arXiv:1011.6077v1 [math.CT] 28 Nov 2010
HEREDITARY UNISERIAL CATEGORIES WITH SERRE DUALITY
ADAM-CHRISTIAAN VAN ROOSMALEN
Abstract. An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes
of subobjects of a given indecomposable object form a linearly ordered poset. In this paper,
we classify the hereditary uniserial categories with Serre duality. They fall into two types: the
first type is given by the representations of the quiver A
n
with linear orientation (and infinite
variants thereof), the second type by tubes (and an infinite variant). These last categories give
a new class of hereditary categories wi th Serre duality, called big tubes.
Contents
1. Introduction 1
2. Preliminarie s 3
3. Grothendieck categories and tubes 5
4. Big tubes 10
5. Paths in hereditary categories 14
6. Some properties of uniserial categories 15
7. Description and classification by 2-colimits 19
References 23
1. Introduction
We will fix an algebraically closed field k, and will only consider k-linear categories. We will
say that a Hom-finite abelian category A is uniserial if for every indecomposable X ∈ Ob A the
sub objects of X are linear ly ordered by inclusion. The uniserial hereditary length categories with
only finitely many (nonisomorphic) simple objects have been classified in [1] (see also [8, 9, 12]).
Theorem 1.1. Let A be a hereditary uniserial length category with finitely many isomorphism
classes of simple objects, then A is equivalent to either
• the category rep
k
A
n
of finite dimensional represent ations of the quiver A
n
with linear
orientation, or
• the category nilp
k
˜
A
n
of finite dimensional nilpotent representations of the quiver
˜
A
n
with
cyclic orientation.
Categorie s of the second type will be referred to as tubes. We wish to replace the condition
on the length of the objects by the exis tence of a Serre functor. Note that the two cla sses of
categories mentioned above do have Se rre duality, so that the exis tence of a Ser re functor is a
(strictly) weaker condition. Our main result is the following (Theorem 7.13 in the text).
Theorem 1.2. Let A be an essentially small k-linear u niserial hereditary category with Serre
duality. Then A is equivalent to one of the following
(1) the category rep
cfp
L of finitely presented and cofinitely presented representations of L
where L is a locally discrete linearly ordered poset, either without minimal or maximal
elements, or with both a minimal and a maximal element, or
(2) a (big) tube.
2010 Mathematics Subject Classification. 18E10, 16G30.
1
2 ADAM-CHRISTIAAN VAN ROOSMALEN
We refer to §4 for definitions. Categories of the first type have been introduced in [28] (see also
[22]); they are dire cted and generalize categories of the form rep A
n
with n ∈ N. Every object in
rep
cfp
L is finitely presented (it is the cokernel of a map between finitely generated projectives in
Rep L) a nd is cofinitely presented (it is the kernel of a map between finitely cogenerated injectives
in Rep L). When L has a minimal and a maximal element, then rep
cfp
L
∼
=
rep L and they
are equivalent to category of finitely presented representations of a thread quiver
·
P
//
·
for
a linearly ordered poset P, possibly empty (s ee [3] for the definition of a thread quiver and its
representations.).
A big tube is an infinite generalization of a tube (A definition is given in §4). It is not a length
category and ha s infinitely many isomorphism classes of indecomposable s imple objects. Every
indecomposable object in a big tube lies in a subcategory of the form nilp
k
˜
A
n
and thus (ro ughly
speaking) we may see a big tube as a union of its s ubtubes. T his approach (to c onsider a big tube
as a filtered 2-colimit of tubes) will be taken in §7 to finish the proof of Theorem 1.2.
We do not know of any mention of big tubes in the literature, and thus believe these to be a
new type of hereditary c ategories with Serre duality. These categories ar e used, for example, in
the construction of Hall alge bras or cluster categories.
Our main motivation fo r introducing and studying big tubes comes from the following. The
category rep
k
Q of finite dimensional representations of a tame quiver Q has tubes as subcate-
gories. More precisely, every regular module lies in a tube and the embedding nilp
k
˜
A
n
→ rep
k
Q
maps Auslander-Reiten sequences to Auslander-Reiten sequences (so that the embedding maps
the Auslander-Reiten quiver of nilp
k
˜
A
n
to a component of the Auslander-Reiten quiver of re p
k
Q;
it is of the form ZA
∞
/hτ
n+1
i).
By replacing the tame quiver Q by a “nice tame infinite variant” one obta ins a category A of
representations which contains a big tube as a full subcategory, such tha t the Auslander-Reiten
translations in A and in the big tube coincide. Here, a “nice tame infinite variant” of Q is given
by a certain thread quiver. We r efer to §4.4 for an example of a big tube in the category of
representations of the thread quiver
Likewise, in the category of coherent s heaves o n weighted pr ojective lines (see [13] or [18]), the
simple objects are contained in tubes. The aforementioned example can be considered as (being
derived equivalent to) an example of a weighted projective line with an infinite weight (Remark
4.5).
The proof of the classification consists of three steps. Let A be any essentially small k-linear
uniserial heredita ry category with Serre duality. The first step (Proposition 6.1) is to prove some
consequences of Serre duality, most importantly that A has “enough simples” in the se nse that
every indecomp osable object has a simple socle and a simple top. We will then take a cofinite
subset of isomorphism classe s of simple objects and co nsider the perpendicular subcategory. The
second step of the proof is to say that this category is a her editary uniserial length category with
finitely many simple objects (Proposition 6.11).
It then follows that A is a filtered 2-colimit of such length subcategories A
i
. To prove that A is
equivalent to a category in Theorem 1.2, we will take an appropiate such category B and similarly
write it as a filtered 2-colimit of length subc ategories B
i
. Finding an equivalence between A and B
is then the same as finding a consistent set of equivalences between the length subcategories A
i
of
A and the corre sp onding ones of B. To find such equivalences, we w ill embed the categories A
i
and
the categories B
i
in their Ind-closures. Such categories are lo c ally finite Grothendieck categories of
finite type and functors between them are described using coalgebras o r (dually) pseudocompact
algebras. We will use this structure to define a consistent set of functors from the subcategories A
i
to the s ubcategories B
i
, inducing an equivalence between A and B, completing the classification.
Relevant definitions and theorems about Grothendieck categories in are given in §3.
Acknowledgments The author wishes to thank Michel Van den Bergh for meaningful discus-
sions, and wishes to thank Jan
ˇ
S
ˇ
tov´ıˇcek and Joost Vercruysse for many useful comments on an
early dra ft. The author also gratefully acknowledges the financial and administrative support of
the Hausdor ff Center for Mathematics in Bonn.
HEREDITARY UNISERIAL CATEGORIES WITH SERRE DUALITY 3
2. Preliminaries
2.1. Notatio ns and conventions. We will assume all categories are k-linear for an algebraically
closed field k. A category C is called Hom-finite if dim
k
Hom
C
(A, B) < ∞ for all A, B ∈ Ob C, and
an abelian category C is called Ext-finite if and only if dim
k
Ext
i
C
(A, B) < ∞ for all A, B ∈ Ob C
and all i ≥ 0. An abelian Ex t-finite category is thus automatically also Hom-finite. An abelian
category will be called semi-simple if Ext
1
(−, −) = 0 and hereditary if Ext
2
(−, −) = 0.
We will also choose a Grothendieck universe U and assume all our categ ories are U-categories,
i.e. every Hom-sets in the category is an element of U. A category is called U-small (or just small)
if the object-set is also an element of U and it is called essentially U-small (or essentially small)
if it is equivalent to a U-small category.
Following [21, Theorem A] we will say that an Ext-finite hereditary category A has Serre duality
[4] if and only if A has almost split sequences and there is a one-one cor respondence between the
indecomposable projective objects P and the indecomposable injective objects I, such that the
simple top of P is isomorphic to the simple socle of I. The Auslander-Reiten translate in A will
be denoted by τ.
2.2. Paths i n Krull-Schmidt categories. Fo r a Krull-Schmidt category A we will denote by
ind A a (chosen) maximal set of nonis omorphic indecomposables of A. For a Krull-Schmidt sub-
category B of A, we will choose ind B as a subset of ind A. We wish to advic e the reader to not
confuse the set ind A with the category Ind A of ind-objects mentioned below.
Let A be a Krull-Schmidt category and A, B ∈ ind A. An unoriented path from A to B is a
sequence of objects A = X
0
, X
1
, . . . , X
n
= B such that Hom(X
i
, X
i+1
) 6= 0 or Hom(X
i+1
, X
i
) 6= 0
for all 0 ≤ i < n. Similarly, an oriented path (also abbreviated to path) from A to B is a sequence
of objects A = X
0
, X
1
, . . . , X
n
= B such that Hom(X
i
, X
i+1
) 6= 0 for all 0 ≤ i < n.
An abelian category A will be called indecomposable if and only if it is nonzer o and not equiv-
alent to the product category of two no nzero categories. If A is Hom-finite, and hence Krull-
Schmidt, then A is indec omp osable if and only if there is an unoriented path between any two
indecomposable objects of A.
2.3. Perpendicul ar subcategori es. Let A be an abelian Ext-finite hereditary category and let
S ⊆ Ob A. We will denote by S
⊥
the full subcateg ory of A consis ting of all objects X with
Hom(S, X) = Ext(S, X) = 0, calle d the category r ight perpendicular to S. It follows from [14,
Propos ition 1.1] that S
⊥
is again an abelian hereditary category and that the embedding S
⊥
→ A
is exact. If S = {E} consists of a single object E ∈ Ob A, then we will also write E
⊥
for S
⊥
.
Let E ∈ Ob A be an exceptiona l object (i.e. Ext(E, E) = 0). It follows from [14, Prop osition
3.2] that the embedding i : E
⊥
→ A has a left adjoint L : A → E
⊥
. If E is furthermore a simple
object, then the left adjoint L : A → E
⊥
is exact (see [14, Proposition 2.2]). Also note that L
maps a simple object of A to either a simple object in E
⊥
or to zero.
For eas y reference, we will combine these results in a proposition.
Proposition 2.1. [14] Let A be an abelian Ext-finite hereditary category. Let S ⊆ Ob A be a finite
set of simple and exceptional objects. The category S
⊥
is abelian and hereditary, the embedding
S
⊥
→ A is exact and has an exact left adjoint.
2.4. 2-colimits. In this article, we will sometimes s ee a category as a union of some suitable
small subcategories. In other words, so me categories of consideration will be 2 -colimits of smaller
categories (all 2-colimits in this ar ticle can be s e en as unions of full subcategories). We will repeat
some definitions and results from [29] (see also [5, 6, 17]). We will work in the strict 2-category
Cat of small categories, thus:
• the 0-cells are given by small categories,
• the 1-cells are functors,
• the 2-cells are natural transformatio ns.
Composition of 1-c ells is denoted by ◦. Following [1 9, 29] we will write ◦ for vertical composition
of 2-cells and • for horizontal composition.
4 ADAM-CHRISTIAAN VAN ROOSMALEN
Definition 2.2. Let P be a small 1-category. A 2-fu nctor (with strict identities) a : P → Cat is
given by the following data:
(1) a 0-cell a(i) of Cat for every i ∈ Ob P,
(2) a 1-cell a(s) : a(i) → a(j) of Cat for every morphism s : i → j in P and a(1
i
) = 1
a(i)
for
all i ∈ Ob P,
(3) a natural equivalence Φ(s, t) : a(t ◦ s)
∼
→ a(t) ◦ a(s) for all compo sable morphisms s, t ∈
Mor P,
satisfying the following condition: for three composable morphisms u, t, s ∈ Mor P, we have the
following commutative diagram
a(u ◦ t ◦ s)
Φ(t◦s,u)
//
Φ(s,u◦t)
a(u)a(t ◦ s)
1
a(u)•Φ(s,t)
a(u ◦ t)a(s)
Φ(t,u)•1
a(s)
//
a(u)a(t)a(s)
A 2-functor is calle d strict if Φ(s, t) = 1, thus a(t ◦ s) = a(t) ◦ a(s) for all composable s, t ∈ Mor P
Remark 2.3. A strict 2-functor is just a functor from I to the underlying 1-category of Cat.
Example 2.4. For every object C of Cat, there is a 2-functor C : P → Cat sending every object
of P to C and sending every morphism of C to the identity on C.
Definition 2.5. Let a, b : P → Cat be two 2-functors. A 2-natural transformation f : a → b
between 2 diagrams consists of the following data :
(1) a 1-cell f
i
: a(i) → b(i) of Cat for every i ∈ Ob P, and
(2) a natural equivalence θ
f
s
: b(s) ◦ f
i
→ f
j
◦ a(s) for every morphism s : i → j in P.
such that for a ny two composable morphisms s : i → j, t : j → k in P, we have the following
commutative diagram
b(t ◦ s) ◦ f
i
Φ
b
(s,t)•1
f
i
//
θ
f
t◦s
b(t) ◦ b(s) ◦ f
i
1
b(t)
•θ
f
s
b(t) ◦ f
i
◦ a(s)
θ
f
t
•1
a(s)
f
k
◦ a(t ◦ s)
f
k
•Φ
a
(s,t)
//
f
k
◦ a(t) ◦ a(s)
Definition 2.6. L et a, b : I → Cat be two 2-functors and f, g : a → b be 2-natural tr ansformations .
A modification Λ : f → g consists in giving a 2 -cell Λ
i
: f
i
→ g
i
for a ll objects i ∈ I such that for
all s : i → j in I the following diagram commutes
b(s) ◦ f
i
θ
f
s
//
1
b(s)
•Λ
i
f
j
◦ a(s)
Λ
j
•1
a(s)
b(s) ◦ g
i
θ
g
s
//
g
j
◦ a(s)
Definition 2.7. The diagrams, 2-natural transformations, and modifications form a (strict) 2-
category called 2F(I, Cat).
We can now give the definition of a 2-c olimit.
Definition 2.8. Let a : I → Cat be a 2 -functor. We say a admits a 2-colimit if and only if there
exist
(1) a category 2 colim a, and
(2) a 2-natural transformation σ : a → 2 colim a,
HEREDITARY UNISERIAL CATEGORIES WITH SERRE DUALITY 5
such that for every categ ory C the functor
(− ◦ σ) : Hom
Cat
(2 colim a, C) → Hom
2F
(a, C)
is an equivalence of categor ie s.
The 2-category Cat of all small categories has 2-co limits ([29, Theorem A.3.4]).
Theorem 2.9. Let I be a small category and a : I → Cat a 2-functor. Then a admits a 2-colimit.
The following result is [29, Proposition A.5 .5]
Proposition 2.10. Let a : P → Cat be a 2-functor where P is a small filtered category. Suppose
that a(i) is an additive (abelian) category for any i ∈ Ob P and that a(s) is an additive (exact)
functor for every morphism s ∈ Mor P. Then 2 colim a is an additive (abelian) category and the
natural funct ors σ
i
: a(i) → 2 colim a are additive (exact).
The next result ([29, Proposition A.3.6]) will be applicable to all 2-co limits we will consider.
Proposition 2.11. Let P be a small filtered category such that between two given objects there is
at most one morphism, and let a : P → Cat be a 2-functor such that every fu nctor a(s) is fully
faithful (s ∈ Mor P). Any object X ∈ 2 colim a is isomorphic to an object of the form σ
i
(X
′
) where
X
′
∈ a(i). For any i, j ∈ Ob P, X ∈ Ob a(i), and Y ∈ Ob a(j), we have that
Hom
2 colim a
(σ
i
X, σ
j
Y )
∼
=
Hom
a(k)
(a(s)X, a(t)Y )
where k ∈ Ob P such that there are morphisms s : i → k and t : j → k. The above isomorphism
is given by ϕ
k
: a(k) → 2 colim a.
3. Grothendieck categories and tubes
In the proof of our main theorem, we wil be interested in functors between catego ries without
injectives. In order to handle such functors better, we will embed such a categor y A first in its
category of Ind-objects Ind A defined below. Such a category will be a Gro thendieck category of
finite type if A is an (essentially small) Hom- finite length category.
In this section, we recall some relevant definitions and results. Our aim is Corollary 3.17 which
describes the functors we will be interested in. We will only use these results when A is a tube
(as the other categories we will cons ider have enough injectives); the category of Ind-objects of a
tube is briefly described in §3.4.
3.1. Locally finite Grothendieck categories. An abelian category is called a Grothendieck
category if it has a generator and exact direct limits. It is well-known that a Grothendieck
category has injective e nvelopes [11, The orem II.2] and an injective cogenerator.
Let A be an essentially small abelian category. We denote by Ind A the full subcategory of
Mod A consisting of a ll left exact contravariant functors A → Mod k. It has been shown in [11]
that Ind A is a Grothendieck category. Every object A ∈ Ind A can be written as a formal small
filtered colimit in A (thus an object of I nd A is given by a functor from a small filtered category
to A) a nd the Hom-sets may be computed by
Hom
Ind A
(lim
−→
i
A
i
, lim
−→
j
B
j
) = lim
←−
i
lim
−→
j
Hom
A
(A
i
, B
j
)
If A and B are essentially small abelian catego ries and F : A → B is a functor, then F lifts to
a functor
F : Ind A → Ind B as follows ([16])
F (lim
−→
i
A
i
) = lim
−→
i
F (A
i
).
The action o n the Hom-spaces is the obvious one. If F is faithful, fully faithful, left exact, or right
exact, then the same holds for
F . Furthermore, it follows easily from the definition that a left
or right adjoint functor L, R : B → A of F lifts to a left or right adjoint functor L, R : B → A,
repsectively.
We have the following.