diff --git a/papers/project_paper_1_relativity/paper_1_relativity.md b/papers/project_paper_1_relativity/paper_1_relativity.md deleted file mode 100644 index c881e8f2..00000000 --- a/papers/project_paper_1_relativity/paper_1_relativity.md +++ /dev/null @@ -1,44 +0,0 @@ ---- -title: "Research Paper: The Thermodynamic Bias Toward Manifolds in Causal Sets: Path Integral Prerequisites for Lorentz Invariance (Letter)" -date: "2026-06-01T08:00:00Z" -draft: false -tags: ["#research", "physics", "intellecton"] ---- - -**Abstract:** The extraction of the Minkowski metric from discrete causal graphs in Causal Set Theory (CST) is complicated by the Kleitman-Rothschild (KR) entropy dominance. While recent path integral formulations (Loomis & Carlip 2018) have shown suppression of non-manifold sets, the exact topological phase boundary remains unclear. We introduce a thermodynamic partition function governed by the discrete Benincasa-Dowker action augmented with an intensive non-local volume penalty. By evaluating the partition function with a controlled $p$-dependent entropy functional, we demonstrate a first-order topological phase transition. A fluctuation analysis confirms the exactness of the mean-field in the thermodynamic limit. This establishes a rigorous statistical mechanical mechanism by which CST dynamically selects phases with stable Myrheim-Meyer dimensions, a prerequisite for macroscopic Lorentz invariance. - -## The Partition Function and the KR Ensemble -Let $\Omega_N$ be the space of causal sets of $N$ elements. The canonical partition function is defined over the Benincasa-Dowker action $S_{BD}$ and an auxiliary volume penalty $V(\mathcal{C}) = \sum_{x \prec y} | \{ z \in \mathcal{C} \mid x \prec z \prec y \} |$: - -$$ -Z = \sum_{\mathcal{C} \in \Omega_N} \exp\left( -S_{BD}^{(d)}(\mathcal{C}) - \beta V(\mathcal{C}) \right) -$$ - -The dominant contribution to $\Omega_N$ are Kleitman-Rothschild (KR) posets (Kleitman & Rothschild 1975), which decompose into three bipartite layers $L_1, L_2, L_3$ with cardinalities $N/4, N/2, N/4$. In the KR phase, the link density between adjacent layers is $p \approx 1/2$. A rigorous continuous entropy density $s(p)$ for this bipartite ensemble is bounded by the Shannon entropy of the edge probabilities: - -$$ -s(p) = -p \ln p - (1-p) \ln(1-p) -$$ - -## Saddle-Point Analysis and First-Order Transition -To properly scale the continuum limit, we normalize the intensive volume penalty $v(p) = \langle V \rangle / N^3$ and absorb the action expectation $\langle S_{BD}^{(d)} \rangle$ into the energy functional. The partition function becomes: - -$$ -Z \approx \int_{0}^{1} dp \, \exp\left[ N^2 s(p) - \langle S_{BD}^{(d)}(p) \rangle - \tilde{\beta} N^3 v(p) \right] -$$ - -where $\tilde{\beta} = \beta / N$ ensures the phase transition survives the thermodynamic limit $N \to \infty$. - -We define the free energy functional $\Phi(p) = -s(p) + \tilde{\beta} N v(p)$. The saddle point condition $\Phi'(p^*) = 0$ yields a highly non-linear gap equation. By computing the Hessian $\Phi''(p^*)$, we find the fluctuations scale as $\sigma_p^2 = 1/|\Phi''(p^*)| = \mathcal{O}(N^{-2})$. Consequently, the mean-field approximation becomes exact as $N \to \infty$. - -At the critical parameter $\tilde{\beta}_c$, the order parameter $p^*(\tilde{\beta})$ undergoes a discontinuous jump $\Delta p^* > 0$, signaling a first-order topological phase transition. Below $\tilde{\beta}_c$, the system resides in the KR phase (undefined dimension). Above $\tilde{\beta}_c$, the system collapses into a sparse, manifold-like phase. - -## Myrheim-Meyer Dimension and Lorentz Invariance -The sparse phase is operationally defined as "manifold-like" if its Myrheim-Meyer dimension $d_{MM}$ matches the target topological dimension $d$ (Surya 2019). This phase exhibits behavior consistent with Poisson sprinklings into Minkowski space (Bombelli et al. 2009), suppressing non-manifold sub-classes identified by Loomis and Carlip (2018). Thus, the volume penalty acts as a topological regularizer, yielding the necessary symmetries for emergent Lorentz invariance. - -## References - -- **[Surya2019]** S. Surya, *Living Rev. Relativ.* **22**, 5 (2019). -- **[Kleitman1975]** D. Kleitman, B. Rothschild, *Trans. Am. Math. Soc.* **205**, 205 (1975). -- **[Loomis2018]** S. P. Loomis, S. Carlip, *Class. Quantum Grav.* **35**, 024002 (2018). -- **[Bombelli2009]** L. Bombelli, J. Henson, R. D. Sorkin, *Mod. Phys. Lett. A* **24**, 2579 (2009). diff --git a/papers/project_paper_1_relativity/references/Bombelli2009.pdf b/papers/project_paper_1_relativity/references/Bombelli2009.pdf deleted file mode 100644 index 28be1f86..00000000 --- a/papers/project_paper_1_relativity/references/Bombelli2009.pdf +++ /dev/null @@ -1,3 +0,0 @@ -version https://git-lfs.github.com/spec/v1 -oid sha256:7314322f3517e9523e98a59a35a263db650a0659edd0df1d174e084606884119 -size 461293 diff --git a/papers/project_paper_1_relativity/references/Bombelli2009.txt b/papers/project_paper_1_relativity/references/Bombelli2009.txt deleted file mode 100644 index 1d7a654e..00000000 --- a/papers/project_paper_1_relativity/references/Bombelli2009.txt +++ /dev/null @@ -1,3668 +0,0 @@ -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. 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