The hyperparameter search was random with the objective of minimizing the loss
function used in training on held out data. The search range for each of the
parameters is displayed in 19. Since certain parameters did not seem to
influence the final score as much we decided to use them as constants and
focus on optimizing the hyperparameters in the table. Therefore the number of
sampled anchors was set to 20 and $u=0.1$ in the augmented TripleRE function,
the values matching those seen in Stargraph [46].

Hyperparameter | Search range
---|---
Dropout | [1e-4,1e-1]
Learning rate | [1e-5,1e-3]
Weight decay | [1e-5,1e-2]
Adversarial temperature | [1,10]
Gamma | [0,30]
Num. of sampled neighbors | 0-10
Dim. of hidden layers | { 64, 128, 256, 512}
Table 19: The hyperparameter optimization search ranges used in the selection
of the optimal model for the generation of knowledge graph node embeddings
that would serve as intervention information for the medical claims dataset.

Our final embeddings were 256-dimensional, the learning rate was 2e-4, the
drop-ratio was 5e-3. We used the self-adversarial negative sampling loss with
$\gamma=8$ and we sampled 4 neighbor nodes for each subgraph.

To additionally evaluate the quality of the embeddings we assigned classes to
drug combinations and then scored them using multiple clustering metrics. We
were interested to see if embeddings of drug combinations used for similar
purposes would be embedded closer together than other drug combinations. For
the class label of single drugs we used the first level of the Anatomical
Therapeutic Chemical (ATC) code, which represents one of the 14 anatomical or
pharmacological groups. Since certain medications have more than one ATC code,
we took the mode of all labels for a specific drug. For multiple drugs we
combined all distinct first level values and took the mode of them as the
label. We used the Silhouette metric, Calinski Harabasz index and Davies
Bouldin index as well as the average classification accuracy over 10 runs of
training a random forest classifier on a random sample of 80% of the dataset
and evaluating on the remaining 20%. Out of all tested embeddings the
hyperparameter optimized StarGraph embeddings performed best (exceeding 93% in
the classification accuracy metric).

###  C.6 Pseudo-outcome estimation

In our experiments, we estimate pseudo-outcomes $\tilde{\tau}$ for a given
intervention $w$ using the RA-learner [14]:

$\displaystyle\tilde{\tau}=W(Y-\hat{\mu}_{0}(X))+(1-W)(\hat{\mu}_{1}(X)-Y)$
(18)

where $\hat{\mu}_{w}$ is an estimate of
$\mu_{w}(X)=\mathbb{E}_{\mathcal{P}}\Big{[}Y\mid X=x,W=w\Big{]}$.

Furthermore, in both settings we only estimate CATE for treated individuals.
We focus on treated individuals in the Claims setting because we care about
the risk of an adverse event for prescribing a sick patients drugs that may
cure their sickness, not the adverse event risk of prescribing healthy
patients drugs (which is of less clinical interest). In the LINCS setting, we
focus on treated cells as for these cell-lines $Y(0)$ is also measured from a
cloned cell-line under similar laboratory conditions, which allows us to
directly estimate CATE prediction performance using the PEHE metric. As we
focus on treated samples, the RA-learner can be simplified to
$\tilde{\tau}=Y-\hat{\mu}_{0}(X)$. We estimate $\hat{\mu}_{0}(X)$ using a
random forest model in the Claims setting, whereas in the LINCS setting we use
the point estimate from the untreated control cell line’s gene expression.

###  C.7 Baselines

Here we provide more details on the baselines used in our experiments.

_Trained on test task:_ These baselines leverage CATE estimators which can
only be trained on a single task (typically these are the strongest baselines,
when there is a large enough dataset for a single task). Thus, we train a
single model for each meta-testing task on its train split, and evaluate
performance on its test split. We use a number of strong baselines for CATE
estimation developed by prior work including both model-agnostic and end-to-
end deep learning approaches: T-learner. Specifically, we use the model-
agnostic CATE estimators: [42], X-learner [42], RA-learner [14], R-learner
[53]. We additionally use the end-to-end deep learning estimators DragonNet
[68], TARNet [66], and FlexTENet [15], using implementations from [15]. For
model-agnostic CATE estimators, we use random forest models following prior
work [12, 76].

_Zero-shot_. These baselines use CATE estimators which incorporate
intervention information ($W$) and are capable of multi-task learning. We
train these baselines on all meta-training tasks. These baselines have no
access to the meta-testing tasks during training. We found in preliminary
experiments that in some cases, baseline models trained with vanilla ERM would
not even converge. To allow for fair comparison to baselines, we allow for all
zero-shot baselines to be trained using Reptile (by training using the same
optimization strategy as Algorithm 1, while allowing for training with ERM by
including $k=1$ in the hyperparameter search space).

Firstly, we use GraphITE [23] and Structured Intervention Networks [33]. These
are, to the best of our knowledge, the only methods from prior work which are
(in principle) capable of zero-shot generalization. We use existing
implementations provided by the authors [33].

Additionally, we implement two strong baselines which estimate CATE by
modeling potential outcomes, rather than via pseudo-outcomes. These are
variants of the S-learner and T-learner [42] with meta-learning, which use the
intervention information as input, rather than one-hot encoded vectors of the
different interventions—such that they also have zero-shot capability.
Specifically, we train MLPs using the same architecture as CaML to estimate
the response function from observed outcomes:

$\displaystyle\mu(x,w)=\mathbb{E}_{\mathcal{P}}\Big{[}Y\mid X=x,W=w\Big{]}$
(19)

and estimate CATE by

$\displaystyle\hat{\tau}_{w}(x)=\hat{\mu}(x,w)-\hat{\mu}(x,\textbf{0})$ (20)

Where $w$ denotes the corresponding intervention information $w$ for an
intervention, and 0 denotes a null intervention vector. In the LINCS setting,
we represent 0 as a vector of zeros, whereas in the Claims setting we
represent 0 as the mean embedding of all drugs (as the estimand is the
increase in adverse event likelihood compared to a randomly chosen drug). The
difference between the T-learner and the S-learner is that the T-learner
estimates two models, one for control units and one for treated units. By
contrast, the S-learner estimates a shared model across all units.


}{}\pgfsys@moveto{4.8369pt}{0.0pt}\pgfsys@moveto{5.91328pt}{0.0pt}\pgfsys@curveto{5.91328pt}{0.59447pt}{5.43137pt}{1.07639pt}{4.8369pt}{1.07639pt}\pgfsys@curveto{4.24243pt}{1.07639pt}{3.76051pt}{0.59447pt}{3.76051pt}{0.0pt}\pgfsys@curveto{3.76051pt}{-0.59447pt}{4.24243pt}{-1.07639pt}{4.8369pt}{-1.07639pt}\pgfsys@curveto{5.43137pt}{-1.07639pt}{5.91328pt}{-0.59447pt}{5.91328pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.8369pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}B$.

######  Definition 6.2 (Phantom tower).

Let $A$ be an object in a triangulated category $\mathfrak{T}$ with a
homological ideal $\mathfrak{I}$. A pre-$\mathfrak{I}$-phantom tower over $A$
is a diagram

${A}$${N_{0}}$${N_{1}}$${N_{2}}$${\cdots}$${P_{0}}$${P_{1}}$${P_{2}}$${\cdots}$$\scriptstyle{=}$$\scriptstyle{\iota_{0}^{1}}$$\scriptstyle{\epsilon_{0}}$$\scriptstyle{\iota_{1}^{2}}$$\scriptstyle{\epsilon_{1}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\delta_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\delta_{2}}$$\scriptstyle{\pi_{2}}$

written $\mathcal{P}\to A$ with
$\mathcal{P}=(P_{\bullet},N_{\bullet},\delta,\pi,\iota,\epsilon)$ or
$(P_{\bullet},N_{\bullet})$ in brief, such that:

  * •

The top triangles are exact, i.e.
$P_{n}\xrightarrow{\pi_{n}}N_{n}\xrightarrow{\iota^{n+1}_{n}}N_{n+1}\leavevmode\hbox
to17.26pt{\vbox to12.26pt{\pgfpicture\makeatletter\hbox{\hskip
8.63013pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.59712pt}{3.84387pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\epsilon_{n}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{ {}}
{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.03014pt}{0.0pt}\pgfsys@lineto{4.63014pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.83014pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {{{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.00002pt}{0.0pt}\pgfsys@moveto{1.07637pt}{0.0pt}\pgfsys@curveto{1.07637pt}{0.59447pt}{0.59445pt}{1.07639pt}{-0.00002pt}{1.07639pt}\pgfsys@curveto{-0.59448pt}{1.07639pt}{-1.0764pt}{0.59447pt}{-1.0764pt}{0.0pt}\pgfsys@curveto{-1.0764pt}{-0.59447pt}{-0.59448pt}{-1.07639pt}{-0.00002pt}{-1.07639pt}\pgfsys@curveto{0.59445pt}{-1.07639pt}{1.07637pt}{-0.59447pt}{1.07637pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-0.00002pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}P_{n}$ is an exact
triangle for each $n$.

  * •

The bottom triangles commute, i.e. $\delta_{n}=\epsilon_{n-1}\circ\pi_{n}$ for
each $n$.

  * •

The top maps are $\mathfrak{I}$-phantom, i.e. $\iota^{n+1}_{n}\in\mathfrak{I}$
for each $n$.

  * •

The sequence $\cdots\leavevmode\hbox to17.14pt{\vbox
to14.52pt{\pgfpicture\makeatletter\hbox{\hskip 8.56854pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.53554pt}{4.2631pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\delta_{2}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{ {}}
{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-4.96855pt}{0.0pt}\pgfsys@lineto{4.56856pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.76855pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {{{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.00003pt}{0.0pt}\pgfsys@moveto{1.07635pt}{0.0pt}\pgfsys@curveto{1.07635pt}{0.59447pt}{0.59444pt}{1.07639pt}{-0.00003pt}{1.07639pt}\pgfsys@curveto{-0.5945pt}{1.07639pt}{-1.07642pt}{0.59447pt}{-1.07642pt}{0.0pt}\pgfsys@curveto{-1.07642pt}{-0.59447pt}{-0.5945pt}{-1.07639pt}{-0.00003pt}{-1.07639pt}\pgfsys@curveto{0.59444pt}{-1.07639pt}{1.07635pt}{-0.59447pt}{1.07635pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-0.00003pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}P_{1}\leavevmode\hbox
to17.14pt{\vbox to14.52pt{\pgfpicture\makeatletter\hbox{\hskip
8.56854pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.53554pt}{4.2631pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{\scriptsize{$\delta_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{ {}}
{}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-4.96855pt}{0.0pt}\pgfsys@lineto{4.56856pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.76855pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {{{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-0.00003pt}{0.0pt}\pgfsys@moveto{1.07635pt}{0.0pt}\pgfsys@curveto{1.07635pt}{0.59447pt}{0.59444pt}{1.07639pt}{-0.00003pt}{1.07639pt}\pgfsys@curveto{-0.5945pt}{1.07639pt}{-1.07642pt}{0.59447pt}{-1.07642pt}{0.0pt}\pgfsys@curveto{-1.07642pt}{-0.59447pt}{-0.5945pt}{-1.07639pt}{-0.00003pt}{-1.07639pt}\pgfsys@curveto{0.59444pt}{-1.07639pt}{1.07635pt}{-0.59447pt}{1.07635pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-0.00003pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}P_{0}\xrightarrow{\pi_{0}}N_{0}\rightarrow
0$ is $\mathfrak{I}$-exact.

The last two conditions are equivalent in the presence of the other
conditions. A tower map $\mathcal{P}\to\mathcal{Q}$ of
pre-$\mathfrak{I}$-phantom towers $\mathcal{P}\to A$ and $\mathcal{Q}\to B$
over a morphism $f\colon A\to B$ is a morphism of diagrams compatible with
$f$. We say that $\mathcal{P}$ is $\mathfrak{I}$-phantom if $P_{n}$ is
$\mathfrak{I}$-projective for each $n$.

Meyer shows that any $\mathfrak{I}$-projective resolution embeds in an
$\mathfrak{I}$-phantom tower (note that the suspension of an
$\mathfrak{I}$-projective object is $\mathfrak{I}$-projective) and that any
morphism lifts to maps of phantom towers over the objects [Mey08, Lemma 3.3].
We must additionally consider pre-phantom towers, but the same proof444using
the slightly modified version 2.17 of [MN10, Proposition 3.26] gives the
following statement:

######  Lemma 6.3.

Any $\mathfrak{I}$-projective resolution $P_{\bullet}\rightarrow A$ can be
embedded in an $\mathfrak{I}$-phantom tower, which is unique up to non-
canonical isomorphism.

Furthermore, if $\cal P$ is a pre-$\mathfrak{I}$-phantom tower over $A$, $\cal
Q$ is a pre-$\mathfrak{I}$-phantom tower over $B$, and $P_{\bullet}$ and
$Q_{\bullet}$ are the $\mathfrak{I}$-resolutions of $A$ and $B$ embedded in
$\cal P$ and $\cal Q$ respectively, then any chain map from $P_{\bullet}$ to
$Q_{\bullet}$ over a map $f\colon A\to B$ extends to a tower map $\cal
P\rightarrow\cal Q$ over $f$. In particular, if $\cal P$ is
$\mathfrak{I}$-phantom, any map $f\colon A\rightarrow B$ lifts (non-
canonically) to a tower map $\cal P\rightarrow\cal Q$.

Given a pre-$\mathfrak{I}$-phantom tower $\mathcal{P}\to A$ we construct an
exact couple $\operatorname{EC}(\mathcal{P})$ as follows. The top collection
of exact triangles in $\mathcal{P}$ forms a single triangle of
$\mathbb{Z}$-graded objects after defining $N_{n}=A$,
$\iota^{n+1}_{n}=\operatorname{id}_{A}$ and $P_{n}=0$ for $n<0$.

${N_{\bullet}}$${N_{\bullet}}$${P_{\bullet}}$$\scriptstyle{\iota}$$\scriptstyle{\epsilon}$$\scriptstyle{\pi}$

We now consider the stable homological functor
$F\colon\mathfrak{T}\to\mathbf{Ab}_{*}$. We use this to construct an exact
triangle of bigraded Abelian groups

(6.1)
${D}$${D}$${D_{p,q}=F_{p+q+1}(N_{p+1})}$${C}$${C_{p,q}=F_{p+q}(P_{p})}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{k}$

with maps $i$, $j$ and $k$ defined at $p,q\in\mathbb{Z}$ by555The notation
$F_{n}(f)$ for the map $F_{n}(A)\to F_{n-1}(B)$ induced by a degree one
morphism $f\colon A\mathrel{\leavevmode\hbox to10.07pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.47383pt}{0.0pt}\pgfsys@invoke{
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}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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standard.

$\displaystyle i_{p,q}$ $\displaystyle:=F_{p+q+1}(\iota_{p+1}^{p+2})\colon
D_{p,q}\rightarrow D_{p+1,q-1}$ $\displaystyle\text{deg }i$
$\displaystyle=(1,-1)$ $\displaystyle j_{p,q}$
$\displaystyle:=F_{p+q+1}(\epsilon_{p})\colon D_{p,q}\rightarrow C_{p,q}$
$\displaystyle\text{deg }j$ $\displaystyle=(0,0)$ $\displaystyle k_{p,q}$
$\displaystyle:=F_{p+q}(\pi_{p})\colon C_{p,q}\rightarrow D_{p-1,q}$
$\displaystyle\text{deg }k$ $\displaystyle=(-1,0).$

This forms an exact couple $\operatorname{EC}(\mathcal{P})=(C,D,i,j,k)$ which
is functorial with respect to tower maps. The construction of a spectral
sequence from an exact couple is standard, see for example [ML95, p336-337],
and yields a spectral sequence $(E,d)$ starting from the first sheet. We list
some important properties:

  * •

The first sheet $E^{1}$ satisfies $E^{1}_{p,q}=F_{p+q}(P_{p})$ for $p\geq 0$,
$q\in\mathbb{Z}$ and vanishes for $p\leq-1$. This implies $E^{r}_{p,q}=0$ for
any $r\geq 1$ and $q\in\mathbb{Z}$ when $p\leq-1$.

  * •

The first differential $d^{(1)}_{p,q}\colon E^{1}_{p,q}\to E^{1}_{p-1,q}$ is
given by $F_{p+q}(\delta_{p})\colon F_{p+q}(P_{p})\to F_{p+q-1}(P_{p-1})$ for
$p\geq 0$, $q\in\mathbb{Z}$.

  * •

The second sheet $E^{2}=\frac{k^{-1}(\operatorname{im}i)}{j(\ker i)}$ is given
at $p\geq 0$, $q\in\mathbb{Z}$ by $\frac{\ker
F_{p+q}(\delta_{p})}{\operatorname{im}F_{p+q+1}(\delta_{p+1})}$.

  * •

For $r\geq 1$ the $r$th sheet
$E^{r}=\frac{k^{-1}(\operatorname{im}i^{r-1})}{j(\ker i^{r-1})}$ has
differential $d^{(r)}\colon E^{r}\to E^{r}$ satisfying $d^{(r)}(x)=j(y)$ for
$x\in E^{r}$ and $y\in D$ with $i^{r-1}(y)=k(x)$.

  * •

The limit sheet $E^{\infty}$ is $\frac{\bigcap_{r\geq
1}k^{-1}(\operatorname{im}i^{r})}{\bigcup_{r\geq 1}j(\ker i^{r})}$.

The ABC spectral sequence $\operatorname{ABC}(\mathfrak{T},\mathfrak{I},F,A)$
is the spectral sequence $(E^{r},d^{(r)})_{r\geq 2}$ starting from the second
sheet, for any $\mathfrak{I}$-phantom tower $\mathcal{P}\to A$. In this case,
the description of the second sheet gives an isomorphism
$E^{2}_{p,q}\cong\mathbb{L}_{p}F_{q}(A)$. For any other $\mathfrak{I}$-phantom
tower $\mathcal{Q}\to A$, there is a tower map $\mathcal{P}\to\mathcal{Q}$
over the identity $\operatorname{id}\colon A\to A$ by 6.3. This induces a
morphism of exact couples
$\operatorname{EC}(\mathcal{P})\to\operatorname{EC}(\mathcal{Q})$ and
therefore a morphism of spectral sequences. On the second sheets, this is
induced by a chain map of $\mathfrak{I}$-projective resolutions
$P_{\bullet}\to Q_{\bullet}$ of $A$, and thus is the identity on
$E^{2}_{p,q}=\mathbb{L}_{p}F_{q}(A)$ by 2.17. Since the later sheets are
subquotients of the second sheet, the spectral sequences induced by
$\mathcal{P}$ and $\mathcal{Q}$ are identical from the second sheet onwards.

######  Lemma 6.4.

Let
$(\Phi,\alpha,f)\colon(\mathfrak{T},\mathfrak{I},F,A)\to(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$
be an ABC cycle and let $\cal P^{\prime}$ be an
$\mathfrak{I}^{\prime}$-phantom tower over $A^{\prime}$. Then $\Phi(\cal
P^{\prime})$ is a pre-$\mathfrak{I}$-phantom tower over $\Phi(A^{\prime})$.

###### Proof.

The triangulated functor $\Phi$ preserves exact triangles and sends
$\mathfrak{I}^{\prime}$ into $\mathfrak{I}$, and thus also sends
$\mathfrak{I}^{\prime}$-exact sequences to $\mathfrak{I}$-exact sequences. ∎

######  Proposition 6.5.

An ABC morphism
$\mathfrak{m}=[\Phi,\alpha,f]\colon(\mathfrak{T},\mathfrak{I},F,A)\to(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$
functorially induces a morphism

$\operatorname{ABC}(\mathfrak{m})\colon\operatorname{ABC}(\mathfrak{T},\mathfrak{I},F,A)\to\operatorname{ABC}(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$

of spectral sequences with the following property. Let $\cal P$ be an
$\mathfrak{I}$-phantom tower over $A$, let $\cal P^{\prime}$ be an
$\mathfrak{I}^{\prime}$-phantom tower over $A^{\prime}$, and let
$\tilde{f}\colon\cal P\to\Phi(\cal P^{\prime})$ be a tower map over $f\colon
A\to\Phi(A^{\prime})$. Let $\alpha_{\cal
P^{\prime}}\colon\operatorname{EC}(\Phi(\cal
P^{\prime}))\to\operatorname{EC}(\cal P^{\prime})$ be the morphism of exact
couples induced by the natural transformation $\alpha$. Then the morphism of
spectral sequences induced by $\alpha_{\cal
P^{\prime}}\circ\operatorname{EC}(\tilde{f})\colon\operatorname{EC}(\cal
P)\to\operatorname{EC}(\cal P^{\prime})$ agrees with
$\operatorname{ABC}(\mathfrak{m})$ from the second sheet onwards. On the
second sheets, we have
$\operatorname{ABC}(\mathfrak{m})^{2}_{p,q}=\mathbb{L}_{p}(\mathfrak{m})_{q}\colon\mathbb{L}_{p}F_{q}(A)\to\mathbb{L}_{p}F^{\prime}_{q}(A^{\prime})$.

${N_{0}}$${N_{1}}$${\phantom{l}\cdots\phantom{l}}$${F_{*}(N_{\bullet})}$${F_{*}(N_{\bullet})}$
${P_{0}}$${P_{1}}$${\phantom{l}\cdots\phantom{l}}$${F_{*}(P_{\bullet})}$${\Phi(N_{0}^{\prime})}$${\Phi(N_{1}^{\prime})}$${\phantom{l}\cdots\phantom{l}}$${F_{*}\Phi(N^{\prime}_{\bullet})}$${F_{*}\Phi(N^{\prime}_{\bullet})}$
${\Phi(P_{0}^{\prime})}$${\Phi(P_{1}^{\prime})}$${\phantom{l}\cdots\phantom{l}}$${{F_{*}\Phi(P^{\prime}_{\bullet})}}$${F^{\prime}_{*}(N^{\prime}_{\bullet})}$${F^{\prime}_{*}(N^{\prime}_{\bullet})}$
${{F^{\prime}_{*}(P^{\prime}_{\bullet})}}$

In red on the left is the tower map $\tilde{f}\colon\cal P\to\Phi(\cal
P^{\prime})$, and on the right are the morphisms
$\operatorname{EC}(\tilde{f})$ and $\alpha_{\cal P^{\prime}}$ of exact
couples.

###### Proof.

To construct $\operatorname{ABC}(\mathfrak{m})$, we consider arbitrary phantom
towers $\cal P$ and $\cal P^{\prime}$ over $A$ and $A^{\prime}$ respectively.
By 6.3, there is a tower map $\tilde{f}\colon\cal P\to\Phi(\cal P^{\prime})$
over $f\colon A\to\Phi(A^{\prime})$. We obtain a morphism of exact couples
$\alpha_{\cal P^{\prime}}\circ EC(\tilde{f})$. We set
$\operatorname{ABC}(\mathfrak{m})$ to be the morphism of spectral sequences
induced by this morphism, starting at the second sheet.

Let $\hat{f}\colon P_{\bullet}\to\Phi(P^{\prime}_{\bullet})$ be the
restriction of $\tilde{f}$ to the $\mathfrak{I}$-exact sequence $P_{\bullet}$
inside $\cal P$. By the description of the exact couple to spectral sequence
functor, the map
$\operatorname{ABC}(\mathfrak{m})^{2}_{p,q}\colon\mathbb{L}_{p}F_{q}(A)\to\mathbb{L}_{p}F_{q}(A^{\prime})$
is induced in homology by the chain map $\alpha_{P^{\prime}}\circ F(\hat{f})$.
5.11 then tells us that
$\operatorname{ABC}(\mathfrak{m})^{2}_{p,q}=\mathbb{L}_{p}(\mathfrak{m})_{q}$.
Morphisms of spectral sequences are determined by their restrictions to the
second sheets (see [Wei94, Mapping Lemma 5.2.4]), and therefore
$\operatorname{ABC}(\mathfrak{m})$ is functorial and independent of the choice
of $\mathfrak{I}$-phantom tower $\cal P$ and tower map $\tilde{f}\colon\cal
P\to\Phi(\cal P^{\prime})$. ∎

Constructing functoriality of spectral sequences alone is not the full story;
we care about the convergence of spectral sequences, which is extra structure.
Recall that a spectral sequence $E$ converges (strongly) to a graded abelian
group $G$ with (ascending) filtration $(\cal F_{k}G)_{k\geq 0}$ with
$\mathcal{F}_{0}G=0$ if the filtration is exhaustive ($\bigcup_{k\geq
0}\mathcal{F}_{k}G=G$) and for each $p,q\in\mathbb{Z}$ there is an isomorphism

$E^{\infty}_{p,q}\cong\frac{\cal F_{p+1}G_{p+q}}{\cal F_{p}G_{p+q}}.$

The ABC spectral sequence of an ABC tuple $(\mathfrak{T},\mathfrak{I},F,A)$
converges strongly to the graded abelian group $\mathbb{L}F(A)$ with the
phantom filtration $(\mathbb{L}F\colon\mathfrak{I}^{k}(A))_{k\geq 0}$ [Mey08,
Theorem 5.1]:

######  Definition 6.6 (Phantom filtration).

Let $\mathfrak{I}$ be a homological ideal in a triangulated category
$\mathfrak{T}$, let $F\colon\mathfrak{T}\to\operatorname{\mathbf{Ab}}_{*}$ be
a stable homological functor and let $A$ be an object of $\mathfrak{T}$. For
each $k\geq 0$, set

$F\colon\mathfrak{I}^{k}(A):=\\{x\in F(A)\mid F(f)(x)=0\;\text{for
all}\;f\in\mathfrak{I}^{k}(A,B),\;B\in\mathfrak{T}\\}.$

This forms an ascending filtration of $F(A)$ called the $\mathfrak{I}$-phantom
filtration of $F$ at $A$.

$0=F\colon\mathfrak{I}^{0}(A)\subseteq F\colon\mathfrak{I}^{1}(A)\subseteq
F\colon\mathfrak{I}^{2}(A)\subseteq\cdots\subseteq F(A)$

To show the convergence of the ABC spectral sequence of the ABC tuple
$\mathfrak{M}=(\mathfrak{T},\mathfrak{I},F,A)$, Meyer shows that
$(\mathbb{L}F\colon\mathfrak{I}^{k}(A))_{k\geq 0}$ is exhaustive and
constructs isomorphisms

$\psi_{p,q}(\mathfrak{M})\colon\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}\to\frac{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p}(A)}.$

Our aim is to show that these isomorphisms are functorial with respect to ABC
morphisms. We collect the following description of the phantom filtration:

######  Lemma 6.7.

Let $(\mathfrak{T},\mathfrak{I},F,A)$ be an ABC tuple with associated
localisation functor $L\colon\mathfrak{T}\to\mathfrak{T}$ and let
$\mathcal{P}\to A$ be an $\mathfrak{I}$-phantom tower over $A$:

${A}$${N_{0}}$${N_{1}}$${N_{2}}$${\cdots}$${P_{0}}$${P_{1}}$${P_{2}}$${\cdots}$$\scriptstyle{=}$$\scriptstyle{\iota_{0}^{1}}$$\scriptstyle{\epsilon_{0}}$$\scriptstyle{\iota_{1}^{2}}$$\scriptstyle{\epsilon_{1}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\delta_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\delta_{2}}$$\scriptstyle{\pi_{2}}$

Then $\ker F(\iota^{k}_{0})=F\colon\mathfrak{I}^{k}(A)$ for every $k\geq 0$,
where
$\iota^{k}_{0}:=\iota^{k}_{k-1}\circ\iota^{k-1}_{k-2}\circ\cdots\circ\iota^{1}_{0}$.
Moreover, the subgroups $\mathbb{L}F\colon\mathfrak{I}^{k}(A)$ and
$F\colon\mathfrak{I}^{k}(LA)$ of $\mathbb{L}F(A)$ are equal.

###### Proof.

That $\ker F(\iota^{k}_{0})=F\colon\mathfrak{I}^{k}(A)$ follows immediately
from [Mey08, Lemmas 3.5-3.7]. The localisation functor $L$ maps
$\mathfrak{I}$-phantom towers to $\mathfrak{I}$-phantom towers, so we may
apply the first part of the lemma to $(\mathbb{L}F,\mathcal{P})$ and
$(F,L(\mathcal{P}))$ respectively to conclude
$\mathbb{L}F\colon\mathfrak{I}^{k}(A)=\ker
F(L(\iota_{0}^{k}))=F\colon\mathfrak{I}^{k}(LA)$. ∎

ABC morphisms respect the phantom filtrations in the following sense:

######  Lemma 6.8.

Let
$[\Phi,\alpha,f]\colon(\mathfrak{T},\mathfrak{I},F,A)\to(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$
be an ABC morphism. Then
$\operatorname{App}(\Phi,\alpha,f)(F\colon\mathfrak{I}^{k}(A))\subseteq
F^{\prime}\colon{\mathfrak{I}^{\prime}}^{k}(A^{\prime})$.

###### Proof.

Let $x\in F\colon\mathfrak{I}^{k}(A)$, and suppose
$g\in{\mathfrak{I}^{\prime}}^{k}(A^{\prime},B^{\prime})$ for some
$B^{\prime}\in\mathfrak{T}^{\prime}$. To show that
$\operatorname{App}(\Phi,\alpha,f)(x)\in
F^{\prime}\colon{\mathfrak{I}^{\prime}}^{k}(A^{\prime})$ we must show that
$F^{\prime}(g)\circ\alpha_{A^{\prime}}\circ F(f)(x)$ vanishes. By naturality
of $\alpha$ at $g$ we may rewrite this as $\alpha_{B^{\prime}}\circ
F(\Phi(g)\circ f)(x)$, which vanishes because $\Phi(g)\circ
f\in\mathfrak{I}^{k}$. ∎

In particular, it follows that
$\mathbb{L}(\Phi,\alpha,f)\colon\mathbb{L}F(A)\to\mathbb{L}F^{\prime}(A^{\prime})$
respects the phantom filtrations. The following lemma will help us to describe
the convergence of the ABC spectral sequence:

######  Lemma 6.9.

Let $(C,D,i,j,k)$ be an exact couple, let $(E,d)$ be the associated spectral
sequence and let $p,q\in\mathbb{Z}$ with $p\geq 0$. Recall that we have the
following descriptions.

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{(-1,0)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-12.8786pt}{-1.30695pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{k}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad
E^{\infty}_{p,q}=\frac{\bigcap_{r\geq
1}(k^{-1}(\operatorname{im}i^{r}))_{p,q}}{\bigcup_{r\geq 1}(j(\ker
i^{r}))_{p,q}}.$

Then there is a homomorphism $\psi_{p,q}\colon E^{\infty}_{p,q}\to\frac{(\ker
i^{p+1})_{-1,p+q}}{(\ker i^{p})_{-1,p+q}}$ which makes the following diagram
commute:

${{\bigcap_{r\geq
1}k^{-1}(\operatorname{im}i^{r})_{p,q}}}$${{D_{p-1,q}}}$${{E^{\infty}_{p,q}}}$${{\frac{(\ker
i^{p+1})_{-1,p+q}}{(\ker
i^{p})_{-1,p+q}}}}$$\scriptstyle{k}$$\scriptstyle{\psi_{p,q}}$$\scriptstyle{i^{p}}$

###### Proof.

The map $k\colon\bigcap_{r\geq 1}k^{-1}(\operatorname{im}i^{r})_{p,q}\to
D_{p-1,q}$ factors through $E^{\infty}_{p,q}$ because $k\circ j=0$. We need
only check that the image is contained in $i^{p}((\ker i^{p+1})_{-1,p+q})$.
For $x\in\bigcap_{r\geq 1}k^{-1}(\operatorname{im}i^{r})_{p,q}$ there is $y\in
D_{-1,p+q}$ such that $k(x)=i^{p}(y)$ by construction. Because $i\circ k=0$,
it follows that $y\in\ker i^{p+1}$. ∎

For an ABC tuple $\mathfrak{M}=(\mathfrak{T},\mathfrak{I},F,A)$, 6.7 tells us
that for the exact couple $\operatorname{EC}(\mathcal{P})$ associated to any
$\mathfrak{I}$-phantom tower $\mathcal{P}\to A$ and $p,q\in\mathbb{Z}$ with
$p\geq 0$ we have

$\frac{(\ker i^{p+1})_{-1,p+q}}{(\ker
i^{p})_{-1,p+q}}=\frac{F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{F_{p+q}\colon\mathfrak{I}^{p}(A)}.$

6.9 gives us the map

$\varphi_{p,q}(\mathfrak{M})\colon\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}\to\frac{F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{F_{p+q}\colon\mathfrak{I}^{p}(A)}$

appearing in [Mey08, Proposition 4.3]. Meyer notes that this does not depend
on the choice of $\mathfrak{I}$-phantom tower and shows that it is an
isomorphism whenever $A\in\mathfrak{P}_{\mathfrak{I}}$ [Mey08, Proposition
4.10]. This applies in particular to
$\mathfrak{L}(\mathfrak{M}):=(\mathfrak{T},\mathfrak{I},F,LA)$, thus
$\varphi_{p,q}(\mathfrak{L}(\mathfrak{M}))$ is an isomorphism. The
$\mathfrak{I}$-equivalence $\mu_{A}\colon LA\to A$ induces an isomorphism
$\operatorname{ABC}(\mu_{A})$ of ABC spectral sequences because the
restriction $\mathbb{L}F(\mu_{A})$ to the second sheets is an isomorphism.
Putting these together forms the desired isomorphism
$\psi(\mathfrak{M})_{p,q}$:

(6.2)
${{\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}}}$${{\frac{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p}(A)}}}$${{\operatorname{ABC}(\mathfrak{L}(\mathfrak{M}))^{\infty}_{p,q}}}$${{\frac{F_{p+q}\colon\mathfrak{I}^{p+1}(LA)}{F_{p+q}\colon\mathfrak{I}^{p}(LA)}}}$$\scriptstyle{\psi(\mathfrak{M})_{p,q}}$$\scriptstyle{\varphi(\mathfrak{L}(\mathfrak{M}))_{p,q}}$$\scriptstyle{\operatorname{ABC}(\mu_{A})^{\infty}_{p,q}}$$\scriptstyle{\cong}$

$\,=$

######  Proposition 6.10.

Let $\mathfrak{M}=(\mathfrak{T},\mathfrak{I},F,A)$ and
$\mathfrak{M}^{\prime}=(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$
be ABC tuples and let $\mathfrak{m}\colon\mathfrak{M}\to\mathfrak{M}^{\prime}$
be an ABC morphism. Then the following diagram commutes:

(6.3)
${\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}}$${\frac{F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{F_{p+q}\colon\mathfrak{I}^{p}(A)}}$${{\operatorname{ABC}(\mathfrak{M}^{\prime})}^{\infty}_{p,q}}$${\frac{F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p+1}(A^{\prime})}{F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p}(A^{\prime})}}$$\scriptstyle{\varphi(\mathfrak{M})_{p,q}}$$\scriptstyle{\operatorname{ABC}(\mathfrak{m})^{\infty}_{p,q}}$$\scriptstyle{\operatorname{App}(\mathfrak{m})_{p,q}}$$\scriptstyle{\varphi(\mathfrak{M}^{\prime})_{p,q}}$

###### Proof.

Let $\mathcal{P}\to A$ and $\mathcal{P}^{\prime}\to A^{\prime}$ be phantom
towers over $A$ and $A^{\prime}$ and let
$\operatorname{EC}(\mathcal{P})=(C,D,i,j,k)$ and
$\operatorname{EC}(\mathcal{Q})=(C^{\prime},D^{\prime},i^{\prime},j^{\prime},k^{\prime})$
be the associated exact couples. Suppose $\mathfrak{m}=[\Phi,\alpha,f]$ and
let $\tilde{f}\colon\cal P\to\Phi(\cal P^{\prime})$ be a tower map over
$f\colon A\to\Phi(A^{\prime})$. Let $S\colon C\to C^{\prime}$ and $T\colon
D\to D^{\prime}$ be the constituent components of the morphism $\alpha_{\cal
P^{\prime}}\circ\operatorname{EC}(\tilde{f})$ of exact couples. Then for each
$n\in\mathbb{Z}$ we have $D_{-1,n}=F_{n}(A)$ and
$D^{\prime}_{-1,n}=F^{\prime}_{n}(A^{\prime})$. Furthermore, $T_{-1,n}\colon
D_{-1,n}\to D^{\prime}_{-1,n}$ is given by the map
$\operatorname{App}(\mathfrak{m})_{n}\colon F_{n}(A)\to
F^{\prime}_{n}(A^{\prime})$.

${D}$${D}$${{D_{-1,n}}}$${F_{n}(A)}$${C}$${D^{\prime}}$${D^{\prime}}$${{D^{\prime}_{-1,n}}}$${F^{\prime}_{n}(A^{\prime})}$${C^{\prime}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}T}$$\scriptstyle{i}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}T}$$\scriptstyle{j}$${=}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}T_{-1,n}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\operatorname{App}(\mathfrak{m})_{n}}$$\scriptstyle{k}$$\scriptstyle{i^{\prime}}$$\scriptstyle{j^{\prime}}$${=}$$\scriptstyle{k^{\prime}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S}$

Recall that
$\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}=\frac{\bigcap_{r\geq
1}(k^{-1}(\operatorname{im}i^{r}))_{p,q}}{\bigcup_{r\geq 1}(j(\ker
i^{r}))_{p,q}}$, let $x\in\bigcap_{r\geq
1}(k^{-1}(\operatorname{im}i^{r}))_{p,q}$ and pick $y\in
F_{p+q}\colon\mathfrak{I}^{p+1}(A)$ with
$\varphi(\mathfrak{M})_{p,q}([x])=[y]$. Since $y\in D_{-1,p+q}=F_{p+q}(A)$, we
may use that $\operatorname{App}(\mathfrak{m})_{p+q}=T_{-1,p+q}$ to calculate
that

$\operatorname{App}(\mathfrak{m})_{p,q}(\varphi(\mathfrak{M})_{p,q}([x]))=\operatorname{App}(\mathfrak{m})_{p,q}([y])=\left[\operatorname{App}(\mathfrak{m})_{p+q}(y)\right]=[T(y)].$

On the other hand,
$\operatorname{ABC}(\mathfrak{m})^{\infty}_{p,q}([x])=[S(x)]$. We may
calculate that ${i^{\prime}}^{p}(T(y))=T(i^{p}(y))=T(k(x))=k^{\prime}(S(x))$,
and therefore by 6.9, $\varphi(\mathfrak{M}^{\prime})_{p,q}([S(x)])=[T(y)]$.
We conclude that (6.3) commutes. ∎

######  Theorem (Theorem 6.1).

An ABC morphism $\mathfrak{m}\colon\mathfrak{M}\to\mathfrak{M}^{\prime}$
functorially induces a morphism of ABC spectral sequences
$\operatorname{ABC}(\mathfrak{m})\colon\operatorname{ABC}(\mathfrak{M})\to\operatorname{ABC}(\mathfrak{M}^{\prime})$,
such that:

  1. (i)

the map on the second sheets is given by
$\mathbb{L}_{p}(\mathfrak{m})_{q}\colon\mathbb{L}_{p}(\mathfrak{M})_{q}\to\mathbb{L}_{p}(\mathfrak{M}^{\prime})_{q}$,

  2. (ii)

the map on the limit sheets agrees with
$\mathbb{L}(\mathfrak{m})\colon\mathbb{L}(\mathfrak{M})\to\mathbb{L}(\mathfrak{M}^{\prime})$.

###### Proof.

Let $\mathfrak{m}\colon\mathfrak{M}\to\mathfrak{M}^{\prime}$ be represented by
an ABC cycle

$(\Phi,\alpha,f)\colon(\mathfrak{T},\mathfrak{I},F,A)\to(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime}).$

The construction of the morphism
$\operatorname{ABC}(\mathfrak{m})\colon\operatorname{ABC}(\mathfrak{M})\to\operatorname{ABC}(\mathfrak{M}^{\prime})$
was given in 6.5, where it was shown to be functorial and agree with
$\mathbb{L}_{p}(\mathfrak{m})_{q}$ on the second sheets. To show compatibility
of the map on the limit sheets with $\mathbb{L}(\mathfrak{m})$ we must show
that

${{\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}}}$${{\frac{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p}(A)}}}$${{\operatorname{ABC}(\mathfrak{M}^{\prime})^{\infty}_{p,q}}}$${{\frac{\mathbb{L}^{\prime}F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p+1}(A^{\prime})}{\mathbb{L}^{\prime}F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p}(A^{\prime})}}}$$\scriptstyle{\cong}$$\scriptstyle{\psi(\mathfrak{M})_{p,q}}$$\scriptstyle{\cong}$$\scriptstyle{\psi(\mathfrak{M}^{\prime})_{p,q}}$$\scriptstyle{\operatorname{ABC}(\mathfrak{m})^{\infty}_{p,q}}$$\scriptstyle{\mathbb{L}(\mathfrak{m})_{p,q}}$

commutes. Let $\mu\colon L\Rightarrow\operatorname{id}$ and
$\mu^{\prime}\colon L^{\prime}\Rightarrow\operatorname{id}$ be the
localisation natural transformations associated to $\mathfrak{M}$ and
$\mathfrak{M}^{\prime}$ and set
$\mathfrak{L}(\mathfrak{M})=(\mathfrak{T},\mathfrak{I},F,LA)$,
$\mathfrak{L}(\mathfrak{M}^{\prime})=(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},L^{\prime}A^{\prime})$
and
$\mathfrak{L}(\mathfrak{m})=[\Phi,\alpha,\mathbb{L}_{\Phi}(A^{\prime})\circ
L(f)]$. By (6.2) the diagram breaks into three parts:

${{\operatorname{ABC}(\mathfrak{M})^{\infty}_{p,q}}}$${{\operatorname{ABC}(\mathfrak{L}(\mathfrak{M}))^{\infty}_{p,q}}}$${{\frac{F_{p+q}\colon\mathfrak{I}^{p+1}(LA)}{F_{p+q}\colon\mathfrak{I}^{p}(LA)}}}$${{\frac{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p+1}(A)}{\mathbb{L}F_{p+q}\colon\mathfrak{I}^{p}(A)}}}$${{\operatorname{ABC}(\mathfrak{M}^{\prime})^{\infty}_{p,q}}}$${{\operatorname{ABC}(\mathfrak{L}(\mathfrak{M}^{\prime}))^{\infty}_{p,q}}}$${{\frac{F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p+1}(L^{\prime}A^{\prime})}{F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p}(L^{\prime}A^{\prime})}}}$${{\frac{\mathbb{L}^{\prime}F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p+1}(A^{\prime})}{\mathbb{L}^{\prime}F^{\prime}_{p+q}\colon{\mathfrak{I}^{\prime}}^{p}(A^{\prime})}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\varphi(\mathfrak{M})_{p,q}}$${=}$$\scriptstyle{\mathbb{L}(\mathfrak{m})_{p,q}}$${=}$$\scriptstyle{\operatorname{App}(\mathfrak{L}(\mathfrak{m}))_{p,q}}$$\scriptstyle{\cong}$$\scriptstyle{\varphi(\mathfrak{M})_{p,q}}$$\scriptstyle{\operatorname{ABC}(\mathfrak{L}(\mathfrak{m}))^{\infty}_{p,q}}$$\scriptstyle{\operatorname{ABC}(\mathfrak{m})^{\infty}_{p,q}}$$\scriptstyle{\cong}$

The left square arises from a square of ABC morphisms:

${{(\mathfrak{T},\mathfrak{I},F,A)}}$${{(\mathfrak{T},\mathfrak{I},F,LA)}}$${{(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})}}$${{(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},L^{\prime}A^{\prime})}}$$\scriptstyle{[\Phi,\alpha,f]}$$\scriptstyle{[\operatorname{id}_{\mathfrak{T}},\operatorname{id}_{F},\mu_{A}]}$$\scriptstyle{\mathfrak{L}(\mathfrak{m})}$$\scriptstyle{[\operatorname{id}_{\mathfrak{T}^{\prime}},\operatorname{id}_{F^{\prime}},\mu^{\prime}_{A^{\prime}}]}$

Commutativity of this square boils down to the equation
$\Phi(\mu^{\prime}_{A^{\prime}})\circ\mathbb{L}_{\Phi}(A^{\prime})\circ
L(f)=f\circ\mu_{A}$, which follows from naturality of $\mu$ applied to $f$ and
$\Phi\mu^{\prime}\circ\mathbb{L}_{\Phi}=\mu\Phi$ applied to $A^{\prime}$. The
middle square commutes by 6.10. Finally, the right square commutes as
$\mathbb{L}(\mathfrak{m})=\operatorname{App}(\mathfrak{L}(\mathfrak{m}))$. ∎

##  7\. Isomorphisms of K-theory groups

A morphism of spectral sequences which is an isomorphism on the second sheet
will be an isomorphism on all later sheets and when the sequences converge,
any compatible map of the target groups must be an isomorphism [Wei94,
Comparison Theorem 5.2.12]. An ABC morphism
$\mathfrak{m}\colon\mathfrak{M}\to\mathfrak{M}^{\prime}$ which induces an
isomorphism

$\mathbb{L}_{n}(\mathfrak{m})\colon\mathbb{L}_{n}(\mathfrak{M})\to\mathbb{L}_{n}(\mathfrak{M}^{\prime})$

of derived functors for each $n\geq 0$ will therefore by 6.1 induce an
isomorphism

$\mathbb{L}(\mathfrak{m})\colon\mathbb{L}(\mathfrak{M})\to\mathbb{L}(\mathfrak{M}^{\prime})$

of localisations. Consider an ABC morphism
$\mathfrak{m}=[\operatorname{Ind}_{\Omega},\alpha_{\Omega},f]$ arising from an
étale correspondence as in Example 5.7. If the families of open embeddings
satisfy condition (P) then by 3.17 the localisations are the topological
K-theories and
$\mathbb{L}(\mathfrak{m})=\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)$ as in
Example 5.10. Piecing together the above with the Baum–Connes conjecture we
obtain a tool for proving isomorphisms of the K-theory groups of reduced
crossed products:

######  Corollary 7.1.

Let $G$ and $H$ be second countable Hausdorff étale groupoids with coefficient
C*-algebras $A\in\operatorname{KK}^{G}$ and $B\in\operatorname{KK}^{H}$, let
$\Omega\colon G\to H$ be a second countable Hausdorff étale correspondence and
let $f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$. Let $\cal E$
and $\cal F$ be families of second countable Hausdorff proper groupoids openly
Morita embedded into $G$ and $H$ respectively which are compatible under
$\Omega$, so that we get an ABC morphism

$[\operatorname{Ind}_{\Omega},\alpha_{\Omega},f]\colon(\operatorname{KK}^{G},\mathfrak{I}_{\cal
F_{G}},\mathrm{K}_{*}(G\ltimes-),A)\to(\operatorname{KK}^{H},\mathfrak{I}_{\cal
F_{H}},\mathrm{K}_{*}(H\ltimes-),B).$

Suppose that for each $n\geq 0$ the derived functor map

$\mathbb{L}_{n}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)\colon\mathbb{L}_{n}\mathrm{K}_{*}(G\ltimes
A)\to\mathbb{L}_{n}\mathrm{K}_{*}(H\ltimes B)$

is an isomorphism. Then

$\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)\colon\mathrm{K}^{\mathrm{top}}_{*}(G;A)\to\mathrm{K}^{\mathrm{top}}_{*}(H;B)$

is an isomorphism. If $(G,A)$ and $(H,B)$ satisfy the Baum–Connes conjecture
we obtain an isomorphism
$\mathrm{K}_{*}(G\ltimes_{r}A)\cong\mathrm{K}_{*}(H\ltimes_{r}B)$. If moreover
the $\mathrm{C}^{\ast}$-correspondence $\Omega\ltimes B\colon
G\ltimes\operatorname{Ind}_{\Omega}B\to H\ltimes B$ descends to the reduced
level, the resulting map
$\mathrm{K}^{\mathrm{red}}_{*}(\Omega;f)\colon\mathrm{K}_{*}(G\ltimes_{r}A)\to\mathrm{K}_{*}(H\ltimes_{r}B)$
equals this isomorphism (see Example 5.10).

Application of this result depends on an understanding of the derived functors
$\mathbb{L}_{n}$. Our understanding is especially good for ample groupoids
with the singleton unit space family. The analysis in Section 4 culminates in
the description of
$\mathbb{L}_{n}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)$ in 5.13 as the
induced map $\mathrm{H}_{n}(\Omega;\mathrm{K}_{*}(f))$ in homology from
[Mil24, Theorem 3.5]. If the groupoids have torsion-free isotropy groups, the
unit space family satisfies (P) by 3.5. Thus we may turn isomorphisms in
homology into isomorphisms in K-theory:

######  Corollary 7.2.

Let $G$ and $H$ be second countable Hausdorff ample groupoids with torsion-
free isotropy groups and which satisfy the Baum–Connes conjecture with
coefficients in $\mathrm{C}^{\ast}$-algebras $A\in\operatorname{KK}^{G}$ and
$B\in\operatorname{KK}^{H}$. Let $\Omega\colon G\to H$ be a second countable
Hausdorff étale correspondence and let
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$. Suppose that the
induced map in homology

$\mathrm{H}_{n}(\Omega;\mathrm{K}_{*}(f))\colon\mathrm{H}_{n}(G;\mathrm{K}_{*}(A))\to\mathrm{H}_{n}(H;\mathrm{K}_{*}(B))$

is an isomorphism for each $n\geq 0$. Then there is an isomorphism
$\mathrm{K}_{*}(G\ltimes_{r}A)\cong\mathrm{K}_{*}(H\ltimes_{r}B)$.

When $\Omega\colon G\to H$ is proper, taking the morphism $f_{\Omega}$ induced
by the proper map $\overline{\rho}\colon\Omega/H\to G^{0}$ gives us a version
of this corollary without coefficients. Note that if $G$ and $H$ satisfy the
Baum–Connes conjecture and $C^{*}(\Omega)\colon C^{*}(G)\to C^{*}(H)$ descends
to a $\mathrm{C}^{\ast}$-correspondence $C^{*}_{r}(\Omega)\colon
C^{*}_{r}(G)\to C^{*}_{r}(H)$ at the reduced level, then the induced map in
K-theory is always
$\mathrm{K}^{\mathrm{red}}_{*}(\Omega;f_{\Omega})=\mathrm{K}_{*}(C^{*}_{r}(\Omega))$.

######  Corollary 7.3.

Let $G$ and $H$ be second countable Hausdorff ample groupoids with torsion-
free isotropy groups and which satisfy the Baum–Connes conjecture and let
$\Omega\colon G\to H$ be a proper second countable Hausdorff étale
correspondence. Suppose that the induced map in homology
$\mathrm{H}_{*}(\Omega)\colon\mathrm{H}_{*}(G)\to\mathrm{H}_{*}(H)$ is an
isomorphism. Then there is an isomorphism
$\mathrm{K}_{*}(C^{*}_{r}(G))\cong\mathrm{K}_{*}(C^{*}_{r}(H))$.

Let $S$ be an inverse semigroup (with $0$) with idempotent semilattice $E$.
There is a proper étale correspondence

$\Omega_{S}\colon S^{\times}\to G_{S}$

which induces an isomorphism in homology groups [Mil24, Example 3.10]. Here
$S^{\times}$ is the discrete groupoid of non-zero elements of $S$, which has
unit space $E^{\times}$, and $G_{S}$ is the universal groupoid.

When $G_{S}$ is Hausdorff it is a groupoid model for the
$\mathrm{C}^{\ast}$-algebra $C^{*}_{\lambda}(S)$ generated by the left regular
representation of $S$ by partial isometries on $\ell^{2}(S^{\times})$ [CELY17,
Theorem 5.5.18]. Hausdorffness of the universal groupoid $G_{S}$ is equivalent
to the natural order on $S$ being a weak semilattice [Ste10, Theorem 5.17].
The isotropy group of $S^{\times}$ at an idempotent $e\in E^{\times}$ is the
stabiliser subgroup $S_{e}=\\{s\in S\mid s^{*}s=ss^{*}=e\\}$. Each stabiliser
subgroup embeds in an isotropy group of $G_{S}$, so if $G_{S}$ satisfies the
Baum–Connes conjecture then so does $S^{\times}$ as a discrete groupoid.

Applying 7.3 to the isomorphism $\mathrm{H}_{*}(\Omega_{S})$ in homology we
obtain a K-theory formula previously only achieved for inverse semigroups
admitting an idempotent partial pure homomorphism into a group satisfying
Baum–Connes [Li21, Theorem 1.1], at the price of adding a torsion-freeness
hypothesis:

######  Corollary 7.4.

Let $S$ be a countable inverse semigroup with idempotent semilattice $E$ such
that for each $e\in E^{\times}$ the stabiliser subgroup $S_{e}$ is torsion-
free and the universal groupoid $G_{S}$ is Hausdorff and satisfies the
Baum–Connes conjecture. Then

$\mathrm{K}_{*}(C^{*}_{\lambda}(S))\cong\bigoplus_{\operatorname{orb}(e)\in
S\backslash E^{\times}}\mathrm{K}_{*}(C^{*}_{\lambda}(S_{e})).$

######  Remark 7.5.

In the presence of torsion, groupoid homology is the wrong tool to utilise for
the purpose of this K-theory formula. Nevertheless, we may still apply 7.1 to
the étale correspondence $\Omega_{S}\colon S^{\times}\to G_{S}$ by considering
the families of subgroupoids defined by the finite subgroups of the stabiliser
groups as in Example 3.21. To remove the torsion-freeness assumption in 7.4 it
therefore suffices to prove that the ABC morphism
$\mathfrak{m}_{S}=[\operatorname{Ind}_{\Omega_{S}},\alpha_{\Omega_{S}},f_{\Omega_{S}}]$
induced by $\Omega_{S}$ induces an isomorphism
$\mathbb{L}_{n}(\mathfrak{m}_{S})$ of derived functors for each $n\geq 0$.

######  Example 7.6 (Right LCM monoids).

The left regular algebra $C^{*}_{\lambda}(P)$ of a countable right LCM left
cancellative monoid $P$ is isomorphic to the left regular algebra
$C^{*}_{\lambda}(I_{\ell}(P))$ of its left inverse hull $I_{\ell}(P)$.
Assuming that the universal groupoid is Hausdorff, satisfies Baum–Connes and
has torsion-free stabilisers, 7.4 says that

$\mathrm{K}_{*}(C^{*}_{\lambda}(P))\cong\mathrm{K}_{*}(C^{*}_{\lambda}(P^{*})),$

where $P^{*}$ is the group of invertible elements of $P$. The main advancement
from [Li21, Corollary 1.4] is that $P$ need not embed in a group. Weakening
the right LCM condition to the independence condition, there is a similar
computation that incorporates the constructible ideals of $P$ (see [Li21,
Corollary 1.3]).

######  Example 7.7 (The Toeplitz algebra of a singly aligned higher rank
graph).

The Toeplitz algebra $\mathcal{T}C^{*}(\Lambda)$ of a countable singly aligned
higher rank graph $\Lambda$ is isomorphic to the left regular algebra
$C^{*}_{\lambda}(S_{\Lambda})$ of the graph inverse semigroup $S_{\Lambda}$.
The universal groupoid is Hausdorff, amenable, and has torsion-free
stabilisers. The non-zero idempotents of $S_{\Lambda}$ may be identified with
$\Lambda$ and their orbits under the action of $S_{\Lambda}$ correspond to the
vertices $\Lambda^{0}$. The stabiliser subgroups are trivial, so 7.4 recovers
the K-theory computation [Fle18, Corollary 2.5.7]

$\displaystyle\mathrm{K}_{0}(\mathcal{T}C^{*}(\Lambda))$
$\displaystyle\cong\bigoplus_{v\in\Lambda^{0}}\mathbb{Z}$
$\displaystyle\mathrm{K}_{1}(\mathcal{T}C^{*}(\Lambda))$ $\displaystyle\cong
0.$

As higher rank graphs need not admit a faithful functor to a group (see
[PQR04, Section 7]), the inverse semigroup $S_{\Lambda}$ may not admit an
idempotent pure partial homomorphism to a group, so this computation is not
covered by [Li21, Theorem 1.1].

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