A universal coefficient theorem for actions of finite cyclic groups of square-free order on \(\mathrm C^*\)-algebras
April 14, 2026
Abstract
We prove a Universal Coefficient Theorem for objects in the bootstrap class in the equivariant Kasparov category for a finite cyclic group of square-free order.
1 Introduction↩︎
The classical Universal Coefficient Theorem (UCT) by Rosenberg and Schochet Rosenberg-Schochet:Kunneth? is a fundamental tool in the study of \(\mathrm C^*\)-algebras. The theorem uses a short exact sequence to compute bivariant \(\mathrm{K}\)-theory groups of separable \(\mathrm C^*\)-algebras in the bootstrap class in terms of their \(\mathrm{K}\)-theory. It implies that two \(\mathrm C^*\)-algebras in the bootstrap class are \(\mathrm{KK}\)-equivalent once they have isomorphic \(\mathrm{K}\)-theory. This is a step towards the classification of Kirchberg algebras.
The UCT by Rosenberg–Schochet has been generalised to various situations involving \(\mathrm C^*\)-algebras with extra structure. Meyer and Nest Meyer-Nest:Filtrated_K? established a UCT for \(\mathrm C^*\)-algebras over a finite sober topological space \(X\) using filtrated \(\mathrm{K}\)-theory. This led to new classification results as well. In the equivariant setting, Manuel Köhler Koehler:Thesis? established a remarkable UCT for \(\mathrm C^*\)-algebras with an action of a cyclic group of prime order. This result has since been applied by Meyer Meyer:Actions_Kirchberg? to the classification of actions of \(\mathbb{Z}/p\) on Kirchberg algebras up to equivariant \(\mathrm{KK}\)-equivalence.
The aim of the present article is a generalisation of Köhler’s UCT for actions of finite cyclic groups \(G\) of square-free order. We rely on the fact that the equivariant bootstrap class \(\mathfrak{B}^G\), which is defined to consist of all \(G\)-\(\mathrm C^*\)-algebras that are \(\mathrm{KK}^G\)-equivalent to an action on a Type I \(\mathrm C^*\)-algebra, is generated by \(\mathrm C(G/H)\) for cyclic subgroups \(H \subseteq G\). This was recently shown in Meyer-Nadareishvili:UCT_actions?, using results from Arano-Kubota:Atiyah-Segal?.
To generalise Köhler’s invariant, we use a larger generating set \(\mathfrak{C}^G\) for \(\mathfrak{B}^G\), which consists of the tensor products of Köhler’s generators for the prime-order subgroups. We prove that the resulting \(\mathbb{Z}/2\)-graded category ring, \(\mathfrak{K}_G\), is isomorphic to the tensor product of individual rings \(\mathfrak{K}_{p_i}\) computed by Köhler. Our main result is:
Theorem 1. Let \(G\) be a finite cyclic group of square-free order. Let \(\mathfrak{B}^G\subseteq \mathrm{KK}^G\) be the equivariant bootstrap class. There is a stable homological functor \(U_{\mathfrak{C}^G}\) from \(\mathrm{KK}^G\) to the category of \(\mathbb{Z}/2\)-graded exact countable modules over a certain ring \(\mathfrak{K}_G\), such that for any \(A\in\mathfrak{B}^G\) and \(C\in \mathrm{KK}^G\), there is a natural short exact sequence \[\mathop{\mathrm{Ext}}^1_{\mathfrak{K}_G}\bigl(U_{\mathfrak{C}^G}(\Sigma A), U_{\mathfrak{C}^G}(C)\bigr) \rightarrowtail \mathrm{KK}^G_*(A,C) \twoheadrightarrow \mathop{\mathrm{Hom}}_{\mathfrak{K}_G}\bigl(U_{\mathfrak{C}^G}(A), U_{\mathfrak{C}^G}(C)\bigr).\] If \(C\in\mathfrak{B}^G\) as well, then every isomorphism \(U_{\mathfrak{C}^G}(A) \cong U_{\mathfrak{C}^G}(C)\) of \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_G\)-modules lifts to a \(\mathrm{KK}^G\)-equivalence in \(\mathrm{KK}^G_0(A,C)\). In particular, \(A\) and \(C\) are \(\mathrm{KK}^G\)-equivalent if and only if \(U_{\mathfrak{C}^G}(A) \cong U_{\mathfrak{C}^G}(C)\) as \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_G\)-modules.
Our proof method also describes the range of the invariant: these are precisely the \(\mathbb{Z}/2\)-graded modules over \(\mathfrak{K}_G\) that are countable and exact. It was shown in Meyer:Actions_Kirchberg? that any \(\mathrm{KK}^G\)-equivalence class in the equivariant bootstrap class is represented by a pointwise outer action on a Kirchberg algebra, still in the bootstrap class. Gabe and Szabó Gabe-Szabo:Dynamical_Kirchberg? have shown that two such actions are cocycle conjugate if and only if they are \(\mathrm{KK}^G\)-equivalent. Thus we get a classification of pointwise outer actions of \(G\) on Kirchberg algebras in the bootstrap class.
We start with some preliminaries on homological algebra in equivariant KK-theory and on Köhler’s UCT in Section 2. Section 3 contains a computation of certain equivariant \(\mathrm{KK}\)-groups when the group is a product and both \(\mathrm C^*\)-algebras are tensor products; this is needed to compute the ring \(\mathfrak{K}_G\). Section 4 describes the functor \(U_{\mathfrak{C}^G}\) and contains the proof of Theorem 1. The key lemma says that any module over \(\mathfrak{K}_G\) in the range of the invariant has a projective resolution of length \(1\). We reduce this to the analogous result for cyclic groups of prime order using that \(\mathfrak{K}_G\) is a tensor product of rings for the prime order factors of \(G\).
2 Some preliminaries↩︎
Let \(G\) be a second countable, locally compact group. The Kasparov category \(\mathrm{KK}^G\) of separable \(G\)-\(\mathrm C^*\)-algebras is a symmetric monoidal triangulated category. Its tensor product is given by the spatial tensor product of \(\mathrm C^*\)-algebras equipped with the diagonal \(G\)-action. Exact triangles arise either from mapping cones of equivariant \(^*\)-homomorphisms, or from extensions of \(G\)-\(\mathrm C^*\)-algebras that admit a \(G\)-equivariant, completely positive, contractive section (see the Appendix of Meyer-Nest:BC?). The suspension functor is defined by \(\Sigma \mathrel{:=}\mathrm C_0(\mathbb{R})\otimes {-}\). By Bott periodicity, this functor is an involutive equivalence. Moreover, \(\mathrm{KK}^G\) admits countable coproducts induced by \(\mathrm C_0\)-direct sums, denoted by \(\oplus\).
2.1 Homological algebra in \(\mathrm{KK}^G\)↩︎
The aim of this article is to extend the Universal Coefficient Theorem of Koehler:Thesis? to finite cyclic groups of square-free order. The framework for this is homological algebra in triangulated categories. Therefore, we recall some basic facts from this theory. The situation that we will need is a special case of the following more general setup.
Let \(\mathfrak{T}\) be a triangulated category with countable coproducts. Let \(\Sigma\) denote its suspension. An object \(C\in\mathfrak T\) is called \(\aleph_1\)-compact if the representable functor \(\mathfrak{T}(C,{-})\colon \mathfrak T\to \mathfrak{Ab}\) to the category of abelian groups commutes with countable coproducts. Now let \(\mathfrak{C}\) be an at most countable set of such objects in \(\mathfrak{T}\), and assume moreover that \(\mathfrak{T}_n(C,A) \mathrel{:=}\mathfrak{T}(\Sigma^n C, A)\) is countable for all \(A \in \mathfrak{T}\) and \(n \in \mathbb{Z}\). Let \(\mathfrak{Ab}^{\mathbb{Z}}\) denote the abelian category of \(\mathbb{Z}\)-graded abelian groups, equipped with the suspension homomorphism that shifts degrees. Define the functor \[F_{\mathfrak{C}}\colon\mathfrak{T} \to \prod_{C\in \mathfrak{C}}\mathfrak{Ab}^{\mathbb{Z}}, \qquad A \mapsto \bigl(\mathfrak{T}_n(C,A)\bigr) _{C \in \mathfrak{C},n\in\mathbb{Z}}.\] This intertwines the suspensions in \(\mathfrak T\) and \(\mathfrak{Ab}^{\mathbb{Z}}\). We call a category with a fixed (suspension) automorphism stable. We call a functor between stable categories stable when it intertwines the suspensions. The kernel of \(F_{\mathfrak{C}}\) on morphisms, \(\mathfrak{I}_{\mathfrak{C}}\), is the prototypical example of a stable homological ideal in a triangulated category. This ideal specifies how to do relative homological algebra in \(\mathfrak T\).
Let \(\langle \mathfrak{C} \rangle \subseteq \mathfrak T\) denote the smallest triangulated subcategory of \(\mathfrak T\) that is closed under countable coproducts and contains \(\mathfrak C\). This subcategory will play the role of the bootstrap category in \(\mathrm{KK}\).
A stable homological functor \(H\colon \mathfrak T\to \mathfrak A\) is called \(\mathfrak{I}\)-exact if it vanishes on the ideal \(\mathfrak{I}\). An \(\mathfrak{I}\)-exact stable homological functor \(U\colon \mathfrak T\to \mathfrak{A}_\mathfrak{I}\) is called universal if any \(\mathfrak{I}\)-exact stable homological functor \(H\colon \mathfrak T\to \mathfrak A\) factors uniquely as \(\bar{H}\circ U\), for a stable exact functor \(\bar{H}\colon \mathfrak{A}_\mathfrak{I}\to \mathfrak A\). Such a universal functor closely relates the \(\mathfrak{I}\)-relative homological algebra in \(\mathfrak{T}\) to the homological algebra in the abelian category \(\mathfrak{A}_\mathfrak{I}\). In particular, the relative derived functors in \(\mathfrak{T}\) are identified with those in \(\mathfrak{A}_\mathfrak{I}\) after composition with \(U\).
We now describe the universal \(\mathfrak{I}_{\mathfrak{C}}\)-exact stable homological functor. Let \(\mathfrak{C}\) also denote the \(\mathbb{Z}\)-graded pre-additive category with objects \(\mathfrak{C}\) and morphisms \(\bigoplus_{n \in \mathbb{Z}} \mathfrak{T}_n(A,B)\) for \(A,B \in \mathfrak{C}\). A right \(\mathfrak{C}\)-module is defined as a contravariant stable additive functor \(\mathfrak{C}^\mathrm{op}\to \mathfrak{Ab}^{\mathbb{Z}}\). These modules form a stable abelian category \(\mathfrak{Mod}(\mathfrak{C}^\mathrm{op})\) with direct sums and enough projective objects. The subcategory of countable modules is denoted by \(\mathfrak{Mod}(\mathfrak{C}^\mathrm{op})_{\aleph_1}\). Equipping \(\bigl(\mathfrak{T}_n(C,A)\bigr)_{n \in \mathbb{Z}}\) with the right \(\mathfrak{C}\)-module structure induced by composition in \(\mathfrak{T}\), we enrich \(F_{\mathfrak{C}}\) to a functor \[U_{\mathfrak{C}} \colon \mathfrak{T} \longrightarrow \mathfrak{Mod}(\mathfrak{C}^\mathrm{op})_{\aleph_1}.\] The category \(\mathfrak{Mod}(\mathfrak{C}^\mathrm{op})_{\aleph_1}\) is isomorphic to the category of left modules over the category ring of \(\mathfrak{C}^\mathrm{op}\).
The following is the abstract Universal Coefficient Theorem (UCT):
Theorem 2. Let \(\mathfrak T\) be a triangulated category with countable coproducts and let \(\mathfrak C \subseteq \mathfrak{T}\) be a set of \(\aleph_1\)-compact objects. The universal \(\mathfrak{I}_{\mathfrak{C}}\)-exact stable homological functor is \(U_\mathfrak{C}\).
Let \(A\in \langle \mathfrak C\rangle\) and \(B \in \mathfrak T\). If the \(\mathfrak{C}\)-module \(U_{\mathfrak{C}}(A)\) has a projective resolution of length \(1\), then there is a natural short exact sequence \[\mathop{\mathrm{Ext}}_{{\mathfrak C}}^1 \bigl(U_{\mathfrak{C}}(\Sigma A),U_{\mathfrak{C}}(B)\bigr) \rightarrowtail\mathfrak{T}(A,B) \twoheadrightarrow\mathop{\mathrm{Hom}}_{{\mathfrak C}} \bigl(U_{\mathfrak{C}}(A),U_{\mathfrak{C}}(B)\bigr).\]
Proof. These statements are contained in Meyer-Nest:Homology_in_KK?*Theorem 4.4. ◻
When \(\mathfrak{T}\) is an equivariant Kasparov category, the suspension functor is an involutive equivalence. Hence the \(\mathbb{Z}\)-graded modules above become \(\mathbb{Z}/2\)-graded.
Example 1. Let \(G\) be a trivial group. Then \(\mathfrak{C} = \{\mathbb{C}\}\) generates the well-known bootstrap class and the functor \(U_{\mathfrak{C}}\) is topological \(\mathrm{K}\)-theory, regarded as a functor to the stable abelian category of countable \(\mathbb{Z}/2\)-graded abelian groups. The latter has global homological dimension \(1\). Hence, Theorem 2 applies. It gives the Universal Coefficient Theorem of Rosenberg and Schochet Rosenberg-Schochet:Kunneth?.
2.2 The equivariant bootstrap class↩︎
The bootstrap class in Example 1 has several equivalent descriptions. Some of these still work in the equivariant case.
Definition 1. Let \(G\) be a finite group. The equivariant bootstrap class \(\mathfrak B^G\) is defined as the class of \(G\)-\(\mathrm C^*\)-algebras that are \(\mathrm{KK}^G\)-equivalent to a \(G\)-action on a Type I \(\mathrm C^*\)-algebra.
The bootstrap class may also be described as the localising subcategory generated by the collection of actions of \(G\) on finite-dimensional \(\mathrm C^*\)-algebras. This generating set, however, is rather large and clearly has some redundancies. It is useful to find a set of generators that is irredundant in the sense that no proper subset is again a generating set. For a finite group \(G\), a smaller generating set was found in Meyer-Nadareishvili:UCT_actions?, and we are going to show that it is irredundant in the above sense in Proposition 4. First, we describe this generating set:
Theorem 3 (Meyer-Nadareishvili:UCT_actions?*Corollary 3.3). Let \(G\) be a finite group. Choose one representative for each conjugacy class of cyclic subgroups and let \(\mathfrak{C}\) be the set of \(G\)-\(\mathrm C^*\)-algebras \(\mathrm C(G/H)\) for these representatives of cyclic subgroups \(H\subseteq G\). The bootstrap class is the smallest localising subcategory that contains \(\mathfrak{C}\).
Lemma 1. Let \(G\) be a finite group and \(L\subseteq G\) a cyclic subgroup. There is \(A\in \mathfrak{B}^G\) such that \(\mathrm{K}_*^L(A)\neq 0\) and \(\mathrm{K}_*^H(A)= 0\) for subgroups \(H\subseteq G\) that are not conjugate to \(L\).
Proof. The object \(A\) will, in fact, be “rational”, that is, it belongs to the localisation \(\mathfrak{B}^G_\mathbb{Q}\) of \(\mathfrak{B}^G\) at the rational numbers. It is shown in Bouc-DellAmbrogio-Martos:Splitting? that there is an equivalence from \(\mathfrak{B}^G_\mathbb{Q}\) to the category of Mackey modules over the rationalised representation Green ring of \(G\) that maps any object \(A\) to the family \(\mathrm{K}_*^L(A)\) for all \(L\subseteq G\) with its canonical Mackey module structure. Therefore, an object with the desired properties exists in \(\mathfrak{B}^G_\mathbb{Q}\) if and only if there is a Mackey module \(\mathcal{M}\) over the rationalised representation Green ring of \(G\) with the property that \(\mathcal{M}(L)\neq 0\) and \(\mathcal{M}(H)= 0\) for all subgroups \(H\subseteq G\) not conjugate to \(L\). Such a Mackey module exists by Thevenaz-Webb:The_structure_of_Mackey_functors?*Example 6.7 and Thevenaz-Webb:The_structure_of_Mackey_functors?*Lemma 6.4. ◻
A similar argument does not work for non-cyclic subgroups because Artin’s Induction Theorem is used in Thevenaz-Webb:The_structure_of_Mackey_functors?*Example 6.7.
Proposition 4. Let \(G\) be a finite group and \(L\subseteq G\) a cyclic subgroup. Then \(\langle \mathrm C(G/H)\mid H\subseteq G \text{ cyclic, not conjugate to } L \rangle\) is strictly smaller than \(\mathfrak{B}^{G}\).
Proof. For a subgroup \(H\), let \(\mathop{\mathrm{Res}}^G_H\colon \mathrm{KK}^G\to\mathrm{KK}^H\) be the functor that restricts a \(G\)-action to \(H\). Recall that the induction functor \(\mathop{\mathrm{Ind}}^G_H\colon \mathrm{KK}^H\to\mathrm{KK}^G\) is left adjoint to it (see Meyer-Nest:BC?) and that \(\mathrm C(G/H) = \mathop{\mathrm{Ind}}_H^G \mathbb{C}\). Let \(A\) be any \(G\)-\(\mathrm C^*\)-algebra. Then \[\mathrm{KK}^G_*(\mathrm C(G/H),A) \cong \mathrm{KK}^H_*(\mathbb{C},A) \cong \mathrm{K}_*^H(A).\] Therefore, the object \(A\) from Lemma 1 satisfies \(\mathrm{KK}^G_*(\mathrm C(G/L),A) \neq 0\) and \(\mathrm{KK}^G_*(\mathrm C(G/H),A) = 0\) for all other cyclic subgroups \(H\subseteq G\) that are not conjugate to \(L\). Then \(\mathrm{KK}^G_*(D,A)=0\) for all objects in the localising subcategory generated by \(\mathrm C(G/H)\) for \(H\subseteq G\) not conjugate to \(L\). So \(A\) cannot belong to this localising subcategory. ◻
2.3 Köhler’s invariant↩︎
Let \(p\) be a prime number and let \(G = \mathbb{Z}/p\). Let \(C_p\) denote the mapping cone of the \(\mathbb{Z}/p\)-equivariant unital embedding \(\mathbb{C}\to \mathrm C(\mathbb{Z}/p)\). Define \[\mathfrak C^{\mathbb{Z}/p} = \left\{ C_p, \mathbb{C}, \mathrm C(\mathbb{Z}/p) \right\}.\] Let \(\mathfrak{K}_p\) be the \(\mathbb{Z}/2\)-graded category ring of \(({\mathfrak C^{\mathbb{Z}/p}})^\mathrm{op}\); that is, \[\mathfrak{K}_p = \mathrm{KK}^{\mathbb{Z}/p}\bigl(C_p\oplus\mathbb{C}\oplus\mathrm C(\mathbb{Z}/p),C_p\oplus \mathbb{C}\oplus \mathrm C(\mathbb{Z}/p)\bigr)^\mathrm{op}.\] The ring \(\mathfrak{K}_p\) was computed by Manuel Köhler in Koehler:Thesis?. He showed that every object in the range of the invariant \(U_{\mathfrak{C}^{\mathbb{Z}/p}}\) admits a projective resolution of length \(1\). Using this, he established an equivariant Universal Coefficient Theorem for \(\mathrm{KK}^G\) (Theorem 6 below). Since we will be concerned with specific properties of \(\mathfrak{K}_p\), we recall its presentation in terms of generators and relations (see Koehler:Thesis?, Meyer:Actions_Kirchberg? for more details).
Theorem 5 (Meyer:Actions_Kirchberg?*Theorem 5.10). The ring \(\mathfrak{K}_p\) is the universal ring generated by elements \(1_{j}\) for \(j=0,1,2\) and \(\alpha_{jk}\) for \(0\le j,k \le 2\) with \(j\neq k\) with the relations \[\begin{align} 1_{j}1_{k}&= \delta_{j,k} 1_{j} \qquad \text{for }j,k\in\{0,1,2\},\\ 1_{0}+1_{1}+1_{2}&= 1.\\ 1_{j} \alpha_{jk} 1_{k}&= \alpha_{jk},\\ \alpha_{jk} \alpha_{km}&= 0\qquad \text{if } \{j,k,m\}=\{0,1,2\}, \\ \alpha_{01}\alpha_{10}&= N(1_{0}- \alpha_{02}\alpha_{20}),\\ \alpha_{10}\alpha_{01}&= N(1_{1}- \alpha_{12}\alpha_{21}),\\ p\cdot 1_{2} &= N(1_{2}- \alpha_{20}\alpha_{02}) + N(1_{2}- \alpha_{21}\alpha_{12}), \end{align}\] where \(N(t) \mathrel{:=}1 + t + \dotsb + t^{p-1}\). The \(\mathbb{Z}/2\)-grading on \(\mathfrak{K}_p\) is such that \(\alpha_{12}\) and \(\alpha_{21}\) are odd and all other generators are even.
This theorem is proven in the above form in Meyer:Actions_Kirchberg?, following the computations in Koehler:Thesis?. It is very convenient to name the following elements of \(\mathfrak{K}_p\): \[\begin{align} t_{0}&\mathrel{:=}1_{0}- \alpha_{02} \alpha_{20},&\qquad s_{1}&\mathrel{:=}1_{1}- \alpha_{12} \alpha_{21},\\ t_{2}&\mathrel{:=}1_{2}- \alpha_{20} \alpha_{02},&\qquad s_{2}&\mathrel{:=}1_{2}- \alpha_{21} \alpha_{12}. \end{align}\] The elements \(t_{j}\) and \(s_{j}\) are even. The relations in Theorem 5 say that \[\begin{align} \tag{1} \alpha_{01} \alpha_{10}&= N(t_{0}),\\ \tag{2} \alpha_{10} \alpha_{01}&= N(s_{1}),\\ \tag{3} p\cdot 1_{2} &= N(t_{2}) + N(s_{2}). \end{align}\] The domain and codomain projections of the elements above are shown in Figure 1.
The definitions of \(t_{j}\) and \(s_{j}\) say that \[\begin{align} \tag{4} \alpha_{02} \alpha_{20}&= 1_{0}- t_{0},\\ \tag{5} \alpha_{12} \alpha_{21}&= 1_{1}- s_{1},\\ \tag{6} \alpha_{20} \alpha_{02}&= 1_{2}- t_{2},\\ \tag{7} \alpha_{21} \alpha_{12}&= 1_{2}- s_{2}. \end{align}\] Equations 1 , 2 and 4 –7 express the products \(\alpha_{jk} \alpha_{kj}\) for all \(j, k \in \{0,1,2\}\) with \(j \neq k\), as polynomials in \(t_{j}\) or \(s_{j}\). All other products of two \(\alpha\)-generators are zero. The following relations follow from those in Theorem 5 and will be used in later proofs: \[\begin{align} \tag{8} t_{j}^p &=1\qquad \text{for }j=0,2,\\ \tag{9} s_{j}^p &=1\qquad \text{for }j=1,2,\\ \tag{10} t_{0} \alpha_{01}&= \alpha_{01},\\ \tag{11} t_{2} \alpha_{20}&= \alpha_{20} t_{0},\\ \tag{12} \alpha_{10} t_{0}&= \alpha_{10},\\ \tag{13} s_{1} \alpha_{10}&= \alpha_{10},\\ \tag{14} s_{2}\alpha_{21}&= \alpha_{21} s_{1},\\ \tag{15} \alpha_{01} s_{1}&= \alpha_{01},\\ \tag{16} N(t_{0}) \alpha_{02}&= 0,\\ \tag{17} \alpha_{20} N(t_{0}) &= 0,\\ \tag{18} N(s_{1})\alpha_{12}&= 0,\\ \tag{19} \alpha_{21} N(s_{1}) &= 0. \end{align}\] We also need to characterise the range of the invariant \(U_{\mathfrak{C}^{\mathbb{Z}/p}}\). Given a left \(\mathfrak{K}_p\)-module \(M\), write \(M= M_0 \oplus M_1 \oplus M_2\), corresponding to the decomposition of \(\mathfrak{K}_p\). Let \(\alpha_{jk}^M\) for \(j\neq k\) also denote the map \(M_k \to M_j\), \(y\mapsto \alpha_{jk} y\).
Definition 2. A left \(\mathfrak{K}_p\)-module \(M\) is exact if the two sequences of abelian groups clockwise and counterclockwise around the following triangle are exact: \[\begin{tikzcd}[row sep =5em, column sep =2.5em] &M_1 \ar[dr, shift right, "\alpha_{21}^M"'] \ar[dl, shift right, "\alpha_{01}^M"']&\\ M_0 \ar[rr, shift right, "\alpha_{20}^M"'] \ar[ur, shift right, "\alpha_{10}^M"']&& M_2 \ar[ul, shift right, "\alpha_{12}^M"'] \ar[ll, shift right, "\alpha_{02}^M"'] \end{tikzcd}\]
Theorem 6 (Koehler:Thesis?). Let \(M\) be a countable \(\mathbb{Z}/2\)-graded left \(\mathfrak{K}_p\)-module. The following are equivalent:
\(M= U_{\mathfrak{C}^{\mathbb{Z}/p}}(A)\) for some \(A\) in \(\mathfrak{B}^{\mathbb{Z}/p}\);
\(M= U_{\mathfrak{C}^{\mathbb{Z}/p}}(A)\) for some \(A\) in \(\mathrm{KK}^{\mathbb{Z}/p}\);
\(M\) is exact;
\(M\) has a projective \(\mathfrak{K}_p\)-module resolution of length \(1\).
This implies the following Universal Coefficient Theorem for \(\mathrm{KK}^{\mathbb{Z}/p}\):
Theorem 7 (Koehler:Thesis?). Let \(A\) and \(C\) be separable \(\mathbb{Z}/p\)-\(\mathrm C^*\)-algebras with \(A\in\mathfrak{B}^{\mathbb{Z}/p}\). Then there is a natural short exact sequence \[\mathop{\mathrm{Ext}}^1_{\mathfrak{K}_p}\bigl(U_{\mathfrak{C}^{\mathbb{Z}/p}}(\Sigma A), U_{\mathfrak{C}^{\mathbb{Z}/p}}(C)\bigr) \rightarrowtail \mathrm{KK}^{\mathbb{Z}/p}_*(A,C) \twoheadrightarrow \mathop{\mathrm{Hom}}_{\mathfrak{K}_p}\bigl(U_{\mathfrak{C}^{\mathbb{Z}/p}}(A),U_{\mathfrak{C}^{\mathbb{Z}/p}}(C)\bigr).\] If \(A,C\in\mathfrak{B}^{\mathbb{Z}/p}\), then every isomorphism \(U_{\mathfrak{C}^{\mathbb{Z}/p}}(A) \cong U_{\mathfrak{C}^{\mathbb{Z}/p}}(C)\) of \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_p\)-modules lifts to a \(\mathrm{KK}^{\mathbb{Z}/p}\)-equivalence in \(\mathrm{KK}^{\mathbb{Z}/p}_0(A,C)\). In particular, \(A\) and \(C\) are \(\mathrm{KK}^{\mathbb{Z}/p}\)-equivalent if and only if \(U_{\mathfrak{C}^{\mathbb{Z}/p}}(A) \cong U_{\mathfrak{C}^{\mathbb{Z}/p}}(C)\) as \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_p\)-modules.
3 Equivariant Kasparov groups for products of groups↩︎
A cyclic group of square-free order is a product of cyclic groups of prime order. To deal with equivariant KK-groups for such groups, we shall use the following proposition, which computes certain equivariant KK-groups for products of groups.
Definition 3. A \(\mathrm C^*\)-algebra \(A\in\mathrm{KK}^G\) is Poincaré dualisable if there is a \(\mathrm C^*\)-algebra \(A'\in\mathrm{KK}^G\), called its Poincaré dual, such that \(\mathrm{KK}^G(B\otimes A,C)\cong\mathrm{KK}^G(B,C\otimes A')\) for all \(B,C\in\mathrm{KK}^G\).
Proposition 8. Let \(H\) and \(G\) be finite groups. Let \(A_1,A_2 \in \mathfrak{B}^H\) and \(B_1,B_2 \in \mathfrak{B}^G\). Assume that \(A_1\) and \(B_1\) are Poincaré dualisable and that the abelian groups \(\mathrm{KK}^H(A_1,A_2)\) and \(\mathrm{KK}^G(B_1,B_2)\) are torsionfree. Then the external tensor product map induces an isomorphism \[\mathrm{KK}^H(A_1,A_2)\otimes_\mathbb{Z}\mathrm{KK}^G(B_1,B_2) \xrightarrow{\cong} \mathrm{KK}^{H\times G}( A_1\otimes B_1, A_2\otimes B_2 ).\]
Proof. Let \(A_1'\) and \(B_1'\) be the Poincaré duals of \(A_1\) and \(B_1\), respectively. Duality gives natural isomorphisms \[\begin{align} \pi_{A_1}\colon \mathrm{KK}^H(A_1,A_2) &\xrightarrow{\cong}\mathrm{KK}^H(\mathbb{C},\,A_2\otimes A_1'),\\ \pi_{B_1}\colon \mathrm{KK}^G(B_1,B_2) &\xrightarrow{\cong}\mathrm{KK}^G(\mathbb{C},\,B_2\otimes B_1'). \end{align}\] Next we apply the Green–Julg isomorphisms for finite groups, \[\begin{align} \mu_H\colon \mathrm{KK}^H(\mathbb{C},\,A_2\otimes A_1') &\xrightarrow{\cong} \mathrm{K}_0\bigl((A_2\otimes A_1')\rtimes H\bigr),\\\ \mu_G\colon \mathrm{KK}^G(\mathbb{C},\,B_2\otimes B_1') &\xrightarrow{\cong} \mathrm{K}_0\bigl((B_2\otimes B_1')\rtimes G\bigr). \end{align}\] It follows that \[\mathrm{KK}^H(A_1,A_2)\cong\mathrm{K}_0\bigl((A_2\otimes A_1')\rtimes H\bigr)\text{ and } \mathrm{KK}^G(B_1,B_2)\cong\mathrm{K}_0\bigl((B_2\otimes B_1')\rtimes G\bigr).\]
We claim that the Poincaré dual of an object in the equivariant bootstrap class also belongs to the equivariant bootstrap class. This follows from dellAmbrogio:Cell_G?*Proposition 2.9, where dualisable objects are called rigid. This is because any object that is dualisable in \(\mathrm{KK}^G\) must be compact, and by dellAmbrogio:Cell_G?*Proposition 2.9 all those compact objects already have a dual in the bootstrap class. The equivariant bootstrap class \(\mathfrak{B}^G\) is closed under tensor products because all tensor products of the generators \(\mathrm C(G/H) \otimes \mathrm C(G/L)\) belong to \(\mathfrak{B}^G\). Therefore, \(B_2\otimes B_1' \in \mathfrak{B}^G\). This implies that the crossed product \((B_2\otimes B_1')\rtimes G\) belongs to the nonequivariant bootstrap class. Similarly, \((A_2\otimes A_1')\rtimes H\) belongs to the bootstrap class. Then the Künneth Theorem applies to the tensor product of these two crossed products. Since the abelian groups \(\mathrm{KK}^H(A_1,A_2)\) and \(\mathrm{KK}^G(B_1,B_2)\) are torsionfree by assumption, the Tor-term in the Künneth formula vanishes. Thus the following map is an isomorphism: \[\begin{gather} \alpha\colon \mathrm{K}_0\bigl(( A_2\otimes A_1')\rtimes H\bigr)\otimes_\mathbb{Z} \mathrm{K}_0 \bigl(( B_2\otimes B_1')\rtimes G\bigr) \\ \to \mathrm{K}_0\Bigl( \bigl(( A_2\otimes A_1')\rtimes H\bigr) \otimes \bigl(( B_2\otimes B_1')\rtimes G\bigr)\Bigr). \end{gather}\] The universal property for crossed products implies that there is a canonical isomorphism \[\Phi\colon \bigl((A_2\otimes A_1')\rtimes H\bigr)\otimes \bigl((B_2\otimes B_1')\rtimes G\bigr) \xrightarrow{\cong}(A_2\otimes B_2\otimes A_1'\otimes B_1')\rtimes(H\times G),\] where the action of \(H\times G\) on \(A_2\otimes B_2\otimes A_1'\otimes B_1'\) is the diagonal one induced by the given actions. The inverse \(\mu^{-1}_{H\times G}\) of the Green–Julg isomorphism and the inverse Poincaré duality isomorphism \(\pi^{-1}_{A_1\otimes B_1}\) give isomorphisms \[\begin{gather} \mathrm{K}_0\bigl(( A_2\otimes B_2 \otimes A_1'\otimes B_1') \rtimes (H\times G)\bigr) \cong \mathrm{KK}^{H\times G}(\mathbb{C},A_2\otimes B_2 \otimes A_1'\otimes B_1') \\ \cong \mathrm{KK}^{H\times G}( A_1\otimes B_1, A_2\otimes B_2 ). \end{gather}\] Summing up, we constructed an isomorphism \[\mathrm{KK}^H(A_1,A_2)\otimes_\mathbb{Z}\mathrm{KK}^G(B_1,B_2) \xrightarrow{\psi} \mathrm{KK}^{H\times G}( A_1\otimes B_1, A_2\otimes B_2 )\] with \(\psi =\pi^{-1}_{A_1\otimes B_1}\circ \mu^{-1}_{H\times G}\circ \Phi_*\circ \alpha\circ \bigl((\mu_H\circ\pi_{A_1})\otimes (\mu_G\circ\pi_{B_1})\bigr)\). It remains to show that this map is the external tensor product map. The external Kasparov product is compatible with Poincaré duality and the Green–Julg isomorphism; that is \[\begin{align} \pi_{A_1\otimes B_1}(x\otimes_{\mathrm{ext}} y) &= \pi_{A_1}(x)\otimes_{\mathrm{ext}}\pi_{B_1}(y),\\ \mu_{H\times G}\bigl(\xi\otimes_{\mathrm{ext}}\eta\bigr) &= \Phi_*\bigl(\mu_H(\xi)\otimes \mu_G(\eta)\bigr). \end{align}\] Therefore, \[\begin{align} (\mu_{H\times G}\circ \pi_{A_1\otimes B_1})(x \otimes_{\mathrm{ext}} y) &= \mu_{H\times G}(\pi_{A_1}(x) \otimes_{\mathrm{ext}} \pi_{B_1}(y)) \\ &= \Phi_*(\alpha(\mu_H(\pi_{A_1}(x)) \otimes \mu_G(\pi_{B_1}(y)))) \\ &= \Phi_*\circ \alpha\circ ((\mu_H\circ\pi_{A_1}) \otimes (\mu_G\circ\pi_{B_1}))(x \otimes y). \end{align}\] Using this identity, we substitute \[\begin{align} \psi(x \otimes y) & = \pi^{-1}_{A_1\otimes B_1}\circ \mu^{-1}_{H\times G} \circ \Phi_* \circ \alpha\circ ((\mu_H\circ\pi_{A_1}) \otimes (\mu_G\circ\pi_{B_1}))(x \otimes y) \\ & = \pi^{-1}_{A_1\otimes B_1}\circ \mu^{-1}_{H\times G} \circ (\mu_{H\times G}\circ \pi_{A_1\otimes B_1}) (x \otimes_{\mathrm{ext}} y) = x\otimes_{\mathrm{ext}}y.\qedhere \end{align}\] ◻
4 Extension to cyclic groups of square-free order↩︎
In this section, we prove a Universal Coefficient Theorem for a cyclic group \(G\) of square-free order by reducing to the case of cyclic groups of prime order. Fix a presentation \[G\cong \mathbb{Z}/p_1\times\mathbb{Z}/p_2\times \dots \times \mathbb{Z}/p_k,\] where \(p_1, \dotsc, p_k\) are distinct prime numbers.
We define an invariant as follows. For each prime \(p_i\), let \(C_{p_i}\) denote the mapping cone of the \(G\)-equivariant unital embedding \(\mathbb{C}\to \mathrm C(\mathbb{Z}/p_i)\). Define \[\mathfrak{C}^G \mathrel{:=} \setgiven[\bigl]{ A_1 \otimes A_2 \otimes \dots \otimes A_k}{A_j \in \{ C_{p_j},\;\mathbb{C},\;\mathrm C(\mathbb{Z}/p_j) \}}.\] Here \(A_1 \otimes A_2 \otimes \dots \otimes A_k\) carries the action of \(G\) induced by the \(\mathbb{Z}/p_j\)-action on \(A_j\) for \(j=1,\dotsc,k\).
If \(H \subseteq G\) is a cyclic subgroup, then \(H= H_1 \times \dotsb \times H_k\) with \(H_j =\{1\}\) or \(H_j = \mathbb{Z}/p_j\) for \(j=1,\dotsc,k\). Then \(\mathrm C(G/H) = A_1 \otimes \dotsb \otimes A_k\) with \(A_j = \mathrm C(\mathbb{Z}/p_j)\) or \(A_j = \mathbb{C}\) for \(j=1,\dotsc,k\). Thus \(\mathrm C(G/H)\in \mathfrak{C}^G\) for all cyclic subgroups \(H\subseteq G\).
Tensoring the exact triangle \[\Sigma \mathrm C(\mathbb{Z}/p_j) \to C_{p_j}\to \mathbb{C}\to \mathrm C(\mathbb{Z}/p_j)\] with \(A_i\) as above shows that \[\bigl\langle \setgiven[\bigl]{ A_1 \otimes A_2 \otimes \dots \otimes A_k}{A_i \in \{\mathbb{C},\, \mathrm C(\mathbb{Z}/p_i) \}} \bigr\rangle = \langle \mathfrak{C}^G \rangle.\] Therefore, \(\mathfrak{C}^G\) generates the \(G\)-equivariant bootstrap class, \(\langle \mathfrak{C}^G \rangle = \mathfrak B^G\).
We denote the \(\mathbb{Z}/2\)-graded category ring of \((\mathfrak{C}^G)^\mathrm{op}\) by \(\mathfrak{K}_G\) or \(\mathfrak{K}_{p_1\dotsm p_k}\). For \(G=\mathbb{Z}/p\), this is the ring introduced by Manuel Köhler (see Section 2.3).
Remark 9. It is crucial that \(G\) has square-free order. If \[G\cong \mathbb{Z}/p_1\times\mathbb{Z}/p_2\times \dots \times \mathbb{Z}/p_k\] but the primes \(p_j\) are not all distinct, then there is a subgroup \(H\) that is not of product type as above. Then the generator \(\mathrm C(G/H)\) of \(\mathfrak{B}^G\) does not belong to \(\mathfrak{C}^G\). By Lemma 1, the localising subcategory generated by the generators in \(\mathfrak{C}^G\) is smaller than the equivariant bootstrap class. Therefore, the invariant \(U_{\mathfrak{C}^G}\) cannot give a Universal Coefficient Theorem for \(G\) that holds on the entire bootstrap class.
Corollary 1. Let \(p_1,\dotsc, p_k\) be distinct primes, and let \(G \cong \mathbb{Z}/p_1 \times \dots \times \mathbb{Z}/p_k\). The external tensor product induces an isomorphism of rings \(\bigotimes_{i=1}^k \mathfrak{K}_{p_i}\cong \mathfrak{K}_{p_1\dotsm p_k}\), where the tensor product is taken over the ring of integers.
Proof. All objects in \(\mathfrak{C}^G\) are Poincaré dualisable by dellAmbrogio:Cell_G?*Proposition 2.9, where these are referred to as rigid objects. Then, Proposition 8 applied \(k-1\) times implies, by induction, that the external tensor product induces a group isomorphism \(\bigotimes_{i=1}^k \mathfrak{K}_{p_i}\cong \mathfrak{K}_{p_1\dotsm p_k}\). This is even a ring isomorphism because the external tensor product preserves Kasparov products. ◻
We now generalise Definition 2 to finite cyclic groups of square-free order.
Definition 4. Let \(Q\) be a left module over \(\mathfrak{K}_G\). The external tensor product map induces canonical maps \(\mathfrak{K}_{p_m} \to \bigotimes_{i=1}^k \mathfrak{K}_{p_i} \cong \mathfrak{K}_G\) for every \(m=1,\dotsc,k\). We call \(Q\) exact if it is exact as a \(\mathfrak{K}_{p_m}\)-module for \(m=1,\dotsc,k\).
Any \(\mathfrak{K}_G\)-module of the form \(U_{\mathfrak{C}^G}(A)\) for \(A\in\mathrm{KK}^G\) is exact: this follows from the corresponding statement for cyclic groups of prime order.
Lemma 2. Let \(Q\) be an exact left \(\mathfrak{K}_G\)-module that is countably generated and free as an abelian group. Then \(\mathop{\mathrm{Ext}}^1_{\mathfrak{K}_{G}}(Q,Q')= 0\) for any left \(\mathfrak{K}_{G}\)-module \(Q'\). Therefore, \(Q\) is a projective module over \(\mathfrak{K}_G\).
Proof. Recall that \(\mathop{\mathrm{Ext}}^1_{\mathfrak{K}_{G}}(Q,Q')= 0\) for any left \(\mathfrak{K}_{G}\)-module \(Q'\) if and only if \(Q\) is a projective \(\mathfrak{K}_{G}\)-module. We prove this by induction on the number \(k\) of prime factors of \(\abs{G}\). The base case for a single prime was proven by Manuel Köhler, see Koehler:Thesis?*Theorem 12.4 or Theorem 6. Let \[\sigma\colon 0\to Q'\to Q'' \xrightarrow{\beta} Q\to 0\] be any extension of left \(\mathfrak{K}_{p_1 \dotsm p_k}\)-modules. We must prove that it splits by a module homomorphism. Corollary 1 implies \[\mathfrak{K}_{p_1\dotsm p_k} \cong \biggl( \bigotimes_{i=1}^{k-1} \mathfrak{K}_{p_i} \biggr) \otimes_{\mathbb{Z}} \mathfrak{K}_{p_k} \cong \mathfrak{K}_{p_1\dotsm p_{k-1}}\otimes_{\mathbb{Z}} \mathfrak{K}_{p_k}.\] By the induction hypothesis, \(Q\) is projective over \(\mathfrak{K}_{p_1 \dotsm p_{k-1}}\). Therefore, \(\sigma\) splits by a \(\mathfrak{K}_{p_1 \dotsm p_{k-1}}\)-linear map \(\gamma \colon Q\to Q''\). We are going to define another map \(\gamma\colon Q\to Q''\) that it is both \(\mathfrak{K}_{p_k}\)- and \(\mathfrak{K}_{p_1 \dotsm p_{k-1}}\)-linear. Then it is a module map over the whole ring \(\mathfrak{K}_{p_1\dotsm p_{k-1}}\otimes_{\mathbb{Z}} \mathfrak{K}_{p_k} \cong \mathfrak{K}_{p_1 \dotsm p_k}\). In the following computations, we abbreviate \(p \mathrel{:=}p_k\) and suppress some tensor products to avoid clutter. The generators \(\alpha_{i j}\), \(t_j\) and \(s_j\) are those in the ring \(\mathfrak{K}_p\). We change \(\gamma\) in three steps.
The first step makes \(\gamma\) compatible with the decomposition \(Q= Q_0\oplus Q_1 \oplus Q_2\), where \(Q_j \mathrel{:=}1_j Q\) with the idempotent generators of \(\mathfrak{K}_p\). The map \(\gamma\) is represented by a \(3\times 3\)-matrix of maps \(1_l \gamma 1_j\colon Q_j \to Q_l''\) for \(j,l\in\{0,1,2\}\). As \(\beta\) is \(\mathfrak{K}_p\)-linear, \[\beta\left(\sum_{i=1}^3 1_i\gamma 1_i\right) = \sum_{i=1}^3 1_i\beta\gamma 1_i = \sum_{i=1}^3 1_i \mathrm{id}_{Q} 1_i = \mathrm{id}_Q.\] So we may replace \(\gamma\) by \(\sum_{i=1}^3 1_i\gamma 1_i\), which makes it diagonal. We assume from now on that \(\gamma\) is diagonal and write \[\gamma = \gamma_0+\gamma_1+\gamma_2\] with maps \(\gamma_i\colon Q_i \to Q_i''\) for \(i=0,1,2\).
In the second step, we define a new map \(\gamma'\) that commutes with \(t_0\) and \(s_1\). We achieve this by employing the relations 8 and 9 to average over \(t_0\) and \(s_1\) at the objects \(0\) and \(1\), respectively (see Figure 1), and multiplying by \(p\) at the object \(2\). More precisely, let \[\gamma' \mathrel{:=} \sum_{i=0}^{p-1}t^i_0\gamma_0 t^{p-i}_0 + \sum_{i=0}^{p-1}s^i_1\gamma_1 s^{p-i}_1 + p \gamma_2.\] This map \(\gamma'\) commutes with \(t_0\) and \(s_1\), so that it also commutes with \(N(t_0)\) and \(N(s_1)\).
In the third step, we replace \(\gamma'\) by a map \(\gamma''\) that commutes with \(\alpha_{ij}\) for all distinct \(i,j\in\{0,1,2\}\). The idea behind this is to think of \(\alpha_{ij}\) as a partial isometry up to the constant \(p\) and to identify the corresponding \(p\)-projection elements in the ring. Note that \[\begin{align} p\cdot 1_0 -N(t_0) &= (1_0-t_0)\sum_{i=0}^{p-2}(p-i-1)t_0^i,\\ p\cdot 1_1 -N(s_1) &= (1_1-s_1)\sum_{i=0}^{p-2}(p-i-1)s_1^i. \end{align}\] Then we define \[\begin{align} \gamma''_0 &\mathrel{:=}p\cdot \gamma'_0,\\ \gamma''_1 &\mathrel{:=}\alpha_{10}\gamma'_0 \alpha_{01} + (p\cdot 1_1-\alpha_{10}\alpha_{01})\gamma'_1,\\ \gamma''_2 &\mathrel{:=}\alpha_{20}\gamma'_0 \left(\sum_{i=0}^{p-2}(p-i-1)t_0^i\right)\alpha_{02} + \alpha_{21}\gamma'_1 \left(\sum_{i=0}^{p-2}(p-i-1)s_1^i\right)\alpha_{12}. \end{align}\] We claim that \(\gamma''\) commutes with \(\alpha_{01}\) and \(\alpha_{10}\). The relations 1 , 2 , 10 , and 13 imply \[\begin{align} \alpha_{01}\alpha_{10}\alpha_{01} &= N(t_0)\alpha_{01}= p\alpha_{01},\\ \alpha_{10}\alpha_{01}\alpha_{10} &= N(s_1)\alpha_{10}= p\alpha_{10}. \end{align}\] Therefore, \[\begin{gather} \gamma''\alpha_{10} = \gamma''_1 \alpha_{10} = \alpha_{10}\gamma'_0\alpha_{01}\alpha_{10} + p\gamma'_1\alpha_{10} - \alpha_{10}\alpha_{01}\gamma_1'\alpha_{10} \\= \alpha_{10}\gamma'_0N(t_0)+p\gamma'_1\alpha_{10} - N(s_1)\gamma'_1\alpha_{10} = \alpha_{10}\alpha_{01}\alpha_{10}\gamma'_0 + p\gamma'_1\alpha_{10} - \gamma'_1\alpha_{10}\alpha_{01}\alpha_{10} \\= p\alpha_{10}\gamma'_0 +p\gamma'_1\alpha_{10} - p\gamma'_1\alpha_{10} = p\alpha_{10}\gamma'_0 = \alpha_{10}\gamma''_0 = \alpha_{10}\gamma''. \end{gather}\] Similar computations show that \(\alpha_{01}\gamma'' = \gamma''\alpha_{01}\) and that \(\gamma''\) commutes with \(s_1\) and \(t_0\).
Next, using the relations 4 , 5 , 16 , 17 , 18 , and 19 and that \(\alpha_{mk}\alpha_{kj}=0\) for \(\{j,k,m\}=\{0,1,2\}\), we compute \[\begin{gather} \gamma''\alpha_{21} = \gamma''_2\alpha_{21} = \alpha_{21}\gamma'_1 \left(\sum_{i=0}^{p-2}(p-i-1)s_1^i\right)\alpha_{12}\alpha_{21} \\= \alpha_{21}\gamma'_1 \left(\sum_{i=0}^{p-2}(p-i-1)s_1^i\right)(1_1-s_1) = \alpha_{21}\gamma'_1(p\cdot 1_1-N(s_1)) \\= p\alpha_{21}\gamma'_1 - \alpha_{21}N(s_1)\gamma'_1 = p\alpha_{21}\gamma'_1 = \alpha_{21}\gamma''_1 = \alpha_{21}\gamma'', \end{gather}\] \[\begin{gather} \gamma''\alpha_{12} = \gamma''_1\alpha_{12} = \alpha_{10}\gamma'_0 \alpha_{01}\alpha_{12} + (p\cdot 1_1-\alpha_{10}\alpha_{01})\gamma'_1\alpha_{12} \\= p\gamma'_1\alpha_{12} - \gamma'_1\alpha_{10}\alpha_{01}\alpha_{12} = p\gamma'_1\alpha_{12} = \gamma'_1 (p\cdot1_1-N(s_1))\alpha_{12} \\= \alpha_{12}\alpha_{21}\gamma'_1 \left(\sum_{i=0}^{p-2}(p-i-1)s_1^i\right)\alpha_{12} = \alpha_{12}\gamma''_2 = \alpha_{12}\gamma''. \end{gather}\] Similarly, \(\gamma''\alpha_{20} = \alpha_{20}\gamma''\) and \(\gamma''\alpha_{02} = \alpha_{02}\gamma''\). As the elements \(\alpha_{ij}\) generate the ring \(\mathfrak{K}_p\), it follows that \(\gamma''\) is \(\mathfrak{K}_p\)-linear. Since it remains \(\mathfrak{K}_{p_1\dotsm p_{k-1}}\)-linear, it is \(\mathfrak{K}_{p_1\dotsm p_{k-1} p}\)-linear.
The map \(\gamma''\) is not a section for \(\sigma\). Instead, \[\beta\gamma' = \sum_{i=0}^{p-1}t^i_0\beta 1_0\gamma t^{p-i}_0 + \sum_{i=0}^{p-1}s^i_1\beta 1_1\gamma s^{p-i}_1 + p \beta1_2 \gamma = p\cdot(1_0+1_1+1_2) = p\cdot 1_Q\] and \[\begin{gather} \beta\gamma'' = \beta(\gamma_0''+\gamma_1''+\gamma_2'') = p\beta \gamma'_0 +\alpha_{10}\beta \gamma'_0 \alpha_{01} + (p\cdot 1_1-\alpha_{10}\alpha_{01})\beta\gamma'_1 \\ +\alpha_{20}\beta\gamma'_0 \left(\sum_{i=0}^{p-2}(p-i-1)t_0^i\right)\alpha_{02} + \alpha_{21}\beta\gamma'_1 \left(\sum_{i=0}^{p-2}(p-i-1)s_1^i\right)\alpha_{12} \\= p^2\cdot 1_0 + p^2\cdot 1_1 + p(p\cdot 1_2-N(t_2))+ p(p\cdot 1_2-N(s_2)) \\= p^2\cdot 1_0 + p^2\cdot 1_1 + p^2\cdot 1_2 = p^2\cdot 1_Q. \end{gather}\] Here we used the relations 3 , 6 , 7 , 11 and 14 .
The above calculations say that \(\gamma''\) is a section for the extension \(p^2\sigma\). Thus \([p^2\sigma]=0\) in the group \(\mathop{\mathrm{Ext}}^1_{\mathfrak{K}_{p_1\dotsm p_{k-1} p}}(Q,Q')\). The same argument may be used for \(p_1\) instead of \(p\). This shows \([p_1^2\sigma]=0\). Since \(p\) and \(p_{1}\) are coprime, there are \(m,n\in\mathbb{Z}\) with \(m p^2 + n p_1^2 =1\). So \[[\sigma] = 1[\sigma] = m[p^2\sigma]+n [p_1^2\sigma] = 0 \quad \text{in }\mathop{\mathrm{Ext}}^1_{\mathfrak{K}_{p_1\dotsm p_{k-1}p}}(Q,Q').\qedhere\] ◻
Proposition 10. Let \(G\) be a finite cyclic group of square-free order. Let \(Q\) be a countable \(\mathbb{Z}/2\)-graded left \(\mathfrak{K}_G\)-module. The following are equivalent:
\(Q= U_{\mathfrak{C}^{G}}(A)\) for some \(A\) in \(\mathfrak{B}^{G}\);
\(Q= U_{\mathfrak{C}^{G}}(A)\) for some \(A\) in \(\mathrm{KK}^{G}\);
\(Q\) is exact;
\(Q\) has a projective \(\mathfrak{K}_G\)-module resolution of length \(1\).
Proof. The implication (1)\(\Rightarrow\)(2) is trivial, (2)\(\Rightarrow\)(3) is an easy consequence of the definition of \(\alpha_{i j}\), and (3)\(\Rightarrow\)(4) is Lemma 2. The implication (4)\(\Rightarrow\)(1) is a general statement in the setting of homological algebra in triangulated categories; see, for example, Bentmann-Meyer:More_general?*Proposition 2.3. ◻
Finally, Theorem 1 follows from Theorem 2 because Proposition 10 gives the required length-\(1\) projective resolutions.
5 Uniquely divisible modules↩︎
We now prove stronger results when a \(\mathfrak{K}_G\)-module is uniquely divisible for some of the primes dividing \(\abs{G}\).
Recall that for a prime \(p\), a module \(Q\) is called uniquely \(p\)-divisible if multiplication by \(p\) on \(Q\) is invertible. For a \(\mathfrak{K}_G\)-module \(Q\), a prime \(p\) dividing the order of \(G\), and \(i \in \{0, 1, 2\}\), let \(Q_i^{(p)}\) denote the direct summand \(1^{(p)}_i\cdot Q\) defined by the projection \(1^{(p)}_i\in\mathfrak{K}_p\). This is still a module over \(\mathfrak{K}_{\abs{G}/p}\), the tensor product of \(\mathfrak{K}_{p_j}\) for the other prime divisors of \(\abs{G}\), and \(Q= Q^{(p)}_0 + Q^{(p)}_1 + Q^{(p)}_2\). By \(\vartheta_p\) we denote a primitive \(p\)th root of unity.
Proposition 11. Let \(G\) be a finite cyclic group of square-free order \(\abs{G} = p_1 \cdot p_2 \dotsm p_n\) with distinct primes \(p_1,\dotsc,p_n\). Let \(Q\) be an exact \(\mathfrak{K}_G\)-module. Let \(1\le k \le n\). Suppose that for each prime \(p\in\{p_1,\dotsc, p_k\}\), one of the components \(Q_0^{(p)}\), \(Q_1^{(p)}\) or \(Q_2^{(p)}\) is uniquely \(p\)-divisible. Then, for each such prime, the remaining two components are also uniquely \(p\)-divisible.
For \(I\in \{X,Y,Z\}^k\), let \(R_I = \mathbb{Z}[1/p_1p_2 \dotsm p_k] [\setgiven{\vartheta_{p_j}}{I_j\in\{Y,Z\}}]\). There are \(3^k\) \(\mathbb{Z}/2\)-graded exact \(\mathfrak{K}_{p_{k+1}\dotsm p_n} \otimes_\mathbb{Z}R_I\)-modules \((A_I)_{I\in \{X,Y,Z\}^k}\) that together determine the module \(Q\) uniquely, in a way explained during the proof. If \(k=n\), then \(A_I\) is simply an \(R_I\)-module.
Proof. We fix \(p\in\{p_1,\dotsc,p_k\}\) and focus on the \(\mathfrak{K}_p\)-module structure on \(Q\) for a moment. By Meyer:Actions_Kirchberg?*Theorem 7.2, the pieces \(Q_0^{(p)}\), \(Q_1^{(p)}\) and \(Q_2^{(p)}\) are uniquely \(p\)-divisible. There are a \(\mathbb{Z}/2\)-graded \(\mathbb{Z}[1/p]\)-module \(A_X\) and \(\mathbb{Z}/2\)-graded \(\mathbb{Z}[1/p, \vartheta_{p}]\)-modules \(A_Y,A_Z\) such that \(Q\) has the following form as a \(\mathfrak{K}_{p}\)-module: \[\begin{align} {3} Q_0^{(p)} &= A_X\oplus A_Y, &\; Q_1^{(p)} &= A_X\oplus A_Z, &\; Q_2^{(p)} &= A_Y\oplus \Sigma A_Z,\\ \alpha_{01}^{Q^{(p)}} &= \begin{pmatrix} 1^{A_X}&0\\0&0 \end{pmatrix}, &\; \alpha_{12}^{Q^{(p)}} &= \begin{pmatrix} 0&0\\0&(1-\vartheta_{p})^{A_Z} \end{pmatrix}, &\; \alpha_{20}^{Q^{(p)}} &= \begin{pmatrix} 0&1^{A_Y}\\0&0 \end{pmatrix},\\ \alpha_{10}^{Q^{(p)}} &= \begin{pmatrix} {p}^{A_X}&0\\0&0 \end{pmatrix}, &\; \alpha_{21}^{Q^{(p)}} &= \begin{pmatrix} 0&0\\0&1^{A_Z} \end{pmatrix}, &\; \alpha_{02}^{Q^{(p)}} &= \begin{pmatrix} 0&0\\(1-\vartheta_{p})^{A_Y}&0 \end{pmatrix}. \end{align}\] Here \(\Sigma A_Z\) means \(A_Z\) with opposite parity. Also, \[\begin{align} {2} N(t_{0}^{Q^{(p)}}) &= \begin{pmatrix} p&0\\0&0 \end{pmatrix}, &\qquad N(s_{1}^{Q^{(p)}}) &= \begin{pmatrix} p&0\\0&0 \end{pmatrix},\\ N(t_{2}^{Q^{(p)}}) &= \begin{pmatrix} 0&0\\0&p \end{pmatrix}, &\qquad N(s_{2}^{Q^{(p)}}) &= \begin{pmatrix} p&0\\0&0 \end{pmatrix}. \end{align}\] We denote this module by \(Q^{(p)}(A_X,A_Y,A_Z)\). The following elements of \(\mathfrak{K}_p[1/p]\) are central idempotents, projecting onto \(Q^{(p)}(A_X,0,0)\), \(Q^{(p)}(0,A_Y,0)\) and \(Q^{(p)}(0,0,A_Z)\), respectively: \[\begin{align} E_X^{Q^{(p)}} &\mathrel{:=}\frac{1}{p}N(t_0^{Q^{(p)}}) + \frac{1}{p}N(s_1^{Q^{(p)}}),\\ E_Y^{Q^{(p)}} &\mathrel{:=} \left(1_0^{Q^{(p)}} - \frac{1}{p}N(t_0^{Q^{(p)}})\right) + \frac{1}{p}N(s_2^{Q^{(p)}}),\\ E_Z^{Q^{(p)}} &\mathrel{:=}\left(1_1^{Q^{(p)}} - \frac{1}{p}N(s_1^{Q^{(p)}})\right) + \frac{1}{p}N(t_2^{Q^{(p)}}). \end{align}\] They are well defined because of the unique \(p\)-divisibility assumption.
Now we consider again the entire \(\mathfrak{K}_G\)-module structure on \(Q\). By the tensor product decomposition of Corollary 1, the actions of \(\mathfrak{K}_{p_i}\) and \(\mathfrak{K}_{p_j}\) on \(Q\) commute for all \(i,j\in\{1,\dotsc,n\}\). Each \(E_I^{Q^{(p_j)}}\) belongs to the centre of \(\mathfrak{K}_G[1/p_j]\) and so these are commuting idempotent \(\mathfrak{K}_G\)-module endomorphisms of \(Q\). Thus, \[\label{eq:comp95idem} E_X^{Q^{(p_j)}} + E_Y^{Q^{(p_j)}} + E_Z^{Q^{(p_j)}} = 1_0^{Q^{(p_j)}} +1_1^{Q^{(p_j)}}+1_2^{Q^{(p_j)}} = 1_{Q^{(p_j)}(A_X,A_Y,A_Z)}\tag{20}\] by 3 , and \[Q^{(p_j)}(A_X,A_Y,A_Z) = Q^{(p_j)}(A_X,0,0) \oplus Q^{(p_j)}(0,A_Y,0) \oplus Q^{(p_j)}(0,0,A_Z)\] is a \(\mathfrak{K}_G\)-module isomorphic to \(Q\), where the action of \(\mathfrak{K}_{p_j}\) is determined by \(\alpha_{i k}^{Q^{(p_j)}}\).
Since \(A_X = \mathop{\mathrm{im}}(\alpha_{01}^{Q^{(p_j)}})\), \(A_Y = \mathop{\mathrm{im}}(\alpha_{02}^{Q^{(p_j)}})\), and \(A_Z=\mathop{\mathrm{im}}(\alpha_{12}^{Q^{(p_j)}})\), these components are \(\mathfrak{K}_{p_i}\)-modules for all \(p_i\neq p_j\) by Corollary 1. Also, since \(A_X\), \(A_Y\) and \(A_Z\) are direct \(\mathfrak{K}_{p_i}\)-module summands of \(Q^{(p_j)}(A_X,0,0)\), \(Q^{(p_j)}(0,A_Y,0)\) and \(Q^{(p_j)}(0,0,A_Z)\), respectively, they are themselves exact \(\mathfrak{K}_{p_i}\)-modules. Of course, \(A_X\), \(A_Y\), \(A_Z\) completely determine \(Q^{(p_j)}(A_X,0,0)\), \(Q^{(p_j)}(0,A_Y,0)\) and \(Q^{(p_j)}(0,0,A_Z)\).
When we iterate the decomposition process above for all the divisible primes \(p_1, p_2,\dots ,p_k\), we get \(3^k\) modules \[Q_I \mathrel{:=} \left(\bigotimes_{j=1}^k E_{I_j}^{Q^{(p_j)}}\right) Q\] for \(I\in\{X,Y,Z\}^k\). Here \(Q_I\) is determined by a \(\mathbb{Z}/2\)-graded module \(A_I\) over the ring \(\mathfrak{K}_{p_{k+1}\dots p_n} \otimes_\mathbb{Z}R_I\). Specifically, \[A_I = \left(\prod_{j=1}^k 1_{c(I_j)}^{(p_j)}\right)Q_I = \bigcap_{j=1}^k(Q_I)_{c(I_j)}^{(p_j)},\] for \(c(X)=0,\) \(c(Y)=0\), and \(c(Z)=1\).
By 20 , there is an isomorphism of \(\mathfrak{K}_G\)-modules \[\bigoplus_{I\in\{X,Y,Z\}^k} Q_I = \bigoplus_{I\in\{X,Y,Z\}^k} \left(\bigotimes_{j=1}^k E_{I_j}^{Q^{(p_j)}}\right) Q\cong Q. \qedhere\] ◻
We now consider the special case when \(G\cong \mathbb{Z}/p \times \mathbb{Z}/q\) for two different primes \(p,q\). Let \(Q\) be a uniquely \(q\)-divisible \(\mathfrak{K}_G\)-module. By Proposition 11, \(Q\) may be specified by a triple of exact \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_p\)-modules \((A_X,A_Y,A_Z)\), where \(A_X\) is also a \(\mathbb{Z}[1/q]\)-module and \(A_Y, A_Z\) are modules over \(\mathbb{Z}[1/q, \vartheta_q]\). Then, as shown in the proof of Proposition 11, \[Q\cong \bigl(E_X^{Q^{(q)}}+ E_Y^{Q^{(q)}}+ E_Z^{Q^{(q)}}\bigr)Q = Q^{(q)}(A_X,A_Y,A_Z)\] as \(\mathfrak{K}_G\)-modules.
One way to get candidates for \(A_X\), \(A_Y\) and \(A_Z\) is to take any exact \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_p\)-modules \(A_X^0\), \(A_Y^0\) and \(A_Z^0\) and form \[A_X \mathrel{:=}A_X^0 \otimes_\mathbb{Z}\mathbb{Z}[1/q],\qquad A_Y \mathrel{:=}A_Y^0 \otimes_\mathbb{Z}\mathbb{Z}[1/q,\vartheta_q],\qquad A_Z \mathrel{:=}A_Z^0 \otimes_\mathbb{Z}\mathbb{Z}[1/q,\vartheta_q].\] Since \(\mathbb{Z}[1/q]\) and \(\mathbb{Z}[1/q,\vartheta_q]\) are flat, these are also exact \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_p\)-modules, and they now gain the extra structure that allows us to turn them into uniquely \(q\)-divisible \(\mathfrak{K}_G\)-modules. In particular, we may let \(A_I^0\) be one of the examples in Meyer:Actions_Kirchberg?*Section 9. These examples are important because, up to an extension by a uniquely \(p\)-divisible exact module, they give all examples of exact \(\mathbb{Z}/2\)-graded \(\mathfrak{K}_p\)-modules \(M\) where the ingredient \(M_1\) is a cyclic group. Except for Examples 9.2 and 9.4 in Meyer:Actions_Kirchberg?, we have \(M_1 = \mathbb{Z}/p^k\) for some integer \(k\ge1\), so that \(M_1 \otimes_\mathbb{Z} \mathbb{Z}[1/q] \cong M_1\). This means that if \(A_X^0\) is one of these examples, then \(A_X^0 \otimes_\mathbb{Z}\mathbb{Z}[1/q]\) also corresponds to a group action on a Cuntz algebra.
In contrast, tensoring with \(\mathbb{Z}[\vartheta_q,1/q]\) always makes the groups bigger, so that \(M_1\otimes_\mathbb{Z}\mathbb{Z}[\vartheta_q,1/q]\) cannot be cyclic any more. Sometimes, we may instead find the required action of \(\mathbb{Z}[1/q,\vartheta_q]\) on \(A_Y^0\) or \(A_Z^0\) itself:
Example 2. In Examples 9.7, 9.9, 9.11, and 9.12 in Meyer:Actions_Kirchberg?, the pieces \(M_0\), \(M_1\) and \(M_2\) in the \(\mathfrak{K}_p\)-module \(M\) are all purely \(p\)-torsion. This makes \(M\) a module over the ring \(\mathbb{Z}_p\) of \(p\)-adic integers in a natural way. By naturality, the \(\mathbb{Z}_p\)-module structure commutes with the action of the ring \(\mathfrak{K}_p\), making \(M\) a module over \(\mathfrak{K}_p \otimes_\mathbb{Z}\mathbb{Z}_p\). If there is a homomorphism \(\mathbb{Z}[1/q,\vartheta_q] \to \mathbb{Z}_p\), then we may use this to define an action of \(\mathbb{Z}[1/q,\vartheta_q] \otimes_\mathbb{Z} \mathfrak{K}_p\) on \(M\), which then allows us to take \(M\) for the pieces \(A_Y\) or \(A_Z\) in our decomposition. Such a homomorphism exists if and only if \(p\neq q\) (which we assume anyway) and \(q \mid p-1\). The latter ensures that there is a primitive \(q\)th root of unity in the multiplicative group of the field \(\mathbb{Z}/p\) and hence also in \(\mathbb{Z}_p\) by Hensel’s Lemma.
When \(Q\) is both uniquely \(p\)- and \(q\)-divisible, then it is \(\abs{G}\)-divisible. This suffices to apply the UCT proven in Meyer-Nadareishvili:UCT_actions?*Theorem 1.1. We also get this information in a more concrete way from Proposition 11. Namely, we may specify \(Q\) by \(9\) \(\mathbb{Z}/2\)-graded abelian groups \[A_{(X,X)}, A_{(X,Y)}, A_{(X,Z)}, A_{(Y,X)}, A_{(Y,Y)}, A_{(Y,Z)}, A_{(Z,X)}, A_{(Z,Y)}, A_{(Z,Z)},\] where \(A_{(X,X)}\) is a module over \(\mathbb{Z}[1/pq]\), \(A_{(X,Y)}\) and \(A_{(X,Z)}\) are modules over \(\mathbb{Z}[1/pq, \vartheta_q]\), \(A_{(Y,X)}\) and \(A_{(Z,X)}\) are modules over \(\mathbb{Z}[1/pq, \vartheta_p]\), whereas \(A_{(Y,Y)}\), \(A_{(Y,Z)}\), \(A_{(Z,Y)}\), and \(A_{(Z,Z)}\) are modules over \(\mathbb{Z}[1/pq, \vartheta_p, \vartheta_q]\).
Separating the exact \(\mathfrak{K}_G\)-module into its \(9\) independent direct summands over \(\mathfrak{K}_q\), we get: \[\begin{align} {3} Q&\cong & \Bigl( & \phantom{{}+{}} E_{X}^{Q^{(p)}} \otimes E_{X}^{Q^{(q)}} + E_{X}^{Q^{(p)}} \otimes E_{Y}^{Q^{(q)}} + E_{X}^{Q^{(p)}} \otimes E_{Z}^{Q^{(q)}} \\ & &&+ E_{Y}^{Q^{(p)}} \otimes E_{X}^{Q^{(q)}} + E_{Y}^{Q^{(p)}} \otimes E_{Y}^{Q^{(q)}} + E_{Y}^{Q^{(p)}} \otimes E_{Z}^{Q^{(q)}} \\ & &&+ E_{Z}^{Q^{(p)}} \otimes E_{X}^{Q^{(q)}} + E_{Z}^{Q^{(p)}} \otimes E_{Y}^{Q^{(q)}} + E_{Z}^{Q^{(p)}} \otimes E_{Z}^{Q^{(q)}}\Bigr) Q\\ &\cong &&\phantom{{}\oplus{}} Q^{(q)}(A_{(X,X)},0,0) \oplus Q^{(q)}(0,A_{(X,Y)},0) \oplus Q^{(q)}(0,0,A_{(X,Z)}) \\ & &&\oplus Q^{(q)}(A_{(Y,X)},0,0) \oplus Q^{(q)}(0,A_{(Y,Y)},0) \oplus Q^{(q)}(0,0,A_{(Y,Z)}) \\ & &&\oplus Q^{(q)}(A_{(Z,X)},0,0) \oplus Q^{(q)}(0,A_{(Z,Y)},0) \oplus Q^{(q)}(0,0,A_{(Z,Z)}). \end{align}\]