A characterization of torsion theories in the cluster category of Dynkin type A_{\infty}

Abstract

Let D be the cluster category of Dynkin type A_{\infty}. This paper provides a bijection between torsion theories in D and certain configurations of arcs connecting non-neighbouring integers.

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arXiv:1005.4364v1 [math.RT] 24 May 2010
A characterization of torsion theories in the
cluster category of Dynkin type A∞
Puiman Ng
Abstract
Let D be the cluster category of Dynkin type A∞. Following the
line of [2], this paper provides a bijection between torsion theories
in D and certain configurations of arcs connecting non-neighbouring
integers.
0Introduction
The cluster category D of Dynkin type A∞was introduced in [2]. One of
its several definitions, which is completely analogous to the definition of the
cluster category of type An, motivates us to say that D is a cluster category
of type A∞. Namely, it is the orbit category Df(modΓ)/SΣ−2. Here Γ is
a quiver of type A∞with zigzag orientation and S and Σ are the Serre and
suspension functors of the finite derived category Df(modΓ).
There are also several other ways to realize the category D. In brief, it is
the algebraic triangulated category generated by a 2-spherical object. It is
also the compact derived category Dc(A) of the differential graded cochain
algebra A = C∗(S2;k) where S2is the 2-sphere and k is a field. Finally, D
is the finite derived category Df(k[T]) where k[T] is viewed as a DG algebra
with T placed in homological degree 1 and zero differential. It is ubiquitous
and the reader can refer to [2, Section 0] for more details.
In [2], the cluster tilting subcategories of D were shown to be in bijection
with certain maximal sets of non-crossing arcs connecting non-neighbouring
integers. One can think of these maximal sets as “triangulations of the ∞-
gon”.
A torsion theory in D is a pair (X,Y) of subcategories such that there are no
non-zero morphisms from any object in X to any object in Y, and that for
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each d in D, there is a distinguished triangle x → d → y → in D, with x in
X and y in Y. Note that if T is a cluster tilting subcategory, then (T , ΣT )
is a torsion theory, but t-structures and co-t-structures are other examples
of torsion theories.
In this paper, we will generalize the results of [2] by giving a bijection be-
tween torsion theories in D and certain configurations of arcs connecting
non-neighbouring integers.
Let us recall some material from [2]. The category D has finite dimensional
Hom spaces over a field k and split idempotents, so it is Krull-Schmidt ([2,
Remark 1.2]). Throughout the paper, a subcategory of D is assumed to
be a full subcategory closed under direct sums and direct summands. For
subcategories X and Y of D, the set of morphisms from any x in X to any
y in Y is denoted by (X,Y). In particular, for objects x and y of D, the set
of morphisms from x to y is denoted by (x,y).
By [2, Remark 1.4], the Auslander-Reiten (AR) quiver of D is ZA∞, and we
will use the following standard coordinate system on the AR quiver.
...
??????
(−4,1)
?????
...
??????
(−3,2)
?????
...
??????
(−2,3)
?????
...
???????
···
???????
??????
(−4,0)
??????
??????
(−3,−1)
??????
?????
(−3,1)
?????
??????
?????
(−2,2)
?????
??????
?????
(−1,3)
?????
···
?????
(−3,0)
?????
?????
(−2,1)
?????
?????
(−1,2)
?????
??????
??????
···
?????
(−2,0)
?????
(−1,1)
?????
(0,2)
···
??????
Let Σ be the suspension functor of D. Since D is 2-Calabi-Yau, its Serre
functor is S = Σ2and the AR translation is τ = SΣ−1= Σ. Note that in
terms of coordinates, the action of Σ = τ is Σ(m,n) = (m−1,n−1), see [2,
Remark 1.4].
Now let us have a few definitions to interpret these coordinate pairs in an
alternative way, as arcs connecting non-neighbouring integers.
Definition 0.1. ([2, Definition 3.1]) An arc is a pair (m,n) of integers with
n − m ≥ 2. The arc (m,n) is said to end in each of the integers m and n.
Two arcs (m1,n1) and (m2,n2) are said to cross if we have either m1< m2<
n1< n2or m2< m1< n2< n1. Note that the action of Σ makes sense on
arcs as well.
In all the diagrams to follow arcs are drawn on number lines which are num-
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bered thus:
???
−2−1
0123
????
For example, in the above diagram, the arc (−1,2) is drawn as a curve
between the integers −1 and 2. Crossing of arcs has been defined to match
this geometrical picture in the obvious way.
Definition 0.2. ([2, Definition 3.2]) Let A be a set of arcs. If, for each
integer n, there are only finitely many arcs in A which end in n, then A is
said to be locally finite. A left (resp. right) fountain of A is an integer n for
which there are infinitely many arcs of the form (m,n) (resp. (n,m)) in A.
A fountain of A is an integer n which is both a left and a right fountain of
A. Finally, A is said to be non-crossing if A contains no pair of crossing arcs.
We can identify coordinate pairs (m,n) such that n − m ≥ 2 with either
indecomposable objects of D or with arcs. Hence a set A of arcs gives a col-
lection of indecomposable objects of D, and add of these gives a subcategory
A of D. This sets up a bijection between sets of arcs and subcategories of D.
Now let us describe the layout of this paper.
In Section 1, we give a few definitions.
In Section 2, we consider the precovering and preenveloping properties of a
subcategory of D and show the following.
Theorem. Let A be a subcategory of D and let A be the corresponding set
of arcs. Then A is precovering if and only if each right fountain of A is in
fact a fountain.
Theorem. Let A be a subcategory of D and let A be the corresponding set
of arcs. Then A is preenveloping if and only if each left fountain of A is in
fact a fountain.
They will be proved in Theorem 2.2 and in Theorem 2.3 respectively.
In Section 3, we will need the following definitions. The first two describe
some constraints on a set of arcs A.
Definition 0.3. A set of arcs A is said to satisfy condition (i) if, for each
pair of crossing arcs (a,b) and (c,d) in A, those of the pairs (a,c), (c,b), (b,d)
and (a,d) which are arcs belong to A (for instance, (a,c) is only an arc if
c − a ≥ 2).
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???
a??
?
??
?
?
?
??? ? ? ? ???
?
?
?
c??
?
??
b??
?
??
d
????
Definition 0.4. A set of arcs A is said to satisfy condition (ii) if it has the
following property: if a is a left fountain but not a right fountain of A, b is a
right fountain but not a left fountain of A and b − a ≥ 2, then the arc (a,b)
is in A.
???
a
???
b
????
Using these, we establish the following main result.
Theorem. Let X be a subcategory of D and let X be the corresponding set
of arcs. Then the following conditions are equivalent.
(i) X satisfies condition (i) and condition (ii), and each right fountain of
X is in fact a fountain,
(ii) The subcategory X is precovering and is closed under extensions,
(iii) (X,Y) is a torsion theory for some subcategory Y of D.
Note that in (iii), Y = {y ∈ D|(X,y) = 0}. This will be proved in Theo-
rem 3.18.
Finally in Section 4, we give a few examples, characterizing all t-structures
and co-t-structures in D.
1Definitions
In this section we give a few definitions and state a few observations.
Definition 1.1. Let U be a subcategory of D. Define U⊥= {d ∈ D|(u,d) = 0
for all u in U} and⊥U = {d ∈ D|(d,u) = 0 for all u in U}.
Definition 1.2. Let U be a set of arcs. Define ort(U) to be the set of arcs
ort(U) = {d | d does not cross any arcs in U}.
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Now let X and Y be two subcategories of D such that X =⊥Y and Y = X⊥.
Let X and Y be the corresponding sets of arcs respectively. By [2, Lemma
3.6], we have X = {d | Σd does not cross any arcs in Y} and Y = {d | Σ−1d
does not cross any arcs in X}.
Let Σ−1Y = W. We can now rewrite the above as X = {d | d does not
cross any arcs in W} = ort(W) and W = {d | d does not cross any arcs in
X} = ort(X).
Lemma 1.3. Let X and X be defined as above. Then X =⊥(X⊥) is equiv-
alent to X = ortort(X).
Proof. This is obvious by the above description.
2Precovering (preenveloping) subcategories
In this section we characterize precovering and preenveloping subcategories
in terms of their corresponding sets of arcs.
Let us recall the following property regarding the morphisms in the category
D. Let x be an indecomposable object of D. See [2, Definition 2.1] for the
definitions of the regions H−(x) and H+(x) in the AR quiver of D, which are
sketched as follows.
???
?
?
?
H−(x)H+(x)
???
Σx
?????????
?????????
x Σ−1x
?????????
?????????
?
?
?
??
???????????
???????????
??
We write H(x) = H−(x) ∪ H+(x).
Property 2.1. ([2, Corollary 2.3]) Let x and y be indecomposable objects
of D. Then the following are equivalent.
(i) (x,y) ?= 0,
(ii) (x,y) = k,
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