## REALIZATION OF OBJECTS IN A TRIANGULATED CATEGORY

KATSUHIKO KURIBAYASHI

Abstract. The level of a module over a differential graded algebra measures
the number of steps to build the module in an appropriate triangulated cat-
egory. We define the levels for spaces and investigate the invariant of spaces
over aK-formal space. In particular, the level of the total space of a bundle
over the 4-dimensional sphere is computed with the aid of Auslander-Reiten
theory over spaces due to Jørgensen. We also discuss a realization problem of
indecomposable objects in the derived category of the*d*-dimensional sphere.

The Hopf invariant brings a criterion for the realization.

1. Introduction

Categorical representation theory provides important technical tools and ideas in the study of many areas of mathematics including finite group theory, algebraic geometry and algebraic topology. Triangles and quivers which appear in Auslander- Reiten theory are indeed such tools; see, for example, [15], [16] and [17].

The singular (co)chain complex functor develops algebraic model theory for topo- logical spaces; see [1], [3], [11], [14] and [30]. Thus we anticipate that the singular cochain functor brings, moreover, new insights into the connection between algebra and topology when it works together with the tools mentioned above in the realm of categorical representation theory; see, for related results, [4], [6], [7], [9] and [23].

In this paper, with such expectation in mind, we introduce and investigate a
topological invariant, which is called *the level of a space*. The notion of levels of
objects in a triangulated category was originally introduced by Avramov, Buch-
weitz, Iyengar and Miller in [2]. Roughly speaking, the level of an object *M* in a
triangulated category*T* counts the number of steps required to build *M* out of a
fixed object via triangles in *T*. We then define the level of a space*X* to be that
of the differential graded module*C*^{∗}(*X*;K) obtained by converting the space with
the singular cochain functor*C*^{∗}( ;K), whereKis a field.

In the rest of this section, we overview briefly our main results. We first give
a reduction theorem (Theorem 2.3) for computing the level of the pullback associ-
ated withK-formal spaces. An explicit calculation with the theorem tells us that
a ‘nice’ space such as the total space *E* of a bundle over the sphere *S*^{d} is of low
level; see Propositions 2.4 and 2.5. This means that the object*C*^{∗}(*E*;K) consists
of indecomposable ones, which is called molecules, with low level in an appropri-
ate triangulated subcategory *T* of the derived category D(*C*^{∗}(*S*^{d};K)). As for the
molecules which constitute*C*^{∗}(*E*;K), these are visualized with black vertices in the

*2000 Mathematics Subject Classification*: 16E45, 18E30, 55R20, 13D07.

*Key words and phrases.*Level, Auslander-Reiten quiver, triangulated category, formal space.

Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan e-mail:kuri@math.shinshu-u.ac.jp

1

Auslander-Reiten quiver of*T* as drawn below. Observe that the vertices are put
on the first and second rows; see Lemma 3.3. For more details, see also Theorem
2.8, which is a remarkable result due to Jørgensen.

*· · ·* *◦*

*Z*_{2}

""

EE EE

E *◦*

""

EE EE

E *◦*

""

EE EE

E *◦*

Σ^{−3(d−1)}*Z*_{2}

""

EE EE

E *· · ·*

*◦*

*Z*_{1}

""

EE EE E

<<

yy yy

y *◦*

Σ^{−(d−1)}*Z*_{1}

""

EE EE E

<<

yy yy

y *◦*

Σ^{−2(d−1)}*Z*_{1}

""

EE EE E

<<

yy yy

y *◦* *· · ·*

*◦*

""

EE EE E

<<

yy yy

y *◦*

""

EE EE E

<<

yy yy

y *•*

""

EE EE E

<<

yy yy

y *•*

Σ^{−2(d−1)}*Z*_{0}

""

EE EE E

<<

yy yy y

*· · ·* *◦*

<<

yy yy

y *•**Z*0

<<

yy yy

y *•*_{Σ}*−*(*d−*1)*Z*_{0}

<<

yy yy

y *•* *· · ·*

Here the vertex*Z*0 denotes the differential graded module*C*^{∗}(*S*^{d};K) and Σ is the
shift operator of*T*. Thus one has a new algebraic aspect of a topological object.

We further deal with a problem of realizing a vertex (molecule) in an Auslander- Reiten quiver by a topological space via the singular cochain functor. In conse- quence, we see that almost all molecules appeared in the quiver over the sphere are not realized by finite CW complexes. In fact Theorem 2.12 states that, in the Auslander-Reiten quiver mentioned above, the arrow

*Z*0 *•* //*•* ^{Σ}^{−}^{(d}^{−}^{1)}^{Z}^{1}

is only realizable. This gives a new topological aspect of the Auslander-Reiten quiver.

2. Results

In this section, we state our main results more precisely. The results, which directly motivate us to work in a triangulated category over a singular cochain algebra, are also described.

Let *A* be a simply-connected differential graded algebra over a field of charac-
teristic zero and D(*A*) the derived category of differential graded modules over*A*.

Recently, Jørgensen [18] has proved that the full subcategory D^{c}(*A*) of D(*A*), which
consists of compact objects, has the Auslander-Reiten triangles if the cohomology
of*A*is a Poincar´e duality algebra. It is also proved in [19] that each component of
the Auslander-Reiten quiver is of the formZ*A*_{∞}.

Very recently, Schmidt [36] has shown that the result on Auslander-Reiten com-
ponents holds even if the characteristic of the underlying field is positive; see also
[20]. Thus if *X* is a simply-connected space whose cohomology *H*^{∗}(*X*;K) with
coefficients in a filed K is a Poincar´e algebra, then the singular cochain complex
functor*C*^{∗}( ;K) makes an appropriate space over*X*into an object in D^{c}(*C*^{∗}(*X*;K))
in which Auslander-Reiten theory is applicable; see below for more details.

To describe our result on the level of a space, we begin by recalling from [2] the
definition of the thickening of a triangulated category*T*. For a given object*G*in*T*,
we first define the 0th thickening by thick^{0}_{T}(*G*) =*{*0*}*and thick^{1}_{T}(*G*) by the smallest
strict full subcategory which contains*G*and is closed under taking finite coproducts,
retracts and all shifts. Moreover for *n >* 1 define inductively the *n*th thickening
thick^{n}_{T}(*G*) by the smallest strict full subcategory of*T* which is closed under retracts
and contains objects*M* admitting a distinguished triangle*M*1*→M* *→M*2*→*Σ*M*1

in *T* for which *M*_{1} and *M*_{2} are in thick^{n}_{T}^{−}^{1}(*G*) and thick^{1}_{T}(*G*), respectively. As
mentioned in [2, 2.2.4], the thickenings provide a filtration of the thick subcategory

Thick_{T}(*G*) of*T* containing the object*G*:

*{*0*}*= thick^{0}_{T}(*G*)*⊂ · · · ⊂*thick^{n}_{T}(*G*)*⊂ · · · ⊂ ∪*^{n}*≥*0thick^{n}_{T}(*G*) = Thick_{T}(*G*)*.*

For an object*M* in*T*, we define a numerical invariant level^{G}_{T}(*M*), which is called
*the level ofM* with respect to*G*in*T*, by

level^{G}_{T}(*M*) := inf*{n∈*N*∪ {*0*} |M* *∈*thick^{n}_{T}(*G*)*}.*
The dimension dim*T* of a triangulated category*T* [5] [35] is defined by

dim*T* = inf*{n∈*N*|*there exists an object*G∈ T* with thick^{n+1}_{T} (*G*) =*T }.*
Thus the notion of levels is closely related to the dimension of *T* and to the thick
subcategory of*T* as well.

Let *A* be a differential graded algebra (abbreviated DGA) over a field K and
D(*A*) the derived category of differential graded right*A*-modules (abbreviated DG
modules), which is viewed as a triangulated category. Observe that a triangle in
D(*A*) comes from a cofibre sequence of the form *M* *→*^{f} *N* *→* *C**f* *→* Σ*M* in the
homotopy category of DG modules over*A*. Here*C**f* denotes the mapping cone and
Σ*M* is the suspension of*M* defined by (Σ*M*)^{n}=*M*^{n+1}. In what follows, we denote
by level_{D(A)}(*M*) the invariant level^{A}_{D(A)}(*M*) for any object*M* in D(*A*).

We shall say that a graded vector space *M* is locally finite if *M*^{i} is of finite
dimension for any*i*. Unless otherwise explicitly stated, it is assumed that a space
has the homotopy type of a CW complex whose cohomology with coefficients in the
underlying field is locally finite. Let*X* be a simply-connected space and*T OP/X*
the category of connected spaces over*X*; that is, objects are maps with the target
*X* and morphisms from *α* : *Y* *→* *X* to *β* : *Z* *→* *X* are maps *f* : *Y* *→* *Z* such
that *βf*=*α*. For an object*α*:*Y* *→X* in*T OP/X*, the singular cochain complex
*C*^{∗}(*Y*;K) is regarded as a DG module over the DGA*C*^{∗}(*X*;K) with the morphism
of DGA’s induced by*α*. Thus we have a functor

(2.1) *C*^{∗}( ;K) :*T OP/X→*D(*C*^{∗}(*X*;K))*.*

Let *Y* *→* *X* be an object in *T OP/X*. We then write level_{D(C}*∗*(*X*;K))(*Y*) for the
invariant level_{D(C}*∗*(*X*;K))(*C*^{∗}(*Y*;K)) and refer to it as*the level*of the space*Y*.

Let *m*_{X} : *T V*_{X} *→*^{'} *C*^{∗}(*X*;K) be a minimal TV-model for a simply-connected
space in the sense of Halperin and Lemaire [14]; that is, *T V*_{X} is a DGA whose
underlyingK-algebra is the tensor algebra generated by a graded vector space*V*_{X}
and, for any element *v* *∈* *V*_{X}, the image of*v* by the differential is decomposable;

see also Appendix.

Recall that a space *X* is K*-formal* if it is simply-connected and there exists a
quasi-isomorphism from a minimal*T V*-model for*X* to the cohomology*H*^{∗}(*X*;K).

Thus, in the case, we have a sequence of quasi-isomorphisms
*H*^{∗}(*X*;K) *T V**X*

*φ*_{X}

oo *'* ^{m}^{X}

*'* //*C*^{∗}(*X*;K)*,*

where *m**X* :*T V**X* *→C*^{∗}(*X*;K) denotes a minimal*T V*-model for*X*. Observe that
spheres*S*^{d} with*d >*1 areK-formal for any fieldK[8][34] and a simply-connected
space whose cohomology with coefficients inK is a polynomial algebra generated
by elements with even degree isK-formal; see [32, Section 7].

**Definition 2.1.** Let*q*:*E→B* and*f* :*X* *→B*be maps betweenK-formal spaces.

The pair (*q, f*) is relativelyK-formalizable if there exists a commutative diagram
up to homotopy

*H*^{∗}(*E*;K) *T V**E*
*φ**E*

oo *'* ^{m}^{E}

*'* //*C*^{∗}(*E*;K)
*H*^{∗}(*B*;K)

*H*^{∗}(*q*)

OO

*H*^{∗}(*f*)

²²

*T V**B*
*φ*_{B}

oo *'* ^{m}^{B}

*'* //

e
*q*

OO

e
*f*

²²

*C*^{∗}(*B*;K)

*q*^{∗}

OO

*f*^{∗}

²²*H*^{∗}(*X*;K) *T V**X*

*φ*_{X}

oo *'* ^{m}^{X}

*'* //*C*^{∗}(*X*;K)*,*
in which horizontal arrows are quasi-isomorphisms.

We here comment on a map between K-formal spaces. In general, for a given
quasi-isomorphisms*φ**E*,*m**E*,*φ**B*and*m**B*as in Definition 2.1, there exist DGA maps
e

*q*1and*q*e2which make the right upper square and left that homotopy commutative,
respectively. However, in general, one cannot choose a map *q*ewhich makes upper
two squares homotopy commutative simultaneously even if the maps*φ**E*, *m**E*, *φ**B*

and*m*_{B} are replaced by other quasi-isomorphisms; see Remark 4.5.

The following proposition, which are deduced from the proof of [26, Theorem 1.1], gives examples of relativelyK-formalizable pairs of maps.

**Proposition 2.2.** *A pair* (*q, f*) *of maps between* K*-formal spaces with the same*
*target is relatively* K*-formalizable if the two maps* *q* *and* *f* *satisfy either of the*
*following three conditions concerning a mapπ*:*S→T* *respectively.*

(i) *H*^{∗}(*S*;K) *and* *H*^{∗}(*T*;K) *are polynomial algebras with at most countably many*
*generators in which the operation* *Sq*_{1} *vanishes when the characteristic of the field*
K*is 2. HereSq*_{1}*x*=*Sq*^{n}^{−}^{1}*xforxwith degreen; see*[32, 4.9]*.*

(ii)*H*e^{i}(*S*;K) = 0 *for anyi* *with*dim*H*e^{i}^{−}^{1}(Ω*T*;K)*−*dim(*QH*^{∗}(*T*;K))^{i}*6*= 0*.*

Let *q* : *E* *→* *B* be a fibration over a space *B* and *f* : *X* *→* *B* a map. Let *F*
denote the pullback diagram

*E×*^{B}*X* //

²²

*E*

*q*

²²*X*

*f* //*B.*

Our main theorem concerning the level of a space is stated as follows.

**Theorem 2.3.** *Suppose that the spacesX,B* *andEin the diagramF* *are*K*-formal*
*and the pair* (*q, f*)*is relatively* K*-formalizable. Then*

level_{D(C}*∗*(*X*;K))(*E×*^{B}*X*) = level_{D(H}*∗*(*X*;K))(*H*^{∗}(*E*;K)*⊗*^{L}*H*^{∗}(*B*;K)*H*^{∗}(*X*;K))*.*

In general, the equality in Theorem 2.3 does not hold even if the spaces*X*, *B*
and*E* in*F* areK-formal; see Example 3.2.

By virtue of Theorem 2.3 and Proposition 2.2, we have

**Proposition 2.4.** *Let* *G* *be a simply-connected Lie group and* *G→* *E*_{f} *→* *S*^{4} *a*
*G-bundle with the classifying map* *f* : *S*^{4} *→* *BG. Suppose that* *H*^{∗}(*BG*;K) *is a*

*polynomial algebra on generators with even degree. Then*
level_{D(C}*∗*(*S*^{4};K))(*E**f*) =

½ 2 *if* *H*^{4}(*f*;K)*6*= 0*,*
1 *otherwise.*

**Proposition 2.5.** *Let* *G* *be a simply-connected Lie group andH* *a maximal rank*
*subgroup. LetG/H→E*_{g} *→S*^{4}*be the pullback of the fibrationG/H→BH→*^{π} *BG*
*by a map* *g*:*S*^{4}*→BG. Suppose that* *H*^{∗}(*BG*;K) *andH*^{∗}(*BH*;K)*are polynomial*
*algebras on generators with even degree. Then*

level_{D(C}*∗*(*S*^{4};K))(*E*_{g}) = 1*.*

Propositions 2.4 and 2.5 make one expect that a ‘nice’ object in *T OP/X* is
almost of low level. The intriguing feature is investigated further in subsequent
work [27]. On the other hand, the following result guarantees existence of an object
in*T OP/S*^{d} with the level greater than given arbitrary number.

**Theorem 2.6.** *Suppose that the underlying field* K*is of characteristic zero. For*
*any integerl≥*1*, there exists an objectP**l**→S*^{d} *inT OP/S*^{d} *such that*

level_{D(C}*∗*(*S*^{d};K))(*P**l*)*≥l.*

In order to describe our result on a realization problem for objects in the category
D(*C*^{∗}(*S*^{d};K)) by topological ones, we here give an overview of Jørgensen’s result
in [18].

Let*T* be a triangulated category. We say that an object in*T* is indecomposable if
it is not a coproduct of nontrivial objects. Recall that a triangle*L→*^{u} *M* *→*^{v} *N* *→*^{w} Σ*L*
in*T* is*an Auslander-Reiten triangle*[15] if the following conditions are satisfied:

(i)*L*and*N* are indecomposable.

(ii)*w6*= 0.

(iii) Each morphism*N*^{0}*→N* which is not a retraction factors through*v*.

As mentioned below, those particular triangles and a quiver associated with them
visualize objects in the triangulated category*T*.

We say that a morphism*f* :*M* *→N* in *T* is*irreducible*if it is neither a section
nor a retraction, but satisfies that in any factorization*f* =*rs*, either*s*is a section
or *r*is a retraction. The category*T* is said to have Auslander-Reiten triangles if,
for each object*N* with local endomorphism ring, there exists an Auslander-Reiten
triangle with*N* as the third term from the left. Recall also that an object*K* in*T*
is*compact*if the functor Hom_{T}(*K,* ) preserves coproducts; see [33, Chapter 4].

**Definition 2.7.** The Auslander-Reiten quiver of*T* has as vertices the isomorphism
classes [*M*] of indecomposable objects. It has one arrow from [*M*] to [*N*] when there
is an irreducible morphism*M* *→N* and no arrow from [*M*] to [*N*] otherwise.

Let *A* be a DGA over a field K. We denote by D^{c}(*A*) the full subcategory of
the derived category D(*A*) consisting of the compact objects. For a DG module*M*
over*A*, let*DM* be the dual Hom_{K}(*M,*K) to*M*.

We assume that*A*is locally finite and simply-connected in the sense that*H*^{0}(*A*) =
K and *H*^{1}(*A*) = 0. Observe that the cochain algebra *C*^{∗}(*X*;K) for a simply-
connected space*X* satisfies the condition for the DGA under our assumption for a
space.

Put *d*:= sup*{i* *|* *H*^{i}*A* *6*= 0*}*. One of the main results in [18] asserts that both
D^{c}(*A*) and D^{c}(*A*^{op}) have Auslander-Reiten triangles if and only if there are iso-
morphisms of graded *HA*-modules *HA*(*DHA*) *∼*= *HA*(Σ^{d}*HA*) and (*DHA*)*HA* *∼*=
(Σ^{d}*HA*)_{HA}; that is, *H*^{∗}(*A*) is a Poincar´e duality algebra. Moreover we observe
that the condition for *A* is equivalent to the Gorensteinness of *A* in the sense of
F´elix, Halperin and Thomas [9].

The form of the Auslander-Reiten quiver of D^{c}(*A*) is clarified in [18] and [19]

for a DGA*A*whose cohomology is a Poincar´e duality algebra. The key lemma [18,
Lemma 8.4] to proving results in [18, Section 8] is obtained by using the rational
formality of the spheres. Since the spheres are also K-formal for any field K, the
assumption of the characteristic of the underlying field is removed from all the
results in [18, Section 8]. In particular, we have

**Theorem 2.8.** [18, Theorem 8.13][18, Proposition 8.10]*LetS*^{d}*be thed-dimensional*
*sphere with* *d >*1 *and*K *an arbitrary field. Then the Auslander-Reiten quiver of*
*the category*D^{c}(*C*^{∗}(*S*^{d};K))*consists ofd−*1 *components, each isomorphic to*Z*A*_{∞}*.*
*The component containingZ*_{0}*∼*=*C*^{∗}(*S*^{d};K) *is of the form*

*...*

*Z*_{3}

*...* *...* *...*

*· · ·* *◦*

ÂÂ@

@@

@@ *◦*

*Z*_{2}

ÂÂ@

@@

@@ *◦*

ÂÂ@

@@

@@ *◦*

Σ^{−2(d−1)}*Z*_{2}

ÂÂ@

@@

@@ *· · ·*

*◦*

ÂÂ@

@@

@@

??~

~~

~~ *◦*

*Z*_{1}

ÂÂ@

@@

@@

??~

~~

~~ *◦*

Σ^{−(d−1)}*Z*_{1}

ÂÂ@

@@

@@

??~

~~

~~ *◦* *· · ·*

*◦*

ÂÂ@

@@

@@

??~

~~

~~ *◦*

ÂÂ@

@@

@@

??~

~~

~~ *◦*

Σ^{d−1}@*Z*@@_{0}@@ÂÂ

??~

~~

~~ *◦*

Σ^{−}^{(d}^{−}^{1)}*Z*0

ÂÂ@

@@

@@

??~

~~

~~

*· · ·* *◦*

??~

~~

~~ *◦*

??~

~~

~~ *◦*_{Z}_{0}

??~

~~

~~ *◦* *· · ·*

*Moreover, the cohomology of the indecomposable object*Σ^{−}^{l}*Z**m**has the form*
*H*^{i}(Σ^{−}^{l}*Z**m*)*∼*=

½ K *fori*=*−m*(*d−*1) +*l* *andd*+*l,*
0 *otherwise.*

*Remark*2.9*.* The latter half of Theorem 2.8 implies that indecomposable objects in
D^{c}(*C*^{∗}(*S*^{d};K)) are characterized by the cohomology. Moreover, those objects are
also classified by*the amplitude*of the cohomology of the objects up to shifts. Here
the amplitude of a DG module*M*, denoted amp*M*, is defined to be

amp*M* := sup*{i∈*Z*|M*^{i}*6*= 0*} −*inf*{i∈*Z*|* *M*^{i}*6*= 0*}.*

We are aware that the cohomology of Σ^{−}^{(d}^{−}^{1)}*Z*1 is isomorphic to*H*^{∗}(*S*^{2d}^{−}^{1};K)
as a graded vector space and that there is an irreducible map which induces
*H*^{∗}(*S*^{d};K) =*H*^{∗}(*Z*0)*→H*^{∗}(Σ^{−}^{(d}^{−}^{1)}*Z*1) a morphism of *H*^{∗}(*S*^{d};K)-modules. Thus
one would expect that realizability of the object by a space is related to the Hopf in-
variant*H* :*π*2*d**−*1(*S*^{d})*→*Z; see below for the explicit definition of the realizability.

In fact, we establish

**Proposition 2.10.** *Letφ*:*S*^{2d}^{−}^{1}*→S*^{d}*be a map. The cochain complexC*^{∗}(*S*^{2d}^{−}^{1};K)
*endowed with the* *C*^{∗}(*S*^{d};K)*-module structure induced by the map* *φ*:*S*^{2d}^{−}^{1}*→S*^{d}
*is in* *D*^{c}(*C*^{∗}(*S*^{d};K)) *if and only if* *H*(*φ*)_{K} *is nonzero, where* *H*(*−*)_{K} *denotes the*
*composite of the Hopf invariant with the reduction* Z *→*Z*⊗*K*. In this case, the*
*induced map* *φ*^{∗} : *C*^{∗}(*S*^{d};K) *→* *C*^{∗}(*S*^{2d}^{−}^{1};K) *coincides with the irreducible map*
*Z*0*→*Σ^{−}^{(d}^{−}^{1)}*Z*1 *up to scalar multiple.*

**Definition 2.11.** An object*M*in the category D^{c}(*C*^{∗}(*X*;K)) is*realizable*by an ob-
ject*f* :*Y* *→X* in*TOP /X* if*M* is isomorphic to the cochain complex*C*^{∗}(*Y*;K) en-
dowed with the*C*^{∗}(*X*;K)-module structure induced by*f*^{∗}:*C*^{∗}(*X*;K)*→C*^{∗}(*Y*;K).

Since the 0th cohomology of a space is nonzero and the negative part of the coho-
mology is zero, the only indecomposable objects of the form Σ^{−}^{m(d}^{−}^{1)}*Z**m*(*m≥*0),
which are on the line connecting*Z*_{0}and Σ^{−}^{(d}^{−}^{1)}*Z*_{1}, may be realizable. However, the
following proposition states that most of indecomposable objects in D^{c}(*C*^{∗}(*X*;K))
are*not*realizable by finite CW complexes.

**Theorem 2.12.** *Suppose that the characteristic of the underlying field is greater*
*than* 2 *or zero. An indecomposable object of the form* Σ^{−}^{i}*Z*_{l} *in* D^{c}(*C*^{∗}(*S*^{d};K))*is*
*realizable by a finite CW complex if and only if* *i* = *d−*1*,* *l* = 1 *and* *d* *even, or*
*i*= 0 *andl*= 0*.*

The rest of this paper is organized as follows. Section 3 is devoted to proving Theorems 2.3 and 2.6. In section 4, we prove Proposition 2.10 and Theorem 2.12.

The explicit computations of levels described in Propositions 2.4 and 2.5 are made in Section 5.

We conclude this section with comments on our work. Let *X* be a simply-
connected space whose cohomology with coefficients in a fieldKis a Poincar´e duality
algebra. The Auslander-Reiten quiver of D^{c}(*C*^{∗}(*X*;K)) then visualizes indecompos-
able objects, which will be called*molecules*of the full subcategory, and irreducible
morphisms between them. Even though a molecule in D^{c}(*C*^{∗}(*X*;K)) is not in the
image of the singular cochain complex functor from *TOP /X*, it may be needed
to construct *C*^{∗}(*Y*;K) for a space *Y* over *X* as a *C*^{∗}(*X*;K)-module. In fact, it
follows from the proof of Propositions 2.4 and 2.5 that some molecules are retracts
of *C*^{∗}(*S*^{4};K)-modules *C*^{∗}(*E**f*;K) and *C*^{∗}(*E**g*;K) while they are not realizable; see
also Example 5.3.

A CW complex*Z*is made of disks, which are called cells, by repeating attachment
of them. The result [12, Theorem 4.18] asserts that a CW complex*Z* has a cellular
chain model. In consequence, the cochain complex *C*^{∗}(*Z*;K) is isomorphic to the
dual to the cellular chain complex of*Z*. Thus*C*^{∗}(*Z*;K) may be also regarded as

‘a set of cells’ and hence it seems a creature in some sense. As mentioned above,
indecomposable objects in D^{c}(*C*^{∗}(*X*;K)) are needed when considering images by
the functor *C*^{∗}(*−*;K) from the viewpoint of Auslander-Reiten theory. Therefore
one might regard such an object as structural one smaller than a cell. This is the
reason why we give indecomposable objects in D^{c}(*C*^{∗}(*X*;K)) the name ‘molecules’.

3. Proofs of Theorems 2.3 and 2.6

Let*N* be a left DG module over a DGA*A*. we first observe that the left derived
functor*− ⊗*^{L}*A**N* is defined by*M⊗*^{L}*A**N* :=*F⊗*^{A}*N* for any right DG module*M* over
*A*, where*F* *→M* *→*0 is a semi-free resolution in the sense of F´elix, Halperin and
Thomas [12,*§*6]; see Appendix. We also recall the fact that the bar resolution of a
DG module is a semi-free resolution. This is extracted from the same argument as
in the proof of [11, Lemma 4.3 (ii)]; see also [12, Lemma 6.3].

Let *X* be a simply-connected formal space and *m*_{X} : *T V*_{X} *→*^{'} *C*^{∗}(*X*;K) be a
minimal model. We then have the following equivalences of triangulated categories;

see [23, Proposition 4.2],
D(*C*^{∗}(*X*;K))

*'*
*m*^{∗}_{X}

//D(*T V**X*)^{−⊗}

L

*T VX**H*^{∗}(*X*;K)

*'* //D(*H*^{∗}(*X*;K))*,*

where *m*^{∗}_{X} is the pullback functor; that is, *m*^{∗}_{X}*M* for a *C*^{∗}(*X*;K)-module *M* is
regarded as a *T V*_{X}-module with the DGA map *m*_{X}. We denote by*F*_{X} the com-
position of the functors. Observe that the functor*F*_{X} leaves the cohomology of an
object unchanged.

**Lemma 3.1.** *Under the same hypothesis as in Theorem 2.3, the differential graded*
*module* *F**X*(*C*^{∗}(*E×*^{B}*X*;K))*is isomorphic to* *H*^{∗}(*E*;K)*⊗*^{L}*H*^{∗}(*B*;K)*H*^{∗}(*X*;K)*in the*
*category*D(*H*^{∗}(*X*;K))*.*

*Proof.* We use the same notation as in Introduction and write*C*^{∗}(*−*) for*C*^{∗}(*−*;K).

Let *H* : *T V*_{B} *∧I* *→* *C*^{∗}(*E*) and *K* : *T V*_{B} *∧I* *→* *C*^{∗}(*X*) be homotopies from
*q*^{∗}*◦m*_{B} to*m*_{E}*◦q*eand from*f*^{∗}*◦m*_{B} to*m*_{E}*◦f*e, respectively. Here*T V*_{B}*∧I*denotes
the cylinder object due to Baues and Lemaire [10] in the category of DGA’s; see
Appendix. The homotopies*H* and*K*make*C*^{∗}(*E*) and*C*^{∗}(*X*) into a right*T V**B**∧I*-
module and a left*T V**B**∧I*-module, respectively. We have a*C*^{∗}(*X*)-module of the
form*C*^{∗}(*E*)*⊗*^{L}*T V*_{B}*∧**I**C*^{∗}(*X*). Then there exists a sequence of quasi-isomorphisms of
*T V**X*-modules

*C*^{∗}(*E×*^{B}*X*) *C*^{∗}(*E*)*⊗*^{L}*C*^{∗}(*B*)*C*^{∗}(*X*)

*'*

oo *EM* *C*^{∗}(*E*)*⊗*^{L}*T V*_{B}*C*^{∗}(*X*)

1*⊗**ε*01
*'*²²

1*⊗*_{mB}1

oo *'*

*T V**E**⊗*^{L}*T V*_{B}*T V**X* *'*
*m**E**⊗*1*m**X*

//*C*^{∗}(*E*)*⊗*^{L}*T V*_{B}*C*^{∗}(*X*) ^{1}^{⊗}^{ε}^{1}^{1}

*'* //*C*^{∗}(*E*)*⊗*^{L}*T V*_{B}*∧**I**C*^{∗}(*X*)*,*
where *EM* denotes the Eilenberg-Moore map; see [37, Theorem 3.2]. Therefore
we see that*m*^{∗}_{X}(*C*^{∗}(*E×*^{B}*X*)) is isomorphic to*T V**E**⊗*^{L}*T V**B**T V**X* in D(*T V**X*). Since
*T V*_{E}*⊗*^{L}*T V**B**T V*_{X}is a free*T V*_{X}-module, it follows that (*T V*_{E}*⊗*^{L}*T V**B**T V*_{X})*⊗*^{L}*T V**X**H*^{∗}(*X*)
is isomorphic to*T V**E**⊗*^{L}*T V*_{B}*H*^{∗}(*X*) in D(*H*^{∗}(*X*)). Then the same sequence of quasi-
isomorphisms as above connects*T V**E**⊗*^{L}*T V**B**H*^{∗}(*X*) with*H*^{∗}(*E*)*⊗*^{L}*H*^{∗}(*B*)*H*^{∗}(*X*) in
D(*H*^{∗}(*X*)). In fact we have quasi-isomorphisms

*T V*_{E}*⊗*^{L}*T V*_{B}*H*^{∗}(*X*)

*'*
*φ*_{E}*⊗*1

//*H*^{∗}(*E*)*⊗*^{L}*T V*_{B}*H*^{∗}(*X*)

*'*
1*⊗**ε*01

//*H*^{∗}(*E*)*⊗*^{L}*T V*_{B}*∧**I**H*^{∗}(*X*)

*H*^{∗}(*E*)*⊗*^{L}*H**∗*(*B*)*H*^{∗}(*X*) ^{'} *H*^{∗}(*E*)*⊗*^{L}*T V*_{B}*H*^{∗}(*X*)*.*

1*⊗*_{φB}1

oo

1*⊗**ε*11
*'*OO

This completes the proof. ¤

*Proof of Theorem 2.3.* The result [2, Proposition 3.4 (1)] enables us to deduce that
level_{D(C}*∗*(*X*;K))(*M*) = level_{D(H}*∗*(*X*;K))(*F**X**M*) for any object *M* in D(*C*^{∗}(*X,*K)).

Thus the result follows from Lemma 3.1. ¤

*Example*3.2*.* Let*ν* :*S*^{7}*→S*^{4}be the Hopf map and*E*_{ν} the pullback of*ν*:*S*^{7}*→S*^{4}
by itself. We have a fibration*S*^{3}*→E*_{ν} *→S*^{7}. Then it follows that

(2*.*1) level_{D(C}*∗*(*S*^{7};K))(*E*_{ν})*6*= level_{D(H}*∗*(*S*^{7};K))(*H*^{∗}(*S*^{7};K)*⊗*^{L}*H*^{∗}(*S*^{4};K)*H*^{∗}(*S*^{7};K))*.*

In fact, we have a Koszul resolution of the form

(Γ[*w*]*⊗ ∧*(*s*^{−}^{1}*x*_{4})*⊗H*^{∗}(*S*^{4};K)*, δ*)*→*K*→*0

with*δ*(*s*^{−}^{1}*x*4) =*x*4 and*δ*(*ω*) =*s*^{−}^{1}*x*4*⊗x*4. This gives rise to a semifree resolution
*H*^{∗}(*S*^{7};K)*⊗*Γ[*w*]*⊗ ∧*(*s*^{−}^{1}*x*4)*⊗H*^{∗}(*S*^{4};K)*→H*^{∗}(*S*^{7};K)*→*0

of*H*^{∗}(*S*^{7};K) as an*H*^{∗}(*S*^{4};K)-module. Thus we have

*M* :=*H*^{∗}(*S*^{7};K)*⊗*^{L}*H**∗*(*S*^{4};K)*H*^{∗}(*S*^{7};K) = (*H*^{∗}(*S*^{7};K)*⊗*Γ[*w*]*⊗∧*(*s*^{−}^{1}*x*4)*⊗H*^{∗}(*S*^{7};K)*,*0)*.*

Since dim*H*(*M*) =*∞*, it follows that*M* is not in thick^{n}_{D(H}*∗*(*S*^{7};K))(*H*^{∗}(*S*^{7};K)) for
any*n≥*0. This implies that the right hand side of the inequality (2.1) is infinite.

On the other hand, by Lemma 4.3 below, we see that*C*^{∗}(*E**ν*;K) is in D^{c}(*C*^{∗}(*S*^{7};K))
and hence levelD(*C**∗*(*S*^{7};K))(*E**ν*) *<∞*; see [21, Theorem 5.3] and [18, Lemma 3.2].

We refer the reader to Example 5.2 for the explicit calculation of the level of*E*_{ν}.
We recall a result on the level of a molecule in D^{c}(*C*^{∗}(*S*^{d};K)) due to Schmidt
before proving Theorem 2.6.

**Lemma 3.3.** [36, Proposition 6.6] *Let* *Z*_{i} *be the molecule in*D^{c}(*C*^{∗}(*S*^{d};K)) *de-*
*scribed in Theorem 2.8. Then* level_{D(C}*∗*(*S*^{d};K))(*Z*_{i}) =*i*+ 1*.*

*Proof of Theorem 2.6.* In the case where*l*= 1, the sphere*S*^{d}is the space we require
in the theorem. In what follows, we assume that*l≥*2. Let*∧*(*x**i*;*i∈J*) denote the
free graded commutative algebra generated by elements *x*_{i} (*i* *∈J*). Let*m* be an
integer sufficiently larger than*ld*.

Assume that*d*is even. We have a minimal model*B*= (*∧*(*x, ξ*)*, δ*) overQfor*S*^{d}
with*δ*(*ξ*) =*x*^{2}, where deg*x*=*d*. Consider a Koszul-Sullivan extension of the form

*B* *→*(*∧*(*x, ξ, ρ, w*_{0}*, ..., w*_{l}_{−}_{1})*, D*)
for which the differential*D* is defined by

*D*(*ρ*) =*x, D*(*w*0) = 0 and*D*(*w**i*) = (*ρx−ξ*)*w**i**−*1

for*i≥*1, where deg*w**i*=*i*(2*d−*1) + (2*m−*1)*−i*. Let*π*:*P**l**→S*^{d} be the bullback
of the fibration*|M**l**| → |B|*=*S*_{Q}^{d}, which is the spacial realization of the extension,
by the localizing map *S*^{d} *→* *S*_{Q}^{d}; see [12, Proposition 7.9]. Let *F**π* be the fibre of
the fibration*π*:*P*_{l}*→S*^{d}. Then we see that

*H*^{∗}(*F*_{π};K)*∼*=*H*^{∗}(*F*_{π};Q)*⊗*QK*∼*=*H*^{∗}(*M*_{l}*⊗*^{B}Q)*⊗*QK

=*H*^{∗}(*∧*(*ρ, w*0*, w*1*, ..., w**l**−*1)*, D*)*⊗*QK*.*
Since the generators of*M**l**⊗*^{B}Qare of odd degree, it follows from Lemma 3.3 below
that *C*^{∗}(*P**l*;K) is in D^{c}(*C*^{∗}(*S*^{d};K)) ; see also Remark 4.4. Moreover, we see that
*H*^{∗}(*P**l*;K)*∼*=*H*^{∗}(*P**l*;Q)*⊗*QK*∼*=*H*^{∗}(*M**l*)*⊗*QK.

By using the manner in [28, Section 7] for computing the homology of a DGA (or
by the direct calculation), we have elements 1,*ξ*,*w*0 and (*ρx−ξ*)*w**l**−*1, which form
a basis of*H*^{∗}(*M*_{l}) with degree less than or equal to *l*(2*d−*1) + (2*m−*1)*−*(*l−*1).

Let*Z* be a decomposable direct summand of*C*^{∗}(*P*_{l};K) in D(*C*^{∗}(*S*^{d};K)) containing
a cocycle of degree zero. By virtue of Theorem 2.8, we see that *Z* = Σ^{−}^{k(d}^{−}^{1)}*Z**k*

for some*k* *≥*0; see Remark 2.9. Suppose that*Z* contains a representative of*w*0,
(*ρx−ξ*)*w**l**−*1 or a cohomology class with degree greater than*l*(2*d−*1) + (2*m−*1)*−*
(*l−*1). Theorem 2.8 implies that *H*^{i}(*Z*) = Kif and only if *i* = (*k*+ 1)*d−k* or
*i*= 0. It follows that (*k*+ 1)*d−k≥*2*m−*1 *≥*2*ld−*1 and hence*k* *≥*2*l−*1 *≥l*.

Then Lemma 3.3 allows us to conclude that level_{D(C}*∗*(*S*^{d};K))(*P**l*)*≥l*+ 1.

Suppose that*Z* contains a representative of the element*ξ*. By Theorem 2.8, we
see that*Z*= Σ^{−}^{(d}^{−}^{1)}*Z*_{1}. In this case, let*Z*^{0} be a decomposable direct summand of

*C*^{∗}(*P**l*;K) containing a representative of*w*0. Observe that*Z*^{0}*6*=*Z*. If*Z*^{0}contains a
representative of the element (*ρx−ξ*)*w*_{l}_{−}_{1}, then*Z*^{0}= Σ^{−}^{(2m}^{−}^{1)}Σ^{−}^{(2l}^{−}^{1)(d}^{−}^{1)}*Z*_{2l}_{−}_{1}
since dim*H*^{∗}(*Z*^{0}) = 2 and the amplitude of*Z*^{0}should be 2*ld−*2*l*+ 1. If*Z*^{0}contains
a representative of the cohomology class with degree greater than*l*(2*d−*1) + (2*m−*
1)*−*(*l−*1), then*Z*^{0} = Σ^{−}^{(2m}^{−}^{1)}Σ^{−}^{(2l}^{−}^{1)(d}^{−}^{1)}*Z**k* for some*k≥*2*l−*1. Lemma 3.3
yields that level_{D(C}*∗*(*S*^{d};K))(*P*_{l})*≥*2*l*.

Suppose that*d*is odd. We have a Koszul-Sullivan extension of the form
(*∧*(*x*)*,*0)*→*(*∧*(*x, w*_{0}*, w*_{1}*, ..., w*_{l}_{−}_{1})*, D*) =:*N*_{l}

for which the differential*D* is defined by *D*(*x*) =*D*(*w*0) = 0 and *D*(*w**i*) =*xw**i**−*1

for *i* *≥* 1, where deg*x* = *d* and deg*w*0 = 2*m−*1. We assume that the integer
*m* is sufficiently larger than *ld*. Observe that deg*w**i* = *id*+ (2*m−*1)*−i*. Let
*π* : *P**l* *→* *S*^{d} be the bullback of the fibration *|N**l**| → |*(*∧*(*x*)*,*0)*|* = *S*_{Q}^{d} by the
localizing map*S*^{d}*→S*_{Q}^{d}. The same argument as above does work well to show that
level_{D(C}*∗*(*S*^{d};K))(*P**l*)*≥l*. This completes the proof. ¤
*Remark* 3.4*.* The Whitehead length WL(*Y*) of a simply-connected space *Y* is re-
garded as one of the homotopy invariants which measures geometrical complexity
of a space.

Let *L**Y* be the rational homotopy Lie algebra *π*_{∗}(Ω*Y*)*⊗*Q. We define a sub-
space [*L*_{Y}*, L*_{Y}]^{(l)}of*L*_{Y} by [*L*_{Y}*,*[*L*_{Y}*,*[*..,*[*L*_{Y}*, L*_{Y}]*...*]] (*l*-times) and [*L*_{Y}*, L*_{Y}]^{(0)} =*L*_{Y},
where [*,* ] denotes the Lie bracket of*L*_{Y}. Then the invariant WL(*Y*) is defined to
be the greatest integer *n*such that [*L**Y**, L**Y*]^{(n)}*6*= 0. The result [29, Theorem 2.5]

implies that WL(*P**l*) =*l−*1 for the space*P**l* constructed in the proof of Theorem
2.6.

4. Realization of molecules in D^{c}(*C*^{∗}(*S*^{d};K))

We recall briefly the Hopf invariant. Let *φ* : *S*^{2d}^{−}^{1} *→* *S*^{d} be a map. Choose
generators [*x*_{2d}_{−}_{1}]*∈H*^{2d}^{−}^{1}(*S*^{2d}^{−}^{1};Z) and [*x*_{d}]*∈H*^{d}(*S*^{d};Z). Let *ρ*be an element
of*C*^{∗}(*S*^{2d}^{−}^{1};Z) such that*φ*^{∗}(*x*_{d}) =*dρ*. Since [*x*_{d}]^{2} = 0 in*H*^{∗}(*S*^{d};Z), there exists
an element *ξ*of*C*^{∗}(*S*^{d};Z) such that *dξ* =*x*^{2}_{d}. We then have a cocycle of the form
*ρφ*^{∗}(*x**d*)*−φ*^{∗}(*ξ*). The Hopf invariant*H*(*φ*)*∈*Zis defined by the equality

[*ρφ*^{∗}(*x**d*)*−φ*^{∗}*ξ*] =*H*(*φ*)[*x*2*d**−*1]*.*

*Remark*4.1*.* If*d*is odd, then*H*(*φ*) is zero in general.

We relate the Hopf invariant with a differential of the Eilenberg-Moore spectral
sequence (EMSS). Let*F**φ* be the homotopy fibre of a map*φ*:*S*^{2d}^{−}^{1}*→S*^{d}. Then
*F**φ* fits into the pullback diagram*F*^{0} :

Ω*S*^{d}

²²

Ω*S*^{d}

²²*F**φ* //

²²

*P S*^{d}

²²*π*

*S*^{2d}^{−}^{1} _{φ} //*S*^{d}*.*

Here Ω*S*^{d}*→P S*^{d π}*→S*^{d}denotes the path-loop fibration. We consider the EMSS as-
sociated with*F*^{0}converging to*H*^{∗}(*F*_{φ};K) with*E*_{2}^{∗}^{,}^{∗}*∼*= Tor^{∗}_{H}^{,}^{∗}_{∗}_{(S}*d*;K)(*H*^{∗}(*S*^{2d}^{−}^{1};K)*,*K).

The Koszul resolution ofKas an *H*^{∗}(*S*^{d};K)-module allows us to conclude that
*E*_{2}^{∗}^{,}^{∗}*∼*=

½ *H*^{∗}(*S*^{2d}^{−}^{1};K)*⊗ ∧*(*s*^{−}^{1}*x**d*)*⊗*Γ[*τ*] if*d*is even*,*
*H*^{∗}(*S*^{2d}^{−}^{1};K)*⊗*Γ[*s*^{−}^{1}*x**d*] if*d*is odd*,*

where bideg*s*^{−}^{1}*x*_{d}= (*−*1*, d*) and bideg*τ* = (*−*2*,*2*d*); see [38, Lemma 3.1] and also
[24, Proposition 1.2].

Observe that the Eilenberg-Moore map induces an isomorphism from the homol-
ogy of the bar complex (*B*(*C*^{∗}(*S*^{2d}^{−}^{1};K)*, C*^{∗}(*S*^{d};K)*,*K)*, δ*1+*δ*2) to*H*^{∗}(*F**φ*;K). By
the definitions of differentials*δ*1 and*δ*2, we see that

*δ*_{1}([*x*_{d}*|x*_{d}]) = (*−*1)^{d}*φ*^{∗}(*x*_{d})[*x*_{d}] + (*−*1)^{d}(*−*1)^{d+1}[*x*^{2}_{d}]

= *δ*_{2}((*−*1)^{d}*ρ*[*x*_{d}] + 1[*ξ*])*,*
*δ*_{1}((*−*1)^{d}*ρ*[*x*_{d}] + 1[*ξ*]) = (*−*1)^{d}*{*(*−*1)^{d}^{−}^{1}*ρφ*^{∗}*x*_{d}*}*+*φ*^{∗}*ξ*

= *−*(*ρφ*^{∗}(*x**d*)*−φ*^{∗}*ξ*)*.*

It follows from [22, Lemma 2.1] that*d*2([*x**d**|x**d*]) =*H*(*φ*)_{K}*x*2*d**−*1 in the*E*2-term of
the EMSS. Moreover, by the same argument as in [24, Lemma 1.5], we have

**Lemma 4.2.** *The element*[*x**d**|x**d*]*in the torsion product*Tor_{H}*∗*(*S*^{d};K)(*H*^{∗}(*S*^{2d}^{−}^{1};K)*,*K)*,*
*which is computed by the bar complex mentioned above, coincides with the element*
*τ* *∈*Γ[*τ*] *if* *dis even and with the element* *γ*2(*s*^{−}^{1}*x**d*)*∈*Γ[*s*^{−}^{1}*x**d*] *ifd* *is odd. Thus*
*one hasd*2(*τ*) =*H*(*φ*)_{K}*x*2*d**−*1 *if* *dis even and* *d*2(*γ*2(*s*^{−}^{1}*x**d*)) =*H*(*φ*)_{K}*x*2*d**−*1 = 0*if*
*dis odd.*

In order to study Auslander-Reiten triangles, in [19], Jørgensen introduced the
function*ϕ*: D(*A*)*→*Z*∪ {∞}*defined by

*ϕ*(*M*) := dim*H*^{∗}RHom*A*(*M,*K) = dim*H*^{∗}(*M* *⊗*^{L}*A*K)*.*

This gives a criterion for a given object in D(*A*) to be compact.

**Lemma 4.3.** [36, Lemma 3.9]*Let* *Abe a simply-connected DGA with*dim*H*(*A*)*<*

*∞. An objectM* *in*D(*A*) *is compact if and only if*dim*H*^{∗}(*M*)*<∞andϕ*(*M*)*<*

*∞. In particular, for a map* *φ* : *Y* *→* *X* *from a space* *Y* *to a simply-connected*
*space* *Xwith* dim*H*^{∗}(*X*;K) *<∞, if the total dimension of the cohomology of the*
*homotopy fibre of the mapφ* *is finite, thenC*^{∗}(*Y*;K)*is in*D^{c}(*C*^{∗}(*X*;K))*.*

*Remark* 4.4*.* Let *F**φ* be the homotopy fibre of a map*φ*: *Y* *→X*. The latter half
of Lemma 4.3 follows from the fact that*H*^{∗}(*C*^{∗}(*Y*;K)*⊗*^{L}*C**∗*(*X*;K)K)*∼*=*H*^{∗}(*F*_{φ};K) as
a graded vector space; see [12, Theorem 7.5]. In view of the Leray-Serre spectral
sequence, we see that the dimension of *H*^{∗}(*Y*;K) is finite if so are the dimensions
of*H*^{∗}(*F*_{φ};K) and*H*^{∗}(*X*;K).

Let (*TOP /X*)^{f,}^{K} be the full subcategory of*TOP /X* consisting of objects *φ* :
*Y* *→* *X* with dim*H*^{∗}(*Y*;*K*) *<* *∞* and dim*H*^{∗}(*F**φ*;K) *<* *∞*. Lemma 4.3 asserts
that the singular cochain functor (1.1) restricted to the full subcategory yields the
functor

*C*^{∗}( ;K) : (*TOP /X*)^{f,}^{K}*→*D^{c}(*C*^{∗}(*X*;K))*.*

By using Lemmas 4.2 and 4.3, we prove Proposition 2.10.

*Proof of Proposition 2.10.* Let*{E*e_{r}^{∗}^{,}^{∗}*,d*e*r**}* be the EMSS converging to*H*^{∗}(Ω*S*^{d};K).

Observe that

e
*E*_{2}^{∗}^{,}^{∗}*∼*=

½ *∧*(*s*^{−}^{1}*x**d*)*⊗*Γ[*τ*] if*d*is even*,*
Γ[*s*^{−}^{1}*x**d*] if*d*is odd*,*

where bideg*s*^{−}^{1}*x**d* = (*−*1*, d*) and bideg*τ* = (*−*2*,*2*d*). The result [10, Theorem
III] implies that the EMSS for the fibre square *F*^{0} is a right DG comodule over
*{E*e^{∗}_{r}^{,}^{∗}*,d*e