Keywords

3.1 Introduction

A far-reaching classification theorem in [7] motivates the computation of Eberhard Kirchberg’s ideal-related Kasparov groups K b@KK(X)-theory KK(X; A, B) for separable C ∗dash-algebras A and B over a non-Hausdorff topological space X by means of K-theoretic invariants. We are interested in the specific case of finite spaces here. In [10, 11], Ralf Meyer and Ryszard Nest laid out a theoretic framework that allows for a generalisation of Jonathan Rosenberg’s and Claude Schochet’s universal coefficient theorem [16] to the equivariant setting. Starting from a set of generators of the equivariant bootstrap class, they define a homology theory with a certain universality property, which computes KK(X)-theory via a spectral sequence. In order for this universal coefficient spectral sequence to degenerate to a short exact sequence, it remains to be checked by hand that objects in the range of the homology theory admit projective resolutions of length 1 in the Abelian target category.

Generalising earlier results from [3, 11, 15] the verification of the condition mentioned above for filtrated K-theory was achieved in [2] for the case that the underlying space is a disjoint union of so-called accordion spaces. A finite connected T 0-space X is an accordion space if and only if the directed graph corresponding to its specialisation pre-order is a Dynkin quiver of type A. Moreover, it was shown in [2, 11] that, if X is a finite T 0-space which is not a disjoint union of accordion spaces, then the projective dimension of filtrated K-theory over X is not bounded by 1 and objects in the equivariant bootstrap class are not classified by filtrated K-theory. The assumption of the separation axiom T 0 is not a loss of generality in this context (see [9, §2.5]).

There are two natural approaches to tackle the problem arising for non-accordion spaces: one can either try to refine the invariant—this has been done with some success in [11] and [1]; or one can hold onto the invariant and try to establish projective resolutions of length 1 on suitable subcategories or localisations of the category \(\mathfrak{K}\mathfrak{K}(X)\), in which X-equivariant KK-theory is organised. The latter is the course we pursue in this note. We state our results in the next section.

3.2 Statement of Results

The definition of filtrated K-theory and related notation are recalled in Sect. 3.3.

Proposition 1.

Let X be a finite topological space. Assume that the ideal \(\mathcal{N}\mathcal{T}_{\mathrm{nil}} \subset {\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) is nilpotent and that the decomposition \({\mathcal{N}\mathcal{T}}^{{\ast}}(X) = \mathcal{N}\mathcal{T}_{\mathrm{nil}} \rtimes \mathcal{N}\mathcal{T}_{\mathrm{ss}}\) holds. Fix \(n \in \mathbb{N}\) . For an \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) -module M, the following assertions are equivalent:

  1. 1.

    M has a projective resolution of length n.

  2. 2.

    The Abelian group \(\mathrm{Tor}_{n}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (X)}(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is free and the Abelian group \(\mathrm{Tor}_{n+1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (X)}(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) vanishes.

The basic idea of this paper is to compute the \(\mathrm{Tor}\)-groups above by writing down projective resolutions for the fixed right-module \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\).

Let Z m be the (m + 1)-point space on the set \(\{1,2,\ldots,m + 1\}\) such that Y ⊆ Z m is open if and only if Y ∋ m + 1 or \(Y = \emptyset\). A \({C}^{{\ast}}\) dash-algebra over \(Z_{m}\) is a \({C}^{{\ast}}\) dash-algebra \(A\) with a distinguished ideal such that the corresponding quotient decomposes as a direct sum of m orthogonal ideals. Let S be the set \(\{1,2,3,4\}\) equipped with the topology \(\{\emptyset,4,24,34,234,1234\}\), where we write \(24\mathop{:}=\{2,4\}\) etc. A C ∗dash-algebra over S is a C ∗dash-algebra together with two distinguished ideals which need not satisfy any further conditions; see [9, Lemma 2.35].

Proposition 2.

Let X be a topological space with at most  4 points. Let \(M =\mathrm{ FK}(A)\) for some C ∗ dash-algebra A over X. Then M has a projective resolution of length  2 and \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M) = 0\) .

Moreover, we can find explicit formulas for \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) ; for instance, \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (Z_{3})}(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is isomorphic to the homology of the complex

$$\displaystyle{ \bigoplus _{j=1}^{3}M(j4)\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i & - i&0\\ - i &0 &i \\ 0 &i & - i \end{array} \right )}\bigoplus _{k=1}^{3}M(1234\setminus k)\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i&i&i \end{array} \right )}M(1234)\;. }$$
(3.1)

A similar formula holds for the space S; see (3.6) .

The situation simplifies if we consider rational K b@KK(X)-theory KK(X)-theory, whose morphism groups are given by \(\mathrm{KK}(X;A,B) \otimes \mathbb{Q}\); see [6]. This is a \(\mathbb{Q}\)-linear triangulated category which can be constructed as a localisation of \(\mathfrak{K}\mathfrak{K}(X)\); the corresponding localisation of filtrated K-theory is given by \(A\mapsto \mathrm{FK}(A) \otimes \mathbb{Q}\) and takes values in the category of modules over the \(\mathbb{Q}\)-linear category \({\mathcal{N}\mathcal{T}}^{{\ast}}(X) \otimes \mathbb{Q}\).

Proposition 3.

Let X be a topological space with at most  4 points. Let A and B be C ∗ dash-algebras over X. If A belongs to the equivariant bootstrap class \(\mathcal{B}(X)\) , then there is a natural short exact universal coefficient sequence

$$\displaystyle\begin{array}{rcl} & & \mathrm{Ext}_{{\mathcal{N}\mathcal{T}}^{{\ast}}(X)\otimes \mathbb{Q}}^{1}{\bigl (\mathrm{FK}_{ {\ast}+1}(A) \otimes \mathbb{Q},\mathrm{FK}_{{\ast}}(B) \otimes \mathbb{Q}\bigr )} \rightarrowtail \mathrm{ KK}_{{\ast}}(X;A,B) \otimes \mathbb{Q} {}\\ & & \qquad \qquad \qquad \qquad \twoheadrightarrow \mathrm{Hom}_{{\mathcal{N}\mathcal{T}}^{{\ast}}(X)\otimes \mathbb{Q}}{\bigl (\mathrm{FK}_{{\ast}}(A) \otimes \mathbb{Q},\mathrm{FK}_{{\ast}}(B) \otimes \mathbb{Q}\bigr )}\;. {}\\ \end{array}$$

In [6], a long exact sequence is constructed which in our setting, by the above proposition, reduces the computation of KK∗(X; A, B), up to extension problems, to the computation of a certain torsion theory \(\mathrm{KK}_{{\ast}}(X;A,B; \mathbb{Q}/\mathbb{Z})\).

The next proposition says that the upper bound of 2 for the projective dimension in Proposition 2 does not hold for all finite spaces.

Proposition 4.

There is an \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{4})\) -module M of projective dimension  2 with free entries and \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\neq 0\) . The module \(M \otimes _{\mathbb{Z}}\mathbb{Z}/k\) has projective dimension  3 for every \(k \in \mathbb{N}_{\geq 2}\) . Both M and \(M \otimes _{\mathbb{Z}}\mathbb{Z}/k\) can be realised as the filtrated K-theory of an object in the equivariant bootstrap class  \(\mathcal{B}(X)\) .

As an application of Proposition 2 we investigate in Sect. 3.10 the obstruction term \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }{\bigl (\mathcal{N}\mathcal{T}_{\mathrm{ss}},\mathrm{FK}(A)\bigr )}\) for certain Cuntz-Krieger algebras with four-point primitive ideal spaces. We find:

Proposition 5.

There is a Cuntz-Krieger algebra with primitive ideal space homeomorphic to Z 3 which fulfills Cuntz’s condition (II) and has projective dimension  2 in filtrated K-theory over Z 3 . The analogous statement for the space S holds as well.

The relevance of this observation lies in the following: if Cuntz-Krieger algebras had projective dimension at most 1 in filtrated K-theory over their primitive ideal space, this would lead to a strengthened version of Gunnar Restorff’s classification result [14] with a proof avoiding reference to results from symbolic dynamics.

3.3 Preliminaries

Let X be a finite topological space. A subset Y ⊆ X is called locally closed if it is the difference U ∖ V of two open subsets U and V of X; in this case, U and V can always be chosen such that V ⊆ U. The set of locally closed subsets of X is denoted by \(\mathbb{L}\mathbb{C}(X)\). By \(\mathbb{L}\mathbb{C}{(X)}^{{\ast}}\), we denote the set of non-empty, connected locally closed subsets of X.

Recall from [9] that a C ∗dash-algebra over \(X\) is pair \((A,\psi )\) consisting of a C ∗dash-algebra \(A\) and a continuous map \(\psi: \mathrm{Prim}(A) \rightarrow X\). A C ∗dash-algebra \((A,\psi )\) over \(X\) is called tight if the map ψ is a homeomorphism. A C ∗dash-algebra \((A,\psi )\) over \(X\) comes with distinguished subquotients A(Y ) for every \(Y \in \mathbb{L}\mathbb{C}(X)\).

There is an appropriate version KK(X) of K b@KK(X)-theory bivariant K-theory for C ∗dash-algebras over X (see [7, 9]). The corresponding category, denoted by \(\mathfrak{K}\mathfrak{K}(X)\), is equipped with the structure of a triangulated category (see [12]); moreover, there is an equivariant analogue \(\mathcal{B}(X) \subseteq \mathfrak{K}\mathfrak{K}(X)\) of the bootstrap class [9].

Recall that a triangulated category comes with a class of distinguished candidate triangles. An anti-distinguished triangle is a candidate triangle which can be obtained from a distinguished triangle by reversing the sign of one of its three morphisms. Both distinguished and anti-distinguished triangles induce long exact \(\mathrm{Hom}\)-sequences.

As defined in [11], for \(Y \in \mathbb{L}\mathbb{C}(X)\), we let \(\mathrm{FK}_{Y }(A)\mathop{:}=\mathrm{K}_{{\ast}}{\bigl (A(Y )\bigr )}\) denote the \(\mathbb{Z}/2\)-graded K-group of the subquotient of A associated to Y. Let \(\mathcal{N}\mathcal{T} (X)\) be the \(\mathbb{Z}/2\)-graded pre-additive category whose object set is \(\mathbb{L}\mathbb{C}(X)\) and whose space of morphisms from \(Y\) to \(Z\) is \(\mathcal{N}\mathcal{T}_{{\ast}}(X)(Y,Z)\)—the \(\mathbb{Z}/2\)-graded Abelian group of all natural transformations \(\mathrm{FK}_{Y } \Rightarrow \mathrm{ FK}_{Z}\). Let \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) be the full subcategory with object set \(\mathbb{L}\mathbb{C}{(X)}^{{\ast}}\). We often abbreviate \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) by \({\mathcal{N}\mathcal{T}}^{{\ast}}\).

Every open subset of a locally closed subset of X gives rise to an extension of distinguished subquotients. The corresponding natural maps in the associated six-term exact sequence yield morphisms in the category \(\mathcal{N}\mathcal{T}\), which we briefly denote by i, r and δ.

A (left-)module over \(\mathcal{N}\mathcal{T} (X)\) is a grading-preserving, additive functor from \(\mathcal{N}\mathcal{T} (X)\) to the category \({\mathfrak{A}\mathfrak{b}}^{\mathbb{Z}/2}\) of \(\mathbb{Z}/2\)-graded Abelian groups. A morphism of \(\mathcal{N}\mathcal{T} (X)\)-modules is a natural transformation of functors. Similarly, we define left-modules over \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\). By \(\mathfrak{M}\mathfrak{o}\mathfrak{d}{\bigl ({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\bigr )}_{\mathrm{c}}\) we denote the category of countable \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\)-modules.

Filtrated K-theory is the functor \(\mathfrak{K}\mathfrak{K}(X) \rightarrow \mathfrak{M}\mathfrak{o}\mathfrak{d}{\bigl ({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\bigr )}_{\mathrm{c}}\) which takes a C ∗dash-algebra A over X to the collection \({\bigl (\mathrm{K}_{{\ast}}(A(Y ))\bigr )}_{Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}}\) equipped with the obvious \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\)-module structure.

Let \(\mathcal{N}\mathcal{T}_{\mathrm{nil}} \subset {\mathcal{N}\mathcal{T}}^{{\ast}}\) be the ideal generated by all natural transformations between different objects, and let \(\mathcal{N}\mathcal{T}_{\mathrm{ss}} \subset {\mathcal{N}\mathcal{T}}^{{\ast}}\) be the subgroup spanned by the identity transformations id Y Y for objects \(Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\). The subgroup \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\) is in fact a subring of \({\mathcal{N}\mathcal{T}}^{{\ast}}\) isomorphic to \({\mathbb{Z}}^{\mathbb{L}\mathbb{C}{(X)}^{{\ast}} }\). We say that \({\mathcal{N}\mathcal{T}}^{{\ast}}\) decomposes as semi-direct product \({\mathcal{N}\mathcal{T}}^{{\ast}} = \mathcal{N}\mathcal{T}_{\mathrm{nil}} \rtimes \mathcal{N}\mathcal{T}_{\mathrm{ss}}\) if \({\mathcal{N}\mathcal{T}}^{{\ast}}\) as an Abelian group is the inner direct sum of \(\mathcal{N}\mathcal{T}_{\mathrm{nil}}\) and \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\); see [2, 11]. We do not know if this fails for any finite space.

We define right-modules over \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) as contravariant, grading-preserving, additive functors \({\mathcal{N}\mathcal{T}}^{{\ast}}(X) \rightarrow { \mathfrak{A}\mathfrak{b}}^{\mathbb{Z}/2}\). If we do not specify between left and right, then we always mean left-modules. The subring \(\mathcal{N}\mathcal{T}_{\mathrm{ss}} \subset {\mathcal{N}\mathcal{T}}^{{\ast}}\) is regarded as an \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-right-module by the obvious action: The ideal \(\mathcal{N}\mathcal{T}_{\mathrm{nil}} \subset {\mathcal{N}\mathcal{T}}^{{\ast}}\) acts trivially, while \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\) acts via right-multiplication in \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\mathop{\cong}{\mathbb{Z}}^{\mathbb{L}\mathbb{C}{(X)}^{{\ast}} }\). For an \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-module M, we set \(M_{\mathrm{ss}}\mathop{:}=M/\mathcal{N}\mathcal{T}_{\mathrm{nil}} \cdot M\).

For \(Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\) we define the free \({\mathcal{N}\mathcal{T}}^{{\ast}}\) -left-module on Y by \(P_{Y }(Z)\mathop{:}=\mathcal{N}\mathcal{T} (Y,Z)\) for all \(Z \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\) and similarly for morphisms Z → Z′ in \({\mathcal{N}\mathcal{T}}^{{\ast}}\). Analogously, we define the free \({\mathcal{N}\mathcal{T}}^{{\ast}}\) -right-module on Y by \(Q_{Y }(Z)\mathop{:}=\mathcal{N}\mathcal{T} (Z,Y )\) for all \(Z \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\). An \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-left/right-module is called free if it is isomorphic to a direct sum of degree-shifted free left/right-modules on objects \(Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\). It follows directly from Yoneda’s Lemma that free \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-left/right-modules are projective.

An \(\mathcal{N}\mathcal{T}\)-module M is called exact if the \(\mathbb{Z}/2\)-graded chain complexes

$$\displaystyle{\cdots \rightarrow M(U)\mathop{\longrightarrow}\limits_{}^{i_{U}^{Y }}M(Y )\mathop{\longrightarrow}\limits_{}^{r_{ Y }^{Y \setminus U}}M(Y \setminus U)\mathop{\longrightarrow}\limits_{}^{\delta _{ Y \setminus U}^{U}}M(U)[1] \rightarrow \cdots }$$

are exact for all \(U,Y \in \mathbb{L}\mathbb{C}(X)\) with U open in Y. An \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-module M is called exact if the corresponding \(\mathcal{N}\mathcal{T}\)-module is exact (see [2]).

We use the notation \(C\in \in \mathcal{C}\) to denote that C is an object in a category \(\mathcal{C}\).

In [11], the functors FK Y are shown to be representable, that is, there are objects \(\mathcal{R}_{Y }\in \in \mathfrak{K}\mathfrak{K}(X)\) and isomorphisms of functors \(\mathrm{FK}_{Y }\mathop{\cong}\mathrm{KK}_{{\ast}}(X;\mathcal{R}_{Y },\text{})\). We let \(\widehat{\mathrm{FK}}\) denote the stable cohomological functor on \(\mathfrak{K}\mathfrak{K}(X)\) represented by the same set of objects \(\{\mathcal{R}_{Y }\mid Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\}\); it takes values in \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-right-modules. We warn that \(\mathrm{KK}_{{\ast}}(X;A,\mathcal{R}_{Y })\) does not identify with the K-homology of A(Y ). By Yoneda’s lemma, we have \(\mathrm{FK}(\mathcal{R}_{Y })\mathop{\cong}P_{Y }\) and \(\widehat{\mathrm{FK}}(\mathcal{R}_{Y })\mathop{\cong}Q_{Y }\).

We occasionally use terminology from [10, 11] concerning homological algebra in \(\mathfrak{K}\mathfrak{K}(X)\) relative to the ideal \(\mathfrak{I}\mathop{:}=\ker (\mathrm{FK})\) of morphisms in \(\mathfrak{K}\mathfrak{K}(X)\) inducing trivial module maps on FK. An object \(A\in \in \mathfrak{K}\mathfrak{K}(X)\) is called \(\mathfrak{I}\) -projective if \(\mathfrak{I}(A,B) = 0\) for every \(B\in \in \mathfrak{K}\mathfrak{K}(X)\). We recall from [10] that FK restricts to an equivalence of categories between the subcategories of \(\mathfrak{I}\)-projective objects in \(\mathfrak{K}\mathfrak{K}(X)\) and of projective objects in \(\mathfrak{M}\mathfrak{o}\mathfrak{d}{\bigl ({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\bigr )}_{\mathrm{c}}\). Similarly, the functor \(\widehat{\mathrm{FK}}\) induces a contravariant equivalence between the \(\mathfrak{I}\)-projective objects in \(\mathfrak{K}\mathfrak{K}(X)\) and projective \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-right-modules.

3.4 Proof of Proposition 1

Recall the following result from [11].

Lemma 1 ([11, Theorem 3.12]).

Let X be a finite topological space. Assume that the ideal \(\mathcal{N}\mathcal{T}_{\mathrm{nil}} \subset {\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) is nilpotent and that the decomposition \({\mathcal{N}\mathcal{T}}^{{\ast}}(X) = \mathcal{N}\mathcal{T}_{\mathrm{nil}} \rtimes \mathcal{N}\mathcal{T}_{\mathrm{ss}}\) holds. Let M be an \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) -module. The following assertions are equivalent:

  1. 1.

    M is a free \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) -module.

  2. 2.

    M is a projective \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\) -module.

  3. 3.

    M ss is a free Abelian group and \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (X)}(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M) = 0\) .

Now we prove Proposition 1. We consider the case n = 1 first. Choose an epimorphism \(f: P \twoheadrightarrow M\) for some projective module P, and let K be its kernel. M has a projective resolution of length 1 if and only if K is projective. By Lemma 1, this is equivalent to K ss being a free Abelian group and \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},K) = 0\). We have \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},K) = 0\) if and only if \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M) = 0\) because these groups are isomorphic. We will show that K ss is free if and only if \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is free. The extension \(K \rightarrowtail P \twoheadrightarrow M\) induces the following long exact sequence:

$$\displaystyle{ 0 \rightarrow \mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M) \rightarrow K_{\mathrm{ss}} \rightarrow P_{\mathrm{ss}} \rightarrow M_{\mathrm{ss}} \rightarrow 0\;. }$$

Assume that K ss is free. Then its subgroup \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is free as well. Conversely, if \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is free, then K ss is an extension of free Abelian groups and thus free. Notice that P ss is free because P is projective. The general case \(n \in \mathbb{N}\) follows by induction using an argument based on syzygies as above. This completes the proof of Proposition 1.

3.5 Free Resolutions for \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\)

The \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-right-module \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}\) decomposes as a direct sum \(\bigoplus _{Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}}S_{Y }\) of the simple submodules S Y which are given by \(S_{Y }(Y )\mathop{\cong}\mathbb{Z}\) and S Y (Z) = 0 for Z ≠ Y. We obtain

$$\displaystyle{\mathrm{Tor}_{n}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M) =\bigoplus _{Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}}\mathrm{Tor}_{n}^{\mathcal{N}\mathcal{T}}(S_{ Y },M)\;.}$$

Our task is then to write down projective resolutions for the \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-right-modules S Y . The first step is easy: we map Q Y onto S Y by mapping the class of the identity in Q Y (Y ) to the generator of S Y (Y ). Extended by zero, this yields an epimorphism \(Q_{Y } \twoheadrightarrow S_{Y }\).

In order to surject onto the kernel of this epimorphism, we use the indecomposable transformations in \({\mathcal{N}\mathcal{T}}^{{\ast}}\) whose range is Y. Denoting these by \(\eta _{i}: W_{i} \rightarrow Y\), 1 ≤ i ≤ n, we obtain the two step resolution

$$\displaystyle{\bigoplus _{i=1}^{n}Q_{ W_{i}}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{llll} \eta _{1} & \eta _{2} & \cdots &\eta _{n} \end{array} \right )}Q_{Y } \twoheadrightarrow S_{Y }\;.}$$

In the notation of [11], the map \(\bigoplus _{i=1}^{n}Q_{W_{i}} \rightarrow Q_{Y }\) corresponds to a morphism \(\phi: \mathcal{R}_{Y } \rightarrow \bigoplus _{i=1}^{n}\mathcal{R}_{W_{i}}\) of \(\mathfrak{I}\)-projectives in \(\mathfrak{K}\mathfrak{K}(X)\). If the mapping cone C ϕ of ϕ is again \(\mathfrak{I}\)-projective, the distinguished triangle \(\varSigma C_{\phi } \rightarrow \mathcal{R}_{Y }\mathop{\longrightarrow}\limits_{}^{\phi }\bigoplus _{i=1}^{n}\mathcal{R}_{W_{i}} \rightarrow C_{\phi }\) yields the projective resolution

$$\displaystyle{\cdots \rightarrow Q_{Y } \rightarrow Q_{\phi }[1] \rightarrow \bigoplus _{i=1}^{n}Q_{ W_{i}}[1] \rightarrow Q_{Y }[1] \rightarrow Q_{\phi } \rightarrow \bigoplus _{i=1}^{n}Q_{ W_{i}} \rightarrow Q_{Y } \twoheadrightarrow S_{Y }\;,}$$

where \(Q_{\phi } =\mathrm{ FK}(C_{\phi })\). We denote periodic resolutions like this by

If the mapping cone C ϕ is not \(\mathfrak{I}\)-projective, the situation has to be investigated individually. We will see examples of this in Sects. 3.7 and 3.9. The resolutions we construct in these cases exhibit a certain six-term periodicity as well. However, they begin with a finite number of “non-periodic steps” (one in Sect. 3.7 and two in Sect. 3.9), which can be considered as a symptom of the deficiency of the invariant filtrated K-theory over non-accordion spaces from the homological viewpoint. We remark without proof that the mapping cone of the morphism \(\phi: \mathcal{R}_{Y } \rightarrow \bigoplus _{i=1}^{n}\mathcal{R}_{W_{i}}\) is \(\mathfrak{I}\)-projective for every \(Y \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\) if and only if X is a disjoint union of accordion spaces.

3.6 Tensor Products with Free Right-Modules

Lemma 2.

Let M be an \({\mathcal{N}\mathcal{T}}^{{\ast}}\) -left-module. There is an isomorphism \(Q_{Y } \otimes _{{\mathcal{N}\mathcal{T}}^{{\ast}}}M\mathop{\cong}M(Y )\) of \(\mathbb{Z}/2\) -graded Abelian groups which is natural in \(Y \in \in {\mathcal{N}\mathcal{T}}^{{\ast}}\) .

Proof.

This is a simple consequence of Yoneda’s lemma and the tensor-hom adjunction.

Lemma 3.

Let

$$\displaystyle\begin{array}{rcl} \varSigma \mathcal{R}_{(3)}\mathop{\longrightarrow}\limits_{}^{\gamma }\mathcal{R}_{(1)}\mathop{\longrightarrow}\limits_{}^{\alpha }\mathcal{R}_{(2)}\mathop{\longrightarrow}\limits_{}^{\beta }\mathcal{R}_{(3)}& & {}\\ \end{array}$$

be a distinguished or anti-distinguished triangle in \(\mathfrak{K}\mathfrak{K}(X)\) , where

$$\displaystyle{\mathcal{R}_{(i)} =\bigoplus _{ j=1}^{m_{i} }\mathcal{R}_{Y _{j}^{i}} \oplus \bigoplus _{k=1}^{n_{i} }\varSigma \mathcal{R}_{Z_{k}^{i}}}$$

for 1 ≤ i ≤ 3, \(m_{i},n_{i} \in \mathbb{N}\) and \(Y _{j}^{i},Z_{k}^{i} \in \mathbb{L}\mathbb{C}{(X)}^{{\ast}}\) . Set \(Q_{(i)} =\widehat{\mathrm{ FK}}(\mathcal{R}_{(i)})\) . If M = FK (A) for some \(A\in \in \mathfrak{K}\mathfrak{K}(X)\) , then the induced sequence

(3.2)

is exact.

Proof.

Using the previous lemma and the representability theorem, we naturally identify \(Q_{(i)} \otimes _{{\mathcal{N}\mathcal{T}}^{{\ast}}}M\mathop{\cong}\mathrm{KK}_{{\ast}}(X;\mathcal{R}_{(i)},A)\). Since, in triangulated categories, distinguished or anti-distinguished triangles induce long exact \(\mathrm{Hom}\)-sequences, the sequence (3.2) is thus exact.

3.7 Proof of Proposition 2

We may restrict to connected T 0-spaces. In [9], a list of isomorphism classes of connected T 0-spaces with three or four points is given. If X is a disjoint union of accordion spaces, then the assertion follows from [2]. The remaining spaces fall into two classes:

  1. 1.

    All connected non-accordion four-point T 0-spaces except for the pseudocircle;

  2. 2.

    The pseudocircle (see Sect. 3.7.2).

The spaces in the first class have the following in common: If we fix two of them, say X, Y, then there is an ungraded isomorphism \(\varPhi:{ \mathcal{N}\mathcal{T}}^{{\ast}}(X) \rightarrow {\mathcal{N}\mathcal{T}}^{{\ast}}(Y )\) between the categories of natural transformations on the respective filtrated K-theories such that the induced equivalence of ungraded module categories

$$\displaystyle{{\varPhi }^{{\ast}}:{ \mathfrak{M}\mathfrak{o}\mathfrak{d}}^{\mathrm{ungr}}{\bigl ({\mathcal{N}\mathcal{T}}^{{\ast}}(Y )\bigr )}_{\mathrm{ c}} \rightarrow { \mathfrak{M}\mathfrak{o}\mathfrak{d}}^{\mathrm{ungr}}{\bigl ({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\bigr )}_{\mathrm{ c}}}$$

restricts to a bijective correspondence between exact ungraded \({\mathcal{N}\mathcal{T}}^{{\ast}}(Y )\)-modules and exact ungraded \({\mathcal{N}\mathcal{T}}^{{\ast}}(X)\)-modules. Moreover, the isomorphism Φ restricts to an isomorphism from \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}(X)\) onto \(\mathcal{N}\mathcal{T}_{\mathrm{ss}}(Y )\) and one from \(\mathcal{N}\mathcal{T}_{\mathrm{nil}}(X)\) onto \(\mathcal{N}\mathcal{T}_{\mathrm{nil}}(Y )\). In particular, the assertion holds for X if and only if it holds for Y.

The above is a consequence of the investigations in [1, 2, 11]; the same kind of relation was found in [2] for the categories of natural transformations associated to accordion spaces with the same number of points. As a consequence, it suffices to verify the assertion for one representative of the first class—we choose Z 3—and for the pseudocircle.

3.7.1 Resolutions for the Space Z 3

We refer to [11] for a description of the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{3})\), which in particular implies, that the space Z 3 satisfies the conditions of Proposition 1. Using the extension triangles from [11, (2.5)], the procedure described in Sect. 3.5 yields the following projective resolutions induced by distinguished triangles as in Lemma 3:

Next we will deal with the modules S jk4, where 1 ≤ j < k ≤ 3. We observe that there is a Mayer-Vietoris type exact sequence of the form

(3.3)

Lemma 4.

The candidate triangle \(\varSigma \mathcal{R}_{4} \rightarrow \mathcal{R}_{jk4} \rightarrow \mathcal{R}_{j4} \oplus \mathcal{R}_{k4} \rightarrow \mathcal{R}_{4}\) corresponding to the periodic part of the sequence (3.3) is distinguished or anti-distinguished ( depending on the choice of signs for the maps in (3.3) ) .

Proof.

We give the proof for j = 1 and k = 2. The other cases follow from cyclicly permuting the indices 1, 2 and 3. We denote the morphism \(\mathcal{R}_{124} \rightarrow \mathcal{R}_{14} \oplus \mathcal{R}_{24}\) by \(\varphi\) and the corresponding map \(Q_{14} \oplus Q_{24} \rightarrow Q_{124}\) in (3.3) by \({\varphi }^{{\ast}}\). It suffices to check that \(\widehat{\mathrm{FK}}(\mathrm{Cone}_{\varphi })\) and Q 4 correspond, possibly up to a sign, to the same element in \(\mathrm{Ext}_{{\mathcal{N}\mathcal{T}}^{{\ast}}{(Z_{3})}^{\mathrm{op}}}^{1}{\bigl (\mathrm{ker}{(\varphi }^{{\ast}}),\mathrm{coker}{(\varphi }^{{\ast}})[1]\bigr )}\). We have \(\mathrm{coker}{(\varphi }^{{\ast}})\mathop{\cong}S_{124}\) and an extension \(S_{124}[1] \rightarrowtail Q_{4} \twoheadrightarrow \mathrm{ker}{(\varphi }^{{\ast}})\). Since \(\mathrm{Hom}(Q_{4},S_{124}[1])\mathop{\cong}S_{124}(4)[1] = 0\) and \({\mathrm{Ext}}^{1}(Q_{4},S_{124}[1]) = 0\) because Q 4 is projective, the long exact \(\mathrm{Ext}\)-sequence yields \({\mathrm{Ext}}^{1}{\bigl (\mathrm{ker}{(\varphi }^{{\ast}}),\mathrm{coker}{(\varphi }^{{\ast}})[1]\bigr )}\mathop{\cong}\mathrm{Hom}(S_{124}[1],S_{124}[1])\mathop{\cong}\mathbb{Z}\). Considering the sequence of transformations \(3\mathop{\longrightarrow}\limits_{}^{\delta }124\mathop{\longrightarrow}\limits_{}^{i}1234\mathop{\longrightarrow}\limits_{}^{r}3\), it is straight-forward to check that such an extension corresponds to one of the generators \(\pm 1 \in \mathbb{Z}\) if and only if its underlying module is exact. This concludes the proof because both \(\widehat{\mathrm{FK}}(\mathrm{Cone}_{\varphi })\) and Q 4 are exact.

Hence we obtain the following projective resolutions induced by distinguished or anti-distinguished triangles as in Lemma 3:

To summarize, by Lemma 3, \(\mathrm{Tor}_{n}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{Y },M) = 0\) for Y ≠ 1234 and n ≥ 1.

As we know from [11], the subset 1234 of Z 3 plays an exceptional role. In the notation of [11] (with the direction of the arrows reversed because we are dealing with right-modules), the kernel of the homomorphism \(Q_{124}\oplus Q_{134}\oplus Q_{234}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i&i&i \end{array} \right )}Q_{1234}\) is of the form

It is the image of the module homomorphism

$$\displaystyle\begin{array}{rcl} Q_{14} \oplus Q_{24} \oplus Q_{34}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{llll} i & - i&0 \\ - i&0 &i \\ 0 &i & - i \end{array} \right )}Q_{124} \oplus Q_{134} \oplus Q_{234},& &{}\end{array}$$
(3.4)

the kernel of which, in turn, is of the form

A surjection from \(Q_{4} \oplus Q_{1234}[1]\) onto this module is given by \(\left (\begin{array}{lll} i &i &i\\ \delta _{ 1234}^{14} & 0&0 \end{array} \right )\), where \(\delta _{1234}^{14}\mathop{:}=\delta _{3}^{14} \circ r_{1234}^{3}\). The kernel of this homomorphism has the form

This module is isomorphic to Syz1234[1], where \(\mathrm{Syz}_{1234}\mathop{:}=\ker (Q_{1234} \twoheadrightarrow S_{1234})\). Therefore, we end up with the projective resolution

(3.5)

The homomorphism from \(Q_{124} \oplus Q_{134} \oplus Q_{234}\) to \(Q_{4} \oplus Q_{1234}[1]\) is given by

$$\displaystyle{\left (\begin{array}{llll} 0&0& -\delta _{234}^{4} \\ i &i &i \end{array} \right ),}$$

where \(\delta _{234}^{4}\mathop{:}=\delta _{2}^{4} \circ r_{234}^{2}\).

Lemma 5.

The candidate triangle in \(\mathfrak{K}\mathfrak{K}(X)\) corresponding to the periodic part of the sequence (3.5) is distinguished or anti-distinguished ( depending on the choice of signs for the maps in (3.5) ) .

Proof.

The argument is analogous to the one in the proof of Lemma 4. Again, we consider the group \(\mathrm{Ext}_{{\mathcal{N}\mathcal{T}}^{{\ast}}{(Z_{3})}^{\mathrm{op}}}^{1}{\bigl (\mathrm{ker}{(\varphi }^{{\ast}}),\mathrm{coker}{(\varphi }^{{\ast}})[1]\bigr )}\) where \({\varphi }^{{\ast}}\) now denotes the map (3.4). We have \(\mathrm{coker}{(\varphi }^{{\ast}})\mathop{\cong}\mathrm{Syz}_{1234}\) and an extension \(Q_{4} \rightarrowtail \mathrm{ker}{(\varphi }^{{\ast}}) \twoheadrightarrow S_{1234}[1]\). Using long exact sequences, we obtain

$$\displaystyle\begin{array}{rcl}{ \mathrm{Ext}}^{1}{\bigl (\mathrm{ker}{(\varphi }^{{\ast}}),\mathrm{coker}{(\varphi }^{{\ast}})[1]\bigr )}& & \mathop{\cong}{\mathrm{Ext}}^{1}(S_{ 1234}[1],\mathrm{Syz}_{1234}[1]) {}\\ & & \qquad \qquad \qquad \mathop{\cong}\mathrm{Hom}(S_{1234}[1],S_{1234}[1])\mathop{\cong}\mathbb{Z}. {}\\ \end{array}$$

Again, an extension corresponds to a generator if and only if its underlying module is exact.

By the previous lemma and Sect. 3.6, computing the tensor product of this complex with M and taking homology shows that \(\mathrm{Tor}_{n}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M) = 0\) for n ≥ 2 and that \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is equal to \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{1234},M)\) and isomorphic to the homology of the complex (3.1).

Example 1.

For the filtrated K-module with projective dimension 2 constructed in [11, §5] we get \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\mathop{\cong}\mathbb{Z}/k\).

Remark 1.

As explicated in the beginning of this section, the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(S)\) corresponding to the four-point space S defined in the introduction is isomorphic in an appropriate sense to the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{3})\). As has been established in [1], the indecomposable morphisms in \({\mathcal{N}\mathcal{T}}^{{\ast}}(S)\) are organised in the diagram

In analogy to (3.1), we have that \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (S)}(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\) is isomorphic to the homology of the complex

$$\displaystyle\begin{array}{rcl} M(12)[1] \oplus M(4) \oplus M(13)[1]\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} \delta & - r&0 \\ - i&0 &i\\ 0 &r &-\delta \end{array} \right )}M(34) \oplus M(1)[1] \oplus M(24)& & \\ \mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i&\delta &i \end{array} \right )}M(234)\;,& &{}\end{array}$$
(3.6)

where M = FK(A) for some separable C ∗dash- algebra A over X.

3.7.2 Resolutions for the Pseudocircle

Let \(C_{2} =\{ 1,2,3,4\}\) with the partial order defined by 1 < 3, 1 < 4, 2 < 3, 2 < 4. The topology on C 2 is thus given by \(\{\emptyset,3,4,34,134,234,1234\}\). Hence the non-empty, connected, locally closed subsets are

$$\displaystyle{\mathbb{L}\mathbb{C}{(C_{2})}^{{\ast}} =\{ 3,4,134,234,1234,13,14,23,24,124,123,1,2\}\;.}$$

The partial order on C 2 corresponds to the directed graph

The space C 2 is the only T 0-space with at most four points with the property that its order complex (see [11, Definition 2.6]) is not contractible; in fact, it is homeomorphic to the circle \({\mathbb{S}}^{1}\). Therefore, by the representability theorem [11, §2.1] we find

$$\displaystyle{\mathcal{N}\mathcal{T}_{{\ast}}(C_{2},C_{2})\mathop{\cong}\mathrm{KK}_{{\ast}}(X;\mathcal{R}_{C_{2}},\mathcal{R}_{C_{2}})\mathop{\cong}\mathrm{K}_{{\ast}}{\bigl (\mathcal{R}_{C_{2}}(C_{2})\bigr )}\mathop{\cong}\mathrm{{K}}^{{\ast}}\left ({\mathbb{S}}^{1}\right )\mathop{\cong}\mathbb{Z} \oplus \mathbb{Z}[1]\;,}$$

that is, there are non-trivial odd natural transformations \(\mathrm{FK}_{C_{2}} \Rightarrow \mathrm{ FK}_{C_{2}}\). These are generated, for instance, by the composition \(C_{2}\mathop{\longrightarrow}\limits_{}^{r}1\mathop{\longrightarrow}\limits_{}^{\delta }3\mathop{\longrightarrow}\limits_{}^{i}C_{2}\). This follows from the description of the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(C_{2})\) below. Note that \(\delta _{C_{2}}^{C_{2}} \circ \delta _{C_{ 2}}^{C_{2}}\) vanishes because it factors through \(r_{13}^{1} \circ i_{3}^{13} = 0\).

Fig. 3.1
figure 1

Indecomposable natural transformations in \({\mathcal{N}\mathcal{T}}^{{\ast}}(C_{2})\)

Full size image

Figure 3.1 displays a set of indecomposable transformations generating the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(C_{2})\) determined in [1, §6.3.2], where also a list of relations generating the relations in the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(C_{2})\) can be found. From this, it is straight-forward to verify that the space C 2 satisfies the conditions of Proposition 1.

Proceeding as described in Sect. 3.5, we find projective resolutions of the following form (we omit explicit descriptions of the boundary maps):

and similarly for S 124;

and similarly for S 2. Again, the periodic part of each of these resolutions is induced by an extension triangle, a Mayer-Vietoris triangle as in Lemma 4 or a more exotic (anti-)distinguished triangle as in Lemma 5 (we omit the analogous computation here).

We get \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{Y },M) = 0\) for every \(Y \in \mathbb{L}\mathbb{C}{(C_{2})}^{{\ast}}\setminus \{123,124,1,2\}\), and further \(\mathrm{Tor}_{n}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{Y },M) = 0\) for all \(Y \in \mathbb{L}\mathbb{C}{(C_{2})}^{{\ast}}\) and n ≥ 2. Therefore,

$$\displaystyle{\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(\mathcal{N}\mathcal{T}_{\mathrm{ss}},M)\mathop{\cong}\bigoplus _{Y \in \{123,124,1,2\}}\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{Y },M)\;.}$$

The four groups \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{Y },M)\) with \(Y \in \{ 123,124,1,2\}\) can be described explicitly as in Sect. 3.7.1 using the above resolutions. This finishes the proof of Proposition 2.

3.8 Proof of Proposition 3

We apply the Meyer-Nest machinery to the homological functor \(\mathrm{FK} \otimes \mathbb{Q}\) on the triangulated category \(\mathfrak{K}\mathfrak{K}(X) \otimes \mathbb{Q}\). We need to show that every \({\mathcal{N}\mathcal{T}}^{{\ast}}\otimes \mathbb{Q}\) module of the form \(M =\mathrm{ FK}(A) \otimes \mathbb{Q}\) has a projective resolution of length 1. It is easy to see that analogues of Propositions 1 and 2 hold. In particular, the term \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} \otimes \mathbb{Q}}(\mathcal{N}\mathcal{T}_{\mathrm{ss}} \otimes \mathbb{Q},M)\) always vanishes. Here we use that \(\mathbb{Q}\) is a flat \(\mathbb{Z}\)-module, so that tensoring with \(\mathbb{Q}\) turns projective \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-module resolutions into projective \({\mathcal{N}\mathcal{T}}^{{\ast}}\otimes \mathbb{Q}\)-module resolutions. Moreover, the freeness condition for the \(\mathbb{Q}\)-module \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} \otimes \mathbb{Q}}(\mathcal{N}\mathcal{T}_{\mathrm{ss}} \otimes \mathbb{Q},M)\) is empty since \(\mathbb{Q}\) is a field.

3.9 Proof of Proposition 4

The computations to determine the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{4})\) are very similar to those for the category \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{3})\) which were carried out in [11]. We summarise its structure in Fig. 3.2. The relations in \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{4})\) are generated by the following:

  • The hypercube with vertices 5, 15, 25, …, 12345 is a commuting diagram;

  • The following compositions vanish:

    $$\displaystyle\begin{array}{rcl} & & \qquad \qquad 1235\mathop{\longrightarrow}\limits_{}^{i}12345\mathop{\longrightarrow}\limits_{}^{r}4\;,\quad 1245\mathop{\longrightarrow}\limits_{}^{i}12345\mathop{\longrightarrow}\limits_{}^{r}3\;, {}\\ & & \qquad \qquad 1345\mathop{\longrightarrow}\limits_{}^{i}12345\mathop{\longrightarrow}\limits_{}^{r}2\;,\quad 2345\mathop{\longrightarrow}\limits_{}^{i}12345\mathop{\longrightarrow}\limits_{}^{r}1\;, {}\\ & & 1\mathop{\longrightarrow}\limits_{}^{\delta }5\mathop{\longrightarrow}\limits_{}^{i}15\;,\quad 2\mathop{\longrightarrow}\limits_{}^{\delta }5\mathop{\longrightarrow}\limits_{}^{i}25\;,\quad 3\mathop{\longrightarrow}\limits_{}^{\delta }5\mathop{\longrightarrow}\limits_{}^{i}35\;,\quad 4\mathop{\longrightarrow}\limits_{}^{\delta }5\mathop{\longrightarrow}\limits_{}^{i}45\;; {}\\ \end{array}$$

  • The sum of the four maps 12345 → 5 via 1, 2, 3, and 4 vanishes.

This implies that the space Z 4 satisfies the conditions of Proposition 1.

Fig. 3.2
figure 2

Indecomposable natural transformations in \({\mathcal{N}\mathcal{T}}^{{\ast}}(Z_{4})\)

Full size image

In the following, we will define an exact \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-left-module M and compute \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{12345},M)\). By explicit computation, one finds a projective resolution of the simple \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-right-module S 12345 of the following form (again omitting explicit formulas for the boundary maps):

Notice that this sequence is periodic as a cyclic six-term sequence except for the first two steps.

Consider the exact \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-left-module M defined by the exact sequence

$$\displaystyle{ \begin{array}{lll} 0 \rightarrow P_{12345}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i\\ i \\ i\\ i \end{array} \right )}\bigoplus _{1\leq i\leq 4}P_{12345\setminus i}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{llll} i & - i&0 &0 \\ - i&0 &i &0 \\ 0 &i & - i&0 \\ i &0 &0 & - i \\ 0 & - i&0 &i \\ 0 &0 &i & - i \end{array} \right )}\bigoplus _{1\leq j<k\leq 4}P_{jk5} \twoheadrightarrow M\;. \end{array} }$$
(3.7)

We have \(\bigoplus _{1\leq l\leq 4}M(l5) \oplus M(12345)[1]\mathop{\cong}0 \oplus {\mathbb{Z}}^{3}\), \(\bigoplus _{1\leq j<k\leq 4}M(jk5)\mathop{\cong}{\mathbb{Z}}^{6}\), and \(M(5) \oplus \bigoplus _{1\leq i\leq 4}M(12345\setminus i)[1]\mathop{\cong}\mathbb{Z}[1] \oplus \mathbb{Z}{[1]}^{8}\). Since

is exact, a rank argument shows that the map

$$\displaystyle{\bigoplus \limits _{1\leq l\leq 4}M(l5) \oplus M(12345)[1] \rightarrow \bigoplus \limits _{1\leq j<k\leq 4}M(jk5)}$$

is zero. On the other hand, the kernel of the map

$$\displaystyle{\bigoplus \limits _{1\leq j<k\leq 4}M(jk5)\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{llllll} i & - i&0 &i &0 &0 \\ - i&0 &i &0 & - i&0 \\ 0 &i & - i&0 &0 &i \\ 0 &0 &0 & - i&i & - i \end{array} \right )}\bigoplus \limits _{1\leq i\leq 4}M(12345\setminus i)}$$

is non-trivial; it consists precisely of the elements in

$$\displaystyle{\bigoplus \limits _{1\leq j<k\leq 4}M(jk5)\mathop{\cong}\bigoplus \limits _{1\leq j<k\leq 4}\mathbb{Z}[\mathrm{id}_{jk5}^{jk5}]}$$

which are multiples of \(([\mathrm{id}_{jk5}^{jk5}])_{1\leq j<k\leq 4}\). This shows \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{12345},M)\mathop{\cong}\mathbb{Z}\). Hence, by Proposition 1, the module M has projective dimension at least 2. On the other hand, (3.7) is a resolution of length 2. Therefore, the projective dimension of M is exactly 2.

Let \(k \in \mathbb{N}_{\geq 2}\) and define \(M_{k} = M \otimes _{\mathbb{Z}}\mathbb{Z}/k\). Since \(\mathrm{Tor}_{2}^{{\mathcal{N}\mathcal{T}}^{{\ast}} }(S_{12345},M_{k})\mathop{\cong}\mathbb{Z}/k\) is non-free, Proposition 1 shows that M k has at least projective dimension 3. On the other hand, if we abbreviate the resolution (3.7) for M by

$$\displaystyle{ 0 \rightarrow {P}^{(5)}\mathop{\longrightarrow}\limits_{}^{\alpha }{P}^{(4)}\mathop{\longrightarrow}\limits_{}^{\beta }{P}^{(3)} \twoheadrightarrow M\;, }$$
(3.8)

a projective resolution of length 3 for M k is given by

$$\displaystyle{0 \rightarrow {P}^{(5)}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} k\\ \alpha \end{array} \right )}{P}^{(5)}\oplus {P}^{(4)}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} \alpha & - k \\ 0&\beta \end{array} \right )}{P}^{(4)}\oplus {P}^{(3)}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} \beta &k \end{array} \right )}{P}^{(3)} \twoheadrightarrow M_{ k}\;,}$$

where k denotes multiplication by k.

It remains to show that the modules M and M k can be realised as the filtrated K-theory of objects in \(\mathcal{B}(X)\). It suffices to prove this for the module M since tensoring with the Cuntz algebra \(\mathcal{O}_{k+1}\) then yields a separable C ∗dash- algebra with filtrated K-theory M k by the Künneth Theorem.

The projective resolution (3.8) can be written as

$$\displaystyle{ 0 \rightarrow \mathrm{ FK}({P}^{2})\mathop{\longrightarrow}\limits_{}^{\mathrm{FK}(f_{ 2})}\mathrm{FK}({P}^{1})\mathop{\longrightarrow}\limits_{}^{\mathrm{FK}(f_{ 1})}\mathrm{FK}({P}^{0}) \twoheadrightarrow M, }$$

because of the equivalence of the category of projective \({\mathcal{N}\mathcal{T}}^{{\ast}}\)-modules and the category of \(\mathfrak{I}\)-projective objects in \(\mathfrak{K}\mathfrak{K}(X)\). Let N be the cokernel of the module map FK(f 2). Using [11, Theorem 4.11], we obtain an object \(A\in \in \mathcal{B}(X)\) with FK(A){≅}N. We thus have a commutative diagram of the form

Since A belongs to the bootstrap class \(\mathcal{B}(X)\) and \(\mathrm{FK}(A)\) has a projective resolution of length 1, we can apply the universal coefficient theorem to lift the homomorphism γ to an element f ∈ KK(X; A, P 0). Now we can argue as in the proof of [11, Theorem 4.11]: since f is \(\mathfrak{I}\)-monic, the filtrated K-theory of its mapping cone is isomorphic to \(\mathrm{coker}(\gamma )\mathop{\cong}M\). This completes the proof of Proposition 4.

3.10 Cuntz-Krieger Algebras with Projective Dimension 2

In this section we exhibit a Cuntz-Krieger algebra A which is a tight C ∗dash-algebra over the space Z 3 and for which the odd part of \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (Z_{3})}{\bigl (\mathcal{N}\mathcal{T}_{\mathrm{ss}},\mathrm{FK}(A)\bigr )}\)—denoted \(\mathrm{Tor}_{1}^{\mathrm{odd}}\) in the following—is not free. By Proposition 2 this C ∗dash-algebra has projective dimension 2 in filtrated K-theory.

In the following we will adhere to the conventions for graph algebras and adjacency matrices from [4]. Let E be the finite graph with vertex set \({E}^{0} =\{ v_{1},v_{2},\ldots,v_{8}\}\) and edges corresponding to the adjacency matrix

$$\displaystyle{ \left (\begin{array}{llll} B_{4} & 0 &0 &0 \\ X_{1} & B_{1} & 0 &0 \\ X_{2} & 0 &B_{2} & 0 \\ X_{3} & 0 &0 &B_{3} \end{array} \right )\mathop{:}=\left (\begin{array}{llll} \left (\begin{array}{lll} 3&2\\ 2 &3 \end{array} \right )&0 &0 &0 \\ \left (\begin{array}{lll} 1&1\\ 1 &1 \end{array} \right )&\left (\begin{array}{lll} 3&2\\ 1 &2 \end{array} \right )&0 &0 \\ \left (\begin{array}{lll} 1&1\\ 1 &1 \end{array} \right )&0 &\left (\begin{array}{lll} 3&2\\ 1 &2 \end{array} \right )&0 \\ \left (\begin{array}{lll} 1&1\\ 1 &1 \end{array} \right )&0 &0 &\left (\begin{array}{lll} 3&2\\ 1 &2 \end{array} \right ) \end{array}\right ) }$$
(3.9)

 . Since this is a finite graph with no sinks and no sources, the associated graph C ∗dash- algebra C ∗(E) is in fact a Cuntz-Krieger algebra (we can replace E with its edge graph; see [13, Remark 2.8]). Moreover, the graph E is easily seen to fulfill condition (K) because every vertex is the base of two or more simple cycles. As a consequence, the adjacency matrix of the edge graph of E fulfills condition (II) from [5]. In fact, condition (K) is designed as a generalisation of condition (II): see, for instance, [8].

Applying [13, Theorem 4.9]—and carefully translating between different graph algebra conventions—we find that the ideals of C ∗(E) correspond bijectively and in an inclusion-preserving manner to the open subsets of the space Z 3. By [9, Lemma 2.35], we may turn A into a tight C ∗dash- algebra over Z 3 by declaring \(A(\{4\}) = I_{\{v_{1},v_{2}\}}\), \(A(\{1,4\}) = I_{\{v_{1},v_{2},v_{3},v_{4}\}}\), \(A(\{2,4\}) = I_{\{v_{1},v_{2},v_{5},v_{6}\}}\) as well as \(A(\{3,4\}) = I_{\{v_{1},v_{2},v_{7},v_{8}\}}\), where I S denotes the ideal corresponding to the saturated hereditary subset S.

It is known how to compute the six-term sequence in K-theory for an extension of graph C ∗dash- algebras: see [4]. Using this and Proposition 2, \(\mathrm{Tor}_{1}^{\mathrm{odd}}\) is the homology of the complex

$$\displaystyle{ \ker (\phi _{0})\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i & - i&0 \\ - i&0 &i \\ 0 &i & - i \end{array} \right )}\ker (\phi _{1})\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lll} i&i&i \end{array} \right )}\ker (\phi _{2})\;, }$$
(3.10)
$$\displaystyle{\mathrm{where}\quad \phi _{0} = \mathrm{diag}\left (\left (\begin{array}{lll} B^{\prime}_{4} & X_{1}^{t} \\ 0 &B^{\prime}_{1} \end{array} \right ),\left (\begin{array}{lll} B^{\prime}_{4} & X_{2}^{t} \\ 0 &B^{\prime}_{2} \end{array} \right ),\left (\begin{array}{lll} B^{\prime}_{4} & X_{3}^{t} \\ 0 &B^{\prime}_{3} \end{array} \right )\right )\;,\quad \phi _{2} = \left (\begin{array}{llll} B^{\prime}_{4} & X_{1}^{t}&X_{2}^{t}&X_{3}^{t} \\ 0 &B^{\prime}_{1} & 0 &0 \\ 0 &0 &B^{\prime}_{2} & 0 \\ 0 &0 &0 &B^{\prime}_{3} \end{array} \right )\;,}$$
$$\displaystyle{\phi _{1} = \mathrm{diag}\left (\left (\begin{array}{lll} B^{\prime}_{4} & X_{1}^{t}&X_{2}^{t} \\ 0 &B^{\prime}_{1} & 0 \\ 0 &0 &B^{\prime}_{2} \end{array} \right ),\left (\begin{array}{lll} B^{\prime}_{4} & X_{1}^{t}&X_{3}^{t} \\ 0 &B^{\prime}_{1} & 0 \\ 0 &0 &B^{\prime}_{3} \end{array} \right ),\left (\begin{array}{lll} B^{\prime}_{4} & X_{2}^{t}&X_{3}^{t} \\ 0 &B^{\prime}_{2} & 0 \\ 0 &0 &B^{\prime}_{3} \end{array} \right )\right )\;,}$$

and \(B^{\prime}_{4} = B_{4}^{t}-\left (\begin{array}{lll} 1&0\\ 0 &1 \end{array} \right ) = \left (\begin{array}{lll} 2&2\\ 2 &2 \end{array} \right )\) and \(B^{\prime}_{j} = B_{j}^{t}-\left (\begin{array}{lll} 1&0\\ 0 &1 \end{array} \right ) = \left (\begin{array}{lll} 2&1\\ 2 &1 \end{array} \right )\) for 1 ≤ j ≤ 3. We obtain a commutative diagram

(3.11)

where f and g have the block forms

$$\displaystyle{f = \left (\begin{array}{lllllll} \mathrm{id} &0 & -\mathrm{ id}&0 &0 &0 \\ 0 &\mathrm{id} &0 &0 &0 &0\\ 0 &0 &0 & -\mathrm{ id } &0 &0 \\ -\mathrm{ id}&0 &0 &0 &\mathrm{id} &0 \\ 0 & -\mathrm{ id}&0 &0 &0 &0 \\ 0 &0 &0 &0 &0 &\mathrm{id}\\ 0 &0 &\mathrm{id } &0 & -\mathrm{ id } &0 \\ 0 &0 &0 &\mathrm{id} &0 &0\\ 0 &0 &0 &0 &0 & -\mathrm{ id} \end{array} \right )\;,\qquad g = \left (\begin{array}{lllllllll} \mathrm{id}&0 &0 &\mathrm{id}&0 &0 &\mathrm{id}&0 &0\\ 0 &\mathrm{id } &0 &0 &\mathrm{id } &0 &0 &0 &0 \\ 0 &0 &\mathrm{id}&0 &0 &0 &0 &\mathrm{id}&0\\ 0 &0 &0 &0 &0 &\mathrm{id } &0 &0 &\mathrm{id} \end{array} \right )\;,}$$

and \(f_{K}\mathop{:}=f\vert _{\ker (\phi _{0})}\), \(f_{I}\mathop{:}=f\vert _{\mathrm{im}(\phi _{0})}\), \(g_{K}\mathop{:}=g\vert _{\ker (\phi _{1})}\), \(g_{I}\mathop{:}=g\vert _{\mathrm{im}(\phi _{1})}\). Notice that f and g are defined in a way such that the restrictions \(f\vert _{\ker (\phi _{0})}\) and \(g\vert _{\ker (\phi _{1})}\) are exactly the maps from (3.10) in the identification made above.

We abbreviate the above short exact sequence of cochain complexes 3.11 as \(K_{\bullet } \rightarrowtail Z_{\bullet } \twoheadrightarrow I_{\bullet }\). The part \(\mathrm{{H}}^{0}(Z_{\bullet }) \rightarrow \mathrm{ {H}}^{0}(I_{\bullet }) \rightarrow \mathrm{ {H}}^{1}(K_{\bullet }) \rightarrow \mathrm{ {H}}^{1}(Z_{\bullet })\) in the corresponding long exact homology sequence can be identified with

$$\displaystyle{\ker (f)\mathop{\longrightarrow}\limits_{}^{\phi _{0}}\ker (f_{I}) \rightarrow \frac{\ker (g_{K})} {\mathrm{im}(f_{K})} \rightarrow 0\;.}$$

Hence

$$\displaystyle{\mathrm{Tor}_{1}^{\mathrm{odd}}\mathop{\cong} \frac{\ker (g_{K})} {\mathrm{im}(f_{K})}\mathop{\cong}\frac{\ker (f_{I})} {\phi _{0}{\bigl (\ker (f)\bigr )}}\mathop{\cong}\frac{\ker (f) \cap \mathrm{im}(\phi _{0})} {\phi _{0}{\bigl (\ker (f)\bigr )}} \;.}$$

We have \(\ker (f) =\{ (v,0,v,0,v,0)\mid v \in {\mathbb{Z}}^{2}\} \subset {({\mathbb{Z}}^{\oplus 2})}^{\oplus (2\cdot 3)}\).

From the concrete form (3.9) of the adjacency matrix, we find that \(\ker (f) \cap \mathrm{im}(\phi _{0})\) is the free cyclic group generated by (1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0), while \(\phi _{0}{\bigl (\ker (f)\bigr )}\) is the subgroup generated by (2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0). We see that \(\mathrm{Tor}_{1}^{\mathrm{odd}}\mathop{\cong}\mathbb{Z}/2\) is not free.

Now we briefly indicate how to construct a similar counterexample for the space S. Consider the integer matrix

$$\displaystyle{ \left (\begin{array}{llll} B_{4} & 0 &0 &0 \\ X_{43} & B_{3} & 0 &0 \\ X_{42} & 0 &B_{2} & 0 \\ X_{41} & X_{31} & X_{21} & B_{1} \end{array} \right )\mathop{:}=\left (\begin{array}{llll} \left (\begin{array}{lll} 3 \end{array} \right )&0 &0 &0 \\ \left (\begin{array}{lll} 2 \end{array} \right )&\left (\begin{array}{lll} 3 \end{array} \right )&0 &0 \\ \left (\begin{array}{lll} 2 \end{array} \right )&0 &\left (\begin{array}{lll} 3 \end{array} \right )&0 \\ \left (\begin{array}{ll} 2\\ 0 \end{array} \right )&\left (\begin{array}{ll} 1\\ 0 \end{array} \right )&\left (\begin{array}{lll} 1\\ 0 \end{array} \right )&\left (\begin{array}{lll} 2&1\\ 1 &2 \end{array} \right ) \end{array}\right )\;. }$$

The corresponding graph F fulfills condition (K) and has no sources or sinks. The associated graph C ∗dash- algebra C ∗(F) is therefore a Cuntz-Krieger algebra satisfying condition (II). It is easily read from the block structure of the edge matrix that the primitive ideal space of C ∗(F) is homeomorphic to S. We are going to compute the even part of \(\mathrm{Tor}_{1}^{{\mathcal{N}\mathcal{T}}^{{\ast}} (S)}{\bigl (\mathcal{N}\mathcal{T}_{\mathrm{ss}},\mathrm{FK}({C}^{{\ast}}(F))\bigr )}\). Since the nice computation methods from the previous example do not carry over, we carry out a more ad hoc calculation.

By Remark 1, the even part of our \(\mathrm{Tor}\)-term is isomorphic to the homology of the complex

where column-wise direct sums are taken. Here \(B^{\prime}_{1} = B_{1}^{t}-\left (\begin{array}{lll} 1&0\\ 0 &1 \end{array} \right ) = \left (\begin{array}{lll} 1&1\\ 1 &1 \end{array} \right )\) and \(B^{\prime}_{j} = B_{j}^{t}-\left (\begin{array}{ll} 1 \end{array} \right ) = \left (\begin{array}{ll} 2 \end{array} \right )\) for 2 ≤ j ≤ 4. This complex can be identified with

$$\displaystyle\begin{array}{rcl} \mathbb{Z} \oplus \mathbb{Z}/2 \oplus \mathbb{Z}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lllllll} 0 &1&0\\ 0 &0 &0 \\ - 2&0&2\\ 0 &1 &0 \\ 0 &0&0 \end{array} \right )}{(\mathbb{Z}/2)}^{2} \oplus \mathbb{Z} \oplus {(\mathbb{Z}/2)}^{2}\mathop{\longrightarrow}\limits_{}^{\left (\begin{array}{lllll} 1&0&0&1&0\\ 0 &1 &1 &0 &0 \\ 0&0&1&0&1 \end{array} \right )}{(\mathbb{Z}/2)}^{3}\;,& & {}\\ \end{array}$$

the homology of which is isomorphic to \(\mathbb{Z}/2\); a generator is given by the class of \((0,1,1,0,1) \in {(\mathbb{Z}/2)}^{2} \oplus \mathbb{Z} \oplus {(\mathbb{Z}/2)}^{2}\). This concludes the proof of Proposition 5.