1. Introduction
In the field of
$C^*$
-algebra classification, K-theory computation has long been a captivating question. An algebra’s K-theory groups will fully classify simple, separable, nuclear and purely infinite
$C^*$
-algebras satisfying the universal coefficient theorem (UCT) by the Kirchberg–Phillips classification theorem [Reference Kirchberg8, Reference Kirchberg and Phillips9]. Further, the Elliott invariant, which consists of the K-theory groups as well as information about the trace simplex, has shown incredible versatility in
$C^*$
-algebra classification [Reference Elliott, Gong, Lin and Niu2, Reference Gong, Lin and Niu4, Reference Gong, Lin and Niu5].
On the other hand, higher-rank graph
$C^*$
-algebras have been an active field of research for the last quarter century. These algebras were introduced in [Reference Kumjian and Pask10], building on work in [Reference Robertson and Steger12]. They have offered a number of creative solutions to long-standing problems in the field. They were used to show that every UCT Kirchberg algebra has nuclear dimension zero [Reference Ruiz, Sims and Sørensen14], they have been related to solutions of the Yang–Baxter equations [Reference Vdovina16, Reference Yang18] and they have reformulated multipullback quantum odd sphere information in terms of universal properties [Reference Hajac, Nest, Pask, Sims and Zieliński6].
With all of these promising results and many more on the horizon, it seems crucial to refine our understanding of higher-rank graph algebra K-theory. A landmark paper in this direction was Evans [Reference Evans3], in which the work of [Reference Kasparov7, Reference Robertson and Steger13, Reference Schochet15] was developed into a spectral sequence converging to the K-theory of a higher-rank graph
$C^*$
-algebra. The goal of this paper is to better understand the tools developed by Evans and to use them to make a number of explicit computations.
2. Preliminaries
2.1. Notation
We assume that
$0\in \mathbb {N}$
. Further for
$1\leq j\leq k$
, we use
$\kappa _j:\mathbb {N} \to \mathbb {N}^k$
to denote the inclusion into the jth coordinate and
$\iota _j:\mathbb {N}^j\to \mathbb {N}^k$
to denote the canonical inclusion onto the first j generators. For a small category,
$\Lambda $
, we denote the object set as
$\Lambda ^0$
. Lastly, for a countable category,
$\Lambda $
, we define the group
$\mathbb {Z}\Lambda ^0:=\bigoplus _{v\in \Lambda ^0} \mathbb {Z}$
.
Definition 2.1 [Reference Kumjian and Pask10, Definition 1.1].
Let
$\Lambda $
be a countable category and let
${d:\Lambda \to \mathbb {N}^k}$
be a functor. If
$(\Lambda , d)$
satisfies the factorisation property, that is, for every morphism
$\lambda \in \Lambda $
and
$n,m\in \mathbb {N}^k$
such that
$d(\lambda )=m+n$
, there exist unique
$\mu ,\nu \in \Lambda $
such that
$d(\mu )=m$
,
$d(\nu )=n$
and
$\lambda = \mu \nu $
, then
$(\Lambda ,d)$
is a k-graph (or graph of rank k).
As mentioned in the Introduction, these objects can be used to build
$C^*$
-algebras. Specifically, this is done using the following relationships.
Definition 2.2 [Reference Kumjian and Pask10, Definition 1.5].
Let
$\Lambda $
be a source-free row-finite k-graph. Then
$C^*(\Lambda )$
is the universal
$C^*$
-algebra generated by a family
$\{s_\lambda : \lambda \in \Lambda \}$
of partial isometries satisfying:
-
(i)
$\{s_v: v\in \Lambda ^0\}$
is a family of mutually orthogonal projections; -
(ii)
$s_{\lambda \mu }=s_\lambda s_\mu $
for all composable
$\lambda ,\mu $
; -
(iii)
$s_\lambda ^*s_\lambda =s_{s(\lambda )}$
for all
$\lambda \in \Lambda $
; and -
(iv)
${s_v=\sum _{\lambda \in r^{-1}(v)\cap d^{-1}(n)} s_\lambda s_\lambda ^* }$
for all
$v\in \Lambda ^0$
and
$n\in \mathbb {N}^k$
.
Many facets of this algebra can be studied by means of the higher-rank graph from which it is built. The focus of this article is on the K-theory groups of
$C^*(\Lambda )$
. These groups have a deep relationship with the coordinate graphs of
$\Lambda $
, which we now define.
Definition 2.3 [Reference Kumjian and Pask10, Definition 1.9].
Let
$f:\mathbb {N}^\ell \to \mathbb {N}^k$
be a monoid morphism. If
$(\Lambda , d)$
is a k-graph, then the pullback is the
$\ell $
-graph
$f^*(\Lambda )$
defined by
$$ \begin{align*}f^*(\Lambda) = \{(\lambda,n):d(\lambda)=f(n)\}\end{align*} $$
with
$d(\lambda ,n)=n$
,
$s(\lambda ,n)=s(\lambda )$
and
$r(\lambda ,n)=r(\lambda )$
.
We call
$\kappa _i^*(\Lambda )$
the ith coordinate graph, and we use
$M_i(\Lambda )$
to denote the adjacency matrix of the ith coordinate graph (simplified to
$M_i$
when there is no ambiguity).
As alluded to in the Introduction, a useful tool for K-theory computation is the spectral sequence. We now introduce basic definitions, notation and results for this tool.
Definition 2.4. A spectral sequence is a set of abelian groups
$E_{p,q}^r$
, together with differentials
$d_{p,q}^r: E_{p,q}^r \to E^r_{p-r,q+r-1}$
, such that
$E_{p,q}^{r+1}\cong \ker (d_{p,q}^r)/\operatorname {im}(d_{p+r,q-r+1})$
Figure 1 The
$E^2$
page for a general spectral sequence.
A helpful metaphor for these objects is a book where r indicates the page you are on. In general, the
$E^2$
page would take the shape shown in Figure 1.
Definition 2.5. A spectral sequence E is said to stabilise if there exists some
$r_0$
such that
$E^r\cong E^{r_0}$
for all
$r\geq r_0$
. This stable page is denoted
$E^\infty $
. A stable spectral sequence is said to converge to
$H_*$
(a family of abelian groups) if, for each n, there exists a filtration
$$ \begin{align*} 0=F_s(H_n)\subseteq \cdots \subseteq F_p(H_n) \subseteq \cdots \subseteq H_n \ \end{align*} $$
such that
$E^\infty _{p,q}\cong F_p(H_{p+q})/F_{p-1}(H_{p+q})$
.
Theorem 2.6 (The Evans chain complex: [Reference Evans3, Theorem 3.15]).
For
$p,k\in \mathbb {N}$
with
${0\leq p \leq k}$
, define the set
$$ \begin{align*}\mathbb{N}_{p,k}:=\{a=(a_1,\ldots,a_p)\in \{1,\ldots, k\}^p: a_i<a_{i+1} \text{ for all } i\}. \end{align*} $$
Equip these sets with the standard map such that, for
$a\in \mathbb {N}_{p,k}$
, the element
$a^i\in \mathbb {N}_{p-1,k}$
is the tuple a with the ith coordinate removed. Additionally,
$N_{0,k}:=\{*\}$
with
$a^1=*$
for all
$a\in N_{1,k}$
.
For a row-finite, source-free k-graph,
$\Lambda $
, there exists a spectral sequence
$E,d$
converging to
$K_*(C^*(\Lambda ))$
with
$E^\infty _{p,q}\cong E^{k+1}_{p,q}$
and
$$ \begin{align*}E^2_{p,q} \cong \begin{cases}H_p(\mathcal{D}) & 0\leq p \leq k \wedge 2|q,\\ 0 &\text{otherwise}, \end{cases}\end{align*} $$
where
$\mathcal {D}$
is the chain complex
$$ \begin{align} \mathcal{D}_p \cong \begin{cases}\bigoplus_{a\in N_{p,k}} \mathbb{Z}\Lambda^0 & 0\leq p \leq k, \\ 0 &\text{otherwise}, \end{cases} \end{align} $$
with differentials
$$ \begin{align} \bigoplus_{a\in N_{p,k}} m_a \mapsto \bigoplus_{b\in N_{p-1,k}} \sum_{a\in N_{p,k}} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i}B_{a_i} (m_a) \end{align} $$
for
$1\leq p \leq k$
and
$B_{a_i}:= I-M_{a_i}^T$
(which we call the co-adjacency matrix).
3. Block matrix presentation
In this section, we investigate the indexing sets
$\mathbb {N}_{p,k}$
more closely. Using this information, we partition the maps
$\partial $
of
$\mathcal {D}$
into recursive block matrices. This recursion is then leveraged to make explicit computations for k-graphs with an invertible co-adjacency matrix and monoid k-graphs.
Proposition 3.1. For a fixed k, there exists a partition of
$N_p= N_p^+\sqcup N_p^-$
with
$N_p^+:= \{a\in N_p: a_p=k\}$
(the k index has been supressed for ease of reading).
-
(i) For
$a\in N_p^-$
and
$1\leq i\leq p$
, we have
$a^i \in N_{p-1}^-$
. -
(ii) Define
$\psi : N_p^+\to N_{p-1}^-$
by
$a\mapsto a^p$
. Then the function
$\psi $
is bijective. -
(iii) There is a natural bijective inclusion
$\varphi : N_{p,k-1}\to N_{p,k}^-$
.
Proof. (i) Let
$a\in N_p^-$
. Since
$a_1<\cdots <a_p$
, we conclude that
$a_i<k$
for all i. Thus, it is not possible for
$a^i$
to have a k in its last position.
(ii) The map
$a\mapsto a^p$
is injective since
$a,b\in N_p^+$
ensures that they agree in the last position and
$a^p=b^p$
ensures that they agree everywhere else. Surjectivity is also immediate since
$b\in N_{p-1}^-$
implies that
$b_i<k$
for all i. Thus,
$(b_1,\ldots ,b_{p-1},k)$
will map to b.
(iii) After observing that this map is well defined, bijectivity can be established quickly from the definition of
$N_{p,k}^-$
.
We now use this information to reindex the image of (2.2).
$$ \begin{align} \bigoplus_{b\in N_{p-1}^+} \sum_{a\in N_{p}} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i}B_{a_i} (m_a) =& \bigoplus_{b\in N_{p-1}^+} \sum_{a\in N_{p}^+} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i}B_{a_i} (m_a)\ \quad \end{align} $$
$$ \begin{align} \bigoplus_{b\in N_{p-1}^-} \sum_{a\in N_{p}} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i}B_{a_i} (m_a) =& \bigoplus_{b\in N_{p-1}^-} (-1)^{p+1}B_k (m_{\psi^{-1}(b)}) \nonumber\\ &\bigoplus_{b\in N_{p-1}^-} \bigg(\sum_{a\in N_{p}^-} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i}B_{a_i} (m_a)\bigg). \end{align} $$
Equalities (3.1) and (3.2) come from Proposition 3.1(i) and (ii), respectively.
Notice that our image has split neatly into three pieces. These three pieces become the blocks of our matrix. Before writing that down explicitly, we notice a recursive relationship between
$\Lambda $
and the pullback,
$\iota ^*_{k-1}(\Lambda )$
.
Proposition 3.2. Suppose
$\Lambda $
is a source-free, row-finite k-graph. Define the subgraphs
$\Lambda _j:=\iota _j^*(\Lambda )$
and call the Evans chain complexes of these subgraphs
$\mathcal {D}^j$
with differentials
$\partial _p^j$
. The differentials of the Evans chain complex on
$\Lambda _{k-1}$
naturally map
$\partial _p^{k-1}: \bigoplus _{N_{p,k}^-} \mathbb {Z}\Lambda ^0 \to \bigoplus _{N_{p-1,k}^-} \mathbb {Z}\Lambda ^0$
and
$\partial _{p-1}^{k-1}: \bigoplus _{N_{p,k}^+} \mathbb {Z}\Lambda ^0 \to \bigoplus _{N_{p-1,k}^+} \mathbb {Z}\Lambda ^0$
.
Figure 2 The differentials
$\partial _3^4$
and
$\partial _4^4$
.
Proof. Since the map of Proposition 3.1(iii) is bijective, we can simply reindex (2.2).
$$ \begin{align*} \bigoplus_{a\in N_{p,k-1}} m_a &\mapsto \bigoplus_{b\in N_{p-1,k-1}} \sum_{a\in N_{p,k-1}} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i} B_{a_i} (m_a) \\ \bigoplus_{\varphi(a)\in N_{p,k}^-} m_a &\mapsto \bigoplus_{\varphi(b)\in N_{p-1,k}^-} \sum_{\varphi(a)\in N_{p,k}^-} \sum_{i=1}^p (-1)^{i+1} \delta_{b,a^i}B_{a_i} (m_a). \end{align*} $$
The mapping of
$\bigoplus _{N_{p,k}^+} \mathbb {Z}\Lambda ^0$
by
$\partial _{p-1}^{k-1}$
follows directly from
$\psi $
preserving the positioning of coordinates when mapping
$N_{p,k}^+ \to N_{p-1,k}^-$
.
Using the above proposition, we can write the image of
$\partial ^k_p$
as
$$ \begin{align*} \partial_{p-1}^{k-1}\bigg(\bigoplus_{a\in N_{p-1}^+} m_a\bigg) + \bigg(\bigoplus_{b\in N_{p-1}^-} (-1)^{p+1} B_k (m_{\psi^{-1}(b)})\bigg) + \partial_p^{k-1}\bigg( \bigoplus_{a\in N_{p,k}^-} m_a\bigg). \end{align*} $$
Our main theorem organises all of this information into the well-studied perspective of block matrices.
Theorem 3.3. For a source-free row-finite k-graph,
$\Lambda $
, let
$\mathcal {D}^j$
denote the Evans chain complex (2.1) for
$\Lambda _j$
. The differentials of
$\mathcal {D}^i$
have the form
$$ \begin{align*}\partial^j_p = \begin{bmatrix} \partial^{j-1}_{p-1} & 0 \\[5pt] (-1)^{p+1} B_j & \partial^{j-1}_{p} \end{bmatrix}\end{align*} $$
with
$\partial _p^\ell $
that are not well defined (that is, for
$\ell>p$
) omitted. Here,
$B_i$
acts coordinate-wise on vectors.
Proof. The proof is simply an application of the reindexing done in earlier propositions combined with the definition of block matrices.
Before moving on, we look at some of the differentials for a
$4$
-graph as an example to better visualise the pattern of these matrices. The differentials depicted in Figure 2 are
$\partial _3^4$
and
$\partial _4^4$
, respectively. Each position is made zero if there is no way to delete an element of the column index and obtain the row index (
$\delta _{b,a^i}$
). If an even position needs to be deleted, a negative is added (
$(-1)^{i+1}$
). Finally, whatever element was deleted determines which
$B_i$
is put in the slot (
$B_{a_i}$
).
Theorem 3.4. If
$\Lambda $
is a source-free row-finite k-graph such that
$B_i$
is bijective for some i with
$1\leq i \leq k$
, then
$K_*(C^*(\Lambda ))\cong 0$
.
Proof. Since
$\Lambda $
is source free and row finite, there exists a spectral sequence
$E,d$
with
$E^2$
page given by the homology groups of the Evans chain complex. Further, there exists a pullback
$f^*(\Lambda )$
that is isomorphic to
$\Lambda $
and swaps
$B_i$
with
$B_k$
. Thus, without loss of generality, we suppose that
$B_k$
is bijective.
We demonstrate that
$\operatorname {im}(\partial _{p+1})=\ker (\partial _p)$
for
$1<p<k$
. The special cases of
$p=1$
and
$p=k$
are not particularly illuminating, but, for completeness, we include them in Lemma A.1.
Since it is already known that
$\operatorname {im}(\partial _{p+1})\subseteq \ker (\partial _p)$
, we demonstrate the reverse inclusion. Since
$B_k$
is injective, it is relatively straightforward to check that
$$ \begin{align*} \ker \left( \begin{bmatrix} (-1)^{p-1}B_k & \partial_{p-1}^{k-1} \\ 0 & I \end{bmatrix} \begin{bmatrix} \partial^{k-1}_{p-1} & 0 \\ (-1)^{p+1} B_k & \partial_p^{k-1}\end{bmatrix} \right) = \ker\left( \begin{bmatrix} -\partial^{k-1}_{p-1} & 0 \\ (-1)^{p+1} B_k & \partial_p^{k-1}\end{bmatrix} \right).\end{align*} $$
Since the adjacency matrices commute [Reference Kumjian and Pask10, Section 6], we see that the left-hand set simplifies to
$$ \begin{align*}\ker\left(\begin{bmatrix} 0 & 0 \\ (-1)^{p+1}B_k &\partial_p^{k-1}\end{bmatrix}\right).\end{align*} $$
This is exactly the set
$\{ (\alpha ,\beta ): (-1)^{p+1}B_k(\alpha )=-\partial _p^{k-1}(\beta ) \}$
. Since
$B_k^{-1}$
exists, we can instead write
$((-1)^{p+2}B_k^{-1}\partial _p^{k-1}(\beta ), \beta )$
.
Lastly, we note that
$B^{-1}_k(\beta ) \in \bigoplus _{N_p^-}(\mathbb {Z}\Lambda ^0) = \bigoplus _{N_{p+1}^+}(\mathbb {Z}\Lambda ^0)$
. In particular,
$$ \begin{align*}\begin{bmatrix} \partial^{k-1}_{p} & 0 \\ (-1)^{p+2} B_k & \partial_{p+1}^{k-1}\end{bmatrix} \begin{bmatrix} B^{-1}_k(\beta)\\ 0\end{bmatrix} = \begin{bmatrix} (-1)^{p+2}B_k^{-1}\partial_p^{k-1}(\beta)\\ \beta\end{bmatrix} = \begin{bmatrix} \alpha \\ \beta\end{bmatrix}.\end{align*} $$
We conclude that
$\ker (\partial _p) = \operatorname {im}(\partial _{p+1})$
. This implies that
$H_p(\mathcal {D})\cong 0$
.
Since
$0$
has no nontrivial quotients, we conclude that
$E^{\infty }_{p,q}\cong 0$
for all
$p,q$
in the Kasparov–Schochet spectral sequence. Then, looking at the definition of convergence, we conclude that the only family
$H_*$
that such a spectral sequence could converge to is the trivial family
$H_*\cong 0$
. Since
$E\implies K_*(C^*(\Lambda ))$
, we deduce that
$K_*(C^*(\Lambda ))\cong 0$
.
Our final result concerns tensor chain complexes. We begin by defining tensor complexes and explicitly stating their differential maps. We will then leverage the similarities between these maps and the Evans differentials.
Definition 3.5. Given two chain complexes of R-modules
$\mathcal {A}$
and
$\mathcal {C}$
, we construct the chain complex
$\mathcal {A}\otimes \mathcal {C}$
by
$$ \begin{align*} (\mathcal{A}\otimes\mathcal{C})_n = \bigoplus_{i+j=n} \mathcal{A}_i \otimes_R \mathcal{C}_j. \end{align*} $$
The differential is then defined on elementary tensors
$a_i\otimes c_j\in \mathcal {A}_i\otimes \mathcal {C}_j$
by
$$ \begin{align*}\partial^\otimes(a_i\otimes c_j) = \partial_i(a_i)\otimes_R c_j \oplus (-1)^ia_i\otimes_R \partial_j(c_j).\end{align*} $$
Theorem 3.6. For a nontrivial monoid k-graph,
$\lambda $
(that is,
$|\Lambda ^0|=1$
and
$B_i>0$
for some i with
$1\leq i \leq k$
), let
$\mathcal {D}$
denote the Evans chain complex. If
$\mathcal {C}^j$
denotes the chain complex
$\mathcal {C}^j: 0 \to \mathbb {Z} \overset {B_j}{\to } \mathbb {Z} \to 0$
, then
$$ \begin{align*}\mathcal{D} \cong \bigotimes_{j=1}^k \mathcal{C}^j.\end{align*} $$
Proof. Since
$k<\infty $
, it is enough to show that, for
$\Lambda _{k-1}:=\iota _{k-1}^*(\Lambda )$
with
$\mathcal {D}^{k-1}$
the Evans chain complex for
$\Lambda _{k-1}$
, we have
$\mathcal {D}^k\cong \mathcal {D}^{k-1} \otimes \mathcal {C}$
with
$\mathcal {C}: 0 \to \mathbb {Z} \overset {B_k}{\to } \mathbb {Z} \to 0$
.
First, we investigate the modules of the complex
$\mathcal {D}^{k-1} \otimes \mathcal {C}$
. From Definition 3.5,
$$ \begin{align*} (\mathcal{A}\otimes\mathcal{C})_p = \mathcal{D}^{k-1}_{p} \otimes_{\mathbb{Z}} \mathbb{Z} \oplus \mathcal{D}^{k-1}_{p-1} \otimes_{\mathbb{Z}} \mathbb{Z} \cong \mathcal{D}_p^{k-1}\oplus \mathcal{D}_{p-1}^{k-1} \cong \mathcal{D}_p^k.\end{align*} $$
The first isomorphism is the natural tensor isomorphism given by multiplication. The second is the reindexing bijections
$\varphi : N_{p,k-1} \to N_{p,k}^-$
and
$\psi ^{-1}: N_{p-1,k}^- \to N_{p,k}^+$
.
It remains to show that these isomorphisms,
$\Psi _p$
, are chain maps, that is, that they commute with the differentials. To avoid ambiguity, we denote the tensor differential by
$\partial ^\otimes $
and the Evans differential by
$\partial $
. Again, we will treat the cases
$p=1$
and
$p=k$
separately because those differentials have a somewhat unique shape (see Lemma A.2).
Let
$1<p<k$
. The map
$\partial _p^k\Psi _p$
acts on elements
$\alpha \otimes x \oplus \beta \otimes y$
with
$\alpha \in \mathcal {D}_{p-1}^{k-1}$
and
$\beta \in \mathcal {D}_{p}^{k-1}$
in the following way.
$$ \begin{align*} \partial_p^k\Psi_p(\alpha \otimes x \oplus \beta \otimes y) & = \partial_p^k\begin{bmatrix} x\cdot \alpha \\[3pt] y\cdot\beta\end{bmatrix} = \begin{bmatrix} \partial_{p-1}^{k-1} & 0 \\[3pt] (-1)^ {p+1} B_k & \partial_p^{k-1}\end{bmatrix}\begin{bmatrix} x\cdot \alpha \\[3pt] y\cdot\beta\end{bmatrix}\\[6pt] & = \begin{bmatrix} x\cdot \partial_{p-1}^{k-1}\alpha \\[3pt] x(-1)^{p+1}B_k\alpha+ y\partial_p^{k-1}\beta\end{bmatrix}. \end{align*} $$
Before moving on to the alternative ordering, we recall that
$\partial ^{\mathcal {C}}_0=0$
and
$\partial ^{\mathcal {C}}_1=B_k$
. This is used to simplify
$\partial ^\otimes $
in the working.
$$ \begin{align*} \Psi_{p-1}\partial_p^\otimes(\alpha \otimes x \oplus \beta \otimes y)&= \Psi_{p-1} (\partial_{p-1}^{k-1}(\alpha) \otimes x \oplus (-1)^{p-1} \alpha \otimes B_k x + \partial_{p}^{k-1}(\beta) \otimes y) \\[6pt] &= \begin{bmatrix} x\cdot \partial_{p-1}^{k-1}(\alpha) \\[3pt] (-1)^{p-1}B_k x \cdot \alpha \end{bmatrix} + \begin{bmatrix} 0 \\[3pt] y\cdot \partial_{p}^{k-1}(\beta) \end{bmatrix}. \end{align*} $$
Finally, since the parity of
$p-1$
and
$p+1$
align, we conclude that these maps are equal.
The benefit of phrasing the Evans complex in terms of a tensor complex is that it allows us to invoke the Künneth theorem
Theorem 3.7 (Künneth theorem; see [Reference Weibel17, Theorem 3.6.3]).
For a chain complex,
$\mathcal {C}$
, with
$\mathcal {C}_p$
and
$\partial _{\mathcal {C}}(\mathcal {C}_p)$
flat for all p,
Footnote
1
and an arbitrary complex
$\mathcal {A}$
, there exists a short exact sequence
$$ \begin{align*} \bigoplus_{i+j=p} H_i(\mathcal{C})\otimes_R H_j(\mathcal{A}) \to H_p(\mathcal{C}\otimes_R\mathcal{A}) \to \bigoplus_{i+j=p-1}\operatorname{Tor}_1^R(H_{i}(\mathcal{C}),H_j(\mathcal{A}))\end{align*} $$
which splits unnaturally.
From this, we can state the
$E^2$
page of the Kasparov–Schochet spectral sequence even more explicitly.
Theorem 3.8. For a nontrivial monoid k-graph,
$\lambda $
, define
$g:= \gcd (B_1,\ldots ,B_k)$
. The
$E^2$
page of the Kasparov–Schochet spectral sequence has the form
$$ \begin{align*} E_{p,q}^2=\begin{cases} \mathbb{Z}_{g}^{\binom{k-1}{p}} & 2\mid q, \\[3pt] 0 & \text{ otherwise. } \end{cases} \end{align*} $$
Proof. The goal is to use Theorem 3.6 which yields
$\mathcal {D} \cong \bigotimes ^k \mathcal {C}^j$
. Then, the Künneth Theorem with some classical finite group theory gives the necessary homology groups. As before, without loss of generality, we suppose that
$B_1>0$
.
We proceed by induction on k. Consider the base case
$H_p(\mathcal {C}^1)$
. Since
$B_1\neq 0$
, we observe that
$E_{1,2q}=H_1(\mathcal {C}^1)=\ker (B_1)=0$
and
$E_{0,2q}=H_0(\mathcal {C}^1)=\mathbb {Z}_{B_1}$
, as desired. Fix some
$j\geq 1$
and define
$g_j:=\gcd (B_1,\ldots ,B_j)$
. Then suppose that
$$ \begin{align*} H_p(\mathcal{C}^1\otimes\cdots\otimes\mathcal{C}^j) \cong (\mathbb{Z}_{g_j})^{\binom{j-1}{p}}.\end{align*} $$
Consider
$(\mathcal {C}^1\otimes \,\cdots \,\otimes \mathcal {C}^j)\otimes \mathcal {C}^{j+1}$
. Since each module in our complex is free, which implies that it is projective (see [Reference Weibel17, Section 2]), we can use the Künneth Theorem and the inductive hypothesis to obtain
$$ \begin{align*} H_p((\mathcal{C}^1\otimes\cdots\otimes \mathcal{C}^j)\otimes \mathcal{C}^{j+1}) & \cong (\mathbb{Z}_{g_j})^{\binom{j-1}{p-1}}\otimes H_1(\mathcal{C}^{j+1}) \oplus (\mathbb{Z}_{g_j})^{\binom{j-1}{p}}\otimes H_0(\mathcal{C}^{j+1}) \\ &\quad \oplus\operatorname{Tor}_1^{\mathbb{Z}}((\mathbb{Z}_{g_j})^{\binom{j-1}{p-2}},H_1(\mathcal{C}^{j+1})) \oplus \operatorname{Tor}_1^{\mathbb{Z}}((\mathbb{Z}_{g_j})^{\binom{j-1}{p-1}},H_0(\mathcal{C}^{j+1})). \end{align*} $$
There are two cases determined by
$B_{j+1}$
. First, suppose that
$B_{j+1}=0$
and thus
$g_{j+1}=g_{j}$
. In this case,
$H_*(\mathcal {C}^{j+1})\cong \mathbb {Z}$
. This would make the torsion groups
$0$
and
$$ \begin{align*} H_p((\mathcal{C}^1\otimes\cdots\otimes \mathcal{C}^j)\otimes \mathcal{C}^{j+1})\cong (\mathbb{Z}_{g_j})^{\binom{j-1}{p}}\oplus (\mathbb{Z}_{g_j})^{\binom{j-1}{p-1}}\cong (\mathbb{Z}_{g_{j+1}})^{\binom{j}{p}}. \end{align*} $$
Next, suppose that
$B_{j+1}\neq 0$
. This would mean that
$H_1(\mathcal {C}^{j+1})=0$
and
$H_0(\mathcal {C}^{j+1})=\mathbb {Z}_{B_{j+1}}$
. In particular, this means that the terms containing
$H_1(\mathcal {C}^{j+1})$
vanish leaving
$$ \begin{align*} (\mathbb{Z}_{g_j})^{\binom{j-1}{p}}\otimes\mathbb{Z}_{B_{j+1}} \oplus \operatorname{Tor}_1^{\mathbb{Z}}((\mathbb{Z}_{g_j})^{\binom{j-1}{p-1}}, \mathbb{Z}_{B_{j+1}} ).\end{align*} $$
It is a well-known fact (see [Reference Massey11, Section 6] that
$$ \begin{align*}\operatorname{Tor}_1^{\mathbb{Z}}((\mathbb{Z}_{g_j})^{\binom{j-1}{p-1}}, \mathbb{Z}_{B_{j+1}} )\cong (\mathbb{Z}_{g_j})^{\binom{j-1}{p-1}}\otimes\mathbb{Z}_{B_{j+1}}\cong (\mathbb{Z}_{g_j}\otimes\mathbb{Z}_{B_{j+1}})^{\binom{j-1}{p-1}} \cong (\mathbb{Z}_{g_{j+1}})^{\binom{j-1}{p-1}},\end{align*} $$
which gives the desired form.
We conclude that
$H_p(\mathcal {D})\cong \mathbb {Z}_{g}^{\binom {k-1}{p}}$
, which allows us to fill in the
$E^2$
page of the Kasparov–Schochet spectral sequence with the desired result.
Remark 3.9. In the case where
$\Lambda \cong \mathbb {N}^k$
(that is, trivial),
$C^*(\Lambda )\cong C(\mathbb {T}^k)$
and thus
$K_1(C^*(\Lambda ))\cong \mathbb {Z}^{2^{k-1}}\cong K_0(C^*(\Lambda ))$
. Because the K-theory is known, we omit the case from Theorem 3.8.
The above result for monoid k-graphs also sheds some light on Theorem 3.4. In particular, we find that an invertible co-adjacency matrix is not necessary for trivial K-theory groups.
Corollary 3.10. Let
$\Lambda $
be a monoid k-graph with
$\gcd (B_1,\ldots ,B_k)=1$
. Then,
$K_*(C^*(\Lambda ))=0$
.
Corollary 3.10 has a very close relationship with the results of [Reference Barlak, Omland and Stammeier1]. However, it is important to note that the results there required additional hypotheses on
$\Lambda $
.
We conclude this article by investigating monoid
$3$
-graphs more generally. Specifically, this will demonstrate that knowing
$E^\infty $
does not always give perfect knowledge of
$K_*(C^*(\Lambda ))$
.
Corollary 3.11. Let
$\Lambda $
be a monoid
$3$
-graph with
$|\Lambda ^{e_i}|>1$
for some i with
$1\leq i\leq 3$
and define
$g=\gcd (B_1,B_2,B_3)$
. Then
$K_1(C^*(\Lambda ))=\mathbb {Z}_g^2$
and there exists a short exact sequence
$$ \begin{align*}\mathbb{Z}_g \to K_0(C^*(\Lambda)) \to \mathbb{Z}_g.\end{align*} $$
Proof. Consider the map
$d_{p,q}^2: E_{p,q}^2 \to E_{p-2,q+1}^2$
. Observe that the parity of q means that
$\operatorname {im}(d_{p,q}^2)=0$
and
$\ker (d_{p,q})=E_{p,q}^2$
. We conclude that
$E^2\cong E^3$
; this is indeed true for any k-graph. Theorem 3.8 gives
$E^3_{p,2q}\cong E^2_{p,2q}\cong \mathbb {Z}_g^{\binom {2}{p}}$
. In particular, this means that
$d^3_{p,q}: E_{p,q}^3 \to E_{p-3,q}$
must be the zero map. We conclude, as before, that
$E^3\cong E^4$
, and thus the sequence has stabilised.
First, we examine the
$K_1$
group. We look at the family
$\ldots ,E_{-1,2}, E_{1,0}, E_{3,-2}, \ldots .$
By Theorem 3.8,
$E_{3,-2}\cong 0$
so all but
$E_{1,0}$
are isomorphic to
$0$
. Moreover, there exists a filtration of
$K_1(C^*(\Lambda ))$
,
$0\leq \cdots \leq F_{-1} \leq F_0\leq F_1 \leq F_2 \leq \cdots \leq K_1(C^*(\Lambda ))$
such that
$F_p/F_{p-1}\cong E_{p,1-p}$
. The zeros in all positions apart from
$E_{1,0}$
allow us to conclude that
$F_{1}\cong K_1(C^*(\Lambda ))$
and
$F_0\cong 0$
. That is,
$K_1(C^*(\Lambda ))\cong F_1/F_0\cong E_{1,0}\cong \mathbb {Z}_g^2$
.
To determine
$K_0$
, we must look at the family with even total degree (
$2\mid (p+q)$
). This family is
$\ldots , 0, 0, \mathbb {Z}_g, 0, \mathbb {Z}_g, 0, 0,\ldots .$
We now consider the filtration
$0\cong F_{-1}\leq F_0 \leq F_1 \leq F_2 \cong K_0(C^*(\Lambda )) $
. Since
${2\nmid 1}$
, Theorem 2.1, ensures that
$E_{1,-1}= 0$
. So
$F_1\cong F_0$
, which allows us to refine to the filtration
$0\leq F_0\leq F_2$
. This ensures that
$F_2/F_0\cong \mathbb {Z}_g$
providing the short exact sequence
$$ \begin{align*} 0 \to \mathbb{Z}_g \to K_0(C^*(\Lambda)) \to \mathbb{Z}_g \to 0.\\[-34pt] \end{align*} $$
Appendix
Lemma A.1. Let
$\Lambda $
be a k-graph with
$B_k$
invertible. Then
$H_0(\mathcal {D})\cong H_k(\mathcal {D})\cong 0$
.
Proof. The differentials
$\partial _1$
and
$\partial _k$
of the Evans chain complex always take the form
$$ \begin{align*} \partial_{1} = \begin{bmatrix} B_k & B_{k-1} & \cdots & B_1 \end{bmatrix}, \quad \partial_k = \begin{bmatrix} B_1 \\ -B_2 \\ \vdots \\ (-1)^{p+1} B_k \end{bmatrix}. \end{align*} $$
Since
$H_0(\mathcal {D})=\operatorname {coker}(\partial _1)$
and
$H_k(\mathcal {D})=\ker (\partial _k)$
, we need only demonstrate surjectivity and injectivity, respectively.
Since
$B_k$
is onto,
$\partial _1 \begin {bmatrix} 0 \\ \vdots \\ \alpha \end {bmatrix}$
is onto
$\mathbb {Z}\Lambda ^0$
. Lastly,
$\partial _k(\alpha )=\partial _k(\beta )$
implies that
$B_k(\alpha )=B_k(\beta )$
and thus, by injectivity of
$B_k$
,
$\alpha =\beta $
.
Lemma A.2. Under the hypotheses of Theorem 3.6, we have equalities
$\partial ^k_k\Psi _k= \Psi _{k-1} \partial ^\otimes $
and
$\partial ^k_1\Psi _1= \Psi _{0} \partial ^\otimes $
.
Proof. Consider
$\partial ^k_k\Psi _k$
and notice that
$N_{k,k}^-=\emptyset $
. Since
$(\mathcal {D}^{k-1} \otimes \mathcal {C})_k = \mathcal {D}^{k-1}_{k-1} \otimes \mathcal {C}_1$
, we let
$\alpha \otimes x\in \mathcal {D}^{k-1}_{k-1} \otimes \mathcal {C}_1$
. Then
$$ \begin{align*} \partial^k_k\Psi_k(\alpha\otimes x) &= \begin{bmatrix} \partial_{k-1}^{k-1}\\ (-1)^{k+1}B_k\end{bmatrix}[x\cdot \alpha] = \begin{bmatrix} x\partial_{k-1}^{k-1}\alpha\\ x(-1)^{k+1}B_k\alpha\end{bmatrix} \\ \Psi_{k-1}\partial^\otimes(\alpha\otimes x) &= \Psi_{k-1}(\partial^{k-1}_{k-1}(\alpha)\otimes x \oplus (-1)^{k-1} B_1 x) = \begin{bmatrix} x\partial_{k-1}^{k-1}\alpha\\ x(-1)^{k-1}B_k\alpha\end{bmatrix}. \end{align*} $$
For the
$p=1$
case, we return to general elements
$\alpha \otimes x \oplus \beta \otimes y \in \mathcal {D}_0^{k-1}\otimes \mathbb {Z} \oplus \mathcal {D}_1^{k-1}\otimes \mathbb {Z}$
. Additionally, we utilise
$\partial ^{k-1}_0=0$
. Then
$$ \begin{align*} \partial^k_1\Psi_1(\alpha\otimes x \oplus \beta\otimes y) &= \begin{bmatrix} (-1)^{k+1}B_k & \partial_{1}^{k-1}\end{bmatrix} \begin{bmatrix} x\cdot\alpha \\ y\cdot \beta \end{bmatrix} = \begin{bmatrix} xB_k\alpha + y\partial_1^{k-1} \beta\end{bmatrix} \\ \Psi_{0}\partial^\otimes(\alpha\otimes x \oplus \beta\otimes y) &= \Psi_{0}((-1)^{0}\otimes B_k x + \partial_1^{k-1} \beta \otimes y ) = \begin{bmatrix} xB_k\alpha + y\partial_1^{k-1} \beta \end{bmatrix}.\\[-36pt] \end{align*} $$
