism we have .
We say that right Auslander–Reiten triangles exist in if for all indecomposables there is a triangle satisfying the conditions above. Existence of left Auslander–Reiten triangles is defined in a similar way, and we say that has Auslander–Reiten triangles if it has both right and left Auslander–Reiten triangles. (Note that in Reference14 one says that has Auslander–Reiten triangles if it has right Auslander–Reiten triangles in our terminology.)
It is shown in Reference14, §4.3 that given the corresponding Auslander–Reiten triangle is unique up to isomorphism of triangles. (By duality a similar result holds if is given.) For a given indecomposable we let be an arbitrary object in isomorphic to , in the Auslander–Reiten triangle corresponding to .
The following characterization of Auslander–Reiten triangles is analogous to the corresponding result on almost split sequences (see Reference5).
Proposition I.2.1
Assume that has right Auslander–Reiten triangles, and assume that we have a triangle in
with and indecomposable and Then the following are equivalent: .
 (1)
The triangle EquationI.2.1 is an Auslander–Reiten triangle.
 (2)
The map is in the socle of as a right and module .
 (3)
The map is in the socle of as a left and module .
Proof.
We will show that 1. and 2. are equivalent. The equivalence of 1. and 3. is similar.
By definition we have Assume that . is a nonautomorphism. Then by (AR3) we have .

Let
be the Auslander–Reiten triangle associated to From the properties of Auslander–Reiten triangles it follows that there is a morphism of triangles .
The fact that together with the fact that is in the socle of implies that must be an isomorphism. But then by the properties of triangles is also an isomorphism. So in fact the triangles EquationI.2.2 and EquationI.2.1 are isomorphic, and hence in particular EquationI.2.1 is an Auslander–Reiten triangle. ■
Corollary I.2.2
Assume that has right Auslander–Reiten triangles. Then the socle of is onedimensional both as a right and as a left module module.
Proof.
It is easy to see that linearly independent elements of the (left or right) socle define different triangles. However Auslander–Reiten triangles are unique. This is a contradiction.
■The following is the basis for the main result of this section.
Proposition I.2.3
The following are equivalent:
 (1)
has a right Serre functor.
 (2)
has right Auslander–Reiten triangles.
If either of these properties holds, then the action of the Serre functor on objects coincides with .
Proof.

Let be an indecomposable object. By Serre duality there is a natural isomorphism
as In particular bimodules. has a one dimensional socle which corresponds to the map Define . and let , be a nonzero element of the socle of We claim that the associated triangle .
is an Auslander–Reiten triangle. Let be indecomposable, and let be a nonisomorphism. We have to show that the composition
is zero. Using Serre duality this amounts to showing that the composition
is zero. Since is a nonisomorphism, this is clear.
This is the interesting direction. As pointed out in Remark I.1.7 it is sufficient to construct the Serre functor on the full subcategory of consisting of the indecomposable objects. For an indecomposable object in we let be the object Let . be a nonzero element of representing the Auslander–Reiten triangle
Sublemma
Let and be indecomposable objects in Then the following hold: .
 (1)
For any nonzero there exists such that .
 (2)
For any nonzero there exists such that .
Proof.
 (1)

Using the properties of Auslander–Reiten triangles there is a morphism between the triangle determined by (the ARtriangle) and the triangle determined by .
The morphism labeled in the above diagram has the required properties.
 (2)

Without loss of generality we may assume that is not an isomorphism. We complete to a triangle
Then and since is nonzero, will not be split. Now we look at the following diagram:
Since is not split we have by (AR3) that Hence by the properties of triangles we have . for a map This proves what we want. .■
Having proved the sublemma we return to the main proof. For any indecomposable object choose a linear map
such that It follows from the sublemma that the pairing .
is nondegenerate. We can now finish our proof by invoking Proposition I.1.4.■
The following is now a direct consequence.
Theorem I.2.4
The following are equivalent:
 (1)
has a Serre functor.
 (2)
has Auslander–Reiten triangles.
Proof.
This follows from applying Proposition I.2.3 together with its dual version for left Serre functors.
■From now on we shall denote by also the equivalence where , is the Serre functor.
I.3. Serre functors on hereditary abelian categories
In this section we investigate the relationship between existence of Serre functors and of almost split sequences for hereditary abelian categories.
If is an finite abelian category, we say that it has a right Serre functor if this is the case for linear .
If has a right Serre functor and and are in then from the fact that , we deduce that only a finite number of can be nonzero.
Before we go on we recall some basic definitions (see [4]). For an indecomposable object in a is right almost split if for any nonisomorphism with indecomposable in there is some with The . is minimal right almost split if in addition is right minimal, that is, any with is an isomorphism. The concepts of left almost split and minimal left almost split are defined similarly. A nonsplit exact sequence is almost split if and are indecomposable and is (minimal) right almost split (or equivalently, is (minimal) left almost split). We say that has right almost split sequences if for every nonprojective indecomposable object there exists an almost split sequence ending in and for each indecomposable projective object , there is a minimal right almost split map Possession of left almost split sequences is defined similarly. We say that . has almost split sequences if it has both left and right almost split sequences.
Now let be an finite hereditary abelian category. linear
The following characterization of when we have a minimal right almost split map to a projective object or a minimal left almost split map from an injective object is easy to see.
Lemma I.3.1
 (1)
There is some minimal right almost split map to an indecomposable projective object if and only if has a unique maximal subobject .(If the conditions are satisfied, then the inclusion map is minimal right almost split.)
 (2)
There is a minimal left almost split map from an indecomposable injective object I if and only if I has a unique simple subobject .(If these conditions are satisfied, then the epimorphism is minimal left almost split.)
Note that if an indecomposable projective object has a maximal subobject, then it must be unique, and if is noetherian, then has a maximal subobject. Similarly if an indecomposable injective object has a simple subobject, then it must be unique.
Lemma I.3.2
Let The following are equivalent: .
 (1)
has right Auslander–Reiten triangles.
 (2)
has right almost split sequences and for every indecomposable projective the simple object possesses an injective hull in .
 (3)
has a right Serre functor.
If any of these conditions holds, then the right Auslander–Reiten triangles in are given by the shifts of the right almost split sequences in together with the shifts of the triangles of the form
where and are as in and is the composition The middle term . of this triangle is isomorphic to .
Proof.
This follows using Proposition I.2.3 and standard properties of almost split sequences and Auslander–Reiten triangles. We leave the proof to the reader.
■We now get the following main result on the connection between Serre duality and almost split sequences.
Theorem I.3.3
Let be an hereditary abelian category. finite
 (1)
has Serre duality if and only if has almost split sequences, and there is a oneone correspondence between indecomposable projective objects and indecomposable injective objects via , .
 (2)
If has no nonzero projective or injective objects, then has Serre duality if and only if it has almost split sequences.
Note that if is the category of finite dimensional representations over of the quiver then , is an hereditary abelian finite with almost split sequences. Since category has nonzero injective objects, but no nonzero projective objects, it does not have Serre duality.
Let and be the full subcategories of whose objects are respectively the projectives and the injectives in If . is a set of objects in then we denote by , the full subcategory of whose objects have no summands in .
When for a finite dimensional hereditary algebra we have an equivalence , where for an indecomposable object , in the object is the left hand term of the almost split sequence with right hand term Also we have the Nakayama functor . which is an equivalence of categories, where , for For the equivalence . where , is the Serre functor, we have and Hence . is in some sense put together from the two equivalences and (see Reference14). Using Lemma I.3.2 we see that the situation is similar in the general case.
Corollary I.3.4
Assume that has a right Serre functor Then the following hold: .
 (1)
defines a fully faithful functor We denote this functor by . “Nakayama functor” the).
 (2)
induces a fully faithful functor which we denote by , .
 (3)
If is indecomposable, then has one dimensional socle, both as left and as right module Let module. be a nonzero element in this socle. Then is simple, and furthermore is a projective cover of and is an injective hull of .
If is a Serre functor, then the functors and defined above are equivalences.
Proof.
That takes the indicated values on objects follows from the nature of the Auslander–Reiten triangles in (given by Lemma I.3.2). The assertion about fully faithfulness of and follows from the corresponding property of Part 3. follows by inspecting the triangle .EquationI.3.1. Finally that and are equivalences in the case that is a Serre functor follows by considering which is a left Serre functor. ,
■In the next section we will use the following result.
Lemma I.3.5
Assume that is a triangulated category with a structure in such a way that every object in lies in some Let . be the heart of the and assume that structure, for and Then . is a hereditary abelian category, and furthermore has a (right) Serre functor if and only if has a (right) Serre functor.
Proof.
To show that is hereditary we have to show that
Now is characterized by the property that it is an effaceable which coincides in degree zero with functor Hence we have to show that . is effaceable. This is clear in degree since there the functor is zero, and for it is also clear since then Reference7, p. 75.
Now standard arguments show that as additive categories and are equivalent to We don’t know if this equivalence yields an exact equivalence between . and but recall that the definition of a (right) Serre functor does not involve the triangulated structure. Hence if , has a (right) Serre functor, then so does and vice versa.
■II. Hereditary noetherian abelian categories with nonzero projective objects
In this section we classify the connected noetherian hereditary abelian categories with Serre functor, in the case where there are nonzero projective objects. finite
Let be a connected hereditary abelian noetherian category with Serre functor and nonzero projective objects. The structure of the projective objects gives rise to an associated quiver finite which satisfies special assumptions, including being locally finite and having no infinite path ending at any vertex. The category , contains the category of finitely presented representations of as a full subcategory, which may be different from Actually . is the full subcategory generated by projective objects. Hence is not in general generated by the projective objects, but it is generated by the preprojective objects. This provides a new interesting phenomenon.
Conversely, starting with a locally finite quiver having no infinite path ending at any vertex, the category is a hereditary abelian category with a right Serre functor (and right almost split sequences), but not necessarily having a Serre functor. We construct a hereditary abelian finite category with Serre functor finite containing as a full subcategory. We describe for which quivers the category is noetherian, and show that in the noetherian case all connected noetherian hereditary abelian categories with Serre duality are of this form. finite
In §II.1 we explain the construction of via inverting a right Serre functor. An alternative approach using derived categories is discussed in §II.3. After learning about our results Claus Ringel found yet another approach towards the construction of Reference24.
In §II.2 we show that if is generated by the preprojective objects, then is equivalent to some That . is generated by preprojective objects when is connected and noetherian is proved in Section II.4, along with showing for which quivers the category is noetherian.
II.1. Hereditary abelian categories constructed from quivers
For a quiver denote by the category of finitely presented representations of Under some additional assumptions on . we construct a hereditary abelian category finite with Serre functor. The category contains as a full subcategory, and is obtained from by formally inverting a right Serre functor.
Let be a quiver with the following properties:
 (P1)
is locally finite, that is, every vertex in is adjacent to only a finite number of other vertices.
 (P2)
There is no infinite path in of the form (in particular there are no oriented cycles).
Note that (P1), (P2) imply the following.
Lemma II.1.1
 (1)
If is a vertex in then there are only a finite number of paths ending in , .
 (2)
If and are vertices in then there are only a finite number of paths from , to .
Proof.
Part 2. is an obvious consequence of 1., so we prove 1. Assume there is an infinite number of paths ending in By (P1) there must be an arrow . such that there is an infinite number of paths ending in Repeating this we obtain an infinite path . such that there is an infinite number of paths ending in every The existence of such an infinite path contradicts (P2). .
■For a vertex in we denote by , and respectively the corresponding projective, injective and simple object in the category of all , By Lemma representations.II.1.1 the objects have finite length. We also have a canonical isomorphism
since both of these vector spaces have as basis the paths from to .
The category is too big for what we want. For example it is not As will be clear from the considerations in the next section the natural subcategory to consider is finite. It is easy to see that this is a hereditary abelian subcategory of . and furthermore by Lemma ,II.1.1 it has finite dimensional Thanks to (P1) the simple and hence the finite dimensional representations are contained in ’s. Hence in particular . for every vertex .
Let and be the full subcategories of consisting respectively of the finite direct sums of the and the finite direct sums of the Let . be the equivalence obtained from additively extending EquationII.1.1. We denote also by the corresponding equivalence Finally denoting by . the composition
we have the following.
Lemma II.1.2
is a right Serre functor.
Proof.
Let Then we need to construct natural isomorphisms .
Since and are finite complexes of projectives we can reduce to the case and So we need natural isomorphisms .
Again both of these vector spaces have a natural basis given by the paths from to This leads to the required isomorphisms. .
■Under the assumptions (P1), (P2) on the quiver we now know that is a hereditary abelian category with a right Serre functor finite such that the image of the projective objects in are injective objects in of finite length. Hence the following result applies to this setting.
Theorem II.1.3
Let be an abelian hereditary category with a right Serre functor finite and enough projectives. Denote the full subcategory of projective objects in by and assume that consists of (injective) objects of finite length. Then there exists an abelian hereditary category finite with the following properties:
 (1)
There exists a full faithful exact embedding .
 (2)
The injectives and projectives in are given by and .
 (3)
possesses a Serre functor which extends in such a way that there is a natural equivalence where we have denoted the derived functor of also by and the extended Serre functor also by .
 (4)
For every indecomposable object there exists such that is defined and lies in where , denotes the functor (see Corollary I.3.4.2) induced by .
Furthermore a quadruple with these properties is unique in the appropriate sense.
Proof.
First we show that if exists satisfying properties then it is unique. This proof will in particular tell us how to construct , We then show that this construction always yields a category . with the required properties.
Since and since the objects in remain projective in by property 2., it follows that the derived functor of is fully faithful. It is also clear that is closed inside under the formation of cones.
Let be the extended Serre functor. If then , can be obtained by starting with objects in and repeatedly taking cones. It then follows from 4. that for we have Hence the triple . satisfies the hypotheses of Proposition I.1.8. Thus we obtain .
According to 4. every indecomposable object in will be of the form with Implicit in the notation . is the assumption that is defined for that is, , for In addition we may assume that . is minimal. Thus either or else for The last case is equivalent to .
for .
If then the same is true for , Hence we have to impose .EquationII.1.4 only for .
We conclude that the indecomposable objects in are of the following form:
 (C1)
The indecomposable objects in .
 (C2)
Objects of the form where and is an indecomposable object in where , .
This completes the determination of as an additive subcategory of and finishes the proof of the uniqueness. ,
Let us now assume that is the additive subcategory of whose indecomposable objects are given by (C1), (C2). We have to show that is a hereditary abelian category satisfying 1.4.
Since has a Serre functor, it has Auslander–Reiten triangles. Below we will need the triangle associated to Using the criterion given in Proposition .I.2.1 we can compute this triangle in So according to Lemma .I.3.2 the requested triangle is of the form
We now define a on structure Using the fact that . is hereditary we easily obtain that as additive categories We now define .
We claim that this is a The only nontrivial axiom we have to verify is that structure.
So this amounts to showing that for and for We separate this into four cases. .
Case 1
and fall under (C1). This case is trivial.
Case 2
falls under (C1) and falls under (C2). Thus we have for some in and We want to show that . for