Continuous Frobenius categories

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by  Kiyoshi Igusa

CONTINUOUS FROBENIUS CATEGORIES
KIYOSHI IGUSA AND GORDANA TODOROV
Dedicated to the memory of Dieter Happel
2 Abstract. We introduce continuous Frobenius categories. These are topological cate-
1 gories which are constructed using representations of the circle over a discrete valuation
0 ring. We show that they are Krull-Schmidt with one indecomposable object for each pair
2 of(notnecessarilydistinct)pointsonthecircle. Byputtingrestrictionsonthesepointswe
p obtainvariousFrobeniussubcategories. ThemainpurposeofconstructingtheseFrobenius
e categories is to give a precise and elementary description of the triangulated structure of
S their stable categories. We show in [7] for which parameters these stable categories have
cluster structure in the sense of [1] and we call these continuous cluster categories.
7
1

] The standard construction of a cluster category of a hereditary algebra is to take the
T
orbit category of the derived category of bounded complexes of finitely generated modules
R
over the algebra:
.
h C ∼= Db(modH)/F
t H
a
where F is a triangulated autoequivalence of Db(modH) [2]. In this paper we construct
m
continuousversionsoftheclustercategoriesoftypeA . Thesecontinuous cluster categories
[ n
are continuously triangulated categories (Sec 0) having uncountably many indecomposable

2
objects and containing the finite and countable cluster categories of type A and A as
v n ∞
subquotients. Cluster categories of type A and A were also studied in [3], [6], [12].
8 n ∞
3 The reason for the term continuous in the names of the categories is the fact that
0 the categories that we define and consider in this paper are topological categories with
0
continuous structure maps (Section 0). The continuity requirement implies that there are
.
9
two possible topologically inequivalent triangulations of the continuous cluster category
0
given by the two 2-fold covering spaces of the Moebius band: connected and disconnected.
2
1 We consider both cases (Remarks 3.1.8, 3.4.2).
: Thetermcluster inthenamesofthecategoriesisjustifiedin[7]whereitisshowthatthe
v
i category C has a cluster structure where cluster mutation is given using the triangulated
X π
structure (see [1]) and that the categories C also have a cluster structure for specific values
c
r
a of c. For the categories Cφ, we have partial results (Cφ has an m-cluster structure in certain
cases). Thispaperisthefirstinaseriesofpapers. Themainpurposeofthispaperistogive
a concrete and self-contained description of the triangulated structures of these continuous
cluster categories being developed in concurrently written papers [7, 8].
We will use representations of the circle over a discrete valuation ring R to construct
continuous Frobenius R-categories F , F and F whose stable categories (triangulated
π c φ
categories by a well-known construction of Happel [4]) are isomorphic to the continuous
categories C , C and C , thus inducing continuous triangulated structure on these topolog-
π c φ
ical K-categories (K = R/m).
2000 Mathematics Subject Classification. 18E30:16G20.
The first author is supported by the National Security Agency.
1
In Section 1. we define representations of the circle; a representation of the circle S1 =
R/2πZ over R is defined to be collection of R-modules V[x] at every point x ∈ S1 and
morphisms V[x] → V[y] associated to any clockwise rotation from x to y with the property
that rotation by 2π is multiplication by the uniformizer t of the ring R. We denote the
projective representations generated at points x by P .
[x]
The Frobenius category F is defined in Section 2: the objects are (V,d) where V is a
π
finitely generated projective representation of S1 over R and d is an endomorphism of V
with square equal to multiplication by t. We show that F is a Frobenius category which
π
has, up to isomorphism, one indecomposable object
(cid:18) (cid:20) (cid:21)(cid:19)
E(x,y) = P (cid:96)P , d = 0 β∗
[x] [y] α 0
∗
for every pair of points 0 ≤ x ≤ y < 2π in S1. (See Definition 2.1.7.) The projective-
injective objects are E(x,x) (i.e. when x = y). The stable category of F is shown to be
π
equivalent to the continuous category C , which is defined in 3.1.5. This construction also
π
works in much greater generality (Proposition 3.4.1).
We also consider F for any positive real number c ≤ π in 2.4; F is defined to be the
c c
additive full subcategory of F generated by all (x,y) where the distance from x to y is at
π
leastπ−c. ObjectsinF areprojective-injectiveifftheyattainthisminimumdistance. The
c
stable category is again triangulated and isomorphic to the continuous category C which
c
has a cluster structure if and only if c = (n+1)π/(n+3) for some positive integer n [7].
In that case we show (in [7]) that C contains a thick subcategory equivalent to the cluster
c
category of type A .
n
The most general version of Frobenius categories that we consider in this paper, are the
categories F , for homeomorphisms φ : S1 → S1 satisfying “orientation preserving” and
φ
some other conditions (see 2.4.1). The categories F , and in particular F , are special cases
c π
of F .
φ
In later papers we will develop other properties of these continuous cluster categories.
We will give recognition principles for (the morphisms in) distinguished triangles in the
continuous cluster categories and other continuous categories. An example of this is given
in 3.3.1. We will also show in later papers that the continuous cluster category C has a
π
unique cluster up to isomorphism and we will find conditions to make the cluster character
into a continuous function. And we will show in later papers how this construction can be
modified to produce continuous Frobenius categories of type D.
The first author would like to thank Maurice Auslander for explaining to him that “The
Krull-SchmidtTheoremisastatementaboutendomorphismrings[ofobjectsinacategory]”.
This observation will be used many times. Also, Maurice told us that each paper should
have only one main result. So, our other results will appear in separate papers. We also
thank Adam-Christiaan van Roosmalen for explaining his work to us.
0. Some remarks on topological R-categories
We recall the definition of a topological category since our constructions are motivated
by our desire to construct continuously triangulated topological categories of type A. By
a “continuously triangulated” category we mean a topological category which is also tri-
angulated so that the defining equivalence T of the triangulated category is a continuous
functor. We also review an easy method for defining the topology on an additive category
out of the topology of a full subcategory of indecomposable objects.
2
Recall that a topological ring is a ring R together with a topology on R so that its
structure maps are continuous. Thus addition + : R × R → R and multiplication · :
R × R → R are required to be continuous mappings. We may sometimes also require
the inverse mapping u (cid:55)→ u−1 to be continuous on the group of units of R. A topological
R-module is an R-module M together with a topology on M so that the structure maps
m : R ×M → M and a : M ×M → M given by m(r,x) = rx and a(x,y) = x+y are
continuous mappings.
Definition 0.0.1. If R is a topological ring, a topological R-category is defined to be a
small R-category C together with a topology on the set of objects Ob(C) and on the set
of all morphisms Mor(C) so that the structure maps of C are continuous mappings. Thus
s,t,id,a,m,c are continuous where
(1) s,t : Mor(C) → Ob(C) are the source and target maps.
(2) m : R × Mor(C) → Mor(C), a : A → Mor(C) are the mappings which give the
R-module structure on each hom set C(X,Y) = (s,t)−1(X,Y). Here A is the subset
of Mor(C)2 consisting of pairs (f,g) of morphisms with the same source and target.
(3) id : Ob(C) → Mor(C) is the mapping which sends each X ∈ Ob(C) to id ∈
X
C(X,X) ⊆ Mor(C).
(4) c : Mor(C)⊕Mor(C) → Mor(C) is composition and Mor(C)⊕Mor(C) is the subset
of Mor(C)×Mor(C) on which composition is defined.
We say that a functor F : C → D between topological categories C,D is continuous if
it is continuous on objects and morphisms. Thus, we require Ob(F) : Ob(C) → Ob(D)
and Mor(F) : Mor(C) → Mor(D) to be continuous mappings. When C,D are topological
R-categories, we usually assume that F is R-linear in the sense that the induced mappings
C(X,Y) → D(FX,FY) are homomorphisms of R-modules for all X,Y ∈ Ob(C).
In this paper we will construct Krull-Schmidt categories C each of which has a natural
topology on the full subcategory D = IndC of carefully chosen representatives of the inde-
composable objects. By the following construction, we obtain a small topological category
addD which is equivalent as an additive category to the entire category C.
Definition 0.0.2. A topological R-category D is called additive if there is a continuous
functor ⊕ : D×D → D which is algebraically a direct sum operation. (D×D is given the
product topology on object and morphism sets.)
Suppose D is a topological R-category which does not contain a zero object (and is thus
notadditive). Thenwedefinetheadditive category addD generatedbyD tobethecategory
of formal ordered direct sums of objects in D. Thus the object space of addD is:
(cid:97)
Ob(addD) = Ob(D)n
n≥0
When n = 0, Ob(D)0 consists of a single object which is the unique zero object of addD.
This is a topological space since it is the disjoint union of Cartesian products of topological
(cid:96)
spaces. We write the object (X ) as the ordered sum X .
i i i
The morphism space is defined analogously:
(cid:97)
Mor(addD) = {((Y ),(f ),(X )) ∈ Ob(D)n×Mor(D)nm×Ob(D)m | f ∈ D(X ,Y )}
i ij j ij j i
n,m≥0
This has the topology of a disjoint union of subspaces of Cartesian products of topological
spaces.
3
Proposition 0.0.3. addD is a topological additive R-category in which direct sum ⊕ is
strictly associative and has a strict unit.
Proof. Direct sum is strictly additive: (A⊕B)⊕C = A⊕(B ⊕C) since objects in addD
are, by definition, equal to ordered sequences of objects in D. Zero is a strict unit for ⊕
since it is the unique empty sequence: 0⊕X = X = X⊕0 for all X. The fact that addD is
a topological category follows easily from the assumption that D is a topological category.
For example, composition of morphisms in addD is given by addition of composites of
morphisms in D and both of these operations are continuous. (cid:3)
1. Representations of the circle S1
In this section we describe the category of representations of the circle over a discrete
valuationring. Specialkindsoffinitelygeneratedprojectiverepresentationsofthecirclewill
be used in section 2 in order to define Frobenius categories. Let R be a discrete valuation
ring with uniformizing parameter t (a fixed generator of the unique maximal ideal m),
quotient field K = R/m = R/(t).
1.1. Representations of S1.
Let S1 = R/2πZ. Let x ∈ R and let [x] denote the corresponding element [x] = x+2πZ
in S1. When we take an element [x] ∈ S1 we mean choose an element of S1 and choose an
arbitrary representative x of this element in R.
Definition 1.1.1. A representation V of S1 over R is defined to be:
(a) an R-module V[x] for every [x] ∈ S1 and
(b) an R-linear map V(x,α) : V[x] → V[x−α] for all [x] ∈ S1 and α ∈ R satisfying
≥0
the following conditions for all [x] ∈ S1:
(1) V(x−β,α)◦V(x,β) = V(x,α+β) for all α,β ∈ R ,
≥0
(2) V(x,2πn) : V[x] → V[x] is multiplication by tn for all n ∈ Z .
≥0
Definition 1.1.2. A morphism f : V → W consists of R-linear maps f : V[x] → W[x]
[x]
for all [x] ∈ S1 so that W(x,α)f = f V(x,α) for all [x] ∈ S1 and α ≥ 0, i.e.,
[x] [x−α]
f[x] (cid:47)(cid:47)
V[x] W[x]
V(x,α) W(x,α)
(cid:15)(cid:15) (cid:15)(cid:15)
f[x−α](cid:47)(cid:47)
V[x−α] W[x−α].
A morphism f : V → W is called a monomorphism or epimorphism if f : V[x] → W[x]
[x]
are monomorphisms or epimorphisms, respectively, for all [x] ∈ S1.
Definition 1.1.3. Let P , for [x] ∈ S1, be the representation of S1 defined as:
[x]
(a) R-module P [x−α] := Reα, the free R-module on one generator eα for each real
[x] x x
number 0 ≤ α < 2π.
(x−α,β)
(b) R-homomorphism P : P [x − α] → P [x − α − β] is the unique R-linear
[x] [x] [x]
homomorphism defined by P(x−α,β)(eα) = eα+β. Here eγ+2πn := tneγ ∈ P [x−γ]
[x] x x x x [x]
for n ∈ Z , and γ ∈ R by definition.
≥0 ≥0
Remark 1.1.4. It follows from the definition that e0 is a generator of the representation
x
P ; we will often denote this generator by e .
[x] x
4
Proposition 1.1.5. Let V be a R-representation of S1. There is a natural isomorphism
∼
P (P ,V) = V[x]
S1 [x]
given by sending f : P → V to f (e ) ∈ V[x]. In particular the ring homomorphism
[x] [x] x
∼ ∼
R → End(P ) = P [x] = R sending r ∈ R to multiplication by r is an isomorphism.
[x] [x]
Proof. Define a homomorphism ϕ : V[x] → P (P ,V) in the following way. For every
S1 [x]
v ∈ V[x] let ϕ(v) ∈ P (P ,V) be given by ϕ(v) (reα) := V(x,α)(rv) ∈ V[x−α] for all
S1 [x] [x−α] x
0 ≤ α < 2π. Then ϕ(v) is the unique morphism P → V such that ϕ(v)(e ) = v.
[x] x
In particular, ϕ(f (e ))(e ) = f (e )) = f(e ). Therefore ϕ(f (e )) = f since both
[x] x x [x] x x [x] x
morphisms send the generator e ∈ P [x] to f (e ). Therefore, ϕ gives an isomorphism
x [x] [x] x
V[x] ∼= P (P ,V) inverse to the map sending f to f (e ). (cid:3)
S1 [x] [x] x
Corollary 1.1.6. Each representation P is projective. In other words, if f : V → W is
[x]
an epimorphism then Hom(P ,V) → Hom(P ,W) is surjective. (cid:3)
[x] [x]
If x ≤ y < x+2π then P [x] = R is generated by eβ where β = y−x. So, we get the
[y] y
following Definition/Corollary.
Definition 1.1.7. Thedepth ofanynonzeromorphismoftheformf : P → P isdefined
[x] [y]
to be the unique nonnegative real number δ(f) = α so that f(e ) = ueα for a unit u ∈ R.
x y
We define the depth of the zero morphism to be ∞.
Lemma 1.1.8. The depth function has the following properties.
(1) For morphisms f : P → P , g : P → P we have δ(g◦f) = δ(g)+δ(f).
[x] [y] [y] [z]
(2) Giventwomorphismf,g : P → P andr,s ∈ Rwehaveδ(rf+sg) ≥ min(δ(f),δ(g)).
[x] [y]
Proof. (1) If f(e ) = ueα and g(e ) = veβ for units u,v ∈ R then gf(e ) = uveα+β making
x y y z x z
δ(g◦f) = α+β = δ(f)+δ(g).
(2) Suppose that f(e ) = ueα and g(e ) = veβ where u,v are units in R. Suppose α =
x y x y
δ(f) ≤ β = δ(g). Then β = α+2πn for some nonnegative integer n and g(e ) = vtneα. So,
x x
rf +sg = (ru+svtn)eα which has depth ≥ α = min(δ(f),δ(g)). (cid:3)
x
(cid:96) (cid:96)
We extend the definition of depth to any morphism f : P → P by
i [xi] j [yj]
δ(f) = min{δ(f ) | f : P → P }.
ji ji [xi] [yj]
Proposition 1.1.9. The extended notion of depth satisfies the following conditions.
(cid:96) (cid:96) (cid:96) (cid:96)
(1) Let f : P → P and g : P → P . Then δ(g◦f) ≥ δ(g)+δ(f).
i [xi] j [yj] j [yj] k [zk]
(2) The depth of f is independent of the choice of decompositions of the domain and
(cid:96) (cid:96)
range of f, i.e. δ(f) = δ(ψ◦f ◦ϕ) for all automorphisms ψ,ϕ of P , P .
j [yj] i [xi]
Proof. (1) implies (2) since δ(ψfϕ) ≥ δ(ψ)+δ(f)+δ(ϕ) ≥ δ(f) and δ(f) ≥ δ(ψfϕ) by
symmetry. Therefore, it suffices to prove (1).
By the extended definition of depth, δ(gf) is equal to the depth of one of its component
functions (gf) : P → P . But this is the sum of composite functions of the form
ki [xi] [zk]
g f : P → P → P . By the lemma above, this gives
kj ji [xi] [yj] [zk]
δ(gf) = min(δ((gf) )) ≥ min(δ(g )+δ(f )) ≥ δ(g)+δ(f). (cid:3)
ki kj ji
5
1.2. Finitely generated projective representations of S1.
It is shown here that finitely generated projective representations of S1 are precisely the
finitely generated torsion free representations.
Definition 1.2.1. ArepresentationV istorsion-free ifeachV[x]isatorsion-freeR-module
and each map V(x,α) : V[x] → V[x−α] is a monomorphism. A representation V is finitely
generated if it is a quotient of a finite sum of projective modules of the form P , i.e. there
[x]
exists an epimorphism (cid:96)n P (cid:16) V.
i=0 [xi]
LetV beafinitelygeneratedtorsion-freerepresentationofS1. Thenthefollowinglemma
shows that a subrepresentation of V generated at any finite set of points on the circle is a
∼ (cid:96)
projective representation P = m P .
i [xi]
Lemma 1.2.2. Let V be as above. Take any finite subset of S1 and represent them with
real numbers
x < x < x < ··· < x < x = x +2π, x ∈ R.
0 1 2 n n+1 0 i
For each 0 ≤ i ≤ n let {v : j = 1,··· ,m } be a subset of V[x ] which maps isomorphically
ij i i
to a basis of the cokernel of V(xi+1,xi+1−xi) : V[xi+1] → V[xi] considered as a vector space
over K = R/(t). Let f : P → V be the morphism defined by f (e ) = v ∈ V[x ] Then
ij [xi] ij xi ij i
(1) f = (cid:80)n (cid:80)mi f : P = (cid:96)n m P → V is a monomorphism.
i=0 j=0 ij i=0 i [xi]
(2) f : P[x ] → V[x ] is an isomorphism for each i.
[xi] i i
Proof. Since V is torsion-free, the maps V(xi,xi−x0) : V[xi] → V[x0] are monomorphisms for
i = 0,1,2,··· ,n. Let Vi = image(V(xi,xi−x0)) ⊂ V[x0]. Then
tV = V ⊆ V ⊆ ··· ⊆ V ⊆ V ⊆ V .
0 n+1 n 2 1 0
∼
Furthermore, V[x ] = V and this isomorphism induces an isomorphism of quotients:
i i
∼
V[x ]/V[x ] = V /V . Let w ∈ V ⊆ V be the image of v ∈ V[x ] and let w =
i i+1 i i+1 ij i 0 ij i ij
∼
w + V ∈ V /V = V[x ]/V[x ]. For each i, the w form a basis for V /V .
ij i+1 i i+1 i i+1 ij i i+1
Taken together, w +tV form a basis for V /tV . Since V is torsion free, it follows from
ij 0 0 0 0
Nakayama’s Lemma, that the w generate V freely. Therefore, the morphism f : P =
ij 0
(cid:96)
m P → V which maps the generators of P to the elements v induces an isomorphism
i xi ij
∼
f : P[x ] = V[x ]. Applying the same argument to the points
[x0] 0 0
x < x < ··· < x < x +2π,x +2π < ··· < x +2π, x ∈ R
i i+1 n 0 1 i i
we see that f : P[x ] → V[x ] is an isomorphism for all i. This proves the second
[xi] i i
condition. The first condition follows. (cid:3)
Proposition 1.2.3. Every finitely generated projective representation of S1 is torsion-free.
Conversely, every finitely generated torsion-free representation of S1 over R is projective
and isomorphic to a direct sum of the form (cid:96)n P .
i=0 [xi]
Proof. The first statement is clear since indecomposable projectives are torsion free and
every direct summand of a torsion-free representation is torsion-free. For the second state-
ment, let V be a finitely generated torsion-free representation of S1. Suppose that V is
generated at n+1 points on the circle: [x ],[x ],··· ,[x ] ∈ S1 where x < x < x < ··· <
0 1 n 0 1 2
x < x = x +2π, with x ∈ R, as in the lemma. Let f : P = (cid:96)n m P → V be the
n n+1 0 i i=0 i [xi]
monomorphism given by the lemma. Then f : P → V is also onto by Condition (2) in the
lemma since V is generated at the points [x ]. Therefore, P ∼= V as claimed. (cid:3)
i
6
1.3. The category P .
S1
Let P be the category of all finitely generated projective (and thus torsion-free) repre-
S1
sentations of S1 over R. By the proposition above, each indecomposable object of P is
S1
isomorphic to P for some [x] ∈ S1.
[x]
Lemma 1.3.1. Any nonzero morphism f : P → P is a categorical epimorphism in P
[x] [y] S1
in the sense that, for any two morphisms g,h : P → V in P , gf = hf implies g = h.
[y] S1
Proof. Letf(e ) = reαforr (cid:54)= 0 ∈ R. Thengf(e ) = g(reα) = g(rV(y,α)e ) = rV(y,α)(g(e )).
x y x y y y
By assumption this is equal to hf(e ) = rV(y,α)(h(e )). Since V is torsion-free, this implies
x y
that g(e ) = h(e ) making g = h. (cid:3)
y y
For the proof of Lemma 2.2.3 and Proposition 2.3.3 below, we need the following easy
observation using the depth δ(f) from Definition 1.2.1.
Proposition 1.3.2. Let f : P → P .
[x] [y]
(1) If g : P → P is a morphism so that δ(f) ≤ δ(g) then there is a unique morphism
[x] [z]
h : P → P so that hf = g.
[y] [z]
(2) If g(cid:48) : P → P is a morphism with δ(g(cid:48)) ≥ δ(f) then there is a unique h(cid:48) : P →
[w] [y] [w]
P so that fh(cid:48) = g(cid:48).
[x]
Proof. We prove the first statement. The second statement is similar. Let α = δ(f),β =
δ(g)−α. Then f(e ) = reα and g(e ) = seα+β where r,s are units in R. Let h : P → P
x y x z [y] [z]
be the morphism given by h(e ) = r−1seβ. Then hf(e ) = rh(eα) = seα+β. So hf = g. (cid:3)
y z x y z
Definition 1.3.3. Since P has one indecomposable object P for every [x] ∈ S1, the
S1 [x]
full subcategory IndP of these objects has a natural topology. The space of objects of
S1
IndP is homeomorphic to the circle S1 and the space of morphisms is the quotient space:
S1
Mor(IndP ) = {(r,x,y) ∈ R×R×R | x ≤ y ≤ x+2π}/ ∼
S1
where the equivalence relation is given by (r,x,y) ∼ (r,x+2πn,y +2πn) for any integer
n and (r,x,x+2π) ∼ (tr,x,x). Here (r,x,y) represents the morphism P → P which
[x] [y]
sends e to rey−x. The second relation comes from the identity re2π = tre0. We give R the
x y x x
m-adic topology.
Remark 1.3.4. (1) The category P is algebraically equivalent to the topological ad-
S1
ditive R-category addIndP given by Definition 0.0.2.
S1
(2) Intheterminologyof[13], P isthefullsubcategoryoffinitelygeneratedprojective
S1
(cid:91)
objects in the big loop KL• where L is the half-open interval [0,2π) considered as
a linearly ordered set.
2. The Frobenius categories F , F , F
π c φ
2.1. Frobenius category F .
π
We define the category F and the set of exact sequences in F (and hence cofibrations
π π
which are the beginning maps and quotient maps which are the end maps in these exact
sequences). Then we show that F is an exact category and that it has enough projectives
π
with respect to the exact structure. Finally, we show that projective and injective objects
in F coincide proving that F is a Frobenius category.
π π
Definition 2.1.1. The category F and the exact sequences in F are defined as:
π π
7
(1) Objects of F are pairs (V,d) where V ∈ P and d : V → V is an endomorphism
π S1
of V so that d2 = t (multiplication by t).
(2) Morphisms in F are f : (V,d) → (W,d) where f : V → W satisfies fd = df.
π
f g f g
(3) Exact sequences in F are (X,d) −→ (Y,d) →− (Z,d) where 0 → X −→ Y →− Z → 0 is
π
exact (and therefore split exact) in P .
S1
FollowingWaldhausen[14]wecallthefirstmorphisminanexactsequenceacofibration and
write it as (X,d) (cid:26) (Y,d) and we call the second morphism a quotient map and denote it
by (Y,d) (cid:16) (Z,d).
Remark 2.1.2. Note that if (V,d) is an object in F , then V cannot be indecomposable
π
since End(P ) = R does not contain an element whose square is t. We will see later that V
x
must have an even number of components.
Theorem 2.1.3. The category F is a Frobenius category.
π
Lemma 2.1.4. f : (V,d) (cid:26) (W,d) is a cofibration (the beginning of an exact sequence) if
and only if f : V → W is a split monomorphism in P . Similarly, f is a quotient map in
S1
F if and only if it is a split epimorphism in P . In particular, all epimorphisms in F
π S1 π
are quotient maps.
Proof. By definition of exactness, the split monomorphism condition is necessary. Con-
versely, suppose that f : V → W is split mono in P . Then the cokernel C is projective,
S1
being a summand of the projective object W. Since fd = df, we have an induced map
d : C → C. Since d2 = t on V and W we must have d2 = t on C. Therefore, f is the
beginning of the exact sequence (V,d) (cid:26) (W,d) (cid:16) (C,d). The other case is similar with
the added comment that all epimorphisms in P are split epimorphisms. (cid:3)
S1
Lemma 2.1.5. F is an exact category.
π
Proof. We verify the dual of the short list of axioms given by Keller [10]. The first two
axioms follow immediately from the lemma above.
(E0) 0 (cid:26) 0 is a cofibration.
(E1) The collection of cofibrations is closed under composition.
(cid:47)(cid:47)f (cid:47)(cid:47) g(cid:47)(cid:47) (cid:47)(cid:47)
(E2) The pushout of an exact sequence (A,d) (B,d) (C,d) along any morphism
h : (A,d) → (A(cid:48),d) exists and gives an exact sequence (A(cid:48),d) (cid:26) (B(cid:48),d) (cid:16) (C,d).
Pf: Since f : A → B is a split monomorphism in P , so is (f,h) : A → B(cid:96)A(cid:48). By
S1
the Lemma, we can let (B(cid:48),d) ∈ F be the cokernel of (f,h). Since the pushout of a split
π
sequence is split, the sequence A(cid:48) → B(cid:48) → C splits in P . Therefore (A(cid:48),d) (cid:26) (B(cid:48),d) (cid:16)
S1
(C,d) is an exact sequence in F . Similarly, we have the dual axiom:
π
(E2)op The pullback of an exact sequence in F exists and is exact.
π
Therefore, F is an exact category. (cid:3)
π
We record the following easy extension of this lemma for future reference.
Proposition 2.1.6. Suppose that A is an additive full subcategory of F with the property
π
that any cofibration in F with both objects in A has cokernel in A and that any quotient
π
map in F with both objects in A has kernel in A. Then A is an exact subcategory of F .
π π
Proof. Under the first condition, cofibrations in A will be closed under composition and
under pushouts since the middle term of the pushout of X (cid:26) Y (cid:16) Z under any morphism
X → X(cid:48) in A is the cokernel of the cofibration X (cid:26) Y (cid:96)X(cid:48). Dually, quotient maps will
8
be closed under pull-backs since the pull-back of a quotient map Y (cid:16) Z along a morphism
W → Z is the kernel of the quotient map Y (cid:96)W (cid:16) Z. So, A is exact. (cid:3)
Definition 2.1.7. Let P be an object of P . We define the object P2 ∈ F to be
S1 π
(cid:18) (cid:20) (cid:21)(cid:19)
P2 := P (cid:96)P, 0 t .
1 0
It is clear that (P (cid:96)Q)2 = P2(cid:96)Q2.
The functor P (cid:55)→ P2 is both left and right adjoint to the forgetful functor (V,d) (cid:55)→ V.
Lemma 2.1.8. F (P2,(V,d)) ∼= P (P,V) and P2 is projective in F .
π S1 π
Proof. A morphism P2 → (V,d) is the same as a pair of morphisms f,g : P → V so that
g = df. So, (f,df) ↔ f gives the desired isomorphism. To see that P2 is projective in F ,
π
consider any quotient map (V,d) (cid:16) (W,d) and a morphism (f,df) : P2 → (W,d). We can
choose a lifting f˜: P → V of f : P → W to get a lifting (f˜,df˜) of (f,df). (cid:3)
Lemma 2.1.9. F ((V,d),P2) ∼= P (V,P) and P2 is injective for cofibrations in F .
π S1 π
Proof. A morphism (V,d) → P2 is the same as a pair of morphisms f,g : V → P so that
f = gd. Therefore, (gd,g) ↔ g gives the isomorphism. To see that P2 is injective for
cofibrations, consider any cofibration (V,d) (cid:26) (W,d) and any morphism (gd,g) : (V,d) →
P2. Then, an extension of (gd,g) to (W,d) is given by (gd,g) where g : W → P is an
extension of g : V → P given by the assumption that V → W is a split monomorphism. (cid:3)
Lemma 2.1.10. The category F has enough projective and injective objects: V2, V ∈ P .
π S1
Proof. For any object (V,d) ∈ F the projective-injective object V2 maps onto (V,d) by
π
the quotient map (1,d) : V2 (cid:16) (V,d). Also (d,1) : (V,d) (cid:26) V2 is a cofibration. (cid:3)
Proof of Theorem 2.1.3. There is only one thing left to prove. We need to show that every
projective object in F is isomorphic to an object of the form P2 for some P ∈ P and is
π S1
therefore injective.
Let (V,d) be a projective object in F . Then the epimorphism (1,d) : V2 → (V,d)
π
splits. Therefore, (V,d) is isomorphic to a direct summand of V2. By Proposition 1.2.3,
the representation V decomposes as V ∼= (cid:96)n P . It follows that V2 ∼= (cid:96)n P2 .
i=0 [xi] i=0 [xi]
Therefore, (V,d) is a direct summand of (cid:96)n P2 . We need a Krull-Schmidt theorem to
i=0 [xi]
let us conclude that (V,d) is isomorphic to a direct sum of a subset of the projective objects
P2 . This follows from the following lemma. (cid:3)
[xi]
Lemma 2.1.11. The endomorphism ring of P2 is a commutative local ring. Therefore,
[x]
every indecomposable component of (cid:96)P2 is isomorphic to one of the terms P2 .
[xi] [xi]
Proof. By the two previous lemmas, an endomorphism of P2 is given by morphism
[x]
(cid:20) (cid:21)
a tb (cid:96) (cid:96)
: P P → P P
b a
where a,b ∈ End(P ) = R. Calculation shows that matrices of this form commute with
[x]
each other. Those matrices with a ∈ (t) form an ideal and, if a ∈/ (t) then
(cid:20) (cid:21)−1 (cid:20) (cid:21)
a tb au −tbu
=
b a −bu au
9
where u is the inverse of a2−tb2 in R. Therefore, End (P2 ) is local. (cid:3)
Fπ [x]
2.2. Indecomposable objects in F .
π
We now describe representations E(x,y) and prove that all indecomposable objects of F
π
are isomorphic to these representations.
Definition 2.2.1. Let [x],[y] be two (not necessarily distinct) elements of S1 and represent
them by real numbers x ≤ y ≤ x+2π. Let α = y−x,β = x+2π−y and let
(cid:18) (cid:20) (cid:21)(cid:19)
E(x,y) = P (cid:96)P , d = 0 β∗
[x] [y] α 0
∗
where α : P → P is the morphism such that α (e ) = eα for the generator e ∈ P [x]
∗ [x] [y] ∗ x y x [x]
and eα ∈ P [x] and, similarly, β : P → P sends e ∈ P [y] to eβ ∈ P [y]. In other
y [y] ∗ [y] [x] y [y] x [x]
words, d(reγ,seδ) = (seδ+β,reγ+α) for all r,s ∈ R and γ,δ ≥ 0.
x y x y
∼
There is an isomorphism E(x,y) = E(y,x+2π) given by switching the two summands
and an equality E(x,y) = E(x + 2πn,y + 2πn) for every integer n. In the special case
x = y, we have α = 0 making α the identity map on P and β = 2π making β equal to
∗ [x] ∗
multiplication by t. Thus, E(x,x) = P2 and E(x,x+2π) ∼= P2 which is projective in F .
[x] [x] π
Lemma 2.2.2. The endomorphism ring of E(x,y) is a commutative local ring. Therefore,
E(x,y) is an indecomposable object of F .
π
(cid:20) (cid:21)
a tb
Proof. Computation shows that endomorphisms of E(x,y) are given by matrices
b a
witha,b ∈ R. ThereforeEnd (E(x,y))isacommutativelocalringasinLemma2.1.11. (cid:3)
Fπ
In order to prove that the category F is Krull-Schmidt we need the following lemma,
π
which uses the notion of depth as defined in 1.1.7
Lemma 2.2.3. Let (V,d) be an object in F and let ϕ : V ∼= (cid:96)n P be a decomposition
π i=0 [xi]
of V into indecomposable summands. Let f : P → P be the component of f = ϕdϕ−1
ji [xi] [xj]
with the smallest depth. Then we may choose i (cid:54)= j, and the representatives x ,x ∈ R so
i j
that x ≤ x ≤ x +π and E(x ,x ) is a direct summand of (V,d).
i j i i j
Proof. We first note that, since the depth of d2 = t is 2π, the depth of d is δ(d) ≤ π.
Therefore δ(f) ≤ π. Next, we show that the minimal depth is attained by an off-diagonal
entry of the matrix (f : P → P ). Suppose that a diagonal entry f has the minimal
ji [xi] [xj] ii
depth. Then δ(f ) = 0 (since it can’t be 2π). But then f is an isomorphism. But f2 is
ii ii
zero modulo t. To cancel the f2 term in f2 there must be some j (cid:54)= i so that f is also an
ii ji
isomorphism, making δ(f ) = 0.
ji
So, we may assume that P and P are distinct components of V and we may choose
[xi] [xj]
the representatives x ,x in R so that x ≤ x ≤ x +π and δ(d) = δ(f) = δ(f ) = x −x .
i j i j i ji j i
Let α = x −x and β = x +2π−x = 2π−α. We now construct a map ρ : E(x ,x ) → V,
j i i j i j
(cid:18) (cid:20) (cid:21)(cid:19)
P (cid:96)P ,d = 0 β∗ −→ρ (V,d);
[xi] [xj] E α∗ 0
α : P → P is defined by α (e ) = eα and β : P → P ] by β (e ) = eα.
∗ [xi] [xj] ∗ xi xj ∗ [xj] [xi] ∗ xj xi
So δ(α ) = α and δ(β ) = β. In order to define ρ consider the following diagram where
∗ ∗
10