intellecton/papers/references/Bombelli2009.txt

arXiv:0810.0096v3  [math.OA]  20 Feb 2012

C∗-ALGEBRAS OVER TOPOLOGICAL SPACES:
FILTRATED K-THEORY

RALF MEYER AND RYSZARD NEST

Abstract. We define the filtrated K-theory of a C∗-algebra over a finite topo-
logical space X and explain how to construct a spectral sequence that computes
the bivariant Kasparov theory over X in terms of filtrated K-theory.
For finite spaces with totally ordered lattice of open subsets, this spectral
sequence becomes an exact sequence as in the Universal Coefficient Theorem,
with the same consequences for classification.
We also exhibit an example where filtrated K-theory is not yet a complete
invariant. We describe two C∗-algebras over a space X with four points that
have isomorphic filtrated K-theory without being KK(X)-equivalent. For this
space X, we enrich filtrated K-theory by another K-theory functor to a com-
plete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient
Theorem.

1. Introduction

1.1. The UCT-problem. One of the main problems in the theory of C∗-algebras
is the computation of the equivariant KK-theory of C∗-algebras endowed with some
extra structure. Here we apply the general techniques developed in [6,9] to the case
of C∗-algebras with a non-trivial ideal lattice. The appropriate version of KK-theory
is Kirchberg’s generalisation of Kasparov theory to C∗-algebras over non-Hausdorff
topological spaces (see [5]). Our goal is to compute it in terms of more manageable
K-theoretic information, generalising the usual Universal Coefficient Theorem that
computes Kasparov’s original theory for C∗-algebras in the bootstrap class by an
exact sequence

(1.1)
Ext
�
K∗+1(A), K∗(B)
�
֌ KK∗(A, B) ։ Hom
�
K∗(A), K∗(B)
�
.

The generalisation of the bootstrap class to the case of C∗-algebras with non-
trivial ideal lattice was introduced and studied in [8]. Let us first recall some of
the notation from [8]. Let X be a (usually non-Hausdorff) topological space. A
C∗-algebra over X is a C∗-algebra A endowed with a continuous map Prim(A) → X.
Let C∗alg(X) be the category of C∗-algebras over X; the morphisms in C∗alg(X) are
given by X-equivariant (in obvious sense) ∗-homomorphisms. Taking Kirchberg’s
KK-groups as morphisms and the same objects, we get the category KK(X). It
has a structure of a triangulated category (see [8]). For finite X, the bootstrap
class B(X) is defined as the smallest subcategory of KK(X) that is closed under
suspension, isomorphism, exact triangles, and direct sums and contains all objects
with underlying C∗-algebra C.
General methods from homological algebra suggest to study a homology the-
ory H∗ for C∗-algebras over X, taking values in some Abelian category C. Under
some mild assumptions, the machinery developed in [6, 9] yields an Adams type
spectral sequence which abuts to KK(X; , ), with an E2-term expressed in terms
of H∗.

2000 Mathematics Subject Classification. 19K35, 46L35, 46L80, 46M18, 46M20.
The second author was supported by the German Research Foundation (Deutsche Forschungs-
gemeinschaft (DFG)) through the Institutional Strategy of the University of G¨ottingen.
1


2
RALF MEYER AND RYSZARD NEST

For classification purposes, we need, instead of a spectral sequence, a short exact
sequence of the type (1.1):

(1.2)
ExtC
�
H∗+1(A), H∗(B)
�
֌ KK∗(X; A, B) ։ HomC
�
H∗(A), H∗(B)
�
,

and a precise description of the range of H∗.
In this case, given two C∗-algebras A and B over X that belong to the bootstrap
class, an isomorphism of H∗(A) to H∗(B) lifts to a KK(X)-equivalence between A
and B. The results of Eberhard Kirchberg then allow to lift this KK(X)-equivalence
to a ∗-isomorphism A ∼= B, provided A and B are tight, purely infinite, stable,
nuclear and separable; here tightness means that the maps Prim(A) → X and
Prim(B) → X are homeomorphisms (see [5]). It is also shown in [8] that, in the
case when X is finite, any object of the bootstrap class is KK(X)-equivalent to a
tight, purely infinite, stable, nuclear, separable C∗-algebra over X.
Hence the existence of an exact sequence of the form (1.2) for all objects of the
bootstrap class leads to a complete classification of the tight, purely infinite, stable,
nuclear, separable C∗-algebras over X in terms of their image under the functor H∗.

1.2. Main results. It is relatively easy to construct filtrations on KK which pro-
duce spectral sequences which converge to KK-groups on the bootstrap category
and whose E2-term involves only the K-theory of the quotients K∗(A/J) for the
ideals J corresponding to minimal open subsets of X; an example is the filtration
used in [8, Section 4.1].
However, this spectral sequence is not very useful for
practical purposes, since it does not degenerate at the E2-level. The second dif-
ferential involves, in particular, the K-theory of various subquotients I/J for the
ideals I ⊂ J ⊂ A and the associated six-term exact sequences in K-theory

(1.3)

K0
�
I
�
� K0
�
J
�
� K0
�
J/I
�

�

K1
�
J/I
�

�

K1
�
J
�
�
K1
�
I
�
.
�

Also higher differentials do not vanish.
To get a short exact sequence instead, we need to consider more sophisticated ho-
mology theories. The homology theory analysed here is “filtrated K-theory,” which
is in some sense the second approximation to this spectral sequence. Roughly speak-
ing, filtrated K-theory comprises the K-theory of various subquotients together with
all canonical maps between these groups. We will make this definition precise later.
The part of it which involves the exact sequences (1.3) appeared previously in the
work of Gunnar Restorff [11] for Cuntz–Krieger algebras and of Mikael Rørdam [13]
and Alexander Bonkat [2] for extensions of C∗-algebras. The UCT theorem in the
case when the ideal structure is given by I1⊳I2⊳A was obtained by Gunnar Restorff
in his phd-thesis [12], where he introduced an invariant which is a particular case
of filtrated K-theory.
In this paper we prove the following

Theorem 1.1. The filtrated K-theory satisfies the Universal Coefficient Theorem
and is a complete invariant for C∗-algebras over those finite topological spaces with
a totally ordered lattice of open subsets.

Note that a C∗-algebra over a space of the type described in this result is essen-
tially the same as a C∗-algebra A together with a finite increasing chain of ideals

{0} = I0 ⊳ I1 ⊳ I2 ⊳ I3 ⊳ · · · ⊳ In−1 ⊳ In = A.

We will also show that the spectral sequence associated to the filtrated K-theory
does not collapse in general. Let (X, <) be the partially ordered set, where X =


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
3

{1, 2, 3, 4} with the partial order given by 1, 2, 3 < 4 and no further strict inequalities
between 1, 2, 3. A C∗-algebra over this space is a C∗-algebra A together with an
ideal I and a decomposition of A/I into a direct sum of three orthogonal ideals.

Theorem 1.2. The filtrated K-theory over (X, <) does not satisfy the Universal
Coefficient Theorem and is not a complete invariant.

In fact, we give an explicit example of two C∗-algebras A and B over X in the
bootstrap class that have isomorphic filtrated K-theory but are not KK(X)-equivalent.
However, for the particular four-point space X, we still get a complete invari-
ant and a Universal Coefficient Theorem as in (1.2), by adding another K-theory
functor to filtrated K-theory.
It is not clear how to construct such an enriched and still computable filtrated
K-theory for general finite spaces.

1.3. The general machinery. Now we explain the general machinery behind our
approach. Let us fix a finite topological space X. The first step is the correct
definition of filtrated K-theory. The filtrated K-theory of a C∗-algebra A over X
comprises the Z/2-graded Abelian groups K∗
�
A(Y )
�
for all locally closed subsets
Y ⊆ X together with all natural transformations between these groups. The main
issue here is to find all natural transformations. These natural transformations
enter in the definition of the target category of the filtrated K-theory functor and
thus influence the Hom and Ext terms that we expect in the Universal Coefficient
Theorem.
We can guess some of these natural transformations. If U is a relatively open
subset of Y , then A(U) is an ideal in A(Y ), with quotient A(Y )/A(U) = A(Y \ U).
This C∗-algebra extension leads to a natural six-term exact sequence

(1.4)

K0
�
A(U)
�
� K0
�
A(Y )
�
� K0
�
A(Y \ U)
�

�

K1
�
A(Y \ U)
�

�

K1
�
A(Y )
�
�
K1
�
A(U)
�
.
�

These exact sequences provide three types of natural transformations associated to
inclusions of open subsets, restriction to closed subset, and boundary maps.
An obvious source for relations between these natural transformations are morph-
isms of C∗-algebra extensions: since the six-term exact sequences in (1.4) are nat-
ural, each natural morphism of extensions provides some commuting diagrams,
which become relations between our generators.
But do these obvious generators and relations already describe all natural trans-
formations? This turns out to be the case for the spaces studied in this article—both
the positive and the negative examples. Although the authors know no counter-
examples, we do not expect this to be so in general.
The starting point for our study of filtrated K-theory is that the covariant func-
tors A �→ K∗
�
A(Y )
�
are representable, that is, they are of the form KK∗(X; RY , A)
for suitable C∗-algebras RY over X—these are the representing objects. Our con-
struction of RY yields commutative C∗-algebras, consisting of C0-functions on suit-
able locally closed subspaces of the order complex of the partial order on X. The
Yoneda Lemma tells us that natural transformations from K∗
�
A(Y )
�
to K∗
�
A(Z)
�

correspond to KK∗(X; RZ, RY ) ∼= K∗
�
RY (Z)
�
. These groups are easy enough to
compute in the examples we consider, and turn out to be definable by the concrete
generators and relations mentioned above.
The natural transformations acting on filtrated K-theory form a Z/2-graded pre-
additive category NT . A (countable) module over NT is, by definition, an additive


4
RALF MEYER AND RYSZARD NEST

functor from NT to the category of (countable) Z/2-graded Abelian groups. By
construction, the filtrated K-theory of any C∗-algebra over X is such a countable
module. Let C be the category of countable NT -modules. This is an Abelian cat-
egory, and filtrated K-theory is a stable homological functor FK from the Kasparov
category KK(X) of C∗-algebras over X to C.
It is easy to check that the functor FK: KK(X) → C is universal in the notation
of [9]. General results on homological ideals in triangulated categories now pro-
duce a cohomological spectral sequence that converges towards KK∗(X; A, B) if A
belongs to the bootstrap class; its E2-term involves Extp
C
�
FK(A), FK(B)
�
.
The main issue is whether the Ext-groups Extp
C
�
FK(A), FK(B)
�
with p ≥ 2
vanish, so that our spectral sequence degenerates to an exact sequence of the desired
form.
This amounts to checking whether FK(A) has a projective resolution of
length 1 in C.
Already for the non-Hausdorff two-point space considered in [2, 13], the cat-
egory C has infinite cohomological dimension, that is, there are objects that admit
no projective resolution of finite length. But these objects do not belong to the
range of the functor FK. If an NT -module A belongs to the range of FK, then
there are exact sequences

(1.5)
· · · → A(U) → A(Y ) → A(Y \ U) → A(U) → · · ·

for any Y ∈ LC(X), U ∈ LC(Y ) because of (1.4). But there are NT -modules
without finite length projective resolutions. For totally ordered spaces, an object
of C has a projective resolution of length 1 if and only if it has a projective resolution
of finite length, if and only if the sequences (1.5) are exact, if and only if it is the
filtrated K-theory of some separable C∗-algebra over X, which we can take in the
bootstrap class.
For the four-point counterexample considered in Section 5, we first find a torsion-
free exact module that is not projective, and then use it to find an exact module
without projective resolutions of length 1. Then we find two non-isomorphic objects
of the bootstrap class with the same filtrated K-theory. The idea here is to consider
a certain exact triangle ΣC → A → B → C, which splits on the level of filtrated
K-theory, so that A ⊕ C and B have the same filtrated K-theory. But we can prove
in our concrete example that A ⊕ C and B are not KK(X)-equivalent.
A C∗-algebra over the four-point space X is a C∗-algebra A with a distinguished
ideal I and a direct sum decomposition of A/I as a direct sum of three orthogonal
ideals.
Since both direct sums and extensions of C∗-algebras can be classified
by filtrated K-theory, it is remarkable that the combination of both provides a
counterexample. Incidentally, the space Xop that corresponds to a C∗-algebra A
with a distinguished ideal I and a direct sum decomposition of I as a direct sum of
three orthogonal ideals also leads to a counterexample in a similar fashion.
For the four-point space X above, there is essentially just one module that ought
to be projective but is not. We can add another invariant to filtrated K-theory that
corresponds to this offending module. Since this changes our whole category, it may
lead to further offending modules, which would have to be added in a second step,
and this could, in principle, go on forever. But in the concrete case at hand, we get
projective resolutions of length 1 for all modules over the enriched filtrated K-theory.
As a result, the enriched filtrated K-theory classifies objects of the bootstrap class
over X up to KK(X)-equivalence, and it classifies purely infinite separable nuclear
stable C∗-algebras with primitive ideal space X and simple subquotients in the
bootstrap class.

1.4. Some basic notation. We shall use the following notation from [8]:
∈∈ we write x ∈∈ C for objects of a category C as opposed to morphisms;


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
5

X topological space, often assumed sober (see [14]);
O(X) set of open subsets of X, partially ordered by ⊆;
LC(X) set of locally closed subsets of X;
LC(X)∗ set of connected, non-empty locally closed subsets of X;
⪯ specialisation preorder on X, defined by x ⪯ y ⇐⇒ {x} ⊆ {y}
A C∗-algebra;
Prim(A) primitive ideal space of A with hull–kernel topology;
I(A) set of closed ∗-ideals in A, partially ordered by ⊆;
C∗alg(X) category of C∗-algebras over X with X-equivariant ∗-homomorphisms
C∗sep(X) full subcategory of separable C∗-algebras over X;
KK(X) Kasparov category of C∗-algebras over X: its objects are separable
C∗-algebras over X, its set of morphisms from A to B is KK0(X; A, B);
B(X) the bootstrap class in KK(X);
iX
Y extension functor C∗alg(Y ) → C∗alg(X) or KK(Y ) → KK(X) for a
subset Y ⊆ X;
ix abbreviation for iX
{x} for x ∈ X;
rY
X restriction functor C∗alg(X) → C∗alg(Y ) or KK(X) → KK(Y ) for a
locally closed subset Y ⊆ X;
Σ suspension ΣA := C0(R, A).

Roughly speaking, a space is sober if it can be recovered from the lattice O(X).
It is explained in [8, §2.5] why we may restrict attention to such spaces. For finite
spaces, sobriety is equivalent to the separation axiom T0, that is, two points are
equal once they have the same closure.
A C∗-algebra over X is pair (A, ψ) consisting of a C∗-algebra A and a continuous
map ψ: Prim(A) → X. If X is sober, this is equivalent to a map

ψ∗ : O(X) → I(A),
U �→ A(U),

that preserves finite infima and arbitrary suprema, that is,

A
� �

U∈F
U
�
=
�

U∈F
A(U),
A
� �

U∈S
U
�
=
�

U∈S
A(U) =

�

U∈S
A(U),

where F ⊆ O(X) is finite and S ⊆ O(X) is arbitrary. In particular, this implies
A(∅) = {0}, A(X) = A, and the monotonicity condition A(U) ⊳ A(V ) for U ⊆ V .
A ∗-homomorphism f : A → B between two C∗-algebras over X is X-equivariant
if f
�
A(U)
�
⊆ B(U) for all U ∈ O(X).
A subset Y ⊆ X is locally closed if and only if Y = U \ V for open subsets
V, U ∈ O(X) with V ⊆ U. Then we define A(Y ) := A(U)/A(V ) for a C∗-algebra A
over X; this does not depend on the choice of U and V by [8, Lemma 2.15].
If Y ⊆ X is locally closed and A is a C∗-algebra over Y , then we extend A to a
C∗-algebra iX
Y A over X by iX
Y A(Z) := A(Y ∩Z) for Z ∈ LC(X). Conversely, we can
restrict a C∗-algebra B over X to a C∗-algebra rY
X(B) over Y by rY
XB(Z) := B(Z)
for all Z ∈ LC(Y ) ⊆ LC(X).
The category KK(X) is triangulated, with exact triangles coming either from
mapping cone triangles of X-equivariant ∗-homomorphisms or, equivalently, from
semi-split C∗-algebra extensions over X (see [7, 8]). Here an extension is called
semi-split if it splits by an X-equivariant completely positive contraction.
The bootstrap class B(X) is the localising subcategory of KK(X) generated by
the objects ixC for all x ∈ X. That is, it is the smallest class of objects containing
these generators that is closed under suspensions, KK(X)-equivalence, semi-split
extensions, and countable direct sums.


6
RALF MEYER AND RYSZARD NEST

2. Filtrated K-theory

Let X be a finite topological space. We do not discuss filtrated K-theory for
C∗-algebras over infinite spaces here.

Definition 2.1. For a locally closed subset Y ⊆ X, we define a functor

FKY : KK(X) → AbZ/2,
FKY (A) := K∗
�
A(Y )
�
.

Here Ab denotes the category of Abelian groups and AbZ/2 denotes the category of
Z/2-graded Abelian groups.

For each Y ∈ LC(X), the functor FKY is stable and homological, that is, it
intertwines the suspension on KK(X) with the translation functor on AbZ/2 (this
functor shifts the grading), and if ΣC → A → B → C is an exact triangle in
KK(X)—this may, for instance, come from a semi-split extension A ֌ B ։ C—
then FKY (A) → FKY (B) → FKY (C) is an exact sequence in AbZ/2.
The functors FKY together form the filtrated K-theory functor. But the latter
also includes its target category, which we now define in a rather abstract way.

Definition 2.2. For Y, Z ∈ LC(X), let NT ∗(Y, Z) be the Z/2-graded Abelian
group of all natural transformations FKY ⇒ FKZ. The composition of natural
transformations provides a product

NT i(Y, Z) × NT j(W, Y ) → NT i+j(W, Z),
f, g �→ f ◦ g,

which is associative and additive in each variable.
We let NT be the Z/2-graded category whose object set is LC and whose morph-
ism space Y → Z is NT ∗(Y, Z). The Abelian group structure on these morphism
spaces turns this into a pre-additive category.

Definition 2.3. A module over NT is a grading preserving, additive functor
G: NT → AbZ/2. That is, it consists of a family of Z/2-graded Abelian groups
GY = (GY,0, GY,1) for Y ∈ LC(X) and product maps

NT i(Y, Z) × GY,j → GZ,i+j
for all Y, Z ∈ LC(X), i, j ∈ Z/2; these product maps are associative, additive in
each variable, and the identity transformations in NT (Y, Y ) act identically on GY
for all Y ∈ LC(X).
Let Mod(NT ) be the category of NT -modules. The morphisms in Mod(NT )
are the natural transformations of functors or, equivalently, families of grading
preserving group homomorphisms GY → G′
Y that commute with the actions of NT .
Let Mod(NT )c be the full subcategory of countable modules.

By construction, the natural transformations FKY ⇒ FKZ in NT ∗(Y, Z) induce
maps FKY (A) → FKZ(A) for all A ∈∈ KK(X). This turns
�
FKY (A)
�

Y ∈LC(X) into
a module over NT . Furthermore, it is well-known that the K-theory of separable
C∗-algebras such as A(Y ) for A ∈∈ KK(X) is countable.

Definition 2.4. Filtrated K-theory is the functor

FK = (FKY )Y ∈LC(X) : KK(X) → Mod(NT )c,
A �→
�
K∗
�
A(Y )
��

Y ∈LC(X).

The target category Mod(NT )c is an important part of this definition because
we will compute groups of morphisms and extensions in this category.
Since A(∅) = {0} for all C∗-algebras over X, we have FK∅ = 0, so that ∅ is a
zero object of NT . Therefore, G∅ vanishes for any NT -module.
If Y ∈ LC(X) is not connected, that is, Y = Y1 ⊔ Y2 with two disjoint relat-
ively open subsets Y1, Y2 ∈ O(Y ) ⊆ LC(X), then A(Y ) ∼= A(Y1) ⊕ A(Y2) for any


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
7

C∗-algebra A over X. Hence FKY (A) ∼= FKY1(A) × FKY2(A). The natural trans-
formations that implement this natural isomorphism correspond to a direct sum
diagram Y ∼= Y1 ⊕ Y2 in NT . Therefore, any NT -module has GY ∼= GY1 ⊕ GY2;
here we use the fact that a functor that is additive on morphisms is also additive
on objects, even if the category in question is only pre-additive.
Since X is finite, any locally closed subset is a disjoint union of its connected
components. This corresponds to a direct sum decomposition Y ∼=
�
j∈π0(Y ) Yj
in NT . Therefore, we lose no information when we replace LC(X) by the subset
LC(X)∗ of non-empty, connected, locally closed subsets.

2.1. The representability theorem. The representability theorem serves two
purposes. We will first use it to describe the category NT . Later, we use it to
construct geometric resolutions in KK(X).

Theorem 2.5. Let X be a finite topological space. The covariant functors FKY for
Y ∈ LC(X) are representable, that is, there are objects RY ∈∈ KK(X) and natural
isomorphisms
KK∗(X; RY , A) ∼= FKY (A) = K∗
�
A(Y )
�

for all A ∈∈ KK(X), Y ∈ LC(X).

Before we prove this theorem in §2.2, we first describe the representing ob-
jects RY explicitly, and we use this to describe the groups of natural transformations
NT ∗(Y, Z) as K-theory groups of certain locally compact spaces.
The construction of RY requires some preparation. We equip X with the spe-
cialisation preorder ⪯ as in [8, §2.7]; recall that x ⪯ y if and only if {x} ⊆ {y}.
Since the topological space X is finite, it carries the Alexandrov topology of the
preorder ⪯, that is, a subset Y ⊆ X is open if and only if x ⪰ y ∈ Y implies x ∈ Y .
Similarly, Y ⊆ X is closed if and only if x ⪯ y ∈ Y implies x ∈ Y , and locally
closed if and only if x ⪯ y ⪯ z and x, z ∈ Y implies y ∈ Y .

Definition 2.6. Let (X, ⪯) be a partially ordered set. Its order complex is the
geometric realisation of the simplicial set Ch(X) whose n-simplices are the chains
x0 ⪯ x1 ⪯ · · · ⪯ xn in X and whose face and degeneracy maps delete or double an
entry of the chain.

Equivalently, Ch(X) is the classifying space of the thin category that has object
set X and a morphism x → y whenever x ⪯ y.
The order complex is the main ingredient in the construction of the representing
objects RY for Y ∈ LC(X).
The non-degenerate n-simplices in Ch(X) are the strict chains x0 ≺ · · · ≺ xn
in X. We let SX be the set of all strict chains. For each I = (x0 ≺ · · · ≺ xn) ∈ SX,
we let ∆I be a copy of ∆n; more formally, ∆I = {(t, I) | t ∈ ∆n}. We also let
∆◦
I ⊆ ∆I be the corresponding open simplex ∆n \ ∂∆n.
The space Ch(X) is obtained from the union �
I∈SX ∆I by identifying ∆I with
the corresponding face in ∆J whenever I, J ∈ SX satisfy I ⊆ J. Thus the underly-
ing set of Ch(X) is a disjoint union

(2.1)
Ch(X) =
�

I∈SX
∆◦
I.

For I ∈ SX, let min I and max I be the (unique) minimal and maximal elements
in SX, respectively. We define two functions

m, M : Ch(X) → X

by mapping points in ∆◦
I to min I and max I, respectively. This well-defines func-
tions on Ch(X) because of (2.1).


8
RALF MEYER AND RYSZARD NEST

Lemma 2.7. If Y ⊆ X is closed, then m−1(Y ) is open and M −1(Y ) is closed in
Ch(X). If Y ⊆ X is open, then m−1(Y ) is closed and M −1(Y ) is open. If Y ⊆ X
is locally closed, then m−1(Y ) and M −1(Y ) are locally closed.

Proof. First we show that M −1(Y ) is closed if Y is closed. If I ∈ SX satisfies
max I ∈ Y , then max J ∈ Y for all J ⊆ I because max J ⪯ max I ∈ Y . Hence
∆I ⊆ M −1(Y ) once M −1(Y ) ∩ ∆◦
I ̸= ∅, so that M −1(Y ) ∩ ∆I is closed for all
I ∈ SX; this implies that M −1(Y ) is closed.
A similar argument shows that m−1(Y ) is closed in Ch(X) if Y is open. Now
the remaining assertions follow easily because the maps m−1 and M −1 commute
with complements, unions, and intersections.
□

More explicitly, if Y ⊆ X is open, then m−1(Y ) is the union of the simplices SX
for all chains x0 ≺ x1 ≺ · · · ≺ xn with x0 ∈ Y and hence x0, . . . , xn ∈ Y . Thus

m−1(Y ) = Ch(Y )
if Y ⊆ X is open.

Similarly,

M −1(Y ) = Ch(Y )
if Y ⊆ X is closed.

Here we identify Ch(Y ) with a subcomplex of Ch(X) in the obvious way.
Let Xop be X with the topology for the reversed partial order ≻; that is, the
open subsets of Xop are the closed subsets of X, and vice versa. We may rephrase
Lemma 2.7 as follows:

Proposition 2.8. The map (m, M): Ch(X) → Xop × X is continuous.

Let
R := C
�
Ch(X)
�

be the C∗-algebra of continuous functions on Ch(X). Since

Prim R = Prim C
�
Ch(X)
� ∼= Ch(X),

the map (m, M) turns R into a C∗-algebra over Xop × X. We abbreviate

S(Y, Z) := m−1(Y ) ∩ M −1(Z) ⊆ Ch(X);

this is a locally closed subset of Ch(X) by Lemma 2.7

Definition 2.9. We let RY be the C∗-algebra over X with

RY (Z) := R(Y op × Z) = C0
�
S(Y, Z)
�

for all Y, Z ∈ LC(X); here Y op denotes Y with the subspace topology from Xop.
Equivalently, we let RY be the restriction of R to Y op × X, viewed as a C∗-algebra
over X via the coordinate projection Y op × X → X.

We will prove the Theorem 2.5 for this choice of RY in §2.2. Taking this for
granted, we use the concrete description of RY to compute the groups of natural
transformations. By the Yoneda Lemma, natural transformations between the func-
tors FKY come from morphisms between the representing objects. More precisely,

(2.2)
NT ∗(Y, Z) ∼= KK∗(X; RZ, RY ) ∼= FKZ(RY ) = K∗
�
RY (Z)
�

= K∗
�
R(Y op × Z)
�
= K∗�
m−1(Y ) ∩ M −1(Z)
�
= K∗�
S(Y, Z)
�
.

By the way, the universal property of Kasparov theory says that it makes no
difference for the natural transformations FKY ⇒ FKZ whether we view these two
functors as defined on C∗sep(X) or KK(X). But since RY only represents FKY on
the level of KK(X), we get KK∗(X; RZ, RY ) and not the space of X-equivariant
∗-homomorphisms RZ → RY .


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
9

We describe S(Y, Z) more explicitly using the closure and boundary operations

Z := {x ∈ X | there is z ∈ Z with x ⪯ z},
∂Z := Z \ Z,

�Y := {x ∈ X | there is y ∈ Y with x ⪰ y},
�∂Y := �Y \ Y.

Of course, Z is the closure of Z in X and �Y is the closure of Y in Xop.

Lemma 2.10. If Y, Z ∈ LC(X), then

S(Y, Z) = Ch(�Y ∩ Z)
� �
Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z)
�
.

In particular,

S(Y, Z) = Ch(Y ∩ Z) \ Ch(Y ∩ ∂Z)
if Y is open,

S(Y, Z) = Ch(�Y ∩ Z) \ Ch(�∂Y ∩ Z)
if Z is closed,

S(Y, Z) = Ch(Y ∩ Z)
if Y is open and Z is closed.

Proof. Let x0 ≺ x1 ≺ · · · ≺ xn be a strict chain in X. The interior of the corres-
ponding simplex belongs to S(Y, Z) if and only if x0 ∈ Y and xn ∈ Z. This implies
xj ∈ �Y and xj ∈ Z for all j, so that the simplex belongs to Ch(�Y ∩Z). Furthermore,
we neither have xj ∈ �∂Y ∩ Z for all j nor xj ∈ �Y ∩ ∂Z for all j because x0 ∈ Y
and xn ∈ Z. Thus the simplex belongs neither to Ch(�Y ∩ ∂Z) nor to Ch(�∂Y ∩ Z).
Conversely, if xj ∈ �Y ∩Z for all j and neither xj ∈ �∂Y ∩Z for all j nor xj ∈ �Y ∩∂Z
for all j, then some xj must belong to Y ∩ Z and some xk must belong to �Y ∩ Z.
Since Y ∩ Z is closed in �Y ∩ Z and �Y ∩ Z is open in �Y ∩ Z, this implies x0 ∈ Y and
xn ∈ Z. This shows that the interior of a simplex belongs to S(Y, Z) if and only if
it is contained in Ch(�Y ∩ Z)
� �
Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z)
�
.
□

Lemma 2.10 and (2.2) yield

NT ∗(Y, Z) ∼= K∗�
S(Y, Z)
� ∼= K∗�
Ch(�Y ∩ Z), Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z)
�
.

This is the K-theory of a finite CW-pair and hence is always finitely generated as
an Abelian group.
If C is any finite simplicial complex, then its barycentric subdivision is of the
form Ch(X), where X is the partially ordered set of non-degenerate simplices in C.
Thus NT ∗(X, X) = K∗(|C|), so that any finitely generated Abelian group arises
as NT ∗(X, X). As a consequence, special properties of the pre-additive category
NT can only be hidden in its composition.
When we identify NT ∗(Y, Z) ∼= KK∗(X; RZ, RY ), then the composition of nat-
ural transformations corresponds to the Kasparov composition product. This gets
somewhat obscured when we follow the isomorphisms

KK∗(X; RZ, RY ) ∼= K∗
�
RY (Z)
�
= K∗�
S(Y, Z)
�
.

To describe the composition of natural transformations in terms of K∗�
S(Y, Z)
�
, we
must first lift elements of K∗�
S(Y, Z)
�
back to KK∗(X; RZ, RY ) and then compose
them. The lifting requires a formula for the natural isomorphism

(2.3)
KK∗(X; RY , A) → K∗
�
A(Y )
�

that occurs in the Representability Theorem. By the Yoneda Lemma, any such
natural transformation is of the form f �→ f∗(ξY ) for a unique

ξY ∈ K0
�
RY (Y )
�
= K0�
S(Y, Y )
�
= K0�
Ch(Y )
�
.

The natural transformation in (2.3) is generated by the class of the 1-dimensional
trivial vector bundle over the compact space Ch(Y ) or, equivalently, the class of
the unit element in K0
�
RY (Y )
�
.


10
RALF MEYER AND RYSZARD NEST

In the examples we consider later, all natural transformations turn out to be
products of obvious ones, coming from the K-theory six-term exact sequences (1.4).
To check this, we only have to verify that a given element α of KK∗(X; RZ, RY ) lifts
a given element of K∗�
S(Y, Z)
�
. The isomorphism (2.3) maps α to [ξZ]⊗RZ(Z)α(Z)
in K∗
�
RY (Z)
�
= K∗�
S(Y, Z)
�
, where α(Z) in KK∗
�
RZ(Z), RY (Z)
�
is obtained
from α by restriction to Z. This product is easy to compute.
To get acquainted with this approach to natural transformations, we compute
some important examples. Let Y ∈ LC(X) and U ∈ O(Y ). Since R is a C∗-algebra
over Xop × X, there is an extension

(2.4)
RY \U ֌ RY ։ RU

of C∗-algebras over X. It contains C∗-algebra extensions

RY \U(Z) ֌ RY (Z) ։ RU(Z)

for all Z ∈ LC(X). Let Z := Y \ U. The extension (2.4) is semi-split in C∗alg(X)
and hence has a class in KK1(X; RU, RZ) and produces an exact triangle

(2.5)
ΣRU → RZ → RY → RU

in KK(X).

Lemma 2.11. The maps in the extension triangle (2.5) correspond to the natural
transformations FKU[1] ⇐ FKZ ⇐ FKY ⇐ FKU in (1.4).

Proof. The natural transformation µY
U : FKU ⇒ FKY in (1.4) is induced by the
natural ∗-homomorphism j : A(U) → A(Y ). For A = RU, this map is invertible
because S(U, Y ) = S(U, U) = Ch(U). Hence j(ξU) ∈ K0�
S(U, Y )
�
is again the
class of the trivial vector bundle on Ch(U); this class corresponds to the natural
transformation µY
U . The restriction map RY ։ RU in (2.4) maps [ξY ] to [ξU]—
recall that both [ξY ] and [ξU] are trivial vector bundles. Hence the restriction map
RY ։ RU and the natural transformation µZ
Y correspond to the same class—the
1-dimensional trivial vector bundle on Ch(U)—in K0�
S(U, Y )
�
.
Similarly, the natural transformation µZ
Y : FKY ⇒ FKZ is induced by the nat-
ural ∗-homomorphism p: A(Y ) ։ A(Z).
For A = RY , this is the restriction
∗-homomorphism C
�
Ch(Y )
�
→ C
�
Ch(Z)
�
because S(Y, Y ) = Ch(Y ) and S(Y, Z) =
Ch(Z). Since the restriction of a trivial bundle remains trivial, µZ
Y corresponds
to the trivial 1-dimensional vector bundle on S(Y, Z) = Ch(Z). The embedding
RZ ։ RY restricts to an identity map on Z because S(Z, Z) = S(Z, Y ) = Ch(Z).
Since this maps [ξZ] to the trivial bundle, the embedding RZ ։ RY and µZ
Y both
correspond to the same class—the 1-dimensional trivial vector bundle on Ch(Z)—in
K0�
S(Y, Z)
�
.
Finally, we study the boundary map δU
Z : FKZ ⇒ FKU[1].
We claim that it
corresponds to the class of the extension RZ ֌ RY ։ RU in KK1(X; RU, RZ).
To prove this, we use that Ch(Y ) is the join of the spaces Ch(U) and Ch(Z), so
that there is a continuous map f : Ch(Y ) → [0, 1] whose fibres over 0 and 1 are
Ch(U) and Ch(Z), respectively.
More precisely, let x0 ≺ x1 ≺ · · · ≺ xn be a strict chain in Y and let ξ be a point
of the corresponding simplex with coordinates (t0, . . . , tn) with t0 + · · · + tn = 1,
that is, ξ = t0x0 + · · · + tnxn. Then there is j ∈ {0, . . . , n} with x0, . . . , xj ∈ U,
xj+1, . . . , xn ∈ Z. We can, therefore, write ξ = tUξU + tZξZ with

ξU = t0x0 + · · · + tjxj

tU
∈ Ch(U),
tU = t0 + · · · + tj,

ξZ = tj+1xj+1 + · · · + tnxn

tZ
∈ Ch(Z),
tZ = tj+1 + · · · + tn.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
11

We define a continuous map f : Ch(Y ) → [0, 1] by ξ �→ tZ. We have

S(U, U) = Ch(U) = f −1(0),
S(Z, Z) = Ch(Z) = f −1(1)

by construction, and hence

S(Z, U) = Ch(Y ) \
�
Ch(U) ⊔ Ch(Z)
�
= f −1�
(0, 1)
�
.

Now we can compute some boundary maps. The boundary map

K0�
S(Z, Z)
� ∼= K0
�
RZ(Z)
�
→ K1
�
RZ(U)
� ∼= K1�
S(Z, U)
�

maps the class of the trivial bundle [ξZ] to f ∗(δ), where δ denotes a generator
of Z ∼= K1�
(0, 1)
�
; this follows from the naturality of the boundary map.
The
boundary map

K0�
S(U, U)
� ∼= K0
�
RU(U)
�
→ K1
�
RZ(U)
� ∼= K1�
S(Z, U)
�

for the extension RZ ֌ RY ։ RU maps the class of the trivial bundle [ξU] to
−f ∗(δ), again by naturality of the boundary map.
□

Remark 2.12. The proof also describes the classes in K0�
S(U, Y )
�
, K0�
S(Y, Z)
�
,
and K1�
S(Z, U)
�
that correspond to the natural transformations in (1.4).
The
natural transformations FKU ⇒ FKY and FKY ⇒ FKZ are represented by the
classes of the trivial vector bundles over the compact spaces S(U, Y ) and S(Y, Z);
the natural boundary map FKZ ⇒ FKU[1] is represented by f ∗(δ) for a generator
of K1�
(0, 1)
�
.

2.2. Proof of Theorem 2.5. We check first that the natural transformation
KK∗(X; RY , A) → K∗
�
A(Y )
�
induced by ξY is an isomorphism if Y is the min-
imal open subset Ux containing some point x ∈ X. The adjointness relation

KK∗(X; ix(A), B) ∼= KK∗
�
A, B(Ux)
�

for all B ∈∈ KK(X) established in [8, Proposition 3.12] yields

KK∗(X; ix(C), B) ∼= KK∗
�
C, B(Ux)
�
= FKUx(B),

that is, ix(C) represents FKUx. To check that RUx does so as well, we must show
that ix(C) and RUx are KK(X)-equivalent.

Recall that ix(C) = (C, x), where x denotes the map Prim(C) ∼= {x}
⊆
−→ X, and

ix(C)(Z) =

�
C
if x ∈ Z,
0
otherwise

for all Z ∈ LC(X).
Since Ux = {y ∈ X | x ⪯ y}, the preordered set Ux has a minimal point,
namely x.
Therefore, the space Ch(Ux) is starlike and hence contractible in a
canonical way towards x. The path from a point in ∆I for I ∈ SUx to the base
point in ∆x lies in ∆I∪{x}. Since max I ∪ {x} = max I, the contraction preserves
the ideals RUx(V ) for V ∈ O(X), so that we get a homotopy equivalence between
C
�
Ch(Ux)
�
and ix(C) in C∗alg(X). Thus RUx corepresents FKUx as well. It is easy
to see that the natural isomorphism KK∗(X; RUx, ) ∼= FKUx is induced by ξUx.
Let Good ⊆ LC(X) be the set of all Z ∈ LC(X) for which the natural trans-
formation KK∗(X; RZ, A) → FKZ(A) induced by ξZ is an isomorphism. We must
show Good = LC(X). We have just seen that Ux ∈ Good for all x ∈ X.
Let Y ∈ LC(X) and U ∈ O(Y ); we claim that all three of U, Y , and Y \ U
are good once two of them are. This follows from the Five Lemma because the


12
RALF MEYER AND RYSZARD NEST

maps induced by ξZ for Z = U, Y, Y \ U intertwine the maps in the six-term exact
sequences (1.4) and

KK0(X; RU, A)
� KK0(X; RY , A)
� KK0(X; RY \U, A)

�

KK1(X; RY \U, A)

�

KK1(X; RY , A)
�
KK1(X; RU, A)
�

for any A ∈∈ KK(X); the latter six-term exact sequence is induced by the semi-
split extension (2.5). The commutativity of the relevant diagrams follows from the
computations in the proof of Lemma 2.11 (which do not depend on Theorem 2.5).
The two-out-of-three property of Good implies:

U, V ∈ O(X),
U, V, U ∩ V ∈ Good
=⇒
U ∪ V ∈ Good

because (U ∪ V ) \ U = V \ (U ∩ V ). By induction on the length of U, this implies
that all open subsets of X belong to Good. Since any locally closed subset is a
difference of two open subsets, we conclude that Good = LC(X). This finishes the
proof of Theorem 2.5.

3. An example

In this section, we restrict our attention to a special class of spaces, namely, the
spaces X = {1, . . ., n} totally ordered by ≤ for n ∈ N. We let

[a, b] := {x ∈ X | a ≤ x ≤ b}.

for a, b ∈ Z. We equip X with the Alexandrov topology, so that the open subsets
are [a, n] for all a ∈ X; the closed subsets are [1, b] with b ∈ X, and the locally
closed subsets are those of the form [a, b] with a, b ∈ X and a ≤ b. Any locally
closed subset of X is connected.

3.1. Computations with the order complex. Since any subset of X is totally
ordered, the space Ch([a, b]) is just a closed simplex of dimension b − a for any
b ≥ a. We denote the corresponding face of Ch(X) by ∆[a,b]. This is understood
to be empty for a > b.
From now on, we let

Y = [a1, b1],
Z = [a2, b2],
with 1 ≤ a1 ≤ b1 ≤ n and 1 ≤ a2 ≤ b2 ≤ n.

Then �Y = [a1, n], �∂Y = [b1 + 1, n], Z = [1, b2], and ∂Z = [1, a2 − 1]. Lemma 2.10
yields

S(Y, Z) = ∆[a1,b2] \
�
∆[a1,a2−1] ∪ ∆[b1+1,b2]
�
.

Now we distinguish three cases:

Case 1: If a2 ≤ a1 ≤ b2 ≤ b1, then S(Y, Z) = ∆[a1,b2] is a non-empty closed simplex.
Hence NT ∗(Y, Z) ∼= K∗�
S(Y, Z)
� ∼= Z[0] (this means Z in degree 0).
Case 2: If a2 − 1 ≤ b1, a1 < a2, and b1 < b2, then S(Y, Z) is obtained from a closed
simplex by removing two disjoint, non-empty closed faces. Excision yields
NT ∗(Y, Z) ∼= K∗�
S(Y, Z)
� ∼= Z[1] (this means Z in degree 1).
Case 3: In all other cases, S(Y, Z) is either empty, a difference of two closed sim-
plices, or a difference σ \ (τ1 ∪ τ2) for two non-empty closed faces τ1 and τ2
of a simplex σ that intersect. Then τ1 ∪ τ2 and σ are both contractible, so
that NT ∗(Y, Z) ∼= K∗�
S(Y, Z)
� ∼= 0.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
13

Summing up, we get

(3.1)
NT ∗(Y, Z) =








Z[0]
if a2 ≤ a1 ≤ b2 ≤ b1,
Z[1]
if a2 − 1 ≤ b1, a1 < a2, and b1 < b2,
0
otherwise.

3.2. Products of natural transformations. Our next task is to identify the
natural transformations that correspond to the generators of the groups in (3.1);
this also allows us to compute products in NT .
First we study the grading preserving transformations that appear in the first
case. We introduce a partial order ≥ and a strict partial order ≫ on LC(X) by

[a1, b1] ≥ [a2, b2]
⇐⇒
a1 ≥ a2 and b1 ≥ b2,

[a1, b1] ≫ [a2, b2]
⇐⇒
a1 > b2.

Our computation shows that NT 0(Y, Z) ̸= {0} if and only if Y ≥ Z but not
Y ≫ Z. This is equivalent to Y ∩ Z being non-empty, closed in Y , and open in Z.
Under these assumptions, there is a natural non-zero ∗-homomorphism given by
the composition
µZ
Y : A(Y ) ։ A(Y ∩ Z) ֌ A(Z)
because A(Y ∩ Z) is a quotient of A(Y ) and an ideal in A(Z). The natural trans-
formation FKY ⇒ FKZ induced by µZ
Y maps ξY ∈ FKY,0(RY ), which is the class
of the trivial line bundle over S(Y, Y ) = ∆[a1,b1], to the trivial line bundle over
S(Y, Z) = ∆[a1,b2]. Since this is the generator of FKZ,0(RY ) = K0�
S(Y, Z)
� ∼= Z[0],
the natural transformation µZ
Y generates NT 0(Y, Z).
If Y ≫ Z, then we let µZ
Y : A(Y ) → A(Z) be the zero map, which induces the zero
transformation FKY ⇒ FKZ. With this convention, we get µZ
Y ◦ µY
W = µZ
W for all
Y, Z, W ∈ LC(X) with W ≥ Y ≥ Z, also if W ≫ Z; this equation holds on the level
of ∗-homomorphisms and, therefore, also for the induced natural transformations.
We can sum this up as follows:

Lemma 3.1. The category NT 0 of grading-preserving natural transformations
FKY ⇒ FKZ for Y, Z ∈ LC(X) is the pre-additive category generated by natural
transformations µZ
Y : FKY ⇒ FKZ for all Y ≥ Z with the relations µZ
Y ◦ µY
W = µZ
W
for W ≥ Y ≥ Z and µZ
Y = 0 for Y ≫ Z.

This list of generators is longer than necessary. Clearly, we can write any µZ
Y
as a product of the transformations µ[a−1,b]
[a,b]
for 2 ≤ a ≤ b ≤ n and µ[a,b−1]
[a,b]
for
1 ≤ a < b ≤ n. Moreover, these transformations themselves are indecomposable,
that is, they cannot be written themselves as products in a non-trivial way.
Now we turn to the natural transformations of degree 1. For any b ∈ X and any
C∗-algebra A over X, we have a natural C∗-algebra extension

A([b, n]) ֌ A([1, n]) ։ A([1, b − 1]),

which generates an odd natural transformation

δb : FK[1,b−1] ⇒ FK[b,n].

Composing with the grading preserving natural transformations µ above, we get a
natural transformation of degree 1

(3.2)
δZ
Y : FKY = FK[a1,b1]
µ=⇒ FK[1,a2−1]
δa2
==⇒ FK[a2,n]
µ=⇒ FK[a2,b2] = FKZ

whenever b1 ≥ a2 − 1.
Equation (3.1) predicts that this transformation vanishes if a1 ≥ a2 or b1 ≥ b2.
This can be verified as follows. Vanishing for a1 ≥ a2 is clear because then [a1, b1] ≫
[1, a2−1]. By the naturality of the boundary map, the transformation in (3.2) agrees


14
RALF MEYER AND RYSZARD NEST

with the composition of µ: FK[a1,b1] ⇒ FK[a1,a2−1] with the boundary map for the
extension

(3.3)
A([a2, b2]) ֌ A([a1, b2]) ։ A([a1, a2 − 1]).

If b1 ≥ b2, then µ[a1,a2−1]
[a1,b1]
factors through the quotient map in (3.3).
But the
composite of two maps in a six-term exact sequence vanishes.
Equation (3.2) produces a natural transformation δZ
Y ∈ NT 1(Y, Z) whenever
a1 < a2, b1 < b2, and a2 − 1 ≤ b1, that is, whenever (3.1) predicts NT 1(Y, Z)
to be non-zero. We claim that δZ
Y generates this group. This follows because the
natural transformation δZ
Y maps the class of the trivial line bundle over S(Y, Y ) to
the generator of K1�
S(Y, Z)
� ∼= Z.
Notice that NT 1([a2, n], Z) = {0} for any Z ∈ LC(X). Since the natural trans-
formation (3.2) above factors through FK[a2,n], any product of two odd natural
transformations vanishes. Thus the category NT is a split extension of NT 0 by
the bimodule NT 1. The bimodule structure on NT 1 is very simple: a product
µZ
Y ◦ δY
W or δZ
Y ◦ µY
W is equal to δZ
W whenever all three natural transformations are
defined, and zero otherwise.

Example 3.2. To make our constructions more concrete, we now consider the ex-
ample n = 2, which corresponds to extensions of C∗-algebras. There are only three
non-empty locally closed subsets: 1 = [1, 1], 12 = [1, 2], and 2 = [2, 2]. The order
complex is an interval; we label its end points 1 and 2. The map (m, M) from
Ch(X) = [1, 2] to Xop × X maps

1 �→ (1, 1),
2 �→ (2, 2),
]1, 2[ �→ (1, 2).

Correspondingly, we have

S(1, 1) = {1},
S(1, 2) = ]1, 2[,
S(1, 12) = [1, 2[,

S(2, 1) = ∅,
S(2, 2) = {2},
S(2, 12) = {2},

S(12, 1) = {1},
S(12, 2) = ]1, 2],
S(12, 12) = [1, 2].

Taking K-theory, we get

NT (1, 1) = Z[0],
NT (1, 2) = Z[1],
NT (1, 12) = 0,

NT (2, 1) = 0,
NT (2, 2) = Z[0],
NT (2, 12) = Z[0],

NT (12, 1) = Z[0],
NT (12, 2) = 0,
NT (12, 12) = Z[0].

3.3. Ring-theoretic properties of the natural transformations. We now ob-
serve some general ring-theoretic properties of NT for X = {1, . . . , n} with the
total order. We exclude the trivial case n = 1. We may replace NT by a Z/2-
graded ring by taking the direct sum of NT ∗(Y, Z) for all Y, Z ∈ LC(X)∗ and
defining the product as usual for a category ring. Then NT -modules become Z/2-
graded modules over this Z/2-graded ring, and ring-theoretic notions such as the
Jacobson radical and the balanced tensor product ⊗NT make sense.

Definition 3.3. Let NT nil ⊆ NT be the subgroup spanned by the natural trans-
formations µZ
Y with Y ̸= Z and δZ
Y with arbitrary Y, Z.
Let NT ss ⊆ NT be the subgroup spanned by the natural transformations µY
Y
with Y ∈ LC(X)∗.

Lemma 3.4. The subgroup NT nil is the maximal nilpotent ideal in NT , it is the
nilradical and the Jacobson radical of NT . The subgroup NT ss is a semi-simple
subring, and NT decomposes as a semi-direct product NT nil ⋊ NT ss.

Proof. Since all µY
Y are idempotent, NT ss is a subring isomorphic to ZLC(X)∗ with
pointwise multiplication. It is easy to see that NT nil is an ideal in NT . It is


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
15

nilpotent, that is, NT k
nil = {0} for some k ∈ N, because LC(X)∗ is finite and ≥
is a partial order on it. Since NT = NT nil ⊕ NT ss as Abelian groups, we get
the desired semi-direct product decomposition. Since the Jacobson radical of NT ss
vanishes, NT nil is both the nilradical and the Jacobson radical of NT .
□

We are going to use Lemma 3.4 to characterise the projective NT -modules. This
characterisation involves the following two definitions.

Definition 3.5. We call an NT -module M exact if the chain complexes

· · · → M(U)
µY
U
−−→ M(Y )
µY \U
Y
−−−→ M(Y \ U)
δU
Y \U
−−−→ M(U) → · · ·

are exact for all Y ∈ LC(X), U ∈ O(Y ) as in (1.5).

Proposition 3.6. Let K ֌ E ։ Q be an extension of NT -modules. If two of the
modules K, E, Q are exact, so is the third one.

Proof. Given U and Y as above and a module M, let C•(M) be the chain complex

· · · → M(U)[m] → M(Y )[m] → M(Y \ U)[m] → M(U)[m − 1] → · · · .

Then C•(K) ֌ C•(E) ։ C•(Q) is an extension of chain complexes. The long
exact homology sequence shows that all three of these chain complexes are exact
once two of them are exact.
□

Definition 3.7. Given an NT -module M, we let

NT nil · M = {x · m | x ∈ NT nil, m ∈ M},
Mss := M/NT nil · M.

We call Mss the semi-simple part of M.

Since the tensor product over NT is right exact, Mss ∼= NT ss ⊗NT M. We need
the following more concrete description of Mss or, equivalently, of NT nil · M.

Lemma 3.8. Let M be an NT -module and let Y = [a, b] with 1 ≤ a ≤ b ≤ n.
Then

(NT nil · M)(Y ) =


















µY
[a+1,b](M[a + 1, b]) + µY
[a,b+1](M[a, b + 1])
if a < b < n,

µY
[a,b+1](M[a, b + 1])
if a = b < n,

µY
[a+1,b](M[a + 1, b]) + δY
[1,a−1](M[1, a − 1])
if 1 < a < b = n,

µY
[a+1,b](M[a + 1, b])
if 1 = a < b = n,

δY
[1,a−1](M[1, a − 1])
if a = b = n.

If M is exact, then

(NT nil · M)(Y ) =

�
ker
�
δ[a+1,b+1]
[a,b]
: M[a, b] → M[a + 1, b + 1]
�
if b < n,

ker
�
µ[1,a]
[a,b] : M[a, b] → M[1, a]
�
if b = n.

Proof. The first assertion holds because any natural transformation FKZ ⇒ FKY
with Z ̸= Y factors through µY
[a+1,b] or µY
[a,b+1] if a < b < n, through µY
[a,b+1] if

a = b < n, and so on. Here we use that the natural transformations µ[a−1,b]
[a,b]
for

2 ≤ a ≤ b ≤ n, µ[a,b−1]
[a,b]
for 1 ≤ a < b ≤ n, and δ[a,n]
[1,a−1] for 2 ≤ a ≤ n already
generate NT ∗, that is, all other transformations µZ
Y or δZ
Y with Y ̸= Z can be
written as products of these generators. By the way, these natural transformations
even form a basis for the subquotient NT nil/NT 2
nil.
Now assume that M is exact. If a = b < n, then

(NT nil · M)[a, a] = range
�
µ[a,a]
[a,a+1]
�
= ker
�
δ[a+1,a+1]
[a,a]
�
.


16
RALF MEYER AND RYSZARD NEST

Similarly, we get

(NT nil · M)[n, n] = ker
�
µ[1,n]
[n,n]
�
,
(NT nil · M)[1, n] = ker
�
µ[1,1]
[1,n]
�
.

Given f1 : A1 → B and f2 : A2 → B and two exact sequences

A1
f1
−→ B
g1
−→ C1,
A2
g1f2
−−−→ C1
g2
−→ C2,

we have

(3.4)
range(f1) + range(f2) = ker(g1) + range(f2)

= {x ∈ B | g1(x) ∈ range(g1 ◦ f2) = ker(g2)} = ker(g2 ◦ g1).

If a < b < n, then we apply this to the maps on M induced by f1 = µY
[a+1,b]
and f2 = µY
[a,b+1] with Y = [a, b].
We get g1 = µ[a,a]
Y
, g1 ◦ f2 = µ[a,a]
[a,b+1], and

hence g2 = δ[a+1,b+1]
[a,a]
and g2 ◦ g1 = δ[a+1,b+1]
[a,b]
. This yields the desired formula for
(NT nil · M)[a, b] for a < b < n, using the exactness of M. If a < b = n, then we
apply the same reasoning to f1 = µY
[a+1,b] and f2 = δY
[1,a−1]. Here we get g1 = µ[a,a]
Y
as above, g1 ◦ f2 = δ[a,a]
[1,a−1], and hence g2 = µ[1,a]
[a,a] and g2 ◦ g1 = µ[1,a]
[a,b]. This yields
the desired formula for (NT nil · M)[a, b] for a < b = n.
□

Remark 3.9. The natural transformation δ[a+1,b+1]
[a,b]
for b < n or µ[1,a]
[a,n] for b = n
is the longest natural transformation out of [a, b] in the following sense: it factors
through δZ
[a,b] or µZ
[a,b] whenever the latter is defined and non-zero. Thus Lemma 3.8
identifies NT nil·M(Y ) with the largest proper subgroup of M(Y ) that is the kernel
of some δZ
[a,b] or µZ
[a,b].

The following proposition is a rather trivial variant of the Nakayama Lemma.
Unlike in the usual Nakayama Lemma, we do not assume the module to be finitely
generated. This is no problem because the relevant ideal NT nil is nilpotent.

Proposition 3.10. Let M be an NT -module with Mss = 0. Then M = 0.

Proof. By assumption, M = NT nil · M. By induction, this implies M = NT j
nil · M
for all j ∈ N. Since NT k
nil = 0 for some k, we get M = 0.
□

3.4. Characterisation of free and projective modules.

Definition 3.11. For Y ∈ LC(X), the free NT -module on Y is defined by

PY (Z) := NT ∗(Y, Z)
for all Z ∈ LC(X).

An NT -module is called free if it is isomorphic to a direct sum of degree-shifted
free modules PY [j], j ∈ Z/2.

Theorem 3.12. Let M be an NT -module. Then the following are equivalent:

(i) M is a free NT -module.
(ii) M is a projective NT -module.
(iii) Mss(Y ) = NT ss ⊗NT M(Y ) is a free Abelian group for all Y ∈ LC(X) and
TorNT
1
(NT ss, M) = 0.
(iv) M(Y ) is a free Abelian group for all Y ∈ LC(X) and M is exact.

Here TorNT
1
denotes the first derived functor of ⊗NT . The first three conditions
remain equivalent when we replace NT by any ring that is a nilpotent extension of
the ring ZN for some N ∈ N.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
17

Proof. The Yoneda Lemma asserts that Hom(PY , M) ∼= M(Y ) for all Y ∈ LC(X)
and all NT -modules M. Hence free modules are projective, that is, (1)=⇒(2). A
functor of the form M �→ R ⊗S M for a ring homomorphism S → R always maps
free modules to free modules and hence maps projective modules to projective mod-
ules. Furthermore, derived functors like TorNT
1
automatically vanish on projective
modules. This yields the implication (2)=⇒(3). We are going to prove that (3)
implies (1).
Since Mss(Y ) is a free Abelian group for all Y , Mss is a free module over NT ss ∼=
ZLC(X)∗. Hence P := NT ⊗NT ssMss is a free NT -module. The canonical projection
M → Mss splits by an NT ss-module homomorphism because Mss is free. This
induces an NT -module homomorphism f : P → M because of the adjointness
relation
HomNT (NT ⊗NT ss X, Y ) ∼= HomNT ss(X, Y ).

We claim that f is invertible, so that M ∼= P is a free module as asserted. We have

Pss = NT ss ⊗NT NT ⊗NT ss Mss ∼= NT ss ⊗NT ss Mss ∼= Mss.

Inspection shows that this isomorphism is induced by f. Since the functor M �→ Mss
is right-exact, this implies coker(f)ss = 0 and hence coker(f) = 0 by the Nakayama
Lemma (Proposition 3.10). That is, f is an epimorphism.
Let K := ker(f), then we get an exact sequence of NT -modules K ֌ P ։ M.
The derived functors of NT ss ⊗NT
provide a long exact sequence

(3.5)
0 → TorNT
1
(NT ss, M) → Kss → Pss
f−→
∼
= Mss → 0.

This exact sequence ends at TorNT
1
(NT ss, P) = 0 because P is projective. Since
TorNT
1
(NT ss, M) = 0 by assumption, we conclude that Kss = 0. Hence another
application of the Nakayama Lemma shows that ker(f) = 0 as well. Thus f is
invertible. This finishes the proof of the implication (3)=⇒(1), showing that the
first three conditions are equivalent. Furthermore, our argument so far works for any
split nilpotent extension of ZN for some N ∈ N because this is the only information
about NT that we have used. Nilpotent extensions of the ring ZN always split
because we can lift orthogonal idempotents in nilpotent extensions.
Free NT -modules are exact, and they consist of free Abelian groups by (3.1).
This yields the implication (1)=⇒(4). We are going to prove that (4) implies (3).
This will finish the proof of the theorem. Since we will use this once again later,
we state half of this argument as a separate lemma:

Lemma 3.13. Let M be an exact NT -module. Then TorNT
1
(NT ss, M) = 0.

Proof. Let π: P → M be an epimorphism with a projective NT -module P, and
let K := ker π. Since projective modules are exact and K ֌ P ։ M is a module
extension, Proposition 3.6 shows that K is exact. We still have an exact sequence
as in (3.5).
Since K and P are exact, Lemma 3.8 identifies Kss(Y ) and Pss(Y ) in a natural
way with subspaces of K(Z) and P(Z) for suitable Z; here we use A/ ker(f) ∼=
range(f) for a group homomorphism f : A → B. Since the map K(Z) → P(Z)
is injective, so is the map Kss(Y ) → Pss(Y ).
Hence the map Kss → Pss is a
monomorphism, forcing TorNT
1
(NT ss, M) = 0 by (3.5).
□

To finish the proof of the implication (4)=⇒(3) in Theorem 3.12, it remains to
check that Mss(Y ) is free for all Y if M is exact and M(Y ) is free for all Y . We
use Lemma 3.8 once again to describe Mss(Y ) as the range of a canonical element
in NT ∗(Y, Z) for a suitable Z. Thus Mss(Y ) is isomorphic to a subgroup of M(Z),
which is a free group by assumption. Hence Mss(Y ) is free as well.
□


18
RALF MEYER AND RYSZARD NEST

4. Homological algebra in KK(X)

Let X be a sober topological space. We are going to apply to KK(X) the gen-
eral machinery for doing homological algebra in triangulated categories discussed
in [9]. This theory goes back to the work on relative homological algebra by Samuel
Eilenberg and John Coleman Moore ([4]), which was carried over to the setting of
triangulated categories by Daniel Christensen [3] and Apostolos Beligiannis [1].

4.1. An ideal in KK(X). Our starting point is a rough idea of the invariant we
want to use. This rough idea is expressed by a homological ideal in the triangulated
category. The ideal I in KK(X) relevant for us is defined by

(4.1)
I(A, B) :=
�
f ∈ KK(X; A, B)
��

f∗ : K∗
�
A(Y )
�
→ K∗
�
B(Y )
�
vanishes for all Y ∈ LC(X)
�
.

It makes no difference if we use LC(X) or LC(X)∗ here.
We claim that I is a homological ideal in the triangulated category KK(X); that
is, it is the kernel (on morphisms) of a stable homological functor from KK(X)
to some stable Abelian category; stability means that the functor intertwines the
suspension automorphism on KK(X) with a given suspension automorphism on the
target Abelian category.
Our starting point is a bare form of filtrated K-theory. Recall the functors

FKY : KK(X) → AbZ/2,
A �→ K∗
�
A(Y )
�

for Y ∈ LC(X) from Definition 2.1 and let

F := (FKY )Y ∈LC(X) : KK(X) →
�

Y ∈LC(X)∗
AbZ/2,
A �→
�
K∗
�
A(Y )
��

Y ∈LC(X)∗.

The target category �
Y ∈LC(X)∗ AbZ/2 of F is Abelian and carries an obvious sus-
pension functor that shifts the Z/2-grading. The functor F is a stable homological
functor, that is, it intertwines the suspension automorphisms and maps exact tri-
angles to long exact sequences. By definition,

(4.2)
I =
�

Y ∈LC(X)∗
ker FKY = ker F,

that is, f ∈ I(A, B) if and only if F(f) = 0. Hence I is a homological ideal with
defining functor F.
We also have I = ker FK with FK as in Definition 2.4: the two functors F
and FK only differ through their target categories. For the time being, we pretend
that we do not yet know anything about filtrated K-theory beyond the ideal I it
defines. The general machinery will automatically lead us to the functor FK.
As explained in [9], the homological ideal I yields various notions of homological
algebra. The following descriptions of these notions follow from [9, Lemmas 3.2
and 3.9, Definition 3.21].
• A morphism f ∈ KK∗(X; A, B) is
– I-epic if the induced maps K∗
�
A(Y )
�
→ K∗
�
B(Y )
�
are surjective for
all Y ∈ LC(X);
– I-monic if the induced maps K∗
�
A(Y )
�
→ K∗
�
B(Y )
�
are injective for
all Y ∈ LC(X);
– an I-equivalence if the induced maps K∗
�
A(Y )
�
→ K∗
�
B(Y )
�
are
bijective for all Y ∈ LC(X).
• A homological functor F : KK(X) → C to some Abelian category C is
I-exact if F(f) = 0 for all f ∈ I; equivalently, F maps I-epimorphisms
to epimorphisms or F maps I-monomorphisms to monomorphisms.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
19

• An object A ∈∈ KK(X) is
– I-contractible if K∗
�
A(Y )
�
= 0 for all Y ∈ LC(X);
– I-projective if the functor KK∗(X; A, ) is I-exact; equivalently, I(A, B) =
0 for all B ∈∈ KK(X), or: any I-epimorphism B → A splits (see [9]
for more equivalent characterisations).
• A chain complex

· · · → An+1
δn+1
−−−→ An
δn
−→ An−1
δn−1
−−−→ An−2 → · · ·

in KK(X)—that is, An ∈∈ KK(X) and δn ∈ KK(X; An, An−1) for all n ∈ Z,
subject to the condition δn−1 ◦ δn = 0—is I-exact (in some degree n) if the
induced chain complexes of Z/2-graded Abelian groups

· · · → K∗
�
An+1(Y )
� (δn+1)∗
−−−−−→ K∗
�
An(Y )
� (δn)∗
−−−→ K∗
�
An−1(Y )
�
→ · · ·

are exact (in degree n) for all Y ∈ LC(X).
• An I-projective resolution of A ∈∈ KK(X) is an I-exact chain complex

· · · → P2
δ2
−→ P1
δ1
−→ P0
δ0
−→ A → 0 → · · ·

with I-projective entries Pn for all n ∈ N.
We shall soon see that there are enough I-projective objects in the sense that any
object of KK(X) has an I-projective resolution. Such resolutions are unique up to
chain homotopy equivalence once they exist.
We use projective resolutions to define derived functors (see [9, Definition 3.27]):
just apply the functor to be derived to an I-projective resolution and take homology.
In particular, this yields extension groups Extn
I(A, B) for all A, B ∈∈ KK(X). Un-
like in usual homological algebra, Ext0
I(A, B) may differ from the morphism space
in KK(X), compare the exact sequence (4.8) in [6].

4.2. Enough projective objects. A strategy to find enough projective objects is
outlined in [9, §3.6]. The idea is to study the left adjoint functor FK⊢
Y of FKY ; this
is defined on P ∈∈ AbZ/2 if there is FK⊢
Y (P) ∈∈ KK(X) and a natural isomorphism

(4.3)
Hom
�
P, FKY (B)
� ∼= KK(X; FK⊢
Y (P), B)

for all B ∈∈ KK(X). Notice that FK⊢
Y need not be defined for all P.
Objects of the form FK⊢
Y (P) are automatically I-projective because the functor
KK(X; FK⊢
Y (P), ) factors through FKY by (4.3) and vanishes on I by (4.2).
The simplest case to look for FK⊢
Y (P) is P = Z[0] (this means Z in degree 0).
The defining property of FK⊢
Y (Z[0]) is a natural isomorphism

KK(X; FK⊢
Y (Z[0]), B) ∼= Hom
�
Z[0], FKY (B)
� ∼= FKY,0(B) = K0
�
B(Y )
�
.

In other words, FK⊢
Y (Z[0]) must represent the covariant functor FKY . Theorem 2.5
provides such representing objects, and yields the following:

Proposition 4.1. For any Y ∈ LC(X), the adjoint functor FK⊢
Y is defined on a
Z/2-graded Abelian group G = G0 ⊕ G1 if G0 and G1 are free and countable. More
precisely,

FK⊢
Y

��

i∈I
Z[εi]

�

=
�

i∈I
RY [εi],

where I is a countable set and εi ∈ Z/2 for all i ∈ I.

Proof. We have just observed that FK⊢
Y (Z[0]) = RY . Since FKY is stable, this
implies FK⊢
Y (Z[1]) = RY [1]. It is a general feature of left adjoint functors that they
commute with direct sums. Since countable direct sums exist in KK(X), we get the
existence of FK⊢
Y on any free countable Z/2-graded Abelian group.
□


20
RALF MEYER AND RYSZARD NEST

Corollary 4.2. There are enough I-projective objects in KK(X), and the class of
I-projective objects in KK(X) is generated by the objects RY for Y ∈ LC(X)∗.
More precisely, any I-projective objects is a retract of a direct sum of suspensions
of these objects.

Proof. This follows from Proposition 4.1 and [9, Proposition 3.37].
□

Often we do not need retracts here, that is, any I-projective object is a direct
sum of suspensions of RY for Y ∈ LC(X)∗; for the totally ordered spaces studied
in §3, this follows from Theorem 3.12.
Since our ideal I is compatible with countable direct sums, the I-contractible
objects form a localising subcategory of KK(X), that is, they form a class NI of
objects that is closed under countable direct sums, retracts, isomorphism, exact
triangles, and suspensions. Furthermore, NI is the complement of the localising
subcategory that is generated by the I-projective objects. These two subcategories
contain much less information than the ideal itself. Roughly speaking, they will be
the same for any reasonable choice of invariant on KK(X) of K-theoretic nature.

Proposition 4.3. The localising subcategory that is generated by the I-projective
objects is the bootstrap category B(X). It consists of all objects of KK(X) that are
KK(X)-equivalent to a tight, nuclear, purely infinite, stable, separable C∗-algebra
over X whose simple subquotients belong to the bootstrap category B ⊆ KK.

Proof. By definition, B(X) is the localising subcategory of KK(X) that is generated
by the objects ix(C) for x ∈ X, see [8]. These generators are I-projective because
they represent the functors FKUx, compare the proof of the Representability The-
orem 2.5. The proof of this theorem also shows that the representing objects RY
belong to the triangulated subcategory of KK(X) generated by RUx for x ∈ X and
hence to B(X). Now Corollary 4.2 shows that all I-projective objects belong to
B(X). Hence the localising subcategory they generate is contained in the bootstrap
class.
Conversely, since the generators of the bootstrap class ix(C) are I-projective,
the localising subcategory generated by the I-projective objects must contain the
whole bootstrap class. This yields the first statement. The second one is contained
in [8, Corollary 5.5].
□

4.3. The universality of filtrated K-theory. The next step in the general pro-
gramme is to determine the universal defining functor for I. This functor is char-
acterised by the universal property that it is I-exact and stable homological and
that any I-exact homological functor on KK(X) factors through it uniquely (up to
natural isomorphism).
The advantage of using the universal functor is that it describes I-projective
resolutions and the associated I-derived functors in KK(X) by projective resolutions
and derived functors in its target Abelian category.
This is the crucial step to
compute these derived functors.
In the presence of enough projective objects, [9, Theorem 3.39] characterises the
universal functor by an adjointness property. In our case, this yields:

Theorem 4.4. The filtrated K-theory functor FK: KK(X) → Mod(NT )c is the
universal I-exact stable homological functor; here Mod(NT )c denotes the category
of all countable graded NT -modules.

The ring of natural transformations NT comes in automatically at this point.

Proof. This is best explained as a special case of a general result on certain homo-
logical ideals. Let T be any triangulated category with countable direct sums, and


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
21

let G be an at most countable set of objects of T. Let IG be the stable homological
ideal defined by the functor

FG : T →
�

G∈G
AbZ,
A �→
�
T∗(G, A)
�

G∈G.

We assume that FG(A) is countable for all A ∈∈ T.
We are dealing with the case where T = KK(X) and G = {RY | Y ∈ LC(X)∗};
Theorem 2.5 identifies T∗(RY , A) = KK∗(X; RY , A) ∼= K∗
�
A(Y )
�
= FKY (A) for
all Y ∈ LC(X)∗, so that IG = I with I as in (4.1).
Viewing G as a full subcategory of T, it becomes a Z-graded pre-additive cat-
egory, so that we get a corresponding category Mod(Gop)c of countable graded right
modules. We can enrich the functor FG to a functor

F ′
G : T → Mod(Gop)c

because the composition in T provides maps

T∗(G′, A) ⊗ T∗(G, G′) → T∗(G, A)

for all G, G′ ∈ G, A ∈∈ T, which form a right G-module structure on
�
T∗(G, A)
�

G∈G.
We claim that the functor F ′
G is the universal IG-exact functor.
In the case at hand, our description of the natural transformations FKY ⇒ FKZ
in §2.1 means that Mod(Gop)c = Mod(NT )c and F ′
G = FK is filtrated K-theory
as defined in Definition 2.4. Hence it suffices to establish the claim above to finish
the proof of Theorem 4.4.
To do this, we check the conditions in [9, Theorem 3.39]. Idempotent morphisms
in KK(X) split because this happens in any triangulated category with countable
direct sums (see [10]).
Call F ′
G(G) = T( , G) for G ∈ G the free Gop-module
on G. Direct sums of free modules are projective, and any object of Mod(Gop)c is a
quotient of a countable direct sum of free modules. Hence Mod(Gop)c has enough
projective objects. Moreover,

HomGop�
F ′
G(G), F ′
G(A)
� ∼= F ′
G(A)(G) = T(G, A)

shows that the left adjoint F ⊢ of F := F ′
G maps F ′
G(G) to G ∈∈ T. Since the
domain of F ⊢ is closed under suspensions, countable direct sums, and retracts, the
adjoint is defined on all projective modules. Furthermore, F ◦ F ⊢(P) ∼= P holds
for free modules and hence for all projective modules P. Having checked all the
hypotheses of [9, Theorem 3.39], we can conclude that F ′
G is indeed universal.
□

Since FK: KK(X) → Mod(NT )c is universal, [9, Theorem 3.41] now tells us,
roughly speaking, that homological algebra in KK(X) with respect to I is equivalent
to homological algebra in the Abelian category Mod(NT )c:
• An object A of KK(X) is I-projective if and only if FK(A) ∈ Mod(NT )c
is projective and

KK∗(X; A, B) ∼= HomNT
�
FK(A), FK(B)
�

for all B ∈∈ KK(X).
Another equivalent condition is that FK(A) ∈ Mod(NT )c is projective
and A belongs to the localising subcategory generated by the I-projective
objects; the latter agrees with the bootstrap class by Proposition 4.3.
• The functor FK and its partially defined left adjoint FK⊢ restrict to an
equivalence of categories between the subcategories of I-projective objects
in KK(X) and of projective objects in Mod(NT )c.
• For any A ∈∈ KK(X), the functors FK and FK⊢ induce bijections between
isomorphism classes of I-projective resolutions of A and isomorphism classes


22
RALF MEYER AND RYSZARD NEST

of projective resolutions of FK(A) in Mod(NT )c. That is, a projective res-
olution in Mod(NT )c lifts to a unique I-projective resolution in KK(X).
This provides the “geometric resolutions” that are used in connection with
the usual Universal Coefficient Theorem for KK.
• For all n ∈ N, there is a natural isomorphism

Extn
I(A, B) ∼= Extn
NT
�
FK(A), FK(B)
�
,

where the right hand side denotes extension groups in the Abelian category
Mod(NT )c.
• For any homological functor G: KK(X) → C, there is a unique right-exact
functor ¯G: Mod(NT )c → C with ¯G ◦ FK(P) = G(P) for all I-projective P.
The left derived functors of G with respect to I are Ln ¯G ◦ FK for n ∈ N,
where Ln ¯G: Mod(NT )c → C denotes the nth left derived functor of ¯G.

4.4. The Universal Coefficient Theorem. In the general theory, the next step
is to construct a spectral sequence whose E2-term involves the extension groups
Extn
I(A[m], B); it converges—in favourable cases—to KK∗(X; A, B). This spectral
sequence is constructed in [3, 6]. Since we aim for an exact sequence, not for a
spectral sequence, we only need the special case considered in [9, Theorem 4.4].
This provides the Universal Coefficient Theorem we want under the assumption
that FK(A) has a projective resolution of length 1 in Mod(NT )c:

Theorem 4.5. Let A, B ∈∈ KK(X). Suppose that FK(A) ∈∈ Mod(NT )c has a
projective resolution of length 1 and that A ∈∈ B(X). Then there are natural short
exact sequences

Ext1
NT
�
FK(A)[j + 1], FK(B)
�
֌ KKj(X; A, B) ։ HomNT
�
FK(A)[j], FK(B)
�

for j ∈ Z/2, where HomNT and Ext1
NT denote the morphism and extension groups
in the Abelian category Mod(NT )c and [j] and [j + 1] denote degree shifts.

The bootstrap class appears here because of Proposition 4.3, which identifies it
with the localising subcategory generated by the I-projective objects.

Corollary 4.6. Let A, B ∈∈ B(X) and suppose that both FK(A) and FK(B) have
projective resolutions of length 1 in Mod(NT )c. Then any morphism FK(A) →
FK(B) in Mod(NT )c lifts to an element in KK0(X; A, B), and an isomorphism
FK(A) ∼= FK(B) lifts to an isomorphism in B(X).

Proof. The lifting of a homomorphism follows from Theorem 4.5. Given an iso-
morphism f : FK(A) → FK(B), we can lift f and f −1 to elements α and β of
KK0(X; A, B) and KK0(X; B, A), respectively. Since β ◦ α lifts the identity map
on FK(A), the difference id − β ◦ α belongs to Ext1
NT
�
FK(A)[j + 1], FK(A)
�
. The
latter is a nilpotent ideal in KK(X; A, A) because of the naturality of the exact
sequence in Theorem 4.5. Hence (id − βα)2 = 0, so that β ◦ α is invertible. The
same argument shows that α ◦ β is invertible, so that α is invertible.
□

This corollary is what is needed for the classification programme, and it depends
on resolutions having length 1. Conversely, if there is A for which FK(A) has no
projective resolution of length 1, then it is likely that there exist non-isomorphic
B, D ∈∈ B(X) with FK(B) ∼= FK(D).
The following theorem provides such a
counterexample, but under a stronger assumption.

Theorem 4.7. Let I be a homological ideal in a triangulated category T with enough
I-projective objects. Let F : T → AIT be a universal I-exact stable homological
functor. Suppose that I2 ̸= 0. Then there exist non-isomorphic objects B, D ∈∈ T
for which F(B) ∼= F(D) in AIT.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
23

Proof. Since I2 ̸= 0, there is A ∈∈ T with I2(A, ) ̸= 0, that is, A is not
I2-projective. The ideal I2 has enough projective objects as well, so that there
is an exact triangle
ΣN2
γ2
−→ ˜A2
α2
−→ A
ι2
−→ N2
with ι2 ∈ I2 and an I2-projective object ˜A2 (this is part of the phantom castle
constructed in [6], where the same notation is used).
Since ι2 ∈ I, this triangle is I-exact and hence provides an extension

F(N2)[1] ֌ F( ˜A2) ։ F(A)

in AIT. Even more, this extension splits because ι2 ∈ I2. This follows because the
canonical map
I(A, N2) → Ext1
I(A, N2[1])
implicitly used above factors through I/I2 and hence annihilates ι2 (see [6, Equa-
tion (4.9)]). As a result, F( ˜A2) ∼= F(A) ⊕ F(N2)[1].
But ˜A2 cannot be isomorphic to A ⊕ N2[1]. If this were the case, then A would
be I2-projective, as a retract of the I2-projective object ˜A2. Then I2(A, ) = 0,
contradicting our choice of A. Hence ˜A2 ̸∼= A ⊕ N2[1].
□

If I2 = 0, then the ABC spectral sequence constructed in [6] degenerates at the
third stage, that is, E3 = E∞. But E2 and E3 differ unless projective resolutions
have length 1. Hence the vanishing of I2 is probably not sufficient for isomorphisms
on the invariant to lift because the boundary map d2 on the second stage of the
ABC spectral sequence may provide further obstructions.
Whether or not filtrated K-theory gives rise to projective resolutions of length 1
depends on the space in question: we will find positive and negative cases below.
Before we turn to examples, we discuss another important issue: does filtrated
K-theory exhaust all of Mod(NT )c?
This is definitely not the case because of
the additional exactness conditions that hold for objects of the form FK(A). The
following result is not optimal but sufficient for our purposes.

Theorem 4.8. Let G ∈∈ Mod(NT )c have a projective resolution of length 1. Then
there is A ∈∈ B(X) with FK(A) ∼= G, and this object is unique up to isomorphism
in B(X).

Proof. Any projective resolution of length 1 in Mod(NT )c is isomorphic to one of
the form
· · · → 0 → FK(P1)
FK(f)
−−−−→ FK(P0) → G
for suitable I-projective objects P1, P0 ∈∈ KK(X) and some f ∈ KK0(X; P1, P0).
Here we use that FK restricts to an equivalence of categories between the subcat-
egories of I-projective objects of KK(X) and of projective objects of Mod(NT )c by
the first paragraph of [9, Theorem 3.41].
We may embed the morphism f in an exact triangle

ΣA
h−→ P1
f−→ P0
g−→ A.

Since FK(f) is injective, the map f is I-monic; thus g is I-epic and h ∈ I. Therefore,
the long exact sequence for FK applied to the above triangle degenerates to a short
exact sequence
FK(P1) ֌ FK(P0) ։ FK(A).
This yields FK(A) ∼= G as desired. The uniqueness of A is already contained in
Corollary 4.6.
□

It remains to understand which objects of the category Mod(NT )c have a pro-
jective resolution of length 1.


24
RALF MEYER AND RYSZARD NEST

4.5. Resolutions of length 1 in the totally ordered case. We return to the
example of the space X = {1, . . ., n} totally ordered by ≤ studied in §3. Let NT be
the graded pre-additive category of natural transformations described in §3, and let
C = Mod(NT )c be the Abelian category of NT -modules. The following theorem
characterises NT -modules with projective resolutions of length 1:

Theorem 4.9. Let M ∈∈ C. The following assertions are equivalent:
(i) M = FK∗(A) for some A ∈∈ KK(X);
(ii) M is exact in the sense of Definition 3.5;
(iii) TorNT
i
(NT ss, M) = 0 for i = 1, 2;
(iv) M has a free resolution of length 1 in C;
(v) M has a projective resolution of length 1 in C;
(vi) M has a projective resolution of finite length in C.

Proof. The exact sequence (1.4) shows that (i) implies (ii). Theorem 4.8 contains
the implication (v)=⇒(i), and the implications (iv)=⇒(v)=⇒(vi) are trivial. We
will show (ii)=⇒(iii)=⇒(iv) and (vi)=⇒(ii), and this will establish the theorem.
First we show that (vi) implies (ii).
Let 0 → Pm → · · · → P0 → M be a
projective resolution of finite length. By a standard “stabilisation” trick, we can
turn this into a free resolution of the same length. Let

Zj = ker(Pj → Pj−1) ∼= range(Pj+1 → Pj).

Thus Zm = 0, P0/Z0 ∼= M, and we have exact sequences Zj ֌ Pj ։ Zj−1
because our chain complex is exact. Since Zm = 0, the exactness of the projective
modules Pm and Proposition 3.6 show recursively that Zj is exact for j = m −
1, m − 2, . . . , 0, so that M is exact. Thus (vi) implies (ii).
Now we prove (ii)=⇒(iii)=⇒(iv). Let P be a countable free module for which
there is an epimorphism π: P ։ M, and let K := ker π. We have an extension
of NT -modules K ֌ P ։ M. Proposition 3.6 shows that K is exact because P
and M are exact. Furthermore, Tori+1(NT ss, M) ∼= Tori(NT ss, K) for all i ≥ 1
because P is projective. Lemma 3.13 applied to M and K yields Tori(NT ss, M) = 0
for i = 1, 2 if M is exact, that is, (ii)=⇒(iii). Now assume (iii). The argument above
yields Tor1(NT ss, K) = 0. Since P is projective, the Abelian groups P(Y ) are free
for all Y ∈ LC(X). The exact sequence in (3.5) yields the same for K(Y ). The
criterion in Theorem 3.12.(3) shows that K is projective.
□

Now we combine the existence of projective resolutions of length 1 with The-
orem 4.5, which still required this as a hypothesis:

Theorem 4.10. Let X be the topological space associated to a totally ordered finite
set, and let A and B be C∗-algebras over X. If A ∈∈ B(X), then there is a natural
short exact sequence

Ext1
NT
�
FK(A)[1], FK(B)
�
֌ KK∗(X; A, B) ։ HomNT
�
FK(A), FK(B)
�
.

In particular, any NT -module morphism FK(A) → FK(B) lifts to an element
in KK∗(X; A, B). If both A and B belong to the bootstrap class B(X), then an
isomorphism FK(A) ∼= FK(B) lifts to a KK-equivalence A ≃ B.

Proof. Use Theorem 4.5 and Corollary 4.6 together with the existence of projective
resolutions of length 1 ensured by Theorem 4.9.
□

Theorem 4.11. Let X be the topological space associated to a totally ordered finite
set, and let A and B be tight, purely infinite, stable, nuclear, separable C∗-algebras
over X whose simple subquotients belong to the bootstrap category. Then an iso-
morphism FK(A) ∼= FK(B) lifts to an X-equivariant ∗-isomorphism A ∼= B.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
25

Furthermore, any countable exact NT -modules is the filtrated K-module of some
tight, purely infinite, stable, nuclear, separable C∗-algebra over X with simple sub-
quotients in the bootstrap category.

Proof. A nuclear C∗-algebras over X belongs to the bootstrap category B(X) if and
only if its fibres belong to the non-equivariant bootstrap category B (see [8, Corol-
lary 4.13]). For a tight C∗-algebra over X, these fibres are the same as the simple
subquotients.
It is also shown in [8, Corollary 5.5] that any object of B(X) is
KK(X)-equivalent to a tight, nuclear, purely infinite, simple, separable C∗-algebra
over X whose simple subquotients belong to the bootstrap category B. A deep clas-
sification result of Eberhard Kirchberg shows that any KK(X)-equivalence between
such objects lifts to an X-equivariant ∗-homomorphism. Now the first assertion fol-
lows from Theorem 4.10. The second assertion also uses Theorem 4.8.
□

5. A counterexample

Now we let X := {1, 2, 3, 4} with the partial order 1, 2, 3 < 4 and no relation
among 1, 2, 3. Hence the open subsets of X are

O(X) =
�
∅, {4}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}
�
,

that is, a non-empty subset is open if and only if it contains 4. The associated
directed graph is
• 1

4 •
�
❑❑ �❑
❑

sss �s
• 2

• 3.

We frequently denote subsets of X simply by 124 := {1, 2, 4}, and so on.
A C∗-algebra over X is a C∗-algebra A with four distinguished ideals

I1 := A(14),
I2 := A(24),
I3 := A(34),
I4 := A(4),

such that I1 + I2 + I3 = A and Ii ∩ Ij = I4 for all 1 ≤ i < j ≤ 3 (see [8, Lemma
2.35]). Equivalently, the ideals Ij/I4 for j = 1, 2, 3 decompose A/I4 into a direct
sum of three orthogonal ideals. The other distinguished ideals are

A(124) = I1 + I2,
A(134) = I1 + I3,
A(234) = I2 + I3.

Any subset of X is locally closed. But a connected locally closed subset is either
open or one of the singletons {1}, {2}, and {3}. Hence the set of connected locally
closed subsets is

LC(X)∗ = {4, 14, 24, 34, 124, 134, 234, 1234, 1, 2, 3}.

The order complex Ch(X) is a graph with four vertices 1, 2, 3, 4 and edges joining
the first three to the last one:

Ch(X) =

����
����
1
▲▲▲▲▲
▲

����
����
2
����
����
4

����
����
3

rrrrr
r

Both maps m, M : Ch(X) → X map the vertices to the corresponding points in X.
Whereas M maps the interior of each edge to 4, the map m maps the interior of
the edge [j, 4] to j for j = 1, 2, 3.
Recall that the space of natural transformations FKY ⇒ FKZ is given by

NT ∗(Y, Z) ∼= K∗�
S(Y, Z)
�
,
S(Y, Z) := m−1(Y ) ∩ M −1(Z) ⊆ Ch(X).


26
RALF MEYER AND RYSZARD NEST

Y \Z
4
14
24
34
124
134
234
1234
1
2
3

4
Z
Z
Z
Z
Z
Z
Z
Z
0
0
0

14
0
Z
0
0
Z
Z
0
Z
Z
0
0

24
0
0
Z
0
Z
0
Z
Z
0
Z
0

34
0
0
0
Z
0
Z
Z
Z
0
0
Z

124
Z[1]
0
0
Z[1]
Z
0
0
Z
Z
Z
0

134
Z[1]
0
Z[1]
0
0
Z
0
Z
Z
0
Z

234
Z[1]
Z[1]
0
0
0
0
Z
Z
0
Z
Z

1234
Z[1]2
Z[1]
Z[1]
Z[1]
0
0
0
Z
Z
Z
Z

1
Z[1]
0
Z[1]
Z[1]
0
0
Z[1]
0
Z
0
0

2
Z[1]
Z[1]
0
Z[1]
0
Z[1]
0
0
0
Z
0

3
Z[1]
Z[1]
Z[1]
0
Z[1]
0
0
0
0
0
Z

Table 1. The ring of natural transformations

It is straightforward to compute these K-theory groups, and the results are listed
in Table 1.
Here the rows are labelled by Y , the columns by Z.
For instance,
the entry Z at (14, 1) means that NT ∗(14, 1) ∼= Z.
The trivial 1-dimensional
bundle over S(14, 1) generates this group.
Hence Remark 2.12 shows that the
generator is the natural transformation that we get from the quotient map A(14) ։
A(1).
Similar arguments show that all the natural transformations of degree 0
are induced by the familiar restriction and extension ∗-homomorphisms for closed
and open subsets. Moreover, the odd natural transformations arise by composing
these ∗-homomorphisms with boundary maps in K-theory long exact sequences. All
relations that they satisfy are predicted by morphisms of extensions and exactness
of the sequences (1.4).
The computations in §3 were based on a description of indecomposable morph-
isms in the category NT ∗. For the space X in question, these are the maps in the
following diagram:

(5.1)

14
i
�

i
■■■■■■■■ �■
124

i
❑❑❑❑❑❑❑❑ �❑
1

◦❊
❊❊
❊

δ

❊❊ �❊
❊

4

i
①①①①①①① �①
①
i
�

i

❋❋❋❋❋❋❋❋ �❋
24

i
✉✉✉✉✉✉✉ �✉
✉

i
■■■■■■■■ �■
134
i
� 1234

r
✉✉✉✉✉✉✉✉ �✉
r
�

r

■■■■■■■■■ �■
2
◦δ
� 4

34
i
�

i
✉✉✉✉✉✉✉✉ �✉
234

i
sssssssss �s
3

②② ◦②
②

δ
②② �②
②

Here we write i for the extension transformation for an open subset, r for the
restriction transformation for a closed subset, and δ for boundary maps in K-theory
long exact sequences.
The indecomposable morphisms in (5.1) provide a minimal set of generators for
the graded ring NT . To describe NT completely, we list the relations. These are
generated by the following:
• the cube with vertices 4, 14, . . . , 1234 is a commuting diagram, that is, all
the commuting squares involving arrows with label i commute;
• the following composite arrows vanish:

124
i−→ 1234
r−→ 3,
134
i−→ 1234
r−→ 2,
234
i−→ 1234
r−→ 1,

1
δ−→ 4
i−→ 14,
2
δ−→ 4
i−→ 24,
3
δ−→ 4
i−→ 34;

• the sum of the three maps 1234 → 4 via 1, 2, and 3 vanishes.


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
27

These relations imply that the diagrams

124
r
�

r
�

−

2

◦δ
�

1
◦δ
� 4

134
r
�

r
�

−

3

◦δ
�

1
◦δ
� 4

234
r
�

r
�

−

2

◦δ
�

3
◦δ
� 4

anti-commute and that the composite of two odd maps vanishes. It is routine to
check that the universal pre-additive category with these generators and relations
is given by the groups listed in Table 1.
Define NT nil and NT ss as in Definition 3.3: NT nil is the linear span of the
groups NT ∗(Y, Z) with Y ̸= Z and NT ss is spanned by the groups NT ∗(Y, Y ).
Then NT nil is a nilpotent ideal in NT and NT ss ∼= ZLC(X)∗ is a semi-simple
ring. Thus NT nil is the maximal nilpotent ideal in NT and we have a semi-direct
product decomposition NT ∼= NT nil ⋊ NT ss as in Lemma 3.4.
The next task is to describe the submodule M ′ := NT nil · M ⊆ M for an exact
NT -module M. The following computations are done as in the proof of Lemma 3.8,
using (3.4) and that the morphisms in (5.1) generate NT .

M ′(14) = range
�
i14
4 : M(4) → M(14)
�
= ker
�
r1
14 : M(14) → M(1)
�
,

and symmetrically for 24 and 34;

M ′(124) = range
�
i124
14 : M(14) → M(124)
�
+
�
i124
24 : M(24) → M(124)
�

= ker
�
δ4
124 : M(124) → M(4)
�
,

where δ4
124 denotes a generator of NT 1(124, 4) ∼= Z; symmetry provides M ′(134)
and M ′(234). We have

M ′(1) = range
�
r1
1234 : M(1234) → M(1)
�
= ker
�
δ234
1
: M(1) → M(234)
�
,

and symmetrically for 2 and 3, and

M ′(4) =

3
�

j=1
range
�
δ4
j : M(j) → M(4)
�
= ker
�
i1234
4
: M(4) → M(1234)
�
.

But something goes wrong with M ′(1234). Equation (3.4) yields

range
�
i1234
124 : M(124) → M(1234)
�
+
�
i1234
134 : M(134) → M(1234)
�

= ker
�
δ14
1234 : M(1234) → M(14)
�
;

to take into account the range of i1234
234 as well, we need an exact sequence containing
δ14
1234 ◦ i1234
234 , which is the generator of NT 1(234, 14) ∼= Z. Since there is no such
exact sequence, our method breaks down at this point.
Another symptom but not a cause of problems is that the map δ4
124 that describes
M ′(124) is not the longest map out of 124: that would be δ34
124.
As we shall see, the analogues of Theorems 3.12 and 4.9 become false for the
space X. First, there is a non-projective exact module M with free Mss; secondly,
there is a module that has no projective resolution of length 1; thirdly, there are
A, B ∈ B(X) with I2(A, B) ̸= 0.
Hence Theorem 4.7 provides non-isomorphic
objects in the bootstrap class B(X) with isomorphic filtrated K-theory. The con-
struction of these counterexamples follows the above pattern: first we find a counter-
example to Theorem 3.12, which we use to find one for Theorem 4.9, which is then
used to find an example as in Theorem 4.7.
We begin with the unexpected non-projective module. Let PY for Y ∈ LC(X)∗

denote the free NT -module on Y , that is,

PY (Z) = NT ∗(Y, Z),
HomNT (PY , N) ∼= N(Y )


28
RALF MEYER AND RYSZARD NEST

for any Y, Z ∈ LC(X)∗ and any NT -module N. A natural transformation FKY ⇒
FKZ corresponds to an element in NT ∗(Y, Z) ∼= PY (Z) ∼= HomNT (Pz, PY ) and
thus induces a module homomorphism PZ → PY in the opposite direction. Hence
the three arrows 124, 134, 234 → 1234 in (5.1) induce a module homomorphism

j : P1234 → P 0 := P124 ⊕ P134 ⊕ P234.

Table 1 shows that there are no module homomorphisms P 0 → P1234, that is, no
non-zero natural transformations from 1234 to 124, 134, or 234.
The crucial observation is that j is a monomorphism, so that P1234 becomes a
submodule of P 0. Since the longest natural transformations out of 1234 are those
to 14, 24 and 34, this follows from the elementary observations that the maps

NT ∗(1234, j4) → NT ∗(1234 \ j, j4)

for j = 1, 2, 3 are, respectively, the identity map on Z.
This follows from the
exactness of free modules because NT ∗(j, j4) = 0 by Table 1.
We describe the quotient

M := P 0/j(P1234)

by its values M(Y ) for Y ∈ LC(X)∗ as in (5.1):

(5.2)

0
i
�

i

❉❉❉❉❉❉❉❉ �❉
Z

i

❋❋❋❋❋❋❋❋ �❋
Z

◦●
●●●
●●●
●

δ
●●●●
●●●
●

Z[1]

i
✇✇✇✇✇✇✇ �✇
✇
i
�

i

●●●●●●●● �●
0

i
③③③③③③③ �③
③

i

❉❉❉❉❉❉❉❉ �❉
Z
i
� Z2

r
①①①①①①① �①
①
r
�

r

❋❋ �❋
❋
❋
❋❋
❋
❋
❋
Z
◦δ
Z[1]

0
i
�

i
③③③③③③③③ �③
Z

i
① �①
①
①
①
①①
①
①
①
Z

✇✇✇◦
✇ ✇
✇✇
✇

δ
✇✇✇
✇ ✇
✇✇
✇

The boundary maps δ act by isomorphisms on M because M(j4) = 0 for j =
1, 2, 3. The other maps can be understood by writing M(1234) = Z3/⟨(1, 1, 1)⟩ and
M(j) = Z2/⟨(1, 1)⟩ for j = 1, 2, 3 as quotients. The three maps Z → Z2 correspond
to the three coordinate embeddings Z ֌ Z3, the maps Z2 → Z to the projections
Z3 ։ Z2 onto coordinate hyperplanes.
The projective resolution

(5.3)
0 → P1234
j−→ P 0 ։ M

does not split because there exist no non-zero morphisms P 0 → P1234. Hence M
is not projective. But Mss is free, and M is exact because the exact modules form
an exact category and P1234 and P 0 are exact. Thus M is a counterexample to
Theorem 3.12.
The module M is directly related to the problem with describing NT nil·M(1234)
encountered above. Since HomNT (PY , N) ∼= N(Y ) for any NT -module N and any
Y ∈ LC(X)∗, the resolution (5.3) provides an exact sequence

0 → HomNT (M, N)

→ N(124) ⊕ N(134) ⊕ N(234) → N(1234) → Ext1
NT (M, N) → 0,

so that
Ext1
NT (M, N) ∼= N(1234)/NT nil · N(1234) ∼= Nss(1234).
Now we use M to construct a counterexample for Theorem 4.9. Let k ∈ N≥2
and let Mk := M/k · M; that is, we replace Z by Z/k everywhere in (5.2). This
module has a projective resolution of length 2 of the form

(5.4)
0 → P1234
(−k,j)
−−−−→ P1234 ⊕ P 0
(j,k)
−−−→ P 0 ։ Mk,


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
29

where k denotes multiplication by k. Using this resolution, we compute

Ext2(Mk, P1234) ∼= Z/k,
Ext1(Mk, P1234) ∼= Hom(Mk, P1234) ∼= 0

because there are no no-zero morphisms P 0 → P1234. Of course, the generator of
Ext2(Mk, P1234) is the class of the projective resolution (5.4). Hence Mk admits no
projective resolution of length 1 and is a counterexample to Theorem 4.9.
Now we claim that Mk is the filtrated K-theory of some C∗-algebra Ak over X in
the bootstrap class B(X). To begin with, M is the filtrated K-theory of some such
C∗-algebra A by Theorem 4.8. Let Bk be a C∗-algebra in the bootstrap class with
K0(Bk) = Z/k and K1(Bk) = 0; for instance, Bk could be the Cuntz algebra Ok+1.
Then Ak := A ⊗ Bk has filtrated K-theory Mk by the K¨unneth Theorem for the
K-theory of tensor products.

Theorem 5.1. Let Ak be a C∗-algebra in the bootstrap class with FK(Ak) ∼= Mk
as constructed above. Then Ak is not I2-projective. Hence there exist B, D ∈ B(X)
that are not KK(X)-equivalent but with the same filtrated K-theory.

Proof. The second assertion follows from the first one using Theorem 4.7 applied
to the bootstrap class B(X) and the restriction of I to B(X).
It remains to prove that Ak cannot be I2-projective. To see this, we lift the
resolution (5.4) to an I-projective resolution

0
◦
�P2
◦
�P1
◦
�P0
�Ak

in B(X) with boundary maps of degree 1, and embed the latter in a phantom tower
(see [6]):

Ak
N0
ι1
0
� N1
ι2
1
�

◦✠✠✠

�✠✠✠

N2
ι3
2
�

◦✠✠✠

�✠✠✠

N3

◦✠✠✠

�✠✠✠

N3

◦☛☛☛

�☛☛☛

· · ·

P0

π0

�✺✺✺✺✺✺
P1

π1

�✺✺✺✺✺✺
�
P2

π2

�✺✺✺✺✺✺
�
0

�✸✸✸✸✸✸
�
· · ·
�

The inductive system (Nj, ιj+1
j
) becomes constant at N3 because Pj = 0 for j ≥ 3.
Since Ak belongs to the bootstrap class, N3 ∼= 0 (see the proof of [6, Proposition
4.5]). This implies N2 ∼= P2.
The composite map ι2
0 : Ak = N0 → N2 ∼= P2 belongs to I2. Suppose that Ak
were I2-projective. Then ι2
0 = ι2
1 ◦ ι1
0 would vanish, and the long exact homology
sequence would yield that the map ι2
1 : N1 → N2 must factor through the map
N1 → P0. But

KK∗(X; P0, P2) ∼= HomNT
�
FK(P0), FK(P2)
�
= HomNT (P 0, P1234) = 0.

Here we have used that filtrated K-theory, by universality, is fully faithful on
I-projective objects and that there are no non-zero module homomorphisms P 0 →
P1234. Since ι2
1 factors through the zero group, it must be the zero map. But then
the map P1 → N1 must be a split surjection, so that N1 is I-projective. Then the
I-exact triangle ΣAk → ΣN1 → P0 → Ak provides an I-projective resolution of Ak
of length 1, which is impossible because FK(Ak) ∼= Mk has no projective resolution
of length 1. As a consequence, Ak is not I2-projective.
□

We can make the two non-equivalent C∗-algebras over X with the same filtrated
K-theory more explicit. One of them is Ak ⊕ ΣR1234, the other one is the mapping
cone of the map ι2
0 : Ak = N0 → N2 ∼= R1234 in the phantom tower above. Both
have Mk ⊕ P1234[1] as their filtrated K-theory.
This counterexample shows that filtrated K-theory does not yet classify purely
infinite stable nuclear separable C∗-algebras in the bootstrap class.


30
RALF MEYER AND RYSZARD NEST

Remark 5.2. Refining filtrated K-theory by taking filtrated K-theory with coeffi-
cients does not help. This gets rid of the counterexample Ak constructed above,
but other objects of B(X) without projective resolution of length 1 remain. An ex-
ample is A ⊗ B, where B is a C∗-algebra in the bootstrap class with K∗(B) = Q[0]
such as an appropriate UHF-algebra. Its filtrated K-theory is M ⊗ Q. This also
has cohomological dimension 2, and this is not affected much by taking K-theory
with coefficients because M ⊗ Q is torsion-free.

5.1. A refined invariant. There are at least two ways to identify the source of
the problem for the space X. The first point of view is that what is missing is an
exact sequence that has the generator α of NT 1(234, 14) as its connecting map.
The map α corresponds to a map ΣR14 → R234 between the representing objects,
which we also denote by α. In the triangulated category KK(X), we can embed the
latter map in an exact triangle

(5.5)
ΣR14
α−→ R234 → R12344 → R14.

The notation R12344 will be explained later. The functors these objects represent
sit in a long exact sequence

(5.6)
· · · → FK14 → FK12344 → FK234
α−→ FK14[1] → · · ·

which is precisely what we want. The second point of view is that the troublemaker
is the non-projective module M. Since M has a projective resolution of length 1,
there is a unique object in the bootstrap class with filtrated K-theory M. Actually,
this yields the same object as the first point of view:

Lemma 5.3. The non-projective module M above agrees with FK(R12344).

Proof. The map FKY (α) vanishes for almost all Y ∈ LC(X)∗ simply because the
graded groups involved have different parity or one of them vanishes. The only
exception is Y = 14. The group FK14(R14) = NT (14, 14) is generated by the
identity natural transformation. Since α is the generator of NT 1(234, 14), the map
FK14(α) is invertible.
Now we apply FK to the long exact sequence for the given exact triangle. Since
FK(α) vanishes on most Y and is invertible for Y = 14, we can easily compute the
groups FKY (R12344). We get the same groups as for the module M. It remains to
check that the isomorphism can be chosen as an NT -module homomorphism. The
main step is to check that the map

Z2 ∼= FK124(R12344) ⊕ FK134(R12344) → FK1234(R12344) ∼= Z2

is invertible. Together with the known relations between the various natural trans-
formations, this implies the assertion. We omit the details of this computation.
□

The representing object R12344 is an algebra of functions on a two-dimensional
simplicial complex, which we do not describe here because it is not illuminating.
The functor that it represents, however, can be described rather nicely as follows.
Let A be a C∗-algebra over X. Pull back the extension A(14) ֌ A(124) ։ A(2)
along the quotient map A(234) ։ A(2) to an extension A(14) ֌ A(12344) ։
A(234). The object R12344 represents the functor

(5.7)
KK∗(X; R12344, A) ∼= K∗
�
A(12344)
�
.

To see this, two observations are necessary. First, K∗
�
R12344(12344)
� ∼= Z; the
generator of this group yields a natural transformation between the two functors
in (5.7). Secondly, this natural transformation is invertible. This follows from the
Five Lemma, once we know that it extends the known natural isomorphisms

KK∗(X; RY , A) ∼= K∗
�
A(Y )
�


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
31

for Y = 14 and Y = 234 to a chain map between the long exact sequences that
we get from (5.5) and from the extension A(14) ֌ A(12344) ։ A(234).
This
extension also explains the notation R12344.
Now we augment filtrated K-theory by adding the covariant functor

B �→ FK12344(B) := K∗
�
A(12344)
� ∼= KK∗(X; R12344, B).

The new invariant takes values in the category of countable NT ′-modules, where
NT ′ is the Z/2-graded category whose object set is LC′ := LC(X)∗ ⊔ {12344} and
whose morphisms are the natural transformations between the various filtrated
K-groups, including now also FK12344. These natural transformations can be com-
puted by the Yoneda Lemma:

NT ′
∗(Y, Z) ∼= KK∗(X; RZ, RY ) ∼= FKZ(RY )

holds for all Y, Z ∈ LC′. The category ring for NT ′
∗ is simply the ring KK∗(X; R, R)
where
R :=
�

Y ∈LC′
RY .

We replace the ideal I in KK(X) studied above by the kernel I′ of the enriched
filtrated K-theory functor

FK′ : KK(X) → Mod(NT ′)c.

The same arguments as above show that there are enough I′-projective objects and
that FK′ is the universal I′-exact stable homological functor.
The passage from I to I′ has improved the situation because R12344 has now
been promoted to an I′-projective object and, therefore, ceases to cause trouble.
In principle, something similar can be done in great generality: whenever we have
an object of the Abelian approximation that has a projective resolution of length 1,
we can lift it uniquely to an object of the triangulated category and refine the
ideal by intersecting it with the kernel of the functor this lifted object represents.
However, the policy to quieten troublemakers by promotion has the tendency to
encourage new troublemakers, so that it is not clear whether this general strategy
always resolves all problems after finitely many steps. But in the relatively simple
example at hand, this turns out to be the case.
To check this, we must describe the category NT ′. If Y, Z ∈ LC(X)∗, then
NT ′
∗(Y, Z) = NT ∗(Y, Z) is given by the table on page 26. Furthermore, if Z ∈
LC(X)∗, then NT ′
∗(12344, Z) ∼= FKZ(R12344) = M(Z) by Lemma 5.3, and this is
described in (5.2). The upshot is:
• there are even natural transformations from FK12344 to FK124, FK134,
FK234—the generators of the respective groups of natural transformations—
such that any natural transformation FK12344 ⇒ FKZ with Z ∈ LC(X)∗

is a sum of natural transformations that factor through one of these three
and a natural transformation FKij4 ⇒ FKZ;
• the sum of the three natural transformations FK12344 ⇒ FK1234 via FK124,
FK134 and FK234 vanishes, and all other relations follow from these and
the already known ones listed after (5.1).
The exact triangle (5.5) yields a long exact sequence

· · · → NT ′
∗+1(Y, 234)
α−→ NT ′
∗(Y, 14) → NT ′
∗(Y, 12344) → NT ′
∗(Y, 234) → · · · ,

which we may use to compute NT ′
∗(Y, 12344) for all Y ∈ LC′. The map α induces
an isomorphism for Y = 234 and the zero map for all other Y because the source
and target have opposite parity or one of them vanishes. Thus

Y
4
14, 24, 34
124, 134, 234
1234
1, 2, 3
12344

NT ′
∗(Y, 12344)
Z2
Z
0
Z[1]
Z[1]
Z


32
RALF MEYER AND RYSZARD NEST

These groups inherit from M their invariance under permutations of 1, 2, 3. Inspect-
ing composition with natural transformations in NT , we arrive at the following:

• there are even natural transformations FKj4 ⇒ FK12344 for j = 1, 2, 3, such
that any natural transformation FKY ⇒ FK12344 with Y ∈ LC(X)∗ factors
through one of them;
• the sum of the three natural transformations FK4 ⇒ FK12344 vanishes,
• the natural transformations FKj4 ⇒ FK1234\j via FK12344 vanish;
• all other relations follow from these and the already known ones.

As one may expect, the basic natural transformations FK14 ⇒ FK12344 ⇒ FK234
are induced by the maps R234 → R12344 → R14 in the exact triangle (5.5).
The indecomposable morphisms of the new category NT ′ are the maps in the
following diagram:

14

❋❋❋❋❋❋❋ �❋
124

❋❋❋❋❋❋❋ �❋
1

◦❃❃❃❃

�❃❃❃❃

4

⑦ �⑦
⑦
⑦
⑦
⑦
⑦
⑦
�

❅❅ �❅
❅
❅❅
❅
24
� 12344
�

✈✈✈✈✈✈✈ �✈
✈

❍❍❍❍❍❍❍❍ �❍
134
� 1234

③③③ �③
③③③③
�

❉ �❉
❉
❉
❉
❉
❉
❉
❉
2
◦
� 4

34

① �①
①①
①①
①①
①
234

①①①①①①① �①
3

◦����

�����

The category ring of NT ′ again has the by now familiar structure: it is a split
nilpotent extension of the semisimple algebra NT ′
ss ∼= ZLC′ spanned by the identity
transformations on the objects and a nilpotent ideal NT ′
nil that is the subgroup
generated by NT ′(Y, Z) with Y ̸= Z.

Definition 5.4. A module over NT ′ is exact if it is exact as an NT -module and
the three sequences

· · · → N∗+1(ij4) → N∗(k4) → N∗(12344) → N∗(ij4) → · · ·

for {i, j, k} = {1, 2, 3} are exact as well.

The range of the invariant FK′ consists of exact NT ′-modules; the three new
exact sequences are, in fact, equivalent for symmetry reasons, and the extension

· · · → N∗+1(234) → N∗(14) → N∗(12344) → N∗(234) → · · ·

is built into the definition of FK12344.
Let N be an exact NT ′-module and let N ′ := NT ′
nil · N. The description of
N ′(14), N ′(1), and N ′(4) is the same as for the category NT , so that these groups
remain kernels of certain maps, as needed. Furthermore, N ′(1234) is the kernel of
the map N(1234) → N(12344)[1] induced by the generator of NT 1(1234, 12344),
so that the problem that appeared for the category NT is cured.
The computation of N ′(124) changes because this group is now the range of the
arrow N(12344) → N(124). But this is part of a long exact sequence because N is
exact, and we get
N ′(124) = ker
�
N(124) → N(34)[1]
�
,

and similarly for N ′(134) and N ′(234).
Finally, N ′(12344) is the sum of the ranges of the maps N(j4) → N(12344) for
j = 1, 2, 3. Using exactness, we identify this in two steps with the kernel of the
map N(12344) → N(4)[1] induced by the generator of NT ′
1(12344, 4).
As a result, the submodule NT ′
nil·N is described using kernels of maps N(Y ) →
N(Z). By the way, these arrows are the longest arrows starting at Y as in Re-
mark 3.9. The same arguments as for totally ordered spaces now show:


C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY
33

Theorem 5.5. An NT ′-module N is free if and only if it is projective, if and only
if it is exact and N(Y ) is a free group for all Y ∈ LC′.

Theorem 5.6. An NT ′-module N has a projective resolution of length 1 if and
only if it is exact.

Theorem 5.7. Let A and B be C∗-algebras over the four-point space X under
consideration.
If A belongs to the bootstrap class B(X), then there is a natural
short exact sequence

Ext1
NT ′
�
FK′(A)[1], FK′(B)
�
֌ KK∗(X; A, B) ։ HomNT ′�
FK′(A), FK′(B)
�
.

In particular, morphisms FK′(A) → FK′(B) lift to elements in KK∗(X; A, B). If
both A and B belong to the bootstrap class, then an isomorphism FK′(A) ∼= FK′(B)
lifts to a KK(X)-equivalence.

Corollary 5.8. The map A �→ FK′(A) is a bijection between the set of isomorphism
classes of tight, stable, purely infinite, separable, nuclear C∗-algebras over X with
simple subquotients in the bootstrap class and the set of isomorphism classes of
countable exact NT ′-modules.

6. Conclusion

We have obtained a Universal Coefficient Theorem that computes KK∗(X; A, B)
for A in the bootstrap class and X of a very special form, namely, {1, . . . , n} with
the Alexandrov topology from the total order. This Universal Coefficient Theorem
can be used to carry over classification results for simple, nuclear, purely infinite
C∗-algebras to nuclear, purely infinite C∗-algebras with primitive ideal space X,
using filtrated K-theory as the invariant.
For general finite topological spaces X, we still get a spectral sequence that
computes KK∗(X; A, B) using filtrated K-theory, but this spectral sequence need
not degenerate to an exact sequence, so that isomorphisms on filtrated K-theory
need not lift to X-equivariant KK-equivalences. In fact, we have found a counter-
example. At the same time, we were able to fix the counterexample by refining
filtrated K-theory. It is unclear whether such a refinement is available for all finite
topological spaces and how it looks like.

References

[1] Apostolos Beligiannis, Relative homological algebra and purity in triangulated categories, J.
Algebra 227 (2000), no. 1, 268–361, doi: 10.1006/jabr.1999.8237. MR 1754234
[2] Alexander
Bonkat,
Bivariante
K-Theorie
f¨ur
Kategorien
projektiver
Systeme
von
C∗-Algebren,
Ph.D.
Thesis,
Westf.
Wilhelms-Universit¨at
M¨unster,
2002,
http://deposit.ddb.de/cgi-bin/dokserv?idn=967387191 (German).
[3] J. Daniel Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv.
Math. 136 (1998), no. 2, 284–339, doi: 10.1006/aima.1998.1735. MR 1626856
[4] Samuel Eilenberg and John Coleman Moore, Foundations of relative homological algebra,
Mem. Amer. Math. Soc. No. 55 (1965), 39. MR 0178036
[5] Eberhard Kirchberg, Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation
nicht-einfacher Algebren, C∗-Algebras (M¨unster, 1999), Springer, Berlin, 2000, pp. 92–141
(German). MR 1796912
[6] Ralf Meyer, Homological algebra in bivariant K-theory and other triangulated categories. II,
Tbil. Math. J. 1 (2008), 165–210. MR 2563811
[7] Ralf Meyer and Ryszard Nest, The Baum–Connes conjecture via localisation of categories,
Topology 45 (2006), no. 2, 209–259, doi: 10.1016/j.top.2005.07.001. MR 2193334
[8]
, C∗-Algebras over topological spaces: the bootstrap class, M¨unster J. Math. 2 (2009),
215–252. MR 2545613
[9]
, Homological algebra in bivariant K-theory and other triangulated categories. I, Tri-
angulated categories (Thorsten Holm, Peter Jørgensen, and Rapha¨el Rouqier, eds.), London
Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 236–289.
MR 2681710


34
RALF MEYER AND RYSZARD NEST

[10] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton
University Press, Princeton, NJ, 2001. MR 1812507
[11] Gunnar Restorff, Classification of Cuntz–Krieger algebras up to stable isomorphism, J. Reine
Angew. Math. 598 (2006), 185–210, doi: 10.1515/CRELLE.2006.074. MR 2270572
[12]
, Classification of Non-Simple C∗-Algebras, Ph.D. Thesis, Københavns Universitet,
2008, http://www.math.ku.dk/~restorff/papers/afhandling_med_ISBN.pdf.
[13] Mikael Rørdam, Classification of extensions of certain C∗-algebras by their six term exact
sequences in K-theory, Math. Ann. 308 (1997), no. 1, 93–117, doi: 10.1007/s002080050067.
MR 1446202
[14] Steven Vickers, Topology via logic, Cambridge Tracts in Theoretical Computer Science, vol. 5,
Cambridge University Press, Cambridge, 1989. MR 1002193

Mathematisches Institut and Courant Research Centre “Higher Order Structures”,
Georg-August Universit¨at G¨ottingen, Bunsenstraße 3–5, 37073 G¨ottingen, Germany
E-mail address: rameyer@uni-math.gwdg.de

Københavns Universitets Institut for Matematiske Fag, Universitetsparken 5, 2100
København, Denmark
E-mail address: rnest@math.ku.dk