American Mathematical Society

Noetherian hereditary abelian categories satisfying Serre duality

By I. Reiten and M. Van den Bergh

Abstract

In this paper we classify upper E x t -finite noetherian hereditary abelian categories over an algebraically closed field k 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 non-zero 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 k will be an algebraically closed field and all rings and categories in this paper will be k -linear.

If upper A is a ring, then mod left-parenthesis upper A right-parenthesis will be the category of finitely generated right upper A -modules. Similarly if upper R is a double-struck upper Z -graded ring, then g r left-parenthesis upper R right-parenthesis will be the category of finitely generated graded right modules with degree zero morphisms, and upper G r left-parenthesis upper R right-parenthesis the category of all graded right upper R -modules. If upper R is noetherian, then following Reference3 t o r s left-parenthesis upper R right-parenthesis will be the full subcategory of g r left-parenthesis upper R right-parenthesis consisting of graded modules with right bounded grading. Also following Reference3 we put q g r left-parenthesis upper R right-parenthesis equals g r left-parenthesis upper R right-parenthesis slash t o r s left-parenthesis upper R right-parenthesis .

For an abelian category script upper C we denote by upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis the bounded derived category of script upper C .

Introduction

One of the goals of non-commutative algebraic geometry is to obtain an understanding of k -linear abelian categories script upper C , for a field k , which have properties close to those of the category of coherent sheaves over a non-singular proper scheme. Hence some obvious properties one may impose on script upper C in this context are the following:

script upper C is upper E x t -finite, i.e. dimension Subscript k Baseline upper E x t Superscript i Baseline left-parenthesis upper A comma upper B right-parenthesis less-than normal infinity

for all upper A comma upper B element-of script upper C and for all i .

script upper C has homological dimension n less-than normal infinity , i.e. upper E x t Superscript i Baseline left-parenthesis upper A comma upper B right-parenthesis equals 0

for upper A comma upper B element-of script upper C and i greater-than n , and n is minimal with this property.

Throughout this paper k will denote a field, and even though it is not always necessary we will for simplicity assume that k is algebraically closed. All categories will be k -linear. When we say that script upper C is upper E x t -finite, it will be understood that this is with respect to the field k .

In most of this paper we will assume that script upper C is an upper E x t -finite abelian category of homological dimension at most 1, in which case we say that script upper C is hereditary.

A slightly more subtle property of non-singular proper schemes is Serre duality. Let upper X be a non-singular proper scheme over k of dimension n , and let coh of left-parenthesis upper X right-parenthesis denote the category of coherent script upper O Subscript upper X -modules. Then the classical Serre duality theorem asserts that for script upper F element-of coh of left-parenthesis upper X right-parenthesis there are natural isomorphisms

upper H Superscript i Baseline left-parenthesis upper X comma script upper F right-parenthesis approximately-equals upper E x t Superscript n minus i Baseline left-parenthesis script upper F comma omega Subscript upper X Baseline right-parenthesis Superscript asterisk

where left-parenthesis minus right-parenthesis Superscript asterisk Baseline equals upper H o m Subscript k Baseline left-parenthesis minus comma k right-parenthesis .

A very elegant reformulation of Serre duality was given by Bondal and Kapranov in Reference8. It says that for any script upper E comma script upper F element-of upper D Superscript b Baseline left-parenthesis coh of left-parenthesis upper X right-parenthesis right-parenthesis there exist natural isomorphisms

upper H o m Subscript upper D Sub Superscript b Subscript left-parenthesis coh of left-parenthesis upper X right-parenthesis right-parenthesis Baseline left-parenthesis script upper E comma script upper F right-parenthesis approximately-equals upper H o m Subscript upper D Sub Superscript b Subscript left-parenthesis coh of left-parenthesis upper X right-parenthesis right-parenthesis Baseline left-parenthesis script upper F comma script upper E circled-times omega Subscript upper X Baseline left-bracket n right-bracket right-parenthesis Superscript asterisk Baseline period

Stated in this way the concept of Serre duality can be generalized to certain abelian categories.

script upper C satisfies Serre duality if it has a so-called Serre functor. The latter is by definition an autoequivalence upper F colon upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis right-arrow upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis such that there are isomorphisms upper H o m left-parenthesis upper A comma upper B right-parenthesis approximately-equals upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk

which are natural in upper A comma upper B .

On the other hand hereditary abelian upper E x t -finite categories script upper C with the additional property of having a tilting object have been important for the representation theory of finite dimensional algebras. Recall that upper T is a tilting object in script upper C if upper E x t Superscript 1 Baseline left-parenthesis upper T comma upper T right-parenthesis equals 0 , and if upper H o m left-parenthesis upper T comma upper X right-parenthesis equals 0 equals upper E x t Superscript 1 Baseline left-parenthesis upper T comma upper X right-parenthesis implies that upper X is 0. These categories script upper C are important in the study of quasitilted algebras, which by definition are the algebras of the form upper E n d Subscript script upper C Baseline left-parenthesis upper T right-parenthesis for a tilting object upper T 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 upper E x t -finite hereditary abelian categories with tilting object have this property Reference16.

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)

script upper C has a Serre functor if and only if upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis has Auslander–Reiten triangles (as defined in Reference14).

(2)

If script upper C is hereditary, then script upper C has a Serre functor if and only if script upper C has almost split sequences and there is a one-one correspondence between the indecomposable projective objects upper P and the indecomposable injective objects upper I , such that the simple top of upper P is isomorphic to the socle of upper I .

Hence upper E x t -finite hereditary abelian categories with Serre duality are of interest both for non-commutative algebraic geometry and for the representation theory of finite dimensional algebras. The main result of this paper is the classification of the noetherian ones.

To be able to state our result we first give a list of hereditary abelian categories satisfying Serre duality.

(a)

If script upper C consists of the finite dimensional nilpotent representations of the quiver upper A overTilde Subscript n or of the quiver upper A Subscript normal infinity Superscript normal infinity , with all arrows oriented in the same direction, then it is classical that script upper C has almost split sequences, and hence Serre duality.

(b)

Let upper X be a non-singular projective connected curve over k with function field upper K , and let script upper O be a sheaf of hereditary script upper O Subscript upper X -orders in upper M Subscript n Baseline left-parenthesis upper K right-parenthesis (see Reference21). Then one proves exactly as in the commutative case that coh of left-parenthesis script upper O right-parenthesis satisfies Serre duality.

(c)

Let upper Q be either upper A Subscript normal infinity Superscript normal infinity or upper D Subscript normal infinity with zig-zag orientation. It is shown in §III.3 that there exists a noetherian hereditary abelian category script upper C which is derived equivalent to the category r e p left-parenthesis upper Q right-parenthesis of finitely presented representations of upper Q , and which has no non-zero projectives or injectives. Depending on upper Q we call this category the double-struck upper Z upper A Subscript normal infinity Superscript normal infinity or the double-struck upper Z upper D Subscript normal infinity category. Since Serre duality is defined in terms of the derived category, it follows that script upper C satisfies Serre duality.

If upper Q equals upper A Subscript normal infinity Superscript normal infinity , then script upper C is nothing but the category g r Subscript double-struck upper Z squared Baseline left-parenthesis k left-bracket x comma y right-bracket right-parenthesis slash left-parenthesis finite length right-parenthesis considered in Reference30. If upper Q equals upper D Subscript normal infinity and char k not-equals 2 , then script upper C is a skew version of the double-struck upper Z upper A Subscript normal infinity Superscript normal infinity category (see §III.3). The double-struck upper Z upper A Subscript normal infinity Superscript normal infinity category and the double-struck upper Z upper D Subscript normal infinity category have also been considered by Lenzing.

(d)

We now come to more subtle examples (see §II). Let upper Q be a connected quiver. Then for a vertex x element-of upper Q we have a corresponding projective representation upper P Subscript x and an injective representation upper I Subscript x . If upper Q is locally finite and there is no infinite path ending at any vertex, the functor upper P Subscript x Baseline right-arrow from bar upper I Subscript x may be derived to yield a fully faithful endofunctor upper F colon upper D Superscript b Baseline left-parenthesis r e p left-parenthesis upper Q right-parenthesis right-parenthesis right-arrow upper D Superscript b Baseline left-parenthesis r e p left-parenthesis upper Q right-parenthesis right-parenthesis . Then upper F behaves like a Serre functor, except that it is not in general an autoequivalence. We call such upper F 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 ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis which satisfies Serre duality. Under the additional hypotheses that upper Q consists of a subquiver upper Q Subscript o with no path of infinite length, with rays attached to vertices of upper Q Subscript o , then ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis turns out to be noetherian (see Theorem II.4.3). Here we mean by a ray an upper A Subscript normal infinity quiver with no vertex which is a sink. An interesting feature of the noetherian categories ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis , 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 script upper C is connected if it cannot be non-trivially written as a direct sum script upper C 1 circled-plus script upper C 2 .

Theorem B

Let script upper C be a connected noetherian upper E x t -finite hereditary abelian category satisfying Serre duality. Then script upper C is one of the categories described in (a)–(d) above.

The cases (a), (b), (c) are those where there are no non-zero projective objects. Those in (a) are exactly the script upper C where all objects have finite length. For the script upper C having some objects of infinite length, then either all indecomposable objects of finite length have finite tau -period (case (b)) or all have infinite tau -period (case (c)). Here the object tau upper C is defined by the almost split sequence 0 right-arrow tau upper C right-arrow upper B right-arrow upper C right-arrow 0 for upper C indecomposable in script upper C , and we have tau upper C equals upper F left-parenthesis upper C right-parenthesis left-bracket negative 1 right-bracket .

Under the additional assumption that script upper C has a tilting object, such a classification was given in Reference19. The only cases are the categories of finitely generated modules mod upper A for a finite dimensional indecomposable hereditary k -algebra upper A and the categories coh of double-struck upper X of coherent sheaves on a weighted projective line double-struck upper X in the sense of Reference12. From the point of view of the above list of examples the first case corresponds to the finite quivers upper Q in (d), in which case ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis equals r e p left-parenthesis upper Q right-parenthesis is equivalent to mod left-parenthesis k upper Q right-parenthesis , where k upper Q is the path algebra of upper Q over k . The categories coh of double-struck upper X are a special case of (b), corresponding to the case where the projective curve is double-struck upper P Superscript 1 (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 ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis , 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 ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis to be noetherian, and furthermore we prove a decomposition theorem which states that an upper E x t -finite 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 non-zero projectives or injectives.

(3)

We are now reduced to the case where there are no non-zero projectives or injectives. The case where all objects have finite length is treated in §III.1.

(4)

The case where there are no non-zero 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:

( alpha )

The simple objects are tau -periodic. In that case, using the results in Reference3, we show that script upper C is of the form q g r left-parenthesis upper R right-parenthesis for upper R a two-dimensional commutative graded ring, where q g r left-parenthesis upper R right-parenthesis is the quotient category g r upper R /finite length. Using Reference1 it then follows that script upper C is of the form (b).

( beta )

The simple objects are not tau -periodic. We show that if such script upper C exists, then it is characterized by the fact that it has either one or two tau -orbits of simple objects. Since the double-struck upper Z upper A Subscript normal infinity Superscript normal infinity and double-struck upper Z upper D Subscript normal infinity category have this property, we are done.

Our methods for constructing the new hereditary abelian categories ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis 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 non-commutative curves considered in Reference30 are noetherian hereditary abelian categories of Krull-dimension one which in general do not satisfy Serre duality (except for the special case listed in (c)). If script upper C 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 script upper C 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 upper H colon upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis right-arrow mod left-parenthesis k right-parenthesis is of finite type if for every upper A element-of upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis only a finite number of upper H left-parenthesis upper A left-bracket n right-bracket right-parenthesis are non-zero. We have already defined what it means for script upper C to have homological dimension n less-than normal infinity . It will be convenient to say more generally that script upper C has finite homological dimension if for any upper A comma upper B in script upper C there is at most a finite number of i with upper E x t Superscript i Baseline left-parenthesis upper A comma upper B right-parenthesis not-equals 0 . In particular, the analogue of this definition makes sense for triangulated categories.

Let script upper C be an upper E x t -finite abelian category of finite homological dimension. Then script upper C is saturated if every cohomological functor upper H colon upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis right-arrow mod left-parenthesis k right-parenthesis of finite type is of the form upper H o m left-parenthesis upper A comma minus right-parenthesis (i.e. upper H is representable).

It is easy to show that a saturated category satisfies Serre duality. It was shown in Reference8 that coh of left-parenthesis upper X right-parenthesis for upper X a non-singular projective scheme is saturated and that saturation also holds for categories of the form mod left-parenthesis normal upper Lamda right-parenthesis with normal upper Lamda a finite dimensional algebra. Inspired by these results we prove the following result in §V.1.

Theorem C

Assume that script upper C is a saturated connected noetherian upper E x t -finite hereditary abelian category. Then script upper C has one of the following forms:

(1)

mod left-parenthesis normal upper Lamda right-parenthesis where normal upper Lamda is a connected finite dimensional hereditary k -algebra.

(2)

coh of left-parenthesis script upper O right-parenthesis where script upper O is a sheaf of hereditary script upper O Subscript upper X -orders (see (b) above) over a non-singular connected projective curve upper X .

It is easy to see that the hereditary abelian categories listed in the above theorem are of the form q g r left-parenthesis upper R right-parenthesis . We refer the reader to Reference9 (see also §V.2) where it is shown in reasonable generality that abelian categories of the form q g r left-parenthesis upper R right-parenthesis 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 upper D left-parenthesis upper E x t Subscript normal upper Lamda Superscript 1 Baseline left-parenthesis upper C comma upper A right-parenthesis right-parenthesis asymptotically-equals ModifyingAbove upper H o m With bar left-parenthesis upper A comma upper D upper T r upper C right-parenthesis for artin algebras (where upper D equals upper H o m Subscript k Baseline left-parenthesis minus comma k right-parenthesis ), 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 upper D Superscript b Baseline left-parenthesis mod normal upper Lamda right-parenthesis when normal upper Lamda is a k -algebra of finite global dimension Reference14. 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 script upper C is hereditary, then script upper C has Serre duality if and only if it has almost split sequences and there is a one-one correspondence between indecomposable projective objects upper P and indecomposable injective objects upper I , such that upper P modulo its unique maximal subobject is isomorphic to the socle of upper I .

I.1. Preliminaries on Serre duality

Let script upper A be a k -linear upper H o m -finite additive category. A right Serre functor is an additive functor upper F colon script upper A right-arrow script upper A together with isomorphisms

StartLayout 1st Row with Label left-parenthesis upper I .1 .1 right-parenthesis EndLabel eta Subscript upper A comma upper B Baseline colon upper H o m left-parenthesis upper A comma upper B right-parenthesis right-arrow upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk EndLayout

for any upper A comma upper B element-of script upper A which are natural in upper A and upper B . A left Serre functor is a functor upper G colon script upper A right-arrow script upper A together with isomorphisms

StartLayout 1st Row with Label left-parenthesis upper I .1 .2 right-parenthesis EndLabel zeta Subscript upper A comma upper B Baseline colon upper H o m left-parenthesis upper A comma upper B right-parenthesis right-arrow upper H o m left-parenthesis upper G upper B comma upper A right-parenthesis Superscript asterisk EndLayout

for any upper A comma upper B element-of script upper A which are natural in upper A and upper B . 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 eta Subscript upper A Baseline colon upper H o m left-parenthesis upper A comma upper F upper A right-parenthesis right-arrow k be given by eta Subscript upper A comma upper A Baseline left-parenthesis i d Subscript upper A Baseline right-parenthesis , and let f element-of upper H o m left-parenthesis upper A comma upper B right-parenthesis . Looking at the commutative diagram (which follows from the naturality of eta Subscript upper A comma upper B in upper B )

StartLayout 1st Row 1st Column upper H o m left-parenthesis upper A comma upper A right-parenthesis 2nd Column right-arrow Overscript eta Subscript upper A comma upper A Endscripts 3rd Column upper H o m left-parenthesis upper A comma upper F upper A right-parenthesis Superscript asterisk 2nd Row 1st Column upper H o m left-parenthesis upper A comma f right-parenthesis down-arrow 2nd Column Blank 3rd Column upper H o m left-parenthesis f comma upper F upper A right-parenthesis Superscript asterisk Baseline down-arrow 4th Column Blank 3rd Row 1st Column upper H o m left-parenthesis upper A comma upper B right-parenthesis 2nd Column right-arrow Overscript eta Subscript upper A comma upper B Endscripts 3rd Column upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk EndLayout

we find for g element-of upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis that

StartLayout 1st Row with Label left-parenthesis upper I .1 .3 right-parenthesis EndLabel eta Subscript upper A comma upper B Baseline left-parenthesis f right-parenthesis left-parenthesis g right-parenthesis equals eta Subscript upper A Baseline left-parenthesis g f right-parenthesis period EndLayout

Similarly by the naturality of eta Subscript upper A comma upper B in upper A we obtain the following commutative diagram:

StartLayout 1st Row 1st Column upper H o m left-parenthesis upper B comma upper B right-parenthesis 2nd Column right-arrow Overscript eta Subscript upper B comma upper B Endscripts 3rd Column upper H o m left-parenthesis upper B comma upper F upper B right-parenthesis Superscript asterisk 2nd Row 1st Column upper H o m left-parenthesis f comma upper B right-parenthesis down-arrow 2nd Column Blank 3rd Column upper H o m left-parenthesis upper B comma upper F f right-parenthesis Superscript asterisk Baseline down-arrow 4th Column Blank 3rd Row 1st Column upper H o m left-parenthesis upper A comma upper B right-parenthesis 2nd Column right-arrow Overscript eta Subscript upper A comma upper B Endscripts 3rd Column upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk EndLayout

This yields for g element-of upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis the formula

StartLayout 1st Row with Label left-parenthesis upper I .1 .4 right-parenthesis EndLabel eta Subscript upper A comma upper B Baseline left-parenthesis f right-parenthesis left-parenthesis g right-parenthesis equals eta Subscript upper B Baseline left-parenthesis upper F left-parenthesis f right-parenthesis g right-parenthesis EndLayout

and we get the following description of the functor upper F .

Lemma I.1.1

The following composition coincides with upper F :

upper H o m left-parenthesis upper A comma upper B right-parenthesis right-arrow Overscript eta Subscript upper A comma upper B Baseline Endscripts upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk Baseline right-arrow Overscript left-parenthesis eta Subscript upper B comma upper F upper A Superscript asterisk Baseline right-parenthesis Superscript negative 1 Baseline Endscripts upper H o m left-parenthesis upper F upper A comma upper F upper B right-parenthesis period

Proof.

To prove this we need to show that for f element-of upper H o m left-parenthesis upper A comma upper B right-parenthesis and g element-of upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis one has eta Subscript upper A comma upper B Baseline left-parenthesis f right-parenthesis left-parenthesis g right-parenthesis equals eta Subscript upper B comma upper F upper A Superscript asterisk Baseline left-parenthesis upper F f right-parenthesis left-parenthesis g right-parenthesis . Thanks to the formulas EquationI.1.3, EquationI.1.4 we obtain eta Subscript upper A comma upper B Baseline left-parenthesis f right-parenthesis left-parenthesis g right-parenthesis equals eta Subscript upper B Baseline left-parenthesis upper F left-parenthesis f right-parenthesis g right-parenthesis and also eta Subscript upper B comma upper F upper A Superscript asterisk Baseline left-parenthesis upper F f right-parenthesis left-parenthesis g right-parenthesis equals eta Subscript upper B comma upper F upper A Baseline left-parenthesis g right-parenthesis left-parenthesis upper F f right-parenthesis equals eta Subscript upper B Baseline left-parenthesis upper F left-parenthesis f right-parenthesis g right-parenthesis . Thus we obtain indeed the correct result.

We have the following immediate consequence.

Corollary I.1.2

If upper F is a right Serre functor, then upper F is fully faithful.

Also note the following basic properties.

Lemma I.1.3

(1)

If upper F and upper F prime are right Serre functors, then they are naturally isomorphic.

(2)

script upper A has a right Serre functor if and only if upper H o m left-parenthesis upper A comma minus right-parenthesis Superscript asterisk is representable for all upper A element-of script upper A .

From the above discussion it follows that there is a lot of redundancy in the data left-parenthesis upper F comma left-parenthesis eta Subscript upper A comma upper B Baseline right-parenthesis Subscript upper A comma upper B Baseline right-parenthesis . In fact we have the following.

Proposition I.1.4

In order to give left-parenthesis upper F comma left-parenthesis eta Subscript upper A comma upper B Baseline right-parenthesis Subscript upper A comma upper B Baseline right-parenthesis it is necessary and sufficient to give the action of upper F on objects, as well as k -linear maps eta Subscript upper A Baseline colon upper H o m left-parenthesis upper A comma upper F upper A right-parenthesis right-arrow k such that the composition

StartLayout 1st Row with Label left-parenthesis upper I .1 .5 right-parenthesis EndLabel upper H o m left-parenthesis upper A comma upper B right-parenthesis times upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis right-arrow upper H o m left-parenthesis upper A comma upper F upper A right-parenthesis right-arrow Overscript eta Subscript upper A Baseline Endscripts k EndLayout

yields a non-degenerate pairing for all upper A comma upper B element-of script upper A . If we are given eta Subscript upper A , then eta Subscript upper A comma upper B is obtained from the pairing EquationI.1.5. Furthermore the action upper F on maps

upper F colon upper H o m left-parenthesis upper A comma upper B right-parenthesis right-arrow upper H o m left-parenthesis upper F upper A comma upper F upper B right-parenthesis

is defined by the property that for f element-of upper H o m left-parenthesis upper A comma upper B right-parenthesis we have eta Subscript upper A Baseline left-parenthesis g f right-parenthesis equals eta Subscript upper B Baseline left-parenthesis upper F left-parenthesis f right-parenthesis g right-parenthesis for all g element-of upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis .

Proof.

It is clear from the previous discussion that the data left-parenthesis upper F comma left-parenthesis eta Subscript upper A comma upper B Baseline right-parenthesis Subscript upper A comma upper B Baseline right-parenthesis gives rise to left-parenthesis eta Subscript upper A Baseline right-parenthesis Subscript upper A with the required properties. So conversely assume that we are given left-parenthesis eta Subscript upper A Baseline right-parenthesis Subscript upper A and the action of upper F on objects. We define left-parenthesis eta Subscript upper A comma upper B Baseline right-parenthesis Subscript upper A comma upper B and the action of upper F on maps as in the statement of the proposition.

We first show that upper F is a functor. Indeed let upper A comma upper B comma upper C element-of script upper A and assume that there are maps g colon upper A right-arrow upper B and h colon upper B right-arrow upper C . Then for all f element-of upper H o m left-parenthesis upper C comma upper F upper A right-parenthesis we have eta Subscript upper A Baseline left-parenthesis f h g right-parenthesis equals eta Subscript upper C Baseline left-parenthesis upper F left-parenthesis h g right-parenthesis f right-parenthesis , but also eta Subscript upper A Baseline left-parenthesis f h g right-parenthesis equals eta Subscript upper B Baseline left-parenthesis upper F left-parenthesis g right-parenthesis f h right-parenthesis equals eta Subscript upper C Baseline left-parenthesis upper F left-parenthesis h right-parenthesis upper F left-parenthesis g right-parenthesis f right-parenthesis . Thus by non-degeneracy we have upper F left-parenthesis h g right-parenthesis equals upper F left-parenthesis h right-parenthesis upper F left-parenthesis g right-parenthesis .

It is easy to see that the pairing EquationI.1.5 defines an isomorphism

eta Subscript upper A comma upper B Baseline colon upper H o m left-parenthesis upper A comma upper B right-parenthesis right-arrow upper H o m left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk Baseline colon f right-arrow from bar eta Subscript upper A Baseline left-parenthesis negative f right-parenthesis

which is natural in upper A and upper B .

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

script upper A 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

script upper A has a Serre functor if and only if the functors upper H o m left-parenthesis upper A comma minus right-parenthesis Superscript asterisk and upper H o m left-parenthesis minus comma upper A right-parenthesis Superscript asterisk are representable for all upper A element-of script upper A .

Remark I.1.7

In the sequel script upper A 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 eta Subscript upper A comma upper B Baseline comma eta Subscript upper A Baseline comma upper F , etc. on the full subcategory of script upper A 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 script upper A be an additive category as above, and let upper U colon script upper A right-arrow script upper A be a fully faithful additive endofunctor. Then there exists an additive category script upper B with the following properties:

(1)

There is a fully faithful functor i colon script upper A right-arrow script upper B .

(2)

There is an autoequivalence upper U overbar colon script upper B right-arrow script upper B together with a natural isomorphism nu colon upper U overbar i right-arrow i upper U .

(3)

For every object upper B element-of script upper B there is some b element-of double-struck upper N such that upper U overbar Superscript b Baseline upper B is isomorphic to i left-parenthesis upper A right-parenthesis with upper A element-of script upper A .

Furthermore a quadruple left-parenthesis script upper B comma i comma upper U overbar comma nu right-parenthesis with these properties is unique (in the appropriate sense).

Proof.

Let us sketch the construction of script upper B . The uniqueness will be clear.

The objects in script upper B are formally written as upper U Superscript negative a Baseline upper A with upper A element-of script upper A and a element-of double-struck upper Z . A morphism upper U Superscript negative a Baseline upper A right-arrow upper U Superscript negative b Baseline upper B is formally written as upper U Superscript negative c Baseline f with f element-of upper H o m Subscript script upper A Baseline left-parenthesis upper U Superscript c minus a Baseline upper A comma upper U Superscript c minus b Baseline upper B right-parenthesis where c element-of double-struck upper Z is such that c greater-than-or-equal-to a , c greater-than-or-equal-to b . We identify upper U Superscript negative c Baseline f with upper U Superscript negative c minus 1 Baseline left-parenthesis upper U f right-parenthesis .

The functor i is defined by i left-parenthesis upper A right-parenthesis equals upper U Superscript 0 Baseline upper A and the functor upper U overbar is defined by ModifyingAbove upper U With bar left-parenthesis upper U Superscript negative a Baseline right-parenthesis left-parenthesis upper A right-parenthesis equals upper U Superscript negative a plus 1 Baseline left-parenthesis upper A right-parenthesis . It is clear that these have the required properties.

The following lemma provides a complement to this proposition in the case that script upper A is triangulated.

Lemma I.1.9

Assume that in addition to the usual hypotheses one has that script upper A is triangulated. Let left-parenthesis script upper B comma i comma upper U overbar comma nu right-parenthesis be as in the previous proposition. Then there is a unique way to make script upper B into a triangulated category such that i and upper U overbar are exact.

Proof.

If we require exactness of i and upper U overbar , then there is only one way to make script upper B into a triangulated category. First we must define the shift functor by left-parenthesis upper U Superscript negative a Baseline upper A right-parenthesis left-bracket 1 right-bracket equals upper U Superscript negative a Baseline left-parenthesis upper A left-bracket 1 right-bracket right-parenthesis and then the triangles in script upper B must be those diagrams that are isomorphic to

upper U Superscript negative c Baseline upper A right-arrow Overscript upper U Superscript negative c Baseline f Endscripts upper U Superscript negative c Baseline upper B right-arrow Overscript upper U Superscript negative c Baseline g Endscripts upper U Superscript negative c Baseline upper C right-arrow Overscript upper U Superscript negative c Baseline h Endscripts upper U Superscript negative c Baseline upper A left-bracket 1 right-bracket

where

upper A right-arrow Overscript f Endscripts upper B right-arrow Overscript g Endscripts upper C right-arrow Overscript h Endscripts upper A left-bracket 1 right-bracket

is a triangle in script upper A (note that the exactness of upper U overbar is equivalent to that of upper U overbar Superscript negative 1 ).

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 upper U overbar we can translate such problems into ones involving only objects in script upper A . Then we use the triangulated structure of script upper A and afterwards we go back to the original problem by applying a negative power of upper U overbar .

In the sequel we will denote by upper U Superscript negative normal infinity Baseline script upper A the category script upper B which was constructed in Proposition I.1.8. Furthermore we will consider script upper A as a subcategory of upper U Superscript negative normal infinity Baseline script upper A through the functor i . Finally we will usually write upper U for the extended functor upper U overbar .

Below we will only be interested in the special case where upper U equals upper F is a right Serre functor on script upper A . In that case we have the following.

Proposition I.1.10

The canonical extension of upper F to upper F Superscript negative normal infinity Baseline script upper A is a Serre functor.

Proof.

By construction upper F is an automorphism on upper F Superscript negative normal infinity Baseline script upper A . To prove that upper F is a Serre functor we have to construct suitable maps eta Subscript upper F Sub Superscript negative a Subscript upper A comma upper F Sub Superscript negative b Subscript upper B . Pick c greater-than-or-equal-to a , c greater-than-or-equal-to b . Then we have

StartLayout 1st Row 1st Column upper H o m Subscript upper F Sub Superscript negative normal infinity Subscript script upper A Baseline left-parenthesis upper F Superscript negative a Baseline upper A comma upper F Superscript negative b Baseline upper B right-parenthesis 2nd Column equals upper H o m Subscript script upper A Baseline left-parenthesis upper F Superscript c minus a Baseline upper A comma upper F Superscript c minus b Baseline upper B right-parenthesis 2nd Row 1st Column Blank 2nd Column approximately-equals Overscript eta Subscript upper F Sub Superscript c minus a Subscript upper A comma upper F Sub Superscript c minus b Subscript upper B Baseline Endscripts upper H o m Subscript script upper A Baseline left-parenthesis upper F Superscript c minus b Baseline upper B comma upper F Superscript c minus a plus 1 Baseline upper A right-parenthesis Superscript asterisk Baseline 3rd Row 1st Column Blank 2nd Column equals upper H o m Subscript upper F Sub Superscript negative normal infinity Subscript script upper A Baseline left-parenthesis upper F Superscript negative b Baseline upper B comma upper F Superscript negative a plus 1 Baseline upper A right-parenthesis Superscript asterisk Baseline period EndLayout

We define eta Subscript upper F Sub Superscript negative a Subscript upper A comma upper F Sub Superscript negative b Subscript upper B as the composition of these maps. It follows easily from Lemma I.1.1 that the constructed map is independent of c , and it is clear that eta Subscript upper F Sub Superscript negative a Subscript upper A comma upper F Sub Superscript negative b Subscript upper B has the required properties.

We shall also need the following easily verified fact.

Lemma I.1.11

If script upper A equals script upper A 1 circled-plus script upper A 2 is a direct sum of additive categories, then a (right) Serre functor on script upper A restricts to (right) Serre functors on script upper A 1 and script upper A 2 .

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 upper H o m -finite Krull–Schmidt k -categories. Hence the existence of a Serre functor is equivalent to the existence of Auslander–Reiten triangles.

In the sequel script upper A is a upper H o m -finite k -linear Krull-Schmidt triangulated category. Following Reference14 a triangle upper A right-arrow Overscript f Endscripts upper B right-arrow Overscript g Endscripts upper C right-arrow Overscript h Endscripts upper A left-bracket 1 right-bracket in script upper A is called an Auslander–Reiten triangle if the following conditions are satisfied:

(AR1)

upper A and upper C are indecomposable.

(AR2)

h not-equals 0 .

(AR3)

If upper D is indecomposable, then for every non-isomorphism t colon upper D right-arrow upper C we have h t equals 0 .

It is shown in Reference14 that, assuming (AR1) and (AR2), then (AR3) is equivalent to

(AR3) prime

If upper D prime is indecomposable, then for every non-isomorphism s colon upper A right-arrow upper D prime we have s h left-bracket negative 1 right-bracket equals 0 .

We say that right Auslander–Reiten triangles exist in script upper A if for all indecomposables upper C element-of script upper A there is a triangle satisfying the conditions above. Existence of left Auslander–Reiten triangles is defined in a similar way, and we say that script upper A has Auslander–Reiten triangles if it has both right and left Auslander–Reiten triangles. (Note that in Reference14 one says that script upper A has Auslander–Reiten triangles if it has right Auslander–Reiten triangles in our terminology.)

It is shown in Reference14, §4.3 that given upper C the corresponding Auslander–Reiten triangle upper A right-arrow upper B right-arrow upper C right-arrow upper A left-bracket 1 right-bracket is unique up to isomorphism of triangles. (By duality a similar result holds if upper A is given.) For a given indecomposable upper C we let tau overTilde upper C be an arbitrary object in script upper A , isomorphic to upper A in the Auslander–Reiten triangle corresponding to upper C .

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 script upper A has right Auslander–Reiten triangles, and assume that we have a triangle in script upper A

StartLayout 1st Row with Label left-parenthesis upper I .2 .1 right-parenthesis EndLabel upper A right-arrow Overscript f Endscripts upper B right-arrow Overscript g Endscripts upper C right-arrow Overscript h Endscripts upper A left-bracket 1 right-bracket EndLayout

with upper A and upper C indecomposable and h not-equals 0 . Then the following are equivalent:

(1)

The triangle EquationI.2.1 is an Auslander–Reiten triangle.

(2)

The map h is in the socle of upper H o m left-parenthesis upper C comma upper A left-bracket 1 right-bracket right-parenthesis as a right upper E n d left-parenthesis upper C right-parenthesis -module and upper A approximately-equals tau overTilde upper C .

(3)

The map h is in the socle of upper H o m left-parenthesis upper C comma upper A left-bracket 1 right-bracket right-parenthesis as a left upper E n d left-parenthesis upper A right-parenthesis -module and upper A approximately-equals tau overTilde upper C .

Proof.

We will show that 1. and 2. are equivalent. The equivalence of 1. and 3. is similar.

1 period right double arrow 2 period

By definition we have upper A approximately-equals tau overTilde upper C . Assume that t element-of upper E n d left-parenthesis upper C right-parenthesis is a non-automorphism. Then by (AR3) we have h t equals 0 .

2 period right double arrow 1 period

Let StartLayout 1st Row with Label left-parenthesis upper I .2 .2 right-parenthesis EndLabel upper A right-arrow Overscript f Superscript prime Baseline Endscripts upper B prime right-arrow Overscript g Superscript prime Baseline Endscripts upper C right-arrow Overscript h Superscript prime Baseline Endscripts upper A left-bracket 1 right-bracket EndLayout

be the Auslander–Reiten triangle associated to upper C . From the properties of Auslander–Reiten triangles it follows that there is a morphism of triangles StartLayout 1st Row 1st Column upper A 2nd Column right-arrow Overscript f Endscripts 3rd Column upper B 4th Column right-arrow Overscript g Endscripts 5th Column upper C 6th Column right-arrow Overscript h Endscripts 7th Column upper A left-bracket 1 right-bracket 2nd Row 1st Column parallel-to 2nd Column Blank 3rd Column s up-arrow 4th Column Blank 5th Column t up-arrow 6th Column Blank 7th Column parallel-to 8th Column Blank 3rd Row 1st Column upper A 2nd Column right-arrow Overscript f prime Endscripts 3rd Column upper B prime 4th Column right-arrow Overscript g prime Endscripts 5th Column upper C 6th Column right-arrow Overscript h prime Endscripts 7th Column upper A left-bracket 1 right-bracket EndLayout

The fact that h prime not-equals 0 together with the fact that h is in the socle of upper H o m left-parenthesis upper C comma upper A left-bracket 1 right-bracket right-parenthesis implies that t must be an isomorphism. But then by the properties of triangles s 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 script upper A has right Auslander–Reiten triangles. Then the socle of upper H o m left-parenthesis upper C comma tau overTilde upper C left-bracket 1 right-bracket right-parenthesis is one-dimensional both as a right upper E n d left-parenthesis upper C right-parenthesis -module and as a left upper E n d left-parenthesis tau overTilde upper C right-parenthesis -module.

Proof.

It is easy to see that u 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)

script upper A has a right Serre functor.

(2)

script upper A has right Auslander–Reiten triangles.

If either of these properties holds, then the action of the Serre functor on objects coincides with ModifyingAbove tau With tilde left-bracket 1 right-bracket .

Proof.

1 period right double arrow 2 period

Let upper C element-of script upper A be an indecomposable object. By Serre duality there is a natural isomorphism upper H o m left-parenthesis upper C comma upper F upper C right-parenthesis approximately-equals upper H o m left-parenthesis upper C comma upper C right-parenthesis Superscript asterisk

as upper E n d left-parenthesis upper C right-parenthesis -bimodules. In particular upper H o m left-parenthesis upper C comma upper F upper C right-parenthesis has a one dimensional socle which corresponds to the map theta Subscript upper C Baseline colon upper H o m left-parenthesis upper C comma upper C right-parenthesis right-arrow upper H o m left-parenthesis upper C comma upper C right-parenthesis slash r a d left-parenthesis upper C comma upper C right-parenthesis equals k . Define tau overTilde upper C equals upper F upper C left-bracket negative 1 right-bracket , and let h be a non-zero element of the socle of upper H o m left-parenthesis upper C comma tau overTilde upper C left-bracket 1 right-bracket right-parenthesis . We claim that the associated triangle tau overTilde upper C right-arrow upper X right-arrow upper C right-arrow Overscript h Endscripts tau overTilde upper C left-bracket 1 right-bracket

is an Auslander–Reiten triangle. Let upper D be indecomposable, and let t colon upper D right-arrow upper C be a non-isomorphism. We have to show that the compositionupper D right-arrow Overscript t Endscripts upper C right-arrow Overscript h Endscripts upper F upper C

is zero. Using Serre duality this amounts to showing that the composition upper H o m left-parenthesis upper C comma upper D right-parenthesis right-arrow Overscript upper H o m left-parenthesis upper C comma t right-parenthesis Endscripts upper H o m left-parenthesis upper C comma upper C right-parenthesis right-arrow Overscript theta Subscript upper C Baseline Endscripts k

is zero. Since t is a non-isomorphism, this is clear.

2 period right double arrow 1 period

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 script upper A consisting of the indecomposable objects. For upper A an indecomposable object in script upper A we let upper F upper A be the object tau overTilde upper A left-bracket 1 right-bracket . Let h Subscript upper A be a non-zero element of upper H o m left-parenthesis upper A comma tau overTilde upper A left-bracket 1 right-bracket right-parenthesis representing the Auslander–Reiten triangle tau overTilde upper A right-arrow upper X right-arrow upper A right-arrow Overscript h Subscript upper A Baseline Endscripts tau overTilde upper A left-bracket 1 right-bracket period

Sublemma

Let upper A and upper B be indecomposable objects in script upper C . Then the following hold:

(1)

For any non-zero f element-of upper H o m left-parenthesis upper B comma tau overTilde upper A left-bracket 1 right-bracket right-parenthesis there exists g element-of upper H o m left-parenthesis upper A comma upper B right-parenthesis such that f g equals h Subscript upper A .

(2)

For any non-zero g element-of upper H o m left-parenthesis upper A comma upper B right-parenthesis there exists f element-of upper H o m left-parenthesis upper B comma tau overTilde upper A left-bracket 1 right-bracket right-parenthesis such that f g equals h Subscript upper A .

Proof.

(1)

Using the properties of Auslander–Reiten triangles there is a morphism between the triangle determined by h Subscript upper A (the AR-triangle) and the triangle determined by f . StartLayout 1st Row 1st Column tau overTilde upper A 2nd Column right-arrow Overscript Endscripts 3rd Column upper Y 4th Column right-arrow Overscript Endscripts 5th Column upper A 6th Column right-arrow Overscript h Subscript upper A Endscripts 7th Column tau overTilde upper A left-bracket 1 right-bracket 2nd Row 1st Column parallel-to 2nd Column Blank 3rd Column down-arrow 4th Column Blank 5th Column g down-arrow 6th Column Blank 7th Column parallel-to 8th Column Blank 3rd Row 1st Column tau overTilde upper A 2nd Column right-arrow Overscript Endscripts 3rd Column upper X 4th Column right-arrow Overscript Endscripts 5th Column upper B 6th Column right-arrow Overscript f Endscripts 7th Column tau overTilde upper A left-bracket 1 right-bracket EndLayout

The morphism labeled g in the above diagram has the required properties.

(2)

Without loss of generality we may assume that g is not an isomorphism. We complete g to a triangleupper Z right-arrow Overscript s Endscripts upper A right-arrow Overscript g Endscripts upper B right-arrow Overscript t Endscripts upper Z left-bracket 1 right-bracket period

Then upper Z not-equals 0 and since g is non-zero, s will not be split. Now we look at the following diagram: StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column upper B 6th Column Blank 7th Column Blank 2nd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column g up-arrow 6th Column Blank 7th Column Blank 8th Column Blank 3rd Row 1st Column tau overTilde upper A 2nd Column right-arrow Overscript Endscripts 3rd Column upper Y 4th Column right-arrow Overscript Endscripts 5th Column upper A 6th Column right-arrow Overscript h Subscript upper A Endscripts 7th Column tau overTilde upper A left-bracket 1 right-bracket 4th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column s up-arrow 6th Column Blank 7th Column Blank 8th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column upper Z EndLayout

Since s is not split we have by (AR3) that h Subscript upper A Baseline s equals 0 . Hence by the properties of triangles we have h Subscript upper A Baseline equals f g for a map f colon upper B right-arrow tau upper A left-bracket 1 right-bracket . This proves what we want.

Having proved the sublemma we return to the main proof. For any indecomposable object upper A element-of script upper C choose a linear map

eta Subscript upper A Baseline colon upper H o m left-parenthesis upper A comma tau overTilde upper A left-bracket 1 right-bracket right-parenthesis right-arrow k

such that eta Subscript upper A Baseline left-parenthesis h Subscript upper A Baseline right-parenthesis not-equals 0 . It follows from the sublemma that the pairing

StartLayout 1st Row with Label left-parenthesis upper I .2 .3 right-parenthesis EndLabel upper H o m left-parenthesis upper A comma upper B right-parenthesis times upper H o m left-parenthesis upper B comma tau overTilde upper A left-bracket 1 right-bracket right-parenthesis right-arrow k given by left-parenthesis g comma f right-parenthesis right-arrow eta Subscript upper A Baseline left-parenthesis f g right-parenthesis EndLayout

is non-degenerate. We can now finish our proof by invoking Proposition EquationI.1.4.

The following is now a direct consequence.

Theorem I.2.4

The following are equivalent:

(1)

script upper A has a Serre functor.

(2)

script upper A 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 tau overTilde also the equivalence upper F left-bracket negative 1 right-bracket , where upper F 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 script upper C is an upper E x t -finite k -linear abelian category, we say that it has a right Serre functor if this is the case for upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis .

If script upper C has a right Serre functor upper F and upper A and upper B are in script upper C , then from the fact that upper E x t Superscript i Baseline left-parenthesis upper A comma upper B right-parenthesis equals upper H o m left-parenthesis upper A comma upper B left-bracket i right-bracket right-parenthesis equals upper H o m left-parenthesis upper B left-bracket i right-bracket comma upper F upper A right-parenthesis Superscript asterisk we deduce that only a finite number of upper E x t Superscript i Baseline left-parenthesis upper A comma upper B right-parenthesis can be non-zero.

Before we go on we recall some basic definitions (see [4]). For an indecomposable object upper C in script upper C a m a p g colon upper B right-arrow upper C is right almost split if for any non-isomorphism h colon upper X right-arrow upper C with upper X indecomposable in script upper C there is some m a p t colon upper B right-arrow upper C with g t equals h . The m a p g colon upper B right-arrow upper C is minimal right almost split if in addition g colon upper B right-arrow upper C is right minimal, that is, any m a p s colon upper B right-arrow upper B with g s equals g is an isomorphism. The concepts of left almost split and minimal left almost split are defined similarly. A non-split exact sequence 0 right-arrow upper A right-arrow upper B right-arrow upper C right-arrow 0 is almost split if upper A and upper C are indecomposable and g colon upper B right-arrow upper C is (minimal) right almost split (or equivalently, f colon upper A right-arrow upper B is (minimal) left almost split). We say that script upper C has right almost split sequences if for every non-projective indecomposable object upper C element-of script upper C there exists an almost split sequence ending in upper C , and for each indecomposable projective object upper P there is a minimal right almost split map upper E right-arrow upper P . Possession of left almost split sequences is defined similarly. We say that script upper C has almost split sequences if it has both left and right almost split sequences.

Now let script upper C be an upper E x t -finite k -linear hereditary abelian category.

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 upper P if and only if upper P has a unique maximal subobject r a d upper P . (If the conditions are satisfied, then the inclusion map i colon r a d upper P right-arrow upper P 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 upper S . (If these conditions are satisfied, then the epimorphism j colon upper I right-arrow upper I slash upper S is minimal left almost split.)

Note that if an indecomposable projective object has a maximal subobject, then it must be unique, and if upper P is noetherian, then upper P has a maximal subobject. Similarly if an indecomposable injective object upper I has a simple subobject, then it must be unique.

Lemma I.3.2

Let script upper A equals upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis . The following are equivalent:

(1)

script upper A has right Auslander–Reiten triangles.

(2)

script upper C has right almost split sequences and for every indecomposable projective upper P element-of script upper C the simple object upper S equals upper P slash r a d left-parenthesis upper P right-parenthesis possesses an injective hull upper I in script upper C .

(3)

script upper C has a right Serre functor.

If any of these conditions holds, then the right Auslander–Reiten triangles in script upper A are given by the shifts of the right almost split sequences in script upper C together with the shifts of the triangles of the form

StartLayout 1st Row with Label left-parenthesis upper I .3 .1 right-parenthesis EndLabel upper I left-bracket negative 1 right-bracket right-arrow upper X right-arrow upper P right-arrow Overscript h Subscript upper P Baseline Endscripts upper I EndLayout

where upper I and upper P are as in 2 period and h Subscript upper P is the composition upper P right-arrow upper S right-arrow upper I . The middle term upper X of this triangle is isomorphic to upper I slash upper S left-bracket negative 1 right-bracket circled-plus r a d upper P .

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 script upper C be an upper E x t -finite hereditary abelian category.

(1)

script upper C has Serre duality if and only if script upper C has almost split sequences, and there is a one-one correspondence between indecomposable projective objects upper P and indecomposable injective objects upper I , via upper P slash r a d upper P approximately-equals s o c upper I .

(2)

If script upper C has no non-zero projective or injective objects, then script upper C has Serre duality if and only if it has almost split sequences.

Note that if script upper C is the category of finite dimensional representations over k of the quiver dot right-arrow dot right-arrow dot right-arrow ellipsis , then script upper C is an upper E x t -finite hereditary abelian k -category with almost split sequences. Since script upper C has non-zero injective objects, but no non-zero projective objects, it does not have Serre duality.

Let script upper P and script upper I be the full subcategories of script upper C whose objects are respectively the projectives and the injectives in script upper C . If upper H is a set of objects in script upper C , then we denote by script upper C Subscript upper H the full subcategory of script upper C whose objects have no summands in upper H .

When script upper C equals mod upper A for a finite dimensional hereditary k -algebra upper A , we have an equivalence tau colon script upper C Subscript script upper P Baseline right-arrow script upper C Subscript script upper I Baseline , where for an indecomposable object upper C in script upper C the object tau upper C is the left hand term of the almost split sequence with right hand term upper C . Also we have the Nakayama functor upper N colon script upper P right-arrow script upper I , which is an equivalence of categories, where upper N left-parenthesis upper P right-parenthesis equals upper H o m Subscript k Baseline left-parenthesis upper H o m Subscript upper A Baseline left-parenthesis upper P comma upper A right-parenthesis comma k right-parenthesis for upper P element-of script upper P . For the equivalence upper F colon upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis right-arrow upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis , where upper F is the Serre functor, we have upper F vertical-bar script upper C Subscript script upper P Baseline equals tau left-bracket 1 right-bracket and upper F vertical-bar script upper P equals upper N . Hence upper F is in some sense put together from the two equivalences tau and upper N (see Reference14). Using Lemma I.3.2 we see that the situation is similar in the general case.

Corollary I.3.4

Assume that script upper C has a right Serre functor upper F . Then the following hold:

(1)

upper F defines a fully faithful functor script upper P right-arrow script upper I . We denote this functor by upper N left-parenthesis the “Nakayama functor”).

(2)

upper F left-bracket negative 1 right-bracket equals tau overTilde induces a fully faithful functor script upper C Subscript script upper P Baseline right-arrow script upper C Subscript script upper I , which we denote by tau .

(3)

If upper P element-of script upper P is indecomposable, then upper H o m left-parenthesis upper P comma upper N upper P right-parenthesis has one dimensional socle, both as left upper H o m left-parenthesis upper N upper P comma upper N upper P right-parenthesis -module and as right upper H o m left-parenthesis upper P comma upper P right-parenthesis -module. Let h Subscript upper P be a non-zero element in this socle. Then upper S equals im h Subscript upper P is simple, and furthermore upper P is a projective cover of upper S and upper N upper P is an injective hull of upper S .

If upper F is a Serre functor, then the functors upper N and tau defined above are equivalences.

Proof.

That upper F takes the indicated values on objects follows from the nature of the Auslander–Reiten triangles in upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis (given by Lemma I.3.2). The assertion about fully faithfulness of tau and upper N follows from the corresponding property of upper F . Part 3. follows by inspecting the triangle EquationI.3.1. Finally that upper N and tau are equivalences in the case that upper F is a Serre functor follows by considering upper F Superscript negative 1 , which is a left Serre functor.

In the next section we will use the following result.

Lemma I.3.5

Assume that script upper A is a triangulated category with a t -structure left-parenthesis script upper A Subscript 0 Baseline comma script upper A Subscript 0 Baseline right-parenthesis in such a way that every object in script upper A lies in some script upper A Superscript left-bracket a comma b right-bracket . Let script upper C be the heart of the t -structure, and assume that upper H o m left-parenthesis upper A comma upper B left-bracket n right-bracket right-parenthesis equals 0 for upper A comma upper B element-of script upper C and n not-equals 0 comma 1 . Then script upper C is a hereditary abelian category, and furthermore script upper A has a (right) Serre functor if and only if script upper C has a (right) Serre functor.

Proof.

To show that script upper C is hereditary we have to show that

upper E x t Subscript script upper C Superscript n Baseline left-parenthesis upper A comma upper B right-parenthesis equals upper H o m Subscript script upper A Baseline left-parenthesis upper A comma upper B left-bracket n right-bracket right-parenthesis period

Now upper E x t Subscript script upper C Superscript m Baseline left-parenthesis upper A comma minus right-parenthesis is characterized by the property that it is an effaceable delta -functor which coincides in degree zero with upper H o m Subscript script upper C Baseline left-parenthesis upper A comma minus right-parenthesis . Hence we have to show that upper H o m Subscript script upper A Baseline left-parenthesis upper A comma minus left-bracket n right-bracket right-parenthesis is effaceable. This is clear in degree 2 since there the functor is zero, and for n less-than-or-equal-to 1 it is also clear since then upper H o m Subscript script upper A Baseline left-parenthesis upper A comma minus left-bracket n right-bracket right-parenthesis equals upper E x t Subscript script upper C Superscript n Baseline left-parenthesis upper A comma minus right-parenthesis Reference7, p. 75.

Now standard arguments show that as additive categories script upper A and upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis are equivalent to circled-plus Underscript n Endscripts script upper C left-bracket n right-bracket . We don’t know if this equivalence yields an exact equivalence between script upper A and upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis , but recall that the definition of a (right) Serre functor does not involve the triangulated structure. Hence if script upper A has a (right) Serre functor, then so does upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis and vice versa.

II. Hereditary noetherian abelian categories with non-zero projective objects

In this section we classify the connected noetherian hereditary upper E x t -finite abelian categories with Serre functor, in the case where there are non-zero projective objects.

Let script upper C be a connected hereditary abelian noetherian upper E x t -finite category with Serre functor and non-zero projective objects. The structure of the projective objects gives rise to an associated quiver upper Q , which satisfies special assumptions, including being locally finite and having no infinite path ending at any vertex. The category script upper C contains the category r e p upper Q of finitely presented representations of upper Q as a full subcategory, which may be different from script upper C . Actually r e p upper Q is the full subcategory generated by projective objects. Hence script upper C 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 upper Q having no infinite path ending at any vertex, the category r e p upper Q is a hereditary abelian upper E x t -finite category with a right Serre functor (and right almost split sequences), but not necessarily having a Serre functor. We construct a hereditary abelian upper E x t -finite category with Serre functor ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis containing r e p upper Q as a full subcategory. We describe for which quivers upper Q the category ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis is noetherian, and show that in the noetherian case all connected noetherian hereditary abelian upper E x t -finite categories with Serre duality are of this form.

In §II.1 we explain the construction of ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis 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 ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis Reference24.

In §II.2 we show that if script upper C is generated by the preprojective objects, then script upper C is equivalent to some ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis . That script upper C is generated by preprojective objects when script upper C is connected and noetherian is proved in Section II.4, along with showing for which quivers upper Q the category ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis is noetherian.

II.1. Hereditary abelian categories constructed from quivers

For a quiver upper Q denote by r e p left-parenthesis upper Q right-parenthesis the category of finitely presented representations of upper Q . Under some additional assumptions on upper Q we construct a hereditary abelian upper E x t -finite category ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis with Serre functor. The category ModifyingAbove r e p With tilde left-parenthesis upper Q right-parenthesis contains r e p left-parenthesis upper Q right-parenthesis as a full subcategory, and is obtained from r e p left-parenthesis upper Q right-parenthesis by formally inverting a right Serre functor.

Let upper Q be a quiver with the following properties:

(P1)

upper Q is locally finite, that is, every vertex in upper Q is adjacent to only a finite number of other vertices.

(P2)

There is no infinite path in upper Q of the form x 0 left-arrow x 1 left-arrow x 2 left-arrow ellipsis (in particular there are no oriented cycles).

Note that (P1), (P2) imply the following.

Lemma II.1.1

(1)

If x is a vertex in upper Q , then there are only a finite number of paths ending in x .

(2)

If x and y are vertices in upper Q , then there are only a finite number of paths from x to y .

Proof.

Part 2. is an obvious consequence of 1., so we prove 1. Assume there is an infinite number of paths ending in x . By (P1) there must be an arrow x 1 right-arrow x such that there is an infinite number of paths ending in x 1 . Repeating this we obtain an infinite path ellipsis right-arrow x 2 right-arrow x 1 right-arrow x such that there is an infinite number of paths ending in every x Subscript n . The existence of such an infinite path contradicts (P2).

For a vertex x in upper Q we denote by upper P Subscript x , upper I Subscript x and upper S Subscript x respectively the corresponding projective, injective and simple object in upper R e p left-parenthesis upper Q right-parenthesis , the category of all upper Q -representations. By Lemma II.1.1 the objects upper I Subscript x have finite length. We also have a canonical isomorphism

StartLayout 1st Row with Label left-parenthesis II .1 .1 right-parenthesis EndLabel upper H o m left-parenthesis upper P Subscript x Baseline comma upper P Subscript y Baseline right-parenthesis approximately-equals upper H o m left-parenthesis upper I Subscript x Baseline comma upper I Subscript y Baseline right-parenthesis EndLayout

since both of these vector spaces have as basis the paths from y to x .

The category upper R e p left-parenthesis upper Q right-parenthesis is too big for what we want. For example it is not upper E x t -finite. As will be clear from the considerations in the next section the natural subcategory to consider is r e p left-parenthesis upper Q right-parenthesis . It is easy to see that this is a hereditary abelian subcategory of upper R e p left-parenthesis upper Q right-parenthesis , and furthermore by Lemma II.1.1 it has finite dimensional upper E x t ’s. Thanks to (P1) the simple and hence the finite dimensional representations are contained in r e p left-parenthesis upper Q right-parenthesis . Hence in particular upper I Subscript x Baseline element-of r e p left-parenthesis upper Q right-parenthesis for every vertex x .

Let script upper P and script upper I be the full subcategories of r e p left-parenthesis upper Q right-parenthesis consisting respectively of the finite direct sums of the upper P Subscript x and the finite direct sums of the upper I Subscript x . Let upper N colon script upper P right-arrow script upper I be the equivalence obtained from additively extending EquationII.1.1. We denote also by upper N the corresponding equivalence upper N colon upper K Superscript b Baseline left-parenthesis script upper P right-parenthesis right-arrow upper K Superscript b Baseline left-parenthesis script upper I right-parenthesis . Finally denoting by upper F the composition

StartLayout 1st Row with Label left-parenthesis II .1 .2 right-parenthesis EndLabel upper D Superscript b Baseline left-parenthesis r e p left-parenthesis upper Q right-parenthesis right-parenthesis approximately-equals upper K Superscript b Baseline left-parenthesis script upper P right-parenthesis right-arrow Overscript upper N Endscripts upper K Superscript b Baseline left-parenthesis script upper I right-parenthesis right-arrow upper D Superscript b Baseline left-parenthesis r e p left-parenthesis upper Q right-parenthesis right-parenthesis EndLayout

we have the following.

Lemma II.1.2

upper F is a right Serre functor.

Proof.

Let upper A comma upper B element-of upper K Superscript b Baseline left-parenthesis script upper P right-parenthesis . Then we need to construct natural isomorphisms

StartLayout 1st Row with Label left-parenthesis II .1 .3 right-parenthesis EndLabel upper H o m Subscript upper K Sub Superscript b Subscript left-parenthesis script upper P right-parenthesis Baseline left-parenthesis upper A comma upper B right-parenthesis right-arrow upper H o m Subscript upper D Sub Superscript b Subscript left-parenthesis r e p left-parenthesis upper Q right-parenthesis right-parenthesis Baseline left-parenthesis upper B comma upper F upper A right-parenthesis Superscript asterisk Baseline period EndLayout

Since upper A and upper B are finite complexes of projectives we can reduce to the case upper A equals upper P Subscript x and upper B equals upper P Subscript y . So we need natural isomorphisms

upper H o m left-parenthesis upper P Subscript x Baseline comma upper P Subscript y Baseline right-parenthesis right-arrow upper H o m left-parenthesis upper P Subscript y Baseline comma upper I Subscript x Baseline right-parenthesis Superscript asterisk Baseline period

Again both of these vector spaces have a natural basis given by the paths from y to x . This leads to the required isomorphisms.

Under the assumptions (P1), (P2) on the quiver upper Q we now know that r e p upper Q is a hereditary abelian upper E x t -finite category with a right Serre functor upper F such that the image of the projective objects in r e p upper Q are injective objects in r e p upper Q of finite length. Hence the following result applies to this setting.

Theorem II.1.3

Let script upper B be an abelian upper E x t -finite hereditary category with a right Serre functor upper F and enough projectives. Denote the full subcategory of projective objects in script upper B by script upper P and assume that upper F left-parenthesis script upper P right-parenthesis equals script upper I consists of (injective) objects of finite length. Then there exists an upper E x t -finite abelian hereditary category script upper C with the following properties:

(1)

There exists a full faithful exact embedding i colon script upper B right-arrow script upper C .

(2)

The injectives and projectives in script upper C are given by i left-parenthesis script upper I right-parenthesis and i left-parenthesis script upper P right-parenthesis .

(3)

script upper C possesses a Serre functor which extends upper F colon upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis right-arrow upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis in such a way that there is a natural equivalence nu colon upper F i right-arrow i upper F where we have denoted the derived functor of i also by i and the extended Serre functor also by upper F .

(4)

For every indecomposable object upper X element-of script upper C there exists t greater-than-or-equal-to 0 such that tau Superscript t Baseline upper X is defined and lies in script upper B , where tau denotes the functor script upper C Subscript script upper P Baseline right-arrow script upper C Subscript script upper I (see Corollary I.3.4.2) induced by upper F left-bracket negative 1 right-bracket equals tau overTilde .

Furthermore a quadruple left-parenthesis script upper C comma i comma upper F comma nu right-parenthesis with these properties is unique in the appropriate sense.

Proof.

First we show that if script upper C exists satisfying properties 1 period comma 2 period comma 3 period comma 4 period , then it is unique. This proof will in particular tell us how to construct script upper C . We then show that this construction always yields a category script upper C with the required properties.

Since upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis equals upper K Superscript b Baseline left-parenthesis script upper P right-parenthesis and since the objects in script upper P remain projective in script upper C by property 2., it follows that the derived functor of i is fully faithful. It is also clear that upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis is closed inside upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis under the formation of cones.

Let upper F colon upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis right-arrow upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis be the extended Serre functor. If upper K element-of upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis , then upper K can be obtained by starting with objects in script upper C and repeatedly taking cones. It then follows from 4. that for t much-greater-than 0 we have upper F Superscript t Baseline upper K element-of upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis . Hence the triple left-parenthesis upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis comma i comma upper F right-parenthesis satisfies the hypotheses of Proposition I.1.8. Thus we obtain upper D Superscript b Baseline left-parenthesis script upper C right-parenthesis equals upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis .

According to 4. every indecomposable object in script upper C will be of the form tau Superscript negative t Baseline upper Y with upper Y element-of script upper B . Implicit in the notation tau Superscript negative t Baseline upper Y is the assumption that tau Superscript negative l Baseline upper Y is defined for l equals 1 comma period period period comma t , that is, tau Superscript negative i Baseline upper Y not-an-element-of script upper I for i equals 0 comma period period period comma t minus 1 . In addition we may assume that t is minimal. Thus either t equals 0 or else tau Superscript negative l Baseline upper Y not-an-element-of script upper B for l equals 1 comma period period period comma t . The last case is equivalent to

StartLayout 1st Row with Label left-parenthesis II .1 .4 right-parenthesis EndLabel tau Superscript negative i Baseline upper Y not-an-element-of tau script upper B Subscript script upper P union script upper I EndLayout

for i equals 0 comma period period period comma t minus 1 .

If upper Y not-an-element-of tau script upper B Subscript script upper P union script upper I , then the same is true for tau Superscript negative 1 Baseline upper Y . Hence we have to impose EquationII.1.4 only for i equals 0 .

We conclude that the indecomposable objects in script upper C are of the following form:

(C1)

The indecomposable objects in script upper B .

(C2)

Objects of the form tau overTilde Superscript negative t Baseline upper Y where t greater-than 0 and upper Y is an indecomposable object in script upper B minus left-parenthesis tau script upper B Subscript script upper P Baseline union script upper I right-parenthesis , where tau overTilde equals upper F left-bracket negative 1 right-bracket .

This completes the determination of script upper C as an additive subcategory of upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis upper B right-parenthesis , and finishes the proof of the uniqueness.

Let us now assume that script upper C is the additive subcategory of upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis whose indecomposable objects are given by (C1), (C2). We have to show that script upper C is a hereditary abelian category satisfying 1.-4.

Since upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis has a Serre functor, it has Auslander–Reiten triangles. Below we will need the triangle associated to upper P Subscript x Baseline left-bracket 1 right-bracket . Using the criterion given in Proposition EquationI.2.1 we can compute this triangle in upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis . So according to Lemma EquationI.3.2 the requested triangle is of the form

StartLayout 1st Row with Label left-parenthesis II .1 .5 right-parenthesis EndLabel upper I Subscript x Baseline right-arrow upper I Subscript x Baseline slash s o c left-parenthesis upper I Subscript x Baseline right-parenthesis circled-plus r a d left-parenthesis upper P Subscript x Baseline right-parenthesis left-bracket 1 right-bracket right-arrow upper P Subscript x Baseline left-bracket 1 right-bracket right-arrow period EndLayout

We now define a t -structure on upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis . Using the fact that script upper B is hereditary we easily obtain that as additive categories upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis equals circled-plus Underscript n Endscripts script upper C left-bracket n right-bracket . We now define

StartLayout 1st Row 1st Column upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis Subscript 0 2nd Column equals circled-plus Underscript n greater-than-or-equal-to 0 Endscripts script upper C left-bracket n right-bracket comma 2nd Row 1st Column upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis Subscript 0 2nd Column equals circled-plus Underscript n less-than-or-equal-to 0 Endscripts script upper C left-bracket n right-bracket period EndLayout

We claim that this is a t -structure. The only non-trivial axiom we have to verify is that

upper H o m left-parenthesis upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis Subscript 0 Baseline comma upper F Superscript negative normal infinity Baseline upper D Superscript b Baseline left-parenthesis script upper B right-parenthesis Subscript 1 Baseline right-parenthesis equals 0 period

So this amounts to showing that upper E x t Superscript negative i Baseline left-parenthesis upper A comma upper B right-parenthesis equals 0 for upper A comma upper B element-of script upper C and for i greater-than 0 . We separate this into four cases.

Case 1

upper A and upper B fall under (C1). This case is trivial.

Case 2

upper A falls under (C1) and upper B falls under (C2). Thus we have upper B equals tau overTilde Superscript negative t Baseline upper X for some upper X in script upper B and t greater-than 0 . We want to show that upper E x t Superscript negative i Baseline left-parenthesis upper A comma upper B right-parenthesis equals 0 for i greater-than 0 for any upper A element-of script upper B , by induction on t .

First let t equals 1 . We can assume that upper A is projective, since otherwise we could reduce to Case 1 by applying tau overTilde . Then ModifyingAbove tau With tilde left-parenthesis upper A right-parenthesis equals upper I left-bracket negative 1 right-bracket for some object upper I in script upper I , so it is sufficient to show upper E x t Superscript negative i Baseline left-parenthesis upper I left-bracket negative 1 right-bracket comma upper X right-parenthesis approximately-equals upper E x t Superscript minus left-parenthesis i minus 1 right-parenthesis Baseline left-parenthesis upper I comma upper X right-parenthesis equals 0 for i greater-than 0 . For i greater-than 1 this follows from Case 1. For i equals 1 we clearly have upper H o m left-parenthesis upper I comma upper X right-parenthesis equals 0 since upper I is injective in script upper B and upper X is not injective.

Assume now that t greater-than 1 and that the claim has been proved for t minus 1 for all objects upper A in script upper B . Then we can again assume that upper A is projective, and consider upper E x t Superscript minus left-parenthesis i minus 1 right-parenthesis Baseline left-parenthesis upper I comma tau overTilde Superscript negative t plus 1 Baseline upper X right-parenthesis , where ModifyingAbove tau With tilde left-parenthesis upper A right-parenthesis equals upper I left-bracket negative 1 right-bracket . We want to prove that

upper E x t Superscript minus left-parenthesis i minus 1 right-parenthesis Baseline left-parenthesis upper I comma tau overTilde Superscript negative t plus 1 Baseline upper X right-parenthesis equals 0 for i greater-than 0 period

By the induction assumption we only need to consider i equals 1 . So we want to prove that upper H o m left-parenthesis upper J comma tau overTilde Superscript negative t plus 1 Baseline upper X right-parenthesis equals 0 for upper J indecomposable in script upper I , and we do this by induction on the length of upper J which by assumption is finite. We have the Auslander–Reiten triangle upper J right-arrow Overscript g Endscripts upper J slash s o c upper J circled-plus left-parenthesis r a d upper P right-parenthesis left-bracket 1 right-bracket right-arrow upper P left-bracket 1 right-bracket right-arrow upper J left-bracket 1 right-bracket