# Isomorphisms in K-theory from isomorphisms in groupoid homology theories

Alistair Miller

###### Abstract.

We prove that for torsion-free amenable ample groupoids, an isomorphism in
groupoid homology induced by an étale correspondence yields an isomorphism in
the K-theory of the associated $\mathrm{C}^{\ast}$-algebras. We apply this to
extend X. Li’s K-theory formula for left regular inverse semigroup
$\mathrm{C}^{\ast}$-algebras. These results are obtained by developing the
functoriality of the ABC spectral sequence.

00footnotetext: This work contains part of the author’s PhD thesis, which was
supported by the Engineering and Physical Sciences Research Council (EPSRC)
through a doctoral studentship. The author has also been supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 817597) and by the
Independent Research Fund Denmark through the Grant 1054-00094B.

##  1\. Introduction

Étale groupoids and their $\mathrm{C}^{\ast}$-algebras are a central topic in
operator algebras, arising naturally from noncommutative geometry and
topological dynamics. Twisted étale groupoids also arise from within the field
of $\mathrm{C}^{\ast}$-algebras via reconstruction [Ren08] and as models for
every Elliott-classifiable $\mathrm{C}^{\ast}$-algebra [Li20]. The homological
and K-theoretic invariants associated to étale groupoids contain crucial
information. Indeed, the power of K-theory to remember the associated
$\mathrm{C}^{\ast}$-algebra is demonstrated by the celebrated classification
programme [Kir95, Phi00, GLN20, TWW17, Win18], and the K-theory furthermore
often retains information prior to the $\mathrm{C}^{\ast}$-algebra, for
example classifying Cantor minimal systems up to orbit equivalence [GPS95].
While the computation of the operator K-theory $\mathrm{K}_{*}(C^{*}_{r}(G))$
associated to an étale groupoid $G$ can prove quite challenging, it is often
possible to make progress by relating the operator K-theory to more computable
homological invariants of the groupoid such as groupoid homology and
topological K-theory. Our main contribution is to show that an isomorphism in
groupoid homology can be converted to an isomorphism in K-theory when it is
induced by an étale correspondence.

A connection between the homology of an ample groupoid and the operator
K-theory was uncovered by Matui in examples coming from topological dynamics
[Mat12, Mat15, Mat16]. This led him to conjecture that for an ample groupoid
$G$ in a certain class, the K-theory groups decompose as a direct sum of
groupoid homology groups of the same parity:

$\mathrm{K}_{*}(C^{*}_{r}(G))\cong\bigoplus_{i\in\mathbb{N}}\mathrm{H}_{*+2i}(G)$

Known as Matui’s HK conjecture, this has been verified for a wide variety of
examples [FKPS19, BDGW23, PY23], but has counterexamples [Sca20, Dee23] which
ultimately point to a more subtle relationship between $\mathrm{H}_{*}(G)$ and
$\mathrm{K}_{*}(C^{*}_{r}(G))$.

The topological K-theory $\mathrm{K}^{\mathrm{top}}_{*}(G)$ of a second
countable Hausdorff étale groupoid $G$ is an alternative K-theoretic invariant
to the operator K-theory. The Baum–Connes conjecture for $G$ [BCH94, Tu99]
asserts the bijectivity of a homomorphism

$\mu_{G}\colon\mathrm{K}^{\mathrm{top}}_{*}(G)\to\mathrm{K}_{*}(C^{*}_{r}(G))$

known as the Baum–Connes assembly map. This is known to hold for groupoids
with the Haagerup property (in particular amenable groupoids) [Tu99], and
while there are counterexamples [HLS02], the conjecture is open for exact
groupoids. The nature of topological K-theory makes it more susceptible to
homological methods. Specialising to groups, there is a spectral sequence
which converges to the topological K-theory whose second sheet is given by
Bredon homology [Mis03]. An instance of Meyer’s ABC spectral sequence [Mey08]
considered by Proietti and Yamashita for ample groupoids with torsion-free
isotropy converges to the topological K-theory, with groupoid homology on the
second sheet [PY22]. Due to work of Bönicke and Proietti [BP24], there is a
more general spectral sequence converging to the topological K-theory of an
arbitrary étale groupoid:

$E^{2}_{p,q}\Rightarrow\mathrm{K}^{\mathrm{top}}_{p+q}(G)$

The second sheet of this spectral sequence is occupied by a homology theory
for the groupoid analogous to Bredon homology for groups and groupoid homology
for torsion-free ample groupoids. Outside of the torsion-free ample setting we
should expect a homology theory which differs from groupoid homology. This
sheds some light on the HK conjecture; the presence of torsion in the isotropy
makes groupoid homology the wrong invariant to compare to K-theory (see
[Sca20]), and while the spectral sequence degenerates to a direct sum when the
homological dimension is low, this should not be expected in higher dimensions
(see [Dee23]).

To get the most out of the above spectral sequences linking these homology
theories to operator/topological K-theory, we study their functoriality with
respect to the groupoid. We work with a class of morphisms of étale groupoids
called étale correspondences, which are an analogue for étale groupoids of
$\mathrm{C}^{\ast}$-correspondences, modelling the behaviour of both
$*$-homomorphisms and Morita equivalences. They encompass many kinds of
morphisms of étale groupoids, including open embeddings, Morita equivalences,
actors111also known as Zakrzewski or algebraic morphisms and étale
homomorphisms. A proper étale correspondence $\Omega\colon G\to H$ induces a
map in K-theory through the associated proper
$\mathrm{C}^{\ast}$-correspondence [AKM22], and for ample groupoids the author
constructed an induced map
$\mathrm{H}_{*}(\Omega)\colon\mathrm{H}_{*}(G)\to\mathrm{H}_{*}(H)$ in
homology [Mil24].

While the situation is more complex outside of the torsion-free ample setting,
the spirit of this paper is captured by the following result:

######  Theorem A (see 7.3).

Let $G$ and $H$ be second countable Hausdorff ample groupoids with torsion-
free isotropy groups and which satisfy the Baum–Connes conjecture. Suppose
$\Omega\colon G\to H$ is a proper second countable Hausdorff étale
correspondence which induces an isomorphism

$\mathrm{H}_{*}(\Omega)\colon\mathrm{H}_{*}(G)\to\mathrm{H}_{*}(H)$

in homology. Then $\Omega$ induces an isomorphism
$\mathrm{K}_{*}(C^{*}_{r}(G))\cong\mathrm{K}_{*}(C^{*}_{r}(H))$ in K-theory.

If furthermore $\Omega\colon G\to H$ induces a
$\mathrm{C}^{\ast}$-correspondence $C^{*}_{r}(\Omega)\colon C^{*}_{r}(G)\to
C^{*}_{r}(H)$ at the reduced level (see Example 5.10), the induced isomorphism
is given by

$\mathrm{K}_{*}(C^{*}_{r}(\Omega))\colon\mathrm{K}_{*}(C^{*}_{r}(G))\to\mathrm{K}_{*}(C^{*}_{r}(H)).$

Groupoid homology is often more tractable than operator K-theory, and it may
be possible to demonstrate an isomorphism in homology directly where there is
no clear direct method for K-theory. For example, for any inverse semigroup
$S$ with universal groupoid $G_{S}$, there is a proper étale correspondence

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

viewing the set $S^{\times}$ of non-zero elements of $S$ as a discrete
groupoid. This induces an isomorphism in homology [Mil24, Example 3.10].
Applying A to $\Omega_{S}$ enables us to reduce the computation of the
K-theory of the reduced $\mathrm{C}^{\ast}$-algebra $C^{*}_{\lambda}(S)$ to
computations for its stabiliser subgroups at non-zero idempotents:

######  Corollary B (see 7.4).

Let $S$ be a countable inverse semigroup with idempotent semilattice $E$ whose
universal groupoid is Hausdorff and satisfies the Baum–Connes conjecture.
Suppose further that the stabiliser subgroup $S_{e}=\\{s\in S\mid
s^{*}s=ss^{*}=e\\}$ of each non-zero idempotent $e\in E^{\times}$ is torsion-
free. Then

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

Aside from our torsion-freeness assumption, this generalises [Li21, Theorem
1.1] to inverse semigroups that need not admit an idempotent pure partial
homomorphism to a group. Let us explain the key differences in our work to
previous approaches [CEL13, CEL15, Li21] that allow us to overcome the
difficulties in extending to this more general setting. When $S$ admits an
idempotent pure partial homomorphism to a group $\Gamma$, the dynamics of $S$
are governed by $\Gamma$ in such a way that the desired map
$\mathrm{K}_{*}(C^{*}_{r}(S^{\times}))\to\mathrm{K}_{*}(C^{*}_{\lambda}(S))$
can be constructed from a $\Gamma$-equivariant Kasparov cycle
$\alpha\in\operatorname{KK}^{\Gamma}$ as
$\mathrm{K}_{*}(\Gamma\ltimes_{r}\alpha)$. Here the previous approaches make
vital use of the going-down principle [CEOO04], which gives a sufficient
condition on an equivariant Kasparov cycle to induce an isomorphism in the
K-theory of the crossed products. However, in general there need be no group
to govern the dynamics of $S$, so we turn to groupoid techniques. Although a
going-down principle for ample groupoids was developed in [Bön20], there is no
obvious way to construct the desired Kasparov cycle equivariantly with respect
to a single groupoid. Instead, we model the desired map in K-theory with the
proper étale correspondence $\Omega_{S}\colon S^{\times}\to G_{S}$ of
groupoids. The sufficient condition given in A to obtain an isomorphism in
K-theory may be compared to the going-down principle, but rather than having
to work equivariantly with respect to a single group or groupoid we can work
with morphisms on the groupoid level.

B yields a K-theory formula for the left regular algebra $C^{*}_{\lambda}(P)$
of many left cancellative monoids $P$ which satisfy the independence condition
but which need not embed in a group (see Example 7.6). It also recovers the
K-theory of the Toeplitz algebra of a general singly aligned higher rank graph
(Example 7.7).

The results in this paper are centred around the ABC spectral sequence [Mey08,
Theorem 5.1], which is constructed in the Meyer–Nest framework of homological
algebra in triangulated categories via homological ideals [MN06, MN10, Mey08].
The main example we focus on is in the setting of a second countable Hausdorff
étale groupoid $G$ with a specified countable family $\cal F$ of proper open
subgroupoids, meant to resemble the family of finite subgroups of a countable
group, and a separable $G$-$\mathrm{C}^{\ast}$-algebra $A$. There is a
homological ideal $\mathfrak{I}_{\cal F}$ in the (triangulated) equivariant
Kasparov category $\operatorname{KK}^{G}$ consisting of the morphisms which
vanish under restriction
$\operatorname{Res}^{H}_{G}\colon\operatorname{KK}^{G}\to KK^{H}$ to every
member $H$ of $\cal F$. With respect to the homological ideal
$\mathfrak{I}_{\cal F}$ we can form the localisation $\mathbb{L}^{\cal
F}\mathrm{K}_{*}(G\ltimes_{r}-)$ and the derived functors $\mathbb{L}^{\cal
F}_{n}\mathrm{K}_{*}(G\ltimes_{r}-)$ for $n\geq 0$ of the stable homological
functor $\mathrm{K}_{*}(G\ltimes_{r}-)$ on $\operatorname{KK}^{G}$. The
localisation is an approximation to $\mathrm{K}_{*}(G\ltimes_{r}-)$ which
vanishes on the objects in $\mathfrak{I}_{\cal F}$, and comes with a natural
transformation $\mathbb{L}^{\cal
F}\mathrm{K}_{*}(G\ltimes_{r}-)\Rightarrow\mathrm{K}_{*}(G\ltimes_{r}-)$ to
the original functor. For suitable $\cal F$ the localisation may be identified
with the topological K-theory, through which the natural transformation
recovers the Baum–Connes assembly map [BP24, Theorem C]. The derived functors
are analogous to classical derived functors on abelian categories; when $G$ is
ample and $\cal F=\\{G^{0}\\}$ they model the groupoid homology groups
$\mathrm{H}_{n}(G;\mathrm{K}_{*}(-))$ [PY22]. The ABC spectral sequence
approximates the localisation of a stable homological functor $F$ on a
triangulated category $\mathfrak{T}$ at an object $A$ with respect to a
suitable homological ideal $\mathfrak{I}\vartriangleleft\mathfrak{T}$ with its
derived functors:

$E^{2}_{p,q}=\mathbb{L}^{\mathfrak{I}}_{p}F_{q}(A)\Rightarrow\mathbb{L}^{\mathfrak{I}}F_{p+q}(A)$

We refer to such quadruples $(\mathfrak{T},\mathfrak{I},F,A)$ as ABC tuples
and introduce ABC morphisms to place them in a category. We construct ABC
morphisms from étale correspondences $\Omega\colon G\to H$ of étale groupoids
which are compatible with respect to families of subgroupoids on either side.
The two main components of this construction are the induction functor
$\operatorname{Ind}_{\Omega}\colon\operatorname{KK}^{H}\to\operatorname{KK}^{G}$
and the induction natural transformation
$\alpha_{\Omega}\colon\mathrm{K}_{*}(G\ltimes\operatorname{Ind}_{\Omega}-)\Rightarrow\mathrm{K}_{*}(H\ltimes-)$
from [Mil23]. From an ABC morphism $\mathfrak{m}$ we construct a morphism
$\mathbb{L}(\mathfrak{m})$ of the associated localisations, morphisms
$\mathbb{L}_{n}(\mathfrak{m})$ of the derived functors and ultimately a
morphism $\operatorname{ABC}(\mathfrak{m})$ of the associated spectral
sequences:

######  Theorem C (see Theorem 6.1).

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

  1. (i)

the map on the second sheet is given by the derived functor maps
$\mathbb{L}_{n}(\mathfrak{m})$,

  2. (ii)

the map on the limit sheet agrees with the localisation map
$\mathbb{L}(\mathfrak{m})$.

An ABC morphism $\mathfrak{m}$ for which the morphism
$\mathbb{L}_{n}(\mathfrak{m})$ of derived functors is an isomorphism for each
$n$ therefore induces an isomorphism $\mathbb{L}(\mathfrak{m})$ of
localisations [Wei94, Comparison Theorem 5.2.12]. Since we wish to apply this
principle to the topological K-theory of étale groupoids, we initiate a
systematic study into ABC tuples we can associate to a second countable
Hausdorff étale groupoid $G$ whose localisation models the topological
K-theory. While suitable families of proper open subgroupoids have been
constructed in order to obtain the topological K-theory in [BP24], there is no
guarantee that a given étale correspondence is compatible with families of
this form on either side. We seek to remedy this through the introduction of a
practical sufficient condition for a family to be suitable. A family $\cal F$
of proper open subgroupoids of $G$ satisfies condition (P) if for each $x\in
G^{0}$ and each finite subgroup $\Gamma\leq G^{x}_{x}$ of the isotropy group
at $x$, there is a member $H\in\cal F$ containing $\Gamma$. Immediately it is
clear that if $G$ has torsion-free isotropy, then the trivial family
$\\{G^{0}\\}$ satisfies condition (P). In Example 3.6 we introduce an explicit
way to construct families satisfying (P) for any étale groupoid.

######  Theorem D (see Propositions 3.2 and 3.5).

Let $G$ be a second countable Hausdorff étale groupoid and suppose a countable
family $\cal F$ of proper open subgroupoids of $G$ satisfies condition (P).
Then for $A\in\operatorname{KK}^{G}$ there is a natural isomorphism

$\mathbb{L}^{\cal
F}\mathrm{K}_{*}(G\ltimes_{r}A)\cong\mathrm{K}^{\mathrm{top}}_{*}(G;A).$

This can be pushed further if we relax the notion of an open subgroupoid to a
groupoid which need only be openly embedded up to Morita equivalence, which we
term an open Morita embedding. The construction of homological ideals giving
ABC tuples generalises to families of open Morita embeddings, as does
condition (P) and subsequently D. We describe constructions of families of
open Morita embeddings satisfying condition (P) compatible with a given étale
correspondence, yielding:

######  Theorem E (see Example 5.10).

Let $G$ and $H$ be second countable Hausdorff étale groupoids with coefficient
$\mathrm{C}^{\ast}$-algebras $A\in\operatorname{KK}^{G}$ and
$B\in\operatorname{KK}^{H}$. Given a second countable Hausdorff étale
correspondence $\Omega\colon G\to H$ and a morphism
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$ there is an
induced map

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

in topological K-theory. Moreover, $\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)$
may be identified with the map $\mathbb{L}(\mathfrak{m})$ of localisations
induced by an ABC morphism $\mathfrak{m}$.

Without coefficients, this specialises to a map
$\mathrm{K}^{\mathrm{top}}_{*}(\Omega)\colon\mathrm{K}^{\mathrm{top}}_{*}(G)\to\mathrm{K}^{\mathrm{top}}_{*}(H)$
induced by a proper étale correspondence $\Omega\colon G\to H$. Combining
Theorems C and E, we have shown that to prove
$\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)\colon\mathrm{K}^{\mathrm{top}}_{*}(G;A)\to\mathrm{K}^{\mathrm{top}}_{*}(H;B)$
to be an isomorphism, it suffices to construct an ABC morphism $\mathfrak{m}$
with $\mathbb{L}(\mathfrak{m})=\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)$ such
that $\mathbb{L}_{n}(\mathfrak{m})$ is an isomorphism for each $n\geq 0$.

While it may be possible to directly prove that the induced maps of derived
functors are isomorphisms, it is desirable nonetheless to achieve a useful
description of $\mathbb{L}_{n}(\mathfrak{m})$. For torsion-free ample
groupoids we may describe $\mathbb{L}_{n}(\mathfrak{m})$ as the module-
theoretic induced map in homology constructed in [Mil24]. This is the
culmination of work in Section 4 on the relationship between the module
category of an ample groupoid and its equivariant Kasparov category.

######  Theorem F (see 3.22 and 5.13).

Let $G$ and $H$ be second countable Hausdorff ample groupoids with coefficient
$\mathrm{C}^{\ast}$-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
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$ be a morphism.
Then the trivial families $\\{G^{0}\\}$ and $\\{H^{0}\\}$ are compatible under
$\Omega$. For each $n\geq 0$ the map of derived functors
$\mathbb{L}_{n}(\mathfrak{m})$ induced by the resulting ABC morphism
$\mathfrak{m}$ may be identified with the induced map in homology

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

We arrive at A through the combination of Theorems C, D and F. Outside of the
torsion-free ample setting where $G$ and $H$ are merely étale and equipped
with families $\cal E$ and $\cal F$ of open Morita embeddings, we may consider
the associated derived functors $\mathbb{L}^{\cal
E}_{n}\mathrm{K}_{*}(G\ltimes-)$ and $\mathbb{L}^{\cal
F}_{n}\mathrm{K}_{*}(H\ltimes-)$ as exotic homology theories for $G$ and $H$.
The establishment of more concrete frameworks for these homology theories
could ultimately lead to new isomorphisms in K-theory.

The author would like to thank Christian Bönicke, Jamie Gabe, Xin Li and
Valerio Proietti for discussions on the content of this paper, and Kevin
Aguyar Brix, Jamie Gabe, Xin Li and Owen Tanner for detailed feedback on
earlier drafts.

##  2\. The Baum–Connes conjecture for étale groupoids via localisation

Throughout this paper, an étale groupoid is a topological groupoid with
locally compact Hausdorff unit space whose range and source maps are local
homeomorphisms. An ample groupoid is an étale groupoid with totally
disconnected unit space.

Let $G$ be a second countable Hausdorff étale groupoid and let $A$ be a
separable $G$-$\mathrm{C}^{\ast}$-algebra. The Baum–Connes conjecture for $G$
with coefficients in $A$ asserts the bijectivity of the Baum–Connes assembly
map

$\mu_{G,A}\colon\mathrm{K}^{\mathrm{top}}_{*}(G;A)\to\mathrm{K}_{*}(G\ltimes_{r}A)$

from the topological K-theory to the operator K-theory. Bönicke and Proietti
use the Meyer–Nest framework of homological ideals in triangulated categories
to describe the topological K-theory and the assembly map as a localisation.
This is the picture of the Baum–Connes assembly map that we will use in this
paper. This section serves primarily as a review of the relevant Meyer–Nest
theory [MN06, MN10, Mey08] and its applications to étale groupoids and the
Baum–Connes conjecture in [PY22, BP24].

We recall from e.g. [BP24, Section 1.1] the triangulated structure on the
equivariant Kasparov category $\operatorname{KK}^{G}$ of separable
$G$-$\mathrm{C}^{\ast}$-algebras. The suspension functor
$\Sigma\colon\operatorname{KK}^{G}\to\operatorname{KK}^{G}$ is given by
$A\mapsto C_{0}((0,1),A)$ and satisfies $\Sigma^{2}\cong\operatorname{id}$ by
Bott periodicity. The mapping cone of a $G$-equivariant $*$-homomorphism
$f\colon A^{\prime}\to B^{\prime}$ is the $G$-$\mathrm{C}^{\ast}$-algebra

$\operatorname{Cone}(f)=\\{(a,b)\in A^{\prime}\oplus
C_{0}((0,1],B^{\prime})\mid f(a)=b(1)\\}.$

The obvious $*$-homomorphisms $\Sigma B^{\prime}\to\operatorname{Cone}(f)$ and
$\operatorname{Cone}(f)\to A^{\prime}$ together with Bott periodicity at
$B^{\prime}$ define the mapping cone triangle

$\Sigma B^{\prime}\to\operatorname{Cone}(f)\to
A^{\prime}\to\Sigma^{2}B^{\prime}.$

A triangle $A\to B\to C\to\Sigma A$ in $\operatorname{KK}^{G}$ is declared
exact if it is isomorphic to a mapping cone triangle. Countable direct sums
exist in $\operatorname{KK}^{G}$ and are given by the
$\mathrm{C}^{\ast}$-algebraic countable direct sums.

For any étale subgroupoid $H\subseteq G$, restriction defines a triangulated
functor
$\operatorname{Res}^{H}_{G}\colon\operatorname{KK}^{G}\to\operatorname{KK}^{H}$.
Given an action $G\mathbin{\curvearrowright}Z$ of $G$ on a locally compact
Hausdorff space $Z$, any action of $G\ltimes Z$ on a
$\mathrm{C}^{\ast}$-algebra $A$ pushes forward to an action of $G$ on $A$. A
$G$-$\mathrm{C}^{\ast}$-algebra $A$ is proper if there is a proper action
$G\mathbin{\curvearrowright}Z$ and an action $G\ltimes
Z\mathbin{\curvearrowright}A$ whose pushforward recovers
$G\mathbin{\curvearrowright}A$.

In this paper a full subcategory of a triangulated category is localising if
it is a triangulated subcategory closed under countable direct sums.
Localising subcategories are thick, i.e. closed under direct summands.

######  Definition 2.1.

Let $G$ be a second countable Hausdorff étale groupoid. A
$G$-$\mathrm{C}^{\ast}$-algbera $A\in\operatorname{KK}^{G}$ is weakly
contractible if $\operatorname{Res}^{H}_{G}A=0$ for each proper open
subgroupoid $H\subseteq G$. We write $\mathfrak{N}_{G}$ for the (localising)
subcategory of $\operatorname{KK}^{G}$ of weakly contractible
$G$-$\mathrm{C}^{\ast}$-algebras and $\mathfrak{P}_{G}$ for the localising
subcategory generated by the proper $G$-$\mathrm{C}^{\ast}$-algebras.

Pairs of subcategories such as $(\mathfrak{P}_{G},\mathfrak{N}_{G})$ are key
to (Bousfield) localisation in triangulated categories.

######  Definition 2.2 (Complementary pair of subcategories).

Let $\mathfrak{P}$ and $\mathfrak{N}$ be thick full triangulated subcategories
of a triangulated category $\mathfrak{T}$. We call the pair
$(\mathfrak{P},\mathfrak{N})$ complementary if

  * •

$\mathfrak{T}(P,N)=0$ for each $P\in\mathfrak{P}$, $N\in\mathfrak{N}$,

  * •

and for each $A\in\mathfrak{T}$ there is an exact triangle $P\to A\to
N\to\Sigma P$ with $P\in\mathfrak{P}$, $N\in\mathfrak{N}$.

We collect some basic results on complementary subcategories, see e.g. [MN06,
Proposition 2.9]:

######  Proposition 2.3.

Let $(\mathfrak{P},\mathfrak{N})$ be a complementary pair of subcategories of
a triangulated category $\mathfrak{T}$.

  * •

The categories determine each other:

$\displaystyle\mathfrak{N}$
$\displaystyle=\\{N\in\mathfrak{T}\mid\mathfrak{T}(P,N)=0\;\text{for
every}\;P\in\mathfrak{P}\\}$ $\displaystyle\mathfrak{P}$
$\displaystyle=\\{P\in\mathfrak{T}\mid\mathfrak{T}(P,N)=0\;\text{for
every}\;N\in\mathfrak{N}\\}$

  * •

There is a functor $\Delta\colon A\mapsto\Delta_{A}=P_{A}\to A\to
N_{A}\to\Sigma P_{A}$ taking an object $A$ of $\mathfrak{T}$ to an exact
triangle $\Delta_{A}$ in $\mathfrak{T}$ with $P_{A}\in\mathfrak{P}$ and
$N_{A}\in\mathfrak{N}$.

  * •

This determines $\Delta$ up to a unique natural isomorphism, and the
components $A\mapsto P_{A}$ and $A\mapsto N_{A}$ are triangulated.

  * •

For $P\in\mathfrak{P}$ and $N\in\mathfrak{N}$, the maps $P_{P}\to P$ and $N\to
N_{N}$ are isomorphisms.

  * •

Any triangulated functor $\Phi\colon\mathfrak{T}\to\mathfrak{T}^{\prime}$
vanishing on $\mathfrak{N}$ factors uniquely up to natural isomorphism through
the functor $A\mapsto P_{A}\colon\mathfrak{T}\to\mathfrak{P}$.

######  Definition 2.4.

With notation as above, the functor $A\mapsto P_{A}$ is known as the222Note
that it is only defined up to unique natural isomorphism. localisation functor
at $\mathfrak{N}$, which we usually denote by
$L\colon\mathfrak{T}\to\mathfrak{T}$. The morphism $P_{A}\to A$ for
$A\in\mathfrak{T}$ defines the localisation natural transformation $\mu\colon
L\Rightarrow\operatorname{id}$. For a functor
$F\colon\mathfrak{T}\to\mathfrak{C}$, the functor $\mathbb{L}F:=F\circ
L\colon\mathfrak{T}\to\mathfrak{C}$ is called the localisation of $F$ at
$\mathfrak{N}$, and the natural transformation $\mathbb{L}F\Rightarrow F$ is
the (localisation) assembly map.

Bönicke and Proietti prove that the Baum–Connes assembly map may be viewed as
the localisation of the reduced operator K-theory functor at
$\mathfrak{N}_{G}$ [BP24, Theorems B and C].

######  Theorem 2.5 (Identification of the Baum–Connes assembly map).

Let $G$ be a second countable Hausdorff étale groupoid. Then
$(\mathfrak{P}_{G},\mathfrak{N}_{G})$ is a complementary pair of subcategories
of $\operatorname{KK}^{G}$ and for each $G$-$\mathrm{C}^{\ast}$-algebra
$A\in\operatorname{KK}^{G}$ the associated localisation assembly map

$\mathbb{L}\mathrm{K}_{*}(G\ltimes_{r}A)\to\mathrm{K}_{*}(G\ltimes_{r}A)$

for the reduced K-theory functor $K_{*}(G\ltimes_{r}-)$ is naturally
isomorphic to the Baum–Connes assembly map

$\mu_{G,A}\colon\mathrm{K}^{\mathrm{top}}_{*}(G;A)\to\mathrm{K}_{*}(G\ltimes_{r}A).$

######  Remark 2.6.

On the localising subcategory $\mathfrak{P}_{G}$ generated by proper
$G$-$\mathrm{C}^{\ast}$-algebras, the reduced and full crossed products agree.
The natural transformation from $G\ltimes-$ to $G\ltimes_{r}-$ applied to
$LA\to A$ gives rise to a commutative diagram

${{\mathbb{L}\mathrm{K}_{*}(G\ltimes A)}}$${{\mathrm{K}_{*}(G\ltimes
A)}}$${{\mathrm{K}^{\mathrm{top}}_{*}(G;A)}}$${{\mathrm{K}_{*}(G\ltimes_{r}A)}.}$$\scriptstyle{\cong}$$\scriptstyle{\mu_{G,A}}$

The Baum–Connes assembly map $\mu_{G,A}$ therefore factors through the
canonical map $\mathrm{K}_{*}(G\ltimes A)\to\mathrm{K}_{*}(G\ltimes_{r}A)$,
which is a surjection when $\mu_{G,A}$ is. It also means that the topological
K-theory $\mathrm{K}^{\mathrm{top}}_{*}(G;A)$ enjoys the greater functoriality
properties of the full crossed product.

To understand the pair $(\mathfrak{P}_{G},\mathfrak{N}_{G})$ we consider not
just the objects but also morphisms which vanish under the restriction
functors $\operatorname{Res}^{H}_{G}$ of proper open subgroupoids $H\subseteq
G$. Such ideals are key to the Meyer–Nest framework for doing homological
algebra in a triangulated category.

######  Definition 2.7 (Homological functor and ideal).

Let $\mathfrak{T}$ be a triangulated category with suspension functor $\Sigma$
and let $\mathfrak{C}$ be an abelian category. A homological functor
$F\colon\mathfrak{T}\to\mathfrak{C}$ is an additive functor which maps exact
triangles to long exact sequences. If $\mathfrak{C}$ is stable with (additive)
automorphism $\Sigma_{\mathfrak{C}}$ a stable homological functor is a
homological functor $F\colon\mathfrak{T}\to\mathfrak{C}$ with a natural
isomorphism $F\circ\Sigma\cong\Sigma_{\mathfrak{C}}\circ F$. An ideal
$\mathfrak{I}\vartriangleleft\mathfrak{T}$ of morphisms is a homological ideal
if it is the kernel of a stable homological functor.

Typically our stable abelian category will just be the category
$\operatorname{\mathbf{Ab}}_{*}$ of $\mathbb{Z}$-graded abelian groups with
the right shift automorphism. By the existence of a universal stable
homological functor for any triangulated category, the kernel of any
triangulated functor is homological (see [MN10, Section 2.5]).

######  Example 2.8.

Let $G$ be a second countable Hausdorff étale groupoid and let $\cal F$ be a
family of proper open subgroupoids of $G$. Then

$\mathfrak{I}_{\cal F}:=\bigcap_{H\in\cal
F}\ker\operatorname{Res}^{H}_{G}=\ker\prod_{H\in\cal
F}\operatorname{Res}^{H}_{G}$

is a homological ideal in $\operatorname{KK}^{G}$.

Many of the standard notions in homological algebra can be imitated relative
to a homological ideal:

######  Definition 2.9.

Let $\mathfrak{I}$ be a homological ideal in a triangulated category
$\mathfrak{T}$, let $f\colon A\to B$ be a morphism in $\mathfrak{T}$ embedded
in an exact triangle

${\Delta}$${A}$${B}$${C}$${\Sigma
A.}$$\scriptstyle{=}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$

We say $f$ is $\mathfrak{I}$-phantom if $f\in\mathfrak{I}$,
$\mathfrak{I}$-epic if $g\in\mathfrak{I}$, $\mathfrak{I}$-monic if
$h\in\mathfrak{I}$ and an $\mathfrak{I}$-equivalence if $g\in\mathfrak{I}$ and
$h\in\mathfrak{I}$. An object is $\mathfrak{I}$-contractible if its identity
morphism is in $\mathfrak{I}$. An object $P\in\mathfrak{T}$ is
$\mathfrak{I}$-projective if $\mathfrak{I}(P,A)=0$ for every
$A\in\mathfrak{T}$, or equivalently for any morphism $f\colon P\to A$ and any
$\mathfrak{I}$-epimorphism $\pi\colon B\to A$ there is a lift $\tilde{f}\colon
P\to B$ of $f$ through $\pi$. We say $\mathfrak{T}$ has enough
$\mathfrak{I}$-projectives or that $\mathfrak{I}$ has enough projectives if
for any object $A$ there is an $\mathfrak{I}$-epimorphism $\pi\colon P\to A$
from an $\mathfrak{I}$-projective object $P$. The exact triangle $\Delta$ is
$\mathfrak{I}$-exact if $h\in\mathfrak{I}$. A chain complex
$C_{\bullet}=(C_{n},d_{n})_{n\in\mathbb{Z}}$ is $\mathfrak{I}$-exact if
whenever we embed $d_{n}$ and $d_{n+1}$ in exact triangles

${C_{n}}$${C_{n-1}}$${X_{n}}$${\Sigma
C_{n}}$${C_{n+1}}$${C_{n}}$${X_{n+1}}$${\Sigma
C_{n},}$$\scriptstyle{d_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{g_{n}}$$\scriptstyle{d_{n+1}}$$\scriptstyle{f_{n+1}}$$\scriptstyle{g_{n+1}}$

we have $\Sigma f_{n+1}\circ g_{n}\in\mathfrak{I}$. An
$\mathfrak{I}$-resolution $P_{\bullet}\to A$ of an object $A$ is an
$\mathfrak{I}$-exact sequence $\cdots\to P_{n}\to\cdots\to P_{0}\to A\to 0$,
which is an $\mathfrak{I}$-projective resolution if each $P_{n}$ is
$\mathfrak{I}$-projective.

A homological ideal $\mathfrak{I}\vartriangleleft\mathfrak{T}$ can be used to
construct a complementary pair of subcategories in $\mathfrak{T}$. We denote
by $\mathfrak{N}_{\mathfrak{I}}$ the (localising) full subcategory of
$\mathfrak{I}$-contractible objects $A\in\mathfrak{T}$, and we write
$\mathfrak{P}_{\mathfrak{I}}$ for the localising subcategory generated by the
$\mathfrak{I}$-projective objects in $\mathfrak{T}$. For the homological ideal
$\mathfrak{I}_{\cal F}$ associated to a family $\cal F$ of open subgroupoids
as in Example 2.8 we may simply write $\mathfrak{N}_{\cal F}$ and
$\mathfrak{P}_{\cal F}$.

######  Theorem 2.10 (Theorem 3.21 in [Mey08]).

Let $\mathfrak{T}$ be a triangulated category with countable direct sums, and
let $\mathfrak{I}$ be a homological ideal of $\mathfrak{T}$ compatible with
countable direct sums and enough projectives. Then
$(\mathfrak{P}_{\mathfrak{I}},\mathfrak{N}_{\mathfrak{I}})$ is a complementary
pair of subcategories of $\mathfrak{T}$.

For clarity we may write $L_{\mathfrak{I}}\colon\mathfrak{T}\to\mathfrak{T}$
for the associated localisation functor at $\mathfrak{N}_{\mathfrak{I}}$ and
$\mu_{\mathfrak{I}}\colon
L_{\mathfrak{I}}\Rightarrow\operatorname{id}_{\mathfrak{T}}\colon\mathfrak{T}\to\mathfrak{T}$
for the localisation natural transformation. We write
$\mathbb{L}^{\mathfrak{I}}F$ for the localisation $F\circ L_{\mathfrak{I}}$ of
a functor $F$, but may write $\mathbb{L}F$ if the ideal is understood.

######  Definition 2.11 (ABC tuple).

An ABC tuple is a quadruple $(\mathfrak{T},\mathfrak{I},F,A)$ consisting of

  * •

a triangulated category $\mathfrak{T}$ with countable direct sums,

  * •

a homological ideal $\mathfrak{I}\vartriangleleft\mathfrak{T}$ closed under
countable direct sums with enough projectives,

  * •

a stable homological functor
$F\colon\mathfrak{T}\to\operatorname{\mathbf{Ab}}_{*}$ that commutes with
countable direct sums,

  * •

and an object $A\in\mathfrak{T}$.

These are named after the ABC spectral sequence, covered at the end of this
section, as an ABC tuple is precisely the input data for the construction of
the ABC spectral sequence.

######  Example 2.12.

Let $G$ be a second countable Hausdorff étale groupoid, let
$A\in\operatorname{KK}^{G}$ be a $G$-$\mathrm{C}^{\ast}$-algebra and let
$F_{G}\colon\operatorname{KK}^{G}\to\operatorname{\mathbf{Ab}}_{*}$ be the
full K-theory functor $\mathrm{K}_{*}(G\ltimes-)$ or the reduced K-theory
functor $\mathrm{K}_{*}(G\ltimes_{r}-)$. Let $\cal F$ be a countable family of
open subgroupoids of $G$ with associated ideal $\mathfrak{I}_{\cal F}$. Then

$(\operatorname{KK}^{G},F_{G},\mathfrak{I}_{\cal F},A)$

is an ABC tuple. As a special case, we may consider the trivial family $\cal
F=\\{G^{0}\\}$, for which we write $\mathfrak{I}_{0}$.

Let us explain why for countable $\cal F$ the homological ideal
$\mathfrak{I}_{\cal F}$ has enough projectives. For any open subgroupoid
$H\subseteq G$ there is a triangulated induction functor
$\operatorname{Ind}^{G}_{H}\colon\operatorname{KK}^{H}\to\operatorname{KK}^{G}$
and an adjunction $\operatorname{Ind}^{G}_{H}\dashv\operatorname{Res}^{H}_{G}$
[BP24, Theorem 2.3]. We therefore obtain an adjunction

$\displaystyle\operatorname{Ind}_{\cal F}\colon\prod_{H\in\cal
F}\operatorname{KK}^{H}$ $\displaystyle\to\operatorname{KK}^{G}$
$\displaystyle\dashv$ $\displaystyle\operatorname{Res}_{\cal
F}\colon\operatorname{KK}^{G}$ $\displaystyle\to\prod_{H\in\cal
F}\operatorname{KK}^{H}$ $\displaystyle(B_{H})_{H\in\cal F}$
$\displaystyle\mapsto\bigoplus_{H\in\cal F}\operatorname{Ind}^{G}_{H}B_{H}$
$\displaystyle A$
$\displaystyle\mapsto(\operatorname{Res}^{H}_{G}(A))_{H\in\cal F}.$

It is straightforward to verify that for any $H\in\cal F$ and
$B\in\operatorname{KK}^{H}$ the induced algebra $\operatorname{Ind}^{G}_{H}B$
is $\mathfrak{I}_{\cal F}$-projective, and that for any
$A\in\operatorname{KK}^{G}$ the counit
$\epsilon_{A}\colon\operatorname{Ind}_{\cal F}\operatorname{Res}_{\cal F}A\to
A$ is $\mathfrak{I}_{\cal F}$-epic. In general, an adjunction $E\dashv F$ of
triangulated functors not only guarantees that $\mathfrak{I}=\ker F$ has
enough projectives but also provides a characterisation of the
$\mathfrak{I}$-projectives and an explicit $\mathfrak{I}$-projective
resolution of any object, see [MN10, Section 3.6] and [PY22, Section 2.1]:

######  Theorem 2.13.

Let $\mathfrak{T}$ and $\mathfrak{T}^{\prime}$ be triangulated categories with
countable direct sums and let $F\colon\mathfrak{T}\to\mathfrak{T}^{\prime}$ be
a triangulated functor with a left adjoint
$E\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ both compatible with countable
direct sums. Then $\mathfrak{I}=\ker F$ has enough projectives, and the
$\mathfrak{I}$-projective objects are precisely the direct summands of $EA$
for $A\in\mathfrak{T}^{\prime}$. Furthermore, for $B\in\mathfrak{T}$ there is
a natural $\mathfrak{I}$-projective resolution

${\cdots}$${(EF)^{n+1}B}$${\cdots}$${(EF)^{2}B}$${EFB}$${B,}$$\scriptstyle{\delta_{n+1}}$$\scriptstyle{\delta_{n}}$$\scriptstyle{\delta_{2}}$$\scriptstyle{\delta_{1}}$$\scriptstyle{\pi_{0}}$

where $\pi_{0}:=\epsilon_{B}\colon EFB\to B$ is given by the counit $\epsilon$
of the adjunction at $B$ and

$\delta_{n}:=\sum_{i=0}^{n}(-EF)^{i}(\epsilon_{(EF)^{n-i}B})\colon(EF)^{n+1}B\to(EF)^{n}B.$

From this description of the $\mathfrak{I}$-projectives it follows that the
localising subcategory $\langle E\rangle$ generated by the image of $E$ is
equal to $\mathfrak{P}_{\mathfrak{I}}$ and complementary to
$\mathfrak{N}_{\mathfrak{I}}$. For a countable family $\cal F$ of open
subgroupoids as in Example 2.12 the associated complementary pair has
description

$(\mathfrak{P}_{\cal F},\mathfrak{N}_{\cal
F})=(\langle\operatorname{Ind}_{\cal F}\rangle,\mathfrak{N}_{\cal F}).$

Even when $\cal F$ is not countable, we write $\langle\operatorname{Ind}_{\cal
F}\rangle$ for the localising subcategory generated by algebras induced by
members of $\cal F$.

######  Proposition 2.14.

Let $G$ be a second countable Hausdorff étale groupoid and let $\cal F$ be a
(not necessarily countable) family of proper open subgroupoids of $G$. Then we
have containments

$\displaystyle\langle\operatorname{Ind}_{\cal F}\rangle$
$\displaystyle\subseteq\mathfrak{P}_{G},$ $\displaystyle\mathfrak{N}_{G}$
$\displaystyle\subseteq\mathfrak{N}_{\cal F}.$

###### Proof.

For the first containment, for each $P\in\cal F$ and
$B\in\operatorname{KK}^{P}$ the induced algebra $\operatorname{Ind}^{G}_{P}B$
is a proper $G$-$\mathrm{C}^{\ast}$-algebra through the proper $G$-space
$G/P$. The containment $\mathfrak{N}_{G}\subseteq\mathfrak{N}_{\cal F}$ is
immediate. ∎

In order to prove 2.5, Bönicke and Proietti construct a countable family $\cal
F$ of proper open subgroupoids of $G$ for which the reverse inclusions also
hold [BP24, Theorem 3.10 and Lemma 3.16], and so

$(\mathfrak{P}_{G},\mathfrak{N}_{G})=(\mathfrak{P}_{\cal F},\mathfrak{N}_{\cal
F}).$

Diving into the details of the construction, the family $\cal F$ may moreover
be chosen to be finer than any pre-specified open cover $\cal U$ of $G^{0}$ in
the sense that for any $H\in\cal F$ there is $U\in\cal U$ with $H^{0}\subseteq
U$. As a consequence, we obtain the following:

######  Corollary 2.15.

Let $G$ be a proper second countable Hausdorff étale groupoid and let $\cal U$
be an open cover of $G^{0}$. Then $A\in\operatorname{KK}^{G}$ vanishes if and
only if $\operatorname{Res}^{G\mathord{\upharpoonright}_{U}}_{G}A$ vanishes
for each $U\in\cal U$.

###### Proof.

Suppose that $\operatorname{Res}^{G\mathord{\upharpoonright}_{U}}_{G}A\cong 0$
for each $U\in\cal U$. Choose a countable family $\cal I$ of proper open
subgroupoids with $\mathfrak{N}_{\cal I}=\mathfrak{N}_{G}$ such that for any
$H\in\cal I$ there is $U\in\cal U$ with $H^{0}\subseteq U$. It follows that
$\operatorname{Res}^{H}_{G}A\cong 0$ for each $H\in\cal I$, and so
$A\in\mathfrak{N}_{G}$. As $G$ itself is a proper open subgroupoid of $G$, we
have $A\cong\operatorname{Res}^{G}_{G}A\cong 0$. ∎

Framing the Baum–Connes assembly map in terms of localisation gives us access
to the ABC spectral sequence, which converges to the localisation. This was
utilised by Proietti and Yamashita to obtain a spectral sequence [PY22,
Theorem 4.3]

$E^{2}_{p,q}=\mathrm{H}_{p}(G;K_{q}(A))\Rightarrow\mathrm{K}_{p+q}(G\ltimes_{r}A)$

relating groupoid homology to operator K-theory for a second countable
Hausdorff ample groupoid $G$ and a separable $G$-$\mathrm{C}^{\ast}$-algebra
$A$, where $G$ has torsion-free isotropy groups and satisfies a strengthening
of the Baum–Connes conjecture. On the second sheet of the general ABC spectral
sequence are the derived functors with respect to a homological ideal:

######  Definition 2.16 (Derived functors).

Let $\mathfrak{I}$ be a homological ideal in a triangulated category
$\mathfrak{T}$ with enough projectives and let
$F\colon\mathfrak{T}\to\operatorname{\mathbf{Ab}}$ be a functor. For each
$n\geq 0$, the $n$th left derived functor
$\mathbb{L}_{n}^{\mathfrak{I}}F\colon\mathfrak{T}\to\operatorname{\mathbf{Ab}}$
of $F$ is given as follows. For $A\in\mathfrak{T}$ pick any
$\mathfrak{I}$-projective resolution

$\cdots\to P_{n}\to\cdots\to P_{0}\to A$

of $A$. The group $\mathbb{L}_{n}^{\mathfrak{I}}F(A)$ is the $n$th homology
group $\mathrm{H}_{n}(F(P_{\bullet}))$ of the chain complex

$\cdots\to F(P_{n})\to\cdots\to F(P_{0})\to 0.$

For $n\leq-1$ we set $\mathbb{L}^{\mathfrak{I}}_{n}F=0$.

When the homological ideal is clear from context we may simply write
$\mathbb{L}_{n}$, and for the homological ideal $\mathfrak{I}_{\cal F}$
associated to a family $\cal F$ of subgroupoids we write $\mathbb{L}^{\cal
F}_{n}$ instead of $\mathbb{L}^{\mathfrak{I}_{\cal F}}_{n}$. Independence from
the choice of $\mathfrak{I}$-projective resolution and the functoriality of
this construction follow from a version of the fundamental lemma of
homological algebra for $\mathfrak{I}$-projective resolutions [MN10,
Proposition 3.26]:

######  Lemma 2.17 (The fundamental lemma of homological algebra).

Let $\mathfrak{T}$ be a triangulated category and $\mathfrak{I}$ be a
homological ideal. If $A$ and $B$ are objects of $\mathfrak{T}$,
$P_{\bullet}\rightarrow A$ is an $\mathfrak{I}$-projective resolution of $A$
and $Q_{\bullet}\rightarrow B$ is an $\mathfrak{I}$-resolution of $B$, then
any morphism $f\colon A\rightarrow B$ lifts to a chain map $\tilde{f}\colon
P_{\bullet}\rightarrow Q_{\bullet}$ uniquely up to chain homotopy.

######  Example 2.18 (Groupoid homology).

Let $G$ be a second countable Hausdorff ample groupoid and consider the ideal
$\mathfrak{I}_{0}\vartriangleleft\operatorname{KK}^{G}$ for the trivial family
$\\{G^{0}\\}$. Then the $p$th derived functor of the K-theory functor
$\mathrm{K}_{q}(G\ltimes_{r}-)$ at a $G$-$\mathrm{C}^{\ast}$-algebra
$A\in\operatorname{KK}^{G}$ is naturally isomorphic to the groupoid homology
$\mathrm{H}_{p}(G;\mathrm{K}_{q}(A))$ of $G$ with coefficients in the
$G$-module $\mathrm{K}_{q}(A)$ [PY22, Proposition 4.2]:

$\mathbb{L}^{\mathfrak{I}_{0}}_{p}\mathrm{K}_{q}(G\ltimes_{r}A)\cong\mathrm{H}_{p}(G;\mathrm{K}_{q}(A))$

This also holds for the full crossed product as it agrees with the reduced
crossed product on $\mathfrak{I}_{0}$-projective objects.

The ABC spectral sequence links together the derived functors
$\mathbb{L}^{\mathfrak{I}}_{n}F(A)$ and the localisation
$\mathbb{L}^{\mathfrak{I}}F(A)$ [Mey08, Theorem 5.1]:

######  Theorem 2.19 (The ABC spectral sequence).

Let $(\mathfrak{T},\mathfrak{I},F,A)$ be an ABC tuple. Then there is a
strongly convergent spectral sequence

$E^{2}_{p,q}=\mathbb{L}^{\mathfrak{I}}_{p}F_{q}(A)\Rightarrow\mathbb{L}^{\mathfrak{I}}F_{p+q}(A).$

######  Example 2.20.

Let $G$ be a second countable Hausdorff étale groupoid, let $\cal F$ be a
countable family of proper open subgroupoids of $G$ and let
$A\in\operatorname{KK}^{G}$ be a $G$-$\mathrm{C}^{\ast}$-algebra. If
$\mathfrak{N}_{\cal F}=\mathfrak{N}_{G}$ (and therefore also
$\langle\operatorname{Ind}_{\cal F}\rangle=\mathfrak{P}_{G}$) then the ABC
spectral sequence for $(\operatorname{KK}^{G},\mathfrak{I}_{\cal
F},\mathrm{K}_{*}(G\ltimes-),A)$ is given by

$\mathbb{L}^{\cal F}_{p}\mathrm{K}_{q}(G\ltimes
A)\Rightarrow\mathrm{K}^{\mathrm{top}}_{p+q}(G;A).$

##  3\. Homological ideals associated to families of openly embedded groupoids

We have seen that we may study the topological K-theory
$\mathrm{K}^{\mathrm{top}}_{*}(G;A)$ of an étale groupoid $G$ with
coefficients in a $G$-$\mathrm{C}^{\ast}$-algebra $A$ through a family $\cal
F$ of proper open subgroupoids of $G$ satisfying

$(\langle\operatorname{Ind}_{\cal F}\rangle,\mathfrak{N}_{\cal
F})=(\mathfrak{P}_{G},\mathfrak{N}_{G}).$

For each family with this property we obtain an instance of the ABC spectral
sequence that converges to the topological K-theory (Example 2.20). This
raises questions of how to construct families with this property, how to check
a family has this property and how families with this property relate to each
other. We introduce the following sufficient condition:

######  Definition 3.1 (Condition (P)).

Let $G$ be an étale groupoid with a family $\cal F$ of proper open
subgroupoids. We say that $\cal F$ satisfies condition (P) if for any proper
open subgroupoid $P\subseteq G$ and any $x\in P^{0}$ there is an open
neighbourhood $U\subseteq P^{0}$ of $x$ and a member $H\in\cal F$ such that
$P\mathord{\upharpoonright}_{U}\subseteq H$.

######  Proposition 3.2.

Let $G$ be a second countable Hausdorff étale groupoid and let $\cal F$ be a
countable family of proper open subgroupoids satisfying condition (P). Then

$(\langle\operatorname{Ind}_{\cal F}\rangle,\mathfrak{N}_{\cal
F})=(\mathfrak{P}_{G},\mathfrak{N}_{G}).$

###### Proof.

Let $A\in\mathfrak{N}_{\cal F}$ and let $P\subseteq G$ be a proper open
subgroupoid. Applying 2.15 we have $\operatorname{Res}^{P}_{G}A=0$ by
construction and so $A\in\mathfrak{N}_{G}$. As complementary subcategories
determine each other, we also get $\langle\operatorname{Ind}_{\cal
F}\rangle=\mathfrak{P}_{G}$. ∎

To check condition (P) in practice we need only check that for each finite
subgroup $\Gamma$ of the isotropy of $G$ there is a member of $\cal F$
containing $\Gamma$. This is due to the following simple observation.

######  Lemma 3.3.

Let $P$ be a proper étale groupoid and let $O\subseteq P$ be an open subset
containing the isotropy $P^{x}_{x}$ at some point $x\in P^{0}$. Then $O$
contains $P\mathord{\upharpoonright}_{U}$ for some open neighbourhood
$U\subseteq P^{0}$ of $x$.

###### Proof.

The image of $P\setminus O$ under the closed map $r\times s\colon P\to
P^{0}\times P^{0}$ is disjoint from $(x,x)$. It therefore misses $U\times U$
for an open neighbourhood $U$ of $x$, and consequently
$P\mathord{\upharpoonright}_{U}\subseteq O$. ∎

Conversely, each finite subgroup of isotropy of $G$ is contained in some
proper open subgroupoid.

######  Lemma 3.4.

Let $S\mathbin{\curvearrowright}X$ be an action of an inverse semigroup $S$ by
partial homeomorphisms on a locally compact Hausdorff space $X$. For any $x\in
X$ and finite subgroup $\Gamma\leq(S\ltimes X)^{x}_{x}$ there is an idempotent
$e\in E(S)$ with $x\in\operatorname{dom}e$ and a finite subgroup $F\leq
S_{e}=\\{s\in S\;|\;s^{*}s=ss^{*}=e\\}$ such that $\Gamma=\\{[s,x]\;|\;s\in
F\\}$. Consequently, $\Gamma$ is contained in the proper open subgroupoid
$\\{[s,y]\;|\;s\in F,\;y\in\operatorname{dom}e\\}$.

###### Proof.

For each $g\in\Gamma$, let $s_{g}\in S$ represent $g$ in the sense that
$g=[s_{g},x]$, with an idempotent representing the identity element of
$\Gamma$. For each pair $g,h\in\Gamma$ let $e_{g,h}\in E(S)$ be an idempotent
with $x\in\operatorname{dom}e_{g,h}$, such that
$s_{g}s_{h}e_{g,h}=s_{gh}e_{g,h}$, $e_{g,h}\leq\operatorname{dom}s_{h}$,
$e_{g,h}\leq\operatorname{dom}s_{gh}$ and
$s_{h}\boldsymbol{\cdot}e_{g,h}\leq\operatorname{dom}s_{g}$. Let
$e_{0}=\prod_{g,h\in\Gamma}e_{g,h}$ and let
$e=\prod_{g\in\Gamma}s_{g}\boldsymbol{\cdot}e_{0}$. For each $g\in\Gamma$, the
element $t_{g}=s_{g}e$ belongs to $S_{e}$ because
$s_{g}\boldsymbol{\cdot}e=e$. The finite set $F=\\{t_{g}\;|\;g\in\Gamma\\}$ is
a subgroup of $S_{e}$ and by construction satisfies $\Gamma=\\{[t,x]\;|\;t\in
F\\}$. ∎

The following characterisation of condition (P) follows straightforwardly from
the previous two lemmas, noting that every étale groupoid may be written as
the transformation groupoid of an inverse semigroup action.

######  Proposition 3.5.

Let $G$ be an étale groupoid and let $\cal F$ be a family of proper open
subgroupoids of $G$. Then $\cal F$ satisfies condition (P) if and only if for
each $x\in G^{0}$ and finite subgroup $\Gamma\leq G^{x}_{x}$, there is a
member $K\in\cal F$ with $\Gamma\subseteq K$.

The construction of proper open subgroupoids containing finite isotropy
subgroups in 3.4 gives us a recipe for constructing families of subgroupoids
satisfying condition (P).

######  Example 3.6.

Let $G=S\ltimes X$ be the transformation groupoid of an action
$S\mathbin{\curvearrowright}X$ of an inverse semigroup $S$ by partial
homeomorphisms on a locally compact Hausdorff space $X$. For each non-zero
idempotent $e\in E(S)^{\times}$ and finite subgroup $F\leq S_{e}$ consider the
proper open subgroupoid $P(e,F)=\\{[s,y]\;|\;s\in
F,\;y\in\operatorname{dom}e\\}$ of $G$. Then the family

$\cal F_{\text{fin}}=\\{P(e,F)\;|\;e\in E(S)^{\times},\;F\leq
S_{e}\;\text{finite subgroup}\\}$

of proper open subgroupoids of $G$ satisfies condition (P) and is countable
when $S$ is countable. The subgroupoid $P(e,F)$ may be identified with the
transformation groupoid $F_{\downarrow}\ltimes X$, where
$F_{\downarrow}=\\{fd\mid d\in E(S),f\in F\\}$ is the inverse semigroup of
restrictions of elements of $F$ in $S$.

######  Remark 3.7.

For a second countable étale groupoid $G$ the existence of a countable family
$\cal E$ of subgroupoids satisfying (P) follows from the above example. It
follows that any other family $\cal F$ satisfying (P) has a countable
subfamily satisfying (P), since we need only check the condition against the
countable family $\cal E$, each member of which needs only countably many
members of $\cal F$ by second countability.

Suppose $G$ and $H$ are Morita equivalent étale groupoids and let $\cal F$ be
a countable family of proper open subgroupoids of $G$. The Morita equivalence
induces an equivalence $\operatorname{KK}^{G}\cong\operatorname{KK}^{H}$ of
Kasparov categories, so we may search for a counterpart in
$\operatorname{KK}^{H}$ to the homological ideal $\mathfrak{I}_{\cal
F}\vartriangleleft\operatorname{KK}^{G}$. However, there is no clear way to
transfer subgroupoids of $G$ to subgroupoids of $H$. Instead we consider a
looser notion of subgroupoid which is morally an open subgroupoid up to Morita
equivalence, or an open embedding in a category of étale groupoids with étale
correspondences as morphisms. We refer the reader to [AKM22] and [Mil23,
Section 3.1] for an introduction to and examples of étale correspondences.

######  Definition 3.8 (Étale correspondence).

Let $G$ and $H$ be étale groupoids. An étale correspondence $\Omega\colon G\to
H$ is a $G$-$H$-bispace $\Omega$ with anchor maps $\rho\colon\Omega\to G^{0}$
and $\sigma\colon\Omega\to H^{0}$ called the range and source such that the
right action $\Omega\mathbin{\curvearrowleft}H$ is free, proper and étale. An
étale correspondence $\Omega\colon G\to H$ is proper if the induced map
$\overline{\rho}\colon\Omega/H\to G^{0}$ is proper and tight if
$\overline{\rho}$ is an homeomorphism. The composition $\Lambda\circ\Omega$ of
étale correspondences $\Omega\colon G\to H$ and $\Lambda\colon H\to K$ is
given by the fibre product $\Omega\times_{H}\Lambda$ over $H$. A slice
$U\subseteq\Omega$ is an open set on which the source $\sigma\colon U\to
H^{0}$ and the orbit map $q\colon U\to\Omega/H$ are injective.

Composition of étale correspondences is associative up to isomorphism, with an
identity up to isomorphism at $G$ given by the $G$-$G$-bispace $G$. Invertible
correspondences are precisely Morita equivalences.

######  Definition 3.9 (Open Morita embedding).

Let $G$ and $P$ be étale groupoids. An open Morita embedding
$P\hookrightarrow_{M}G$ of $P$ into $G$ is a tight étale correspondence
$\Omega\colon P\to G$ with a free left action
$P\mathbin{\curvearrowright}\Omega$. We say that $P$ is openly Morita embedded
into $G$.

The correspondence $G^{P^{0}}\colon P\to G$ associated to an open inclusion
$P\subseteq G$ is an open Morita embedding, as is a Morita equivalence. The
composition of open Morita embeddings is again an open Morita embedding. These
compositions describe all open Morita embeddings thanks to the following
construction which appears in [MZ15, Proposition 7.16]. For an étale groupoid
$H$ and a free, proper étale right $H$-space $\Omega$, we define an étale
groupoid structure on the space $\Omega\times_{H}\Omega^{*}$. Here
$\Omega^{*}$ is a formal opposite topological space to $\Omega$ defined by a
homeomorphism

$\omega\mapsto\omega^{*}\colon\Omega\to\Omega^{*},$

equipped with a left action $H\mathbin{\curvearrowright}\Omega^{*}$ determined
by $h\boldsymbol{\cdot}\omega^{*}:=(\omega\boldsymbol{\cdot}h^{-1})^{*}$ for
$\omega\in\Omega$ and $h\in H_{\sigma(\omega)}$. Define range, source,
composition and inversion maps on $\Omega\times_{H}\Omega^{*}$ by

$\displaystyle r([\omega_{1},\omega_{2}^{*}]_{H})$
$\displaystyle=[\omega_{1},\omega_{1}^{*}]_{H}$ $\displaystyle
s([\omega_{1},\omega_{2}^{*}]_{H})$
$\displaystyle=[\omega_{2},\omega_{2}^{*}]_{H}$
$\displaystyle[\omega_{1},\omega_{2}^{*}]_{H}[\omega_{2},\omega_{3}^{*}]_{H}$
$\displaystyle=[\omega_{1},\omega_{3}^{*}]_{H}$
$\displaystyle[\omega_{1},\omega_{2}^{*}]_{H}^{-1}$
$\displaystyle=[\omega_{2},\omega_{1}^{*}]_{H}$

for $\omega_{1},\omega_{2},\omega_{3}\in\Omega$ with
$\sigma(\omega_{1})=\sigma(\omega_{2})=\sigma(\omega_{3})$. This defines an
étale groupoid whose unit space may be identified with $\Omega/H$.
Furthermore, $\Omega$ serves as a Morita equivalence
$\Omega\times_{H}\Omega^{*}\sim_{M}H\mathord{\upharpoonright}_{\sigma(\Omega)}$
through an action $\Omega\times_{H}\Omega^{*}\mathbin{\curvearrowright}\Omega$
determined by

$[\omega_{1},\omega_{2}^{*}]_{H}\boldsymbol{\cdot}\omega_{2}=\omega_{1}$

for $\omega_{1},\omega_{2}\in\Omega$ with
$\sigma(\omega_{1})=\sigma(\omega_{2})$. We note that if $\Omega$ is Hausdorff
or second countable then so is $\Omega\times_{H}\Omega^{*}$.

######  Proposition 3.10.

Let $\Omega\colon G\to H$ be an étale correspondence. Then there is an étale
groupoid $K$ and a decomposition of $\Omega$ into

  * •

an actor $G\mathbin{\curvearrowright}\Omega\times_{H}\Omega^{*}$,

  * •

the Morita equivalence $\Omega\colon\Omega\times_{H}\Omega^{*}\to
H\mathord{\upharpoonright}_{\sigma(\Omega)}$, and

  * •

the open inclusion $H\mathord{\upharpoonright}_{\sigma(\Omega)}\subseteq H$.

If $\Omega$ is an open Morita embedding then the actor
$G\mathbin{\curvearrowright}\Omega\times_{H}\Omega^{*}$ arises from an open
embedding which is a homeomorphism on unit spaces.

###### Proof.

Define an actor $G\mathbin{\curvearrowright}\Omega\times_{H}\Omega^{*}$ by
$g\boldsymbol{\cdot}[\omega_{1},\omega_{2}^{*}]_{H}:=[g\boldsymbol{\cdot}\omega_{1},\omega_{2}^{*}]_{H}$
for $[\omega_{1},\omega_{2}^{*}]_{H}\in\Omega\times_{H}\Omega^{*}$ and $g\in
G_{\rho(\omega_{1})}$. The composition of the actor
$G\mathbin{\curvearrowright}\Omega\times_{H}\Omega^{*}$, the Morita
equivalence
$\Omega\times_{H}\Omega^{*}\sim_{M}H\mathord{\upharpoonright}_{\sigma(\Omega)}$
and the open inclusion $H\mathord{\upharpoonright}_{\sigma(\Omega)}\subseteq
H$ is isomorphic to $\Omega\colon G\to H$ via the isomorphism

$\displaystyle(\Omega\times_{H}\Omega^{*})\times_{\Omega\times_{H}\Omega^{*}}\Omega\times_{H\mathord{\upharpoonright}_{\sigma(\Omega)}}H\mathord{\upharpoonright}_{\sigma(\Omega)}$
$\displaystyle\to\Omega$
$\displaystyle[[\omega_{1},\omega_{2}^{*}],\omega_{2},h]$
$\displaystyle\mapsto\omega_{1}\boldsymbol{\cdot}h$

of $G$-$H$-bispaces, where $\omega_{1},\omega_{2}\in\Omega$ and $h\in H$
satisfy $\sigma(\omega_{1})=\sigma(\omega_{2})=r(h)$.

If $\Omega\colon G\to H$ is an open Morita embedding, then so is the actor
$G\mathbin{\curvearrowright}\Omega\times_{H}\Omega^{*}$. By tightness the left
anchor map induces a homeomorphism of unit spaces $\varphi\colon
G^{0}\to\Omega/H$. The continuous homomorphism $g\mapsto
g\boldsymbol{\cdot}\varphi(s(g))\colon G\to\Omega\times_{H}\Omega^{*}$ is
étale as it restricts to $\varphi$ on the unit spaces, and it is injective by
freeness of $G\mathbin{\curvearrowright}\Omega\times_{H}\Omega^{*}$. ∎

This also tells us that an open Morita embedding arises from an open embedding
if the Morita equivalence
$\Omega\times_{H}\Omega^{*}\sim_{M}H\mathord{\upharpoonright}_{\sigma(\Omega)}$
is an isomorphism. This happens precisely if $\Omega=U\boldsymbol{\cdot}H$ for
a slice $U\subseteq\Omega$, in which case we may identify $U$ with
$\sigma(\Omega)$ and $\Omega$ with
$H\mathord{\upharpoonright}_{\sigma(\Omega)}$. However, even the open Morita
embeddings which are not open embeddings resemble embeddings locally. In
general, tight étale correspondences locally resemble groupoid homomorphisms
which are open embeddings on the level of unit spaces:

######  Proposition 3.11.

Let $\Omega\colon G\to H$ be a tight étale correspondence. Then for each slice
$U\subseteq\Omega$ there is an étale homomorphism $\varphi_{U}\colon
G\mathord{\upharpoonright}_{\rho(U)}\to H$ such that
$\varphi_{U}(\rho(u))=\sigma(u)$ for each $u\in U$ and
$g\boldsymbol{\cdot}u=u\boldsymbol{\cdot}\varphi_{U}(g)$ for each pair
$(g,u)\in G\times_{G^{0}}U$. This étale homomorphism induces the composition

$G\mathord{\upharpoonright}_{\rho(U)}\xhookrightarrow{\iota}G\xrightarrow{\Omega}H$

on the level of correspondences. When $\Omega\colon G\to H$ is further an open
Morita embedding, then $\varphi_{U}$ is an open embedding. In particular, if
$U\boldsymbol{\cdot}H=\Omega$ then $\Omega$ is induced by an open embedding.

###### Proof.

Let $\rho_{U}\colon U\to G^{0}$ be the restriction to $U$ of the anchor map
for the action $G\mathbin{\curvearrowright}\Omega$ and consider the following
continuous maps.

$\displaystyle G\mathord{\upharpoonright}_{\rho(U)}$
$\displaystyle\to\Omega^{*}\times_{\Omega/H}\Omega$
$\displaystyle\Omega^{*}\times_{\Omega/H}\Omega$ $\displaystyle\to H$
$\displaystyle g$
$\displaystyle\mapsto(\rho_{U}^{-1}(r(g))^{*},g\boldsymbol{\cdot}\rho_{U}^{-1}(s(g)))$
$\displaystyle(\omega^{*},\omega\boldsymbol{\cdot}h)$ $\displaystyle\mapsto h$

The first is well-defined by tightness of $\Omega$ and the second is well-
defined and continuous because $\Omega\mathbin{\curvearrowleft}H$ is free,
proper and étale. We set $\varphi_{U}\colon
G\mathord{\upharpoonright}_{\rho(U)}\to H$ to be the composition of these
maps. By construction this is a homomorphism and satisfies
$\varphi_{U}(\rho(u))=\sigma(u)$ for each $u\in U$ and
$g\boldsymbol{\cdot}u=u\boldsymbol{\cdot}\varphi_{U}(g)$ for each pair
$(g,u)\in G\times_{G^{0}}U$. The correspondence associated to $\varphi_{U}$
has bispace $\rho(U)\times_{H^{0}}H$, while the composition $\Omega\circ\iota$
has bispace $G^{\rho(U)}\times_{G}\Omega$. The homeomorphism

$\displaystyle\rho(U)\times_{H^{0}}H$ $\displaystyle\to
G^{\rho(U)}\times_{G}\Omega$ $\displaystyle(x,h)$
$\displaystyle\mapsto[x,\rho_{U}^{-1}(x)\boldsymbol{\cdot}h]_{G}$

is an isomorphism of correspondences. Note that these spaces are both
homeomorphic to $U\boldsymbol{\cdot}H\subseteq\Omega$.

If we now assume that $\Omega\colon G\to H$ is an open Morita embedding then
so is the correspondence associated to $\varphi_{U}$, which implies that
$\varphi_{U}$ is an étale injection and thus an open embedding. When we
further have $U\boldsymbol{\cdot}H=\Omega$ then by tightness $\rho(U)=G^{0}$
and so $\varphi_{U}$ induces $\Omega$. ∎

To define a restriction functor for an openly Morita embedded groupoid we
utilise the induction functor
$\operatorname{Ind}_{\Omega}\colon\operatorname{KK}^{H}\to\operatorname{KK}^{G}$
associated to a second countable Hausdorff étale correspondence $\Omega\colon
G\to H$ constructed in [Mil23, Section 6]. This is a triangulated functor
because it sends mapping cones of $H$-equivariant $*$-homomorphisms to mapping
cones of $G$-equivariant $*$-homomorphisms. The induction functor of the étale
correspondence $H^{G^{0}}\colon G\to H$ associated to an open inclusion
$G\subseteq H$ is naturally isomorphic to the restriction functor
$\operatorname{Res}^{G}_{H}$. This motivates the following definition:

######  Definition 3.12 (Restriction for open Morita embeddings).

Let $G$ and $P$ be second countable Hausdorff étale groupoids and let
$P\hookrightarrow_{M}G$ be an open Morita embedding. The restriction functor
$\operatorname{Res}^{P}_{G}\colon\operatorname{KK}^{P}\to\operatorname{KK}^{G}$
is the induction functor of the étale correspondence $P\hookrightarrow_{M}G$.

The induction functors associated to Morita equivalences are themselves
equivalences. As an open Morita embedding $P\hookrightarrow_{M}G$ is the
composition of open inclusions and Morita equivalences, the restriction
functor
$\operatorname{Res}^{P}_{G}\colon\operatorname{KK}^{G}\to\operatorname{KK}^{P}$
has a left adjoint which we term the induction functor
$\operatorname{Ind}^{G}_{P}\colon\operatorname{KK}^{P}\to\operatorname{KK}^{G}$.
Thus by 2.13, for a countable family $\cal F$ of second countable Hausdorff
groupoids openly Morita embedded into $G$, the localising subcategory
$\langle\operatorname{Ind}_{\cal F}\rangle$ generated by these induced
algebras is complementary to the full subcategory $\mathfrak{N}_{\cal F}$ of
$G$-$\mathrm{C}^{\ast}$-algebras $B$ for which
$\operatorname{Res}^{H}_{G}B\cong 0$ for each $H\in\cal F$. This enables us to
build ABC tuples from families of open Morita embeddings, extending Example
2.12:

######  Example 3.13.

Let $G$ be a second countable Hausdorff étale groupoid, let
$A\in\operatorname{KK}^{G}$ be a $G$-$\mathrm{C}^{\ast}$-algebra and let
$F_{G}\colon\operatorname{KK}^{G}\to\operatorname{\mathbf{Ab}}_{*}$ be the
full K-theory functor $\mathrm{K}_{*}(G\ltimes-)$ or the reduced K-theory
functor $\mathrm{K}_{*}(G\ltimes_{r}-)$. Let $\cal F$ be a countable family of
second countable Hausdorff groupoids openly Morita embedded into $G$ and set

$\mathfrak{I}_{\cal F}=\bigcap_{H\in\cal
F}\ker\operatorname{Res}^{H}_{G}\vartriangleleft\operatorname{KK}^{G}.$

Then $(\operatorname{KK}^{G},F_{G},\mathfrak{I}_{\cal F},A)$ is an ABC tuple.

######  Proposition 3.14.

Let $G$ be a second countable Hausdorff étale groupoid and let $\cal F$ be a
family of proper second countable Hausdorff groupoids openly Morita embedded
into $G$. Then

$\displaystyle\langle\operatorname{Ind}_{\cal F}\rangle$
$\displaystyle\subseteq\mathfrak{P}_{G},$ $\displaystyle\mathfrak{N}_{G}$
$\displaystyle\subseteq\mathfrak{N}_{\cal F}.$

###### Proof.

It suffices to consider singleton families. In this case
$(\langle\operatorname{Ind}_{\cal F}\rangle,\mathfrak{N}_{\cal F})$ is
complementary, so we need only demonstrate one inclusion. Let $\Omega\colon
P\hookrightarrow_{M}G$ be an open Morita embedding with $P$ proper and let
$B\in\operatorname{KK}^{P}$. Recall that the induction functor
$\operatorname{Ind}^{P}_{G}$ is a composition of induction functors for the
decomposition of $P\hookrightarrow_{M}G$ into open inclusions and Morita
equivalences (3.10):

$P\hookrightarrow\Omega\times_{H}\Omega^{*}\sim_{M}G\mathord{\upharpoonright}_{\sigma(\Omega)}\hookrightarrow
G$

For both of these open inclusions the included groupoid acts properly on the
including groupoid. For open inclusions $H\subseteq K$ with this property, the
induced algebra $\operatorname{Ind}^{K}_{H}A$ of an
$H$-$\mathrm{C}^{\ast}$-algebra $A$ is $\operatorname{KK}^{K}$-equivalent to
the generalised fixed-point algebra $s^{*}A^{H}$, where $s\colon K_{H^{0}}\to
H^{0}$ is the source map [BP24, Remark 2.1]. The fixed point algebra
$s^{*}A^{H}$ is none other than the induced algebra
$\operatorname{Ind}_{\Lambda}A$ for the étale correspondence
$\Lambda=K_{H^{0}}\colon K\to H$. It follows that
$\operatorname{Ind}^{G}_{P}B\cong\operatorname{Ind}_{\Omega^{*}}B$ in
$\operatorname{KK}^{G}$ for any $B\in\operatorname{KK}^{P}$, where
$\Omega^{*}\colon G\to P$ is the étale correspondence given by the opposite
bispace. This correspondence factors through the action correspondence
associated to the proper $G$-space $\Omega^{*}/P$, which induces the forgetul
functor $\operatorname{KK}^{G\ltimes\Omega^{*}/P}\to\operatorname{KK}^{G}$.
Therefore $\operatorname{Ind}^{G}_{P}B$ is $\operatorname{KK}^{G}$-equivalent
to a proper $G$-$\mathrm{C}^{\ast}$-algebra, and so
$\operatorname{Ind}^{G}_{P}B\in\mathfrak{P}_{G}$. ∎

######  Definition 3.15.

Let $G$ be a second countable Hausdorff étale groupoid with a family $\cal F$
of proper openly Morita embedded groupoids. We say that $\cal F$ satisfies
condition (P) if for any proper open subgroupoid $P\subseteq G$ and any $x\in
P^{0}$ there is an open neighbourhood $U\subseteq P^{0}$ of $x$ and a member
$H\in\cal F$ such that $P\mathord{\upharpoonright}_{U}\hookrightarrow G$
factors through $H\hookrightarrow_{M}G$.

It is straightforward to see that this agrees with the previous definition in
the case that $\cal F$ consists entirely of genuine subgroupoids. This
condition can also be checked at each finite isotropy subgroup:

######  Proposition 3.16.

Let $G$ be an étale groupoid with a family $\cal F$ of proper openly Morita
embedded groupoids. Then $\cal F$ satisfies condition (P) if and only if for
each $x\in G^{0}$ and finite subgroup $\Gamma\leq G^{x}_{x}$, there is a
member $H\in\cal F$ with embedding $\Omega\colon H\hookrightarrow_{M}G$ and an
element $\omega\in\Omega$ with $\sigma(\omega)=x$ and
$\omega\boldsymbol{\cdot}\Gamma\subseteq H\boldsymbol{\cdot}\omega$.

###### Proof.

Suppose that $\cal F$ satisfies condition (P) and let $x\in G^{0}$ and
$\Gamma\subseteq G^{x}_{x}$ be as in the statement. Then by 3.4 there is a
proper open subgroupoid $P\subseteq G$ containing $\Gamma$. By hypothesis
there is an open neighbourhood $U\subseteq P^{0}$ of $x$ and a member
$H\in\cal F$ such that $P\mathord{\upharpoonright}_{U}\to G$ factors through
the embedding $\Omega\colon H\hookrightarrow_{M}G$ via an étale correspondence
$\Lambda\colon P\mathord{\upharpoonright}_{U}\to H$. We obtain an isomorphism
$\Phi\colon G^{U}\cong\Lambda\times_{H}\Omega$ of
$P\mathord{\upharpoonright}_{U}$-$G$-bispaces through which
$\Phi(x)=[\lambda,\omega]_{H}$ for some $\lambda\in\Lambda$ and
$\omega\in\Omega$. For each $g\in\Gamma$ we may calculate that

$[\lambda,\omega\boldsymbol{\cdot}g]_{H}=\Phi(x\boldsymbol{\cdot}g)=\Phi(g\boldsymbol{\cdot}x)=[g\boldsymbol{\cdot}\lambda,\omega]_{H}$

and consequently there is $h\in H$ such that
$h\boldsymbol{\cdot}\omega=\omega\boldsymbol{\cdot}g$.

For the converse, let $P\subseteq G$ be a proper open subgroupoid and let
$x\in P^{0}$. Then $\Gamma=P^{x}_{x}$ is finite so by hypothesis we obtain a
member $H\in\cal F$ with embedding $\Omega\colon H\hookrightarrow_{M}G$ and
$\omega\in\Omega$ with $\sigma(\omega)=x$ and
$\omega\boldsymbol{\cdot}\Gamma\subseteq H\boldsymbol{\cdot}\omega$. Let
$V\subseteq\Omega$ be a slice containing $\omega$. By 3.11 the composition of
$\Omega\colon H\hookrightarrow_{M}G$ with the inclusion
$H\mathord{\upharpoonright}_{\rho(V)}\hookrightarrow H$ is induced by an open
embedding $\varphi_{V}\colon
H\mathord{\upharpoonright}_{\rho(V)}\hookrightarrow G$ satisfying
$\varphi_{V}(\rho(v))=\sigma(v)$ for each $v\in V$ and
$h\boldsymbol{\cdot}v=v\boldsymbol{\cdot}\varphi_{V}(h)$ for each pair
$(h,v)\in H\times_{H^{0}}V$. For each $g\in\Gamma$ pick $h\in H$ such that
$\omega\boldsymbol{\cdot}g=h\boldsymbol{\cdot}\omega$, and thus $h\in
H^{\rho(\omega)}_{\rho(\omega)}$. By construction
$\omega\boldsymbol{\cdot}\varphi_{V}(h)=\omega\boldsymbol{\cdot}g$ and so
$g=\varphi_{V}(h)$ is in the image of $\varphi_{V}$. This image
$\operatorname{im}\varphi_{V}\subseteq G$ is open and contains $\Gamma$ and
thus by 3.3 contains the restriction of $P$ to some open neighbourhood $U$ of
$x$. By inverting we obtain an open embedding $\varphi_{V}^{-1}\colon
P\mathord{\upharpoonright}_{U}\hookrightarrow H$ which composes with the open
Morita embedding $H\hookrightarrow_{M}G$ to recover the inclusion
$P\mathord{\upharpoonright}_{U}\hookrightarrow G$. ∎

######  Proposition 3.17.

Let $G$ be a second countable Hausdorff étale groupoid with a family $\cal F$
of proper second countable Hausdorff openly Morita embedded groupoids
satisfying condition (P). Then

$(\langle\operatorname{Ind}_{\cal F}\rangle,\mathfrak{N}_{\cal
F})=(\mathfrak{P}_{G},\mathfrak{N}_{G}).$

###### Proof.

We may without loss of generality assume that $\cal F$ is countable as we may
pass to a countable subfamily $\cal E$ which also satisfies (P). The pair
$(\langle\operatorname{Ind}_{\cal F}\rangle,\mathfrak{N}_{\cal F})$ is then
complementary, so it suffices to prove that $\mathfrak{N}_{\cal
F}=\mathfrak{N}_{G}$. Let $A\in\mathfrak{N}_{\cal F}$, let $P\subseteq G$ be a
proper open subgroupoid and let $x\in P^{0}$. Pick an open neighbourhood
$U\subseteq P^{0}$ of $x$ and a member $Q\in\cal F$ such that
$P\mathord{\upharpoonright}_{U}\hookrightarrow G$ factors through
$Q\hookrightarrow_{M}G$. It follows that
$\operatorname{Res}^{P\mathord{\upharpoonright}_{U}}_{G}A\cong 0$. Such $U$
cover $P^{0}$, so by 2.15 we may conclude that
$\operatorname{Res}^{P}_{G}A\cong 0$. ∎

######  Example 3.18.

Let $G$ be a second countable Hausdorff étale groupoid with a countable family
$\cal F$ of proper openly Morita embedded groupoids and let
$A\in\operatorname{KK}^{G}$ be a $G$-$\mathrm{C}^{\ast}$-algebra. If $\cal F$
satisfies condition (P) then the ABC spectral sequence for
$(\operatorname{KK}^{G},\mathfrak{I}_{\cal F},\mathrm{K}_{*}(G\ltimes-),A)$ is
given by

$\mathbb{L}^{\cal F}_{p}\mathrm{K}_{q}(G\ltimes
A)\Rightarrow\mathrm{K}^{\mathrm{top}}_{p+q}(G;A).$

If $G$ has torsion-free stabilisers then any open cover of the unit space
$G^{0}$ satisfies (P). If $G$ is ample and $\cal F=\\{G^{0}\\}$ then Example
2.18 gives us an identification

$\mathbb{L}^{\cal F}_{p}\mathrm{K}_{q}(G\ltimes
A)=\mathrm{H}_{p}(G;\mathrm{K}_{q}(A)).$

######  Remark 3.19.

Instead of considering the proper openly Morita embedded groupoids it may be
desirable to consider a family $\cal F$ of open Morita embeddings of proper
principal groupoids or of spaces. A condition on $\cal F$ analogous to (P)
which considers open subsets of $G^{0}$ instead of arbitrary proper open
subgroupoids of $G$ will ensure that $\cal F$ induces the same complementary
pair as the family $\\{G^{0}\\}$. When $G$ has torsion in its isotropy this
need no longer relate well to the topological K-theory
$\mathrm{K}^{\mathrm{top}}_{*}(G)$ but for ample $G$ it will still be
connected to the groupoid homology.

######  Definition 3.20.

Let $\Omega\colon G\to H$ be an étale correspondence and let $\cal E$ and
$\cal F$ be families of open Morita embeddings into $G$ and $H$ respectively.
We say $\cal E$ and $\cal F$ are compatible under $\Omega$ if for each
embedding $K\hookrightarrow_{M}G$ in $\cal E$ there is an embedding
$L\hookrightarrow_{M}H$ in $\cal F$ and an étale correspondence $\Lambda\colon
K\to L$ such that

${G}$${H}$${K}$${L}$$\scriptstyle{\Omega}$$\scriptstyle{\Lambda}$

commutes up to isomorphism.

An immediate consequence of compatibility of families $\cal E$ and $\cal F$
under $\Omega\colon G\to H$ is that
$\operatorname{Ind}_{\Omega}(\mathfrak{I}_{\cal F})\subseteq\mathfrak{I}_{\cal
E}$.

######  Example 3.21.

Let $S$ be an inverse semigroup and let $Z\colon X\to Y$ be an $S$-equivariant
topological correspondence [Mil23, Definition 3.8]. Explicitly, this consists
of spaces $X$, $Y$ and $Z$ with actions of $S$, an $S$-equivariant continuous
map $\rho\colon Z\to X$ and an $S$-equivariant local homeomorphism
$\sigma\colon Z\to Y$ such that
$\rho^{-1}(\operatorname{dom}_{X}s)=\operatorname{dom}_{Z}s$ holds for each
$s\in S$. The resulting étale homomorphism $S\ltimes Z\to S\ltimes Y$ and
action correspondence $S\ltimes X\to S\ltimes Z$ induce an étale
correspondence $\Omega_{Z,S}\colon S\ltimes X\to S\ltimes Y$. Consider the
families $\cal F^{X}_{\text{fin}}$ and $\cal F^{Y}_{\text{fin}}$ in $S\ltimes
X$ and $S\ltimes Y$ respectively which satisfy (P) as in Example 3.6. Their
members are indexed by idempotents $e\in E(S)^{\times}$ and finite inverse
subgroups $F\leq S_{e}$. For each such $(e,F)$ the associated members are the
transformation groupoids $F_{\downarrow}\ltimes X$ and $F_{\downarrow}\ltimes
Y$ by the inverse semigroup $F_{\downarrow}=\\{fd\mid d\in E(S),f\in F\\}$ of
restrictions of $F$ in $S$. Compatibility with $\Omega_{S,Z}$ follows from the
commutativity up to isomorphism of

${S\ltimes X}$${S\ltimes Y}$${F_{\downarrow}\ltimes X}$${F_{\downarrow}\ltimes
Y.}$$\scriptstyle{\Omega_{S,Z}}$$\scriptstyle{\Omega_{F_{\downarrow},Z}}$

######  Proposition 3.22.

Let $\Omega\colon G\to H$ be a second countable Hausdorff étale correspondence
of ample groupoids. Then $\\{G^{0}\\}$ and $\\{H^{0}\\}$ are compatible under
$\Omega$.

###### Proof.

The local homeomorphism $q\colon\Omega\to\Omega/H$ has a continuous section
$\lambda\colon\Omega/H\to\Omega$ because $\Omega$ is totally disconnected,
second countable and Hausdorff. We may then construct an étale correspondence
$\Lambda\colon G^{0}\to H^{0}$ with bispace $\Omega/H$ with anchor maps
$\overline{\rho}\colon\Omega/H\to G^{0}$ and
$\sigma\circ\lambda\colon\Omega/H\to H^{0}$. This makes

${G}$${H}$${G^{0}}$${H^{0}}$$\scriptstyle{\Omega}$$\scriptstyle{\Lambda}$

commute up to isomorphism by construction. ∎

With open Morita embeddings rather than just open subgroupoids, we can arrange
for a compatible pair of families with respect to any étale correspondence.

######  Proposition 3.23.

Let $\Omega\colon G\to H$ be an étale correspondence and let
$K\hookrightarrow_{M}G$ be an open Morita embedding into $G$. Then there is an
open Morita embedding $L\hookrightarrow_{M}H$ and an actor
$K\mathbin{\curvearrowright}L$ with $K\boldsymbol{\cdot}L^{0}=L$ such that

${G}$${H}$${K}$${L}$$\scriptstyle{\Omega}$

commutes. If $K$ is proper then so is $L$.

###### Proof.

Let $\Lambda\colon K\to H$ be the composition
$K\hookrightarrow_{M}G\xrightarrow{\Omega}H$. By 3.10 we may decompose
$\Lambda$ into an actor $K\mathbin{\curvearrowright}M$ and an open Morita
embedding $M\hookrightarrow_{M}H$. The orbit $L=K\boldsymbol{\cdot}M^{0}$ of
$M^{0}$ under the action $K\mathbin{\curvearrowright}M$ is an open subgroupoid
of $M$. The étale correspondence $\Lambda\colon K\to H$ therefore factorises
into an actor $K\mathbin{\curvearrowright}L$ and an open Morita embedding
$L\hookrightarrow_{M}H$. Now suppose $K$ is proper and let $\rho\colon
L^{0}\to K^{0}$ be the anchor map for the actor. Then for any compact
$C\subseteq L^{0}$, the restriction
$L\mathord{\upharpoonright}_{C}=(K\boldsymbol{\cdot}L^{0})\mathord{\upharpoonright}_{C}=K\mathord{\upharpoonright}_{\rho(C)}\boldsymbol{\cdot}C$
is compact. ∎

We note that if everything we started with was second countable or Hausdorff
then so are the groupoids and correspondences constructed in this proof.

######  Corollary 3.24.

Let $G$ and $H$ be second countable Hausdorff étale groupoids and let
$\Omega\colon G\to H$ be a second countable Hausdorff étale correspondence.
Then $\operatorname{Ind}_{\Omega}(\mathfrak{N}_{H})\subseteq\mathfrak{N}_{G}$.
Furthermore, for any (countable) family $\cal E$ of proper second countable
Hausdorff groupoids openly Morita embedded into $G$ satisfying (P) there is a
(countable) family $\cal F$ of proper second countable Hausdorff groupoids
openly Morita embedded into $H$ satisfying (P) which is compatible with $\cal
E$ under $\Omega$. In particular,
$\operatorname{Ind}_{\Omega}(\mathfrak{I}_{\cal F})\subseteq\mathfrak{I}_{\cal
E}$.

###### Proof.

We first note that a countable family $\cal E$ of proper open subgroupoids
satisfying (P) always exists by Example 3.6, so
$\operatorname{Ind}_{\Omega}(\mathfrak{N}_{H})\subseteq\mathfrak{N}_{G}$
follows from the final part of the statement, by way of 3.17.

For each $K\in\cal E$ use 3.23 to pick a proper openly Morita embedded
groupoid $L_{K}\hookrightarrow_{M}H$ and an étale correspondence $K\to L_{K}$
such that

${G}$${H}$${K}$${L_{K}}$$\scriptstyle{\Omega}$

commutes. Let $\cal F_{0}$ be a countable family of proper groupoids openly
Morita embedded into $H$ satisfying (P) and set $\cal F=\\{L_{K}\mid K\in\cal
E\\}\cup\cal F_{0}$. Then $\cal F$ is compatible with $\cal E$ under $\Omega$
by construction, thus $\operatorname{Ind}(\mathfrak{I}_{\cal
F})\subseteq\mathfrak{I}_{\cal E}$, and $\cal F$ satisfies (P) because $\cal
F_{0}$ does. ∎

This tells us that for any étale correspondence $\Omega\colon G\to H$ we can
always build a compatible family $\cal F$ of embeddings into $H$ from a family
$\cal E$ of embeddings into $G$, and to ensure that $\cal F$ satisfies (P) we
can always artificially enlarge the family. In practice we might start with a
pair of families we know are compatible, and ask when $\cal F$ inherits
condition (P) from $\cal E$.

######  Proposition 3.25.

Let $\Omega\colon G\to H$ be an étale correspondence with
$G\mathbin{\curvearrowright}\Omega$ free such that for each $x\in H^{0}$ and
finite isotropy subgroup $\Gamma\subseteq H^{x}_{x}$ there is
$\omega\in\Omega$ with $\sigma(\omega)=x$ and
$\omega\boldsymbol{\cdot}\Gamma\subseteq G\boldsymbol{\cdot}\omega$. Let $\cal
F_{G}$ and $\cal F_{H}$ be families of proper groupoids openly Morita embedded
into $G$ and $H$ which are compatible under $\Omega$. If $\cal F_{G}$
satisfies (P) then $\cal F_{H}$ satisfies (P).

###### Proof.

We will use 3.16 to translate (P) for $\cal F_{G}$ into (P) for $\cal F_{H}$.
Let $x\in H^{0}$ and let $\Gamma\leq H^{x}_{x}$ be a finite subgroup. By
assumption there is $\omega_{1}\in\Omega$ with $\sigma(\omega_{1})=x$ and
$\omega_{1}\boldsymbol{\cdot}\Gamma\subseteq G\boldsymbol{\cdot}\omega$. By
freeness of $G\mathbin{\curvearrowright}\Omega$ there is a finite subgroup
$\Gamma_{1}\leq G^{\rho(\omega_{1})}_{\rho(\omega_{1})}$ with
$\omega_{1}\boldsymbol{\cdot}\Gamma=\Gamma_{1}\boldsymbol{\cdot}\omega_{1}$.
Using condition (P) for $\cal F_{H}$ there is a member $K\in\cal F_{H}$ with
embedding $\Omega_{K}\colon K\hookrightarrow_{M}H$ and an element
$\omega_{2}\in\Omega_{K}$ such that $\sigma(\omega_{2})=\rho(\omega_{1})$ and
$\omega_{2}\boldsymbol{\cdot}\Gamma_{1}\subseteq
K\boldsymbol{\cdot}\omega_{2}$. There is therefore a finite subgroup
$\Gamma_{2}\leq K^{\rho(\omega_{2})}_{\rho(\omega_{2})}$ with
$\omega_{2}\boldsymbol{\cdot}\Gamma_{1}=\Gamma_{2}\boldsymbol{\cdot}\omega_{2}$.
Pick an open Morita embedding $\Omega_{L}\colon L\hookrightarrow_{M}G$ and an
étale correspondence $\Lambda\colon K\to L$ such that
$\Omega\circ\Omega_{K}\cong\Omega_{L}\circ\Lambda$ and let
$\Phi\colon\Omega_{K}\times_{G}\Omega\to\Lambda\times_{L}\Omega_{L}$ be the
isomorphism of $K$-$H$-bispaces. Choose $\lambda\in\Lambda$ and
$\omega_{3}\in\Omega_{L}$ such that
$[\lambda,\omega_{3}]_{L}=\Phi([\omega_{2},\omega_{1}]_{G})$. By construction,
$[\Gamma_{2}\boldsymbol{\cdot}\lambda,\omega_{3}]_{L}=[\lambda,\omega_{3}\boldsymbol{\cdot}\Gamma]_{L}$.
It follows that for each $h\in\Gamma$ there is $l\in L$ such that
$\omega_{3}\boldsymbol{\cdot}h=l\boldsymbol{\cdot}\omega_{3}$, i.e.
$\omega_{3}\boldsymbol{\cdot}\Gamma\subseteq L\boldsymbol{\cdot}\omega_{3}$. ∎

##  4\. The interaction between Kasparov theory and groupoid homology

In Example 2.18 we see that for an ample groupoid $G$ the derived functors of
crossed product K-theory functors on $\operatorname{KK}^{G}$ with respect to
$\mathfrak{I}_{0}=\ker\operatorname{Res}^{G^{0}}_{G}$ are given by groupoid
homology:

$\mathbb{L}_{p}\mathrm{K}_{q}(G\ltimes-)\cong\mathrm{H}_{p}(G;\mathrm{K}_{q}(-))$

The goal of this section is to prepare ourselves to relate homological algebra
in $\operatorname{KK}^{G}$ with respect to $\mathfrak{I}_{0}$ to more
classical homological algebra over $G$ through an analysis of the relationship
between $G$-$\mathrm{C}^{\ast}$-algebras $A\in\operatorname{KK}^{G}$ and the
$G$-modules $\mathrm{K}_{*}(A)\in G\text{-}\mathbf{Mod}$. For other ABC tuples
$(\mathfrak{T},\mathfrak{I},F,A)$ it would be desirable to carry out a similar
analysis to enable computation and handling of the derived functors
$\mathbb{L}_{n}F(A)$.

######  Definition 4.1 (Module over an ample groupoid).

Let $G$ be an ample groupoid. The groupoid ring $\mathbb{Z}[G]$ is the abelian
group of functions $G\to\mathbb{Z}$ generated by the indicator functions
$\chi_{U}$ of compact open Hausdorff subsets $U\subseteq G$, with convolution
as the ring333When $G^{0}$ is not compact $\mathbb{Z}[G]$ is not unital, but
it is always locally unital. multiplication. A (left) $G$-module $M$ is a
(left) $\mathbb{Z}[G]$-module such that $\mathbb{Z}[G]\boldsymbol{\cdot}M=M$.

For any totally disconnected locally LCH space $X$ we may consider the abelian
group $\mathbb{Z}[X]$ as above, and an action $G\mathbin{\curvearrowright}X$
induces a (left) $G$-module structure on $\mathbb{Z}[X]$. For an introduction
to the category $G\text{-}\mathbf{Mod}$ of $G$-modules we refer the reader to
[Mil24, Section 2]. For an ample groupoid $G$ and a
$G$-$\mathrm{C}^{\ast}$-algebra $A=\Gamma_{0}(G^{0},\mathcal{A})$, the
K-theory groups $\mathrm{K}_{*}(A)$ have the following $G$-module structure.
For each clopen subset $V\subseteq G^{0}$ consider the subalgebra
$VA:=C_{0}(V)A=\Gamma_{0}(V,\mathcal{A})$. Inclusion and restriction maps
$VA\to A\to VA$ induce injective and surjective maps
$\mathrm{K}_{*}(VA)\to\mathrm{K}_{*}(A)\to\mathrm{K}_{*}(VA)$. By continuity
of K-theory, $\mathrm{K}_{*}(A)$ is generated by the subgroups
$\mathrm{K}_{*}(VA)$ for $V$ in any open cover of $G^{0}$. Let $\alpha\colon
G\mathbin{\curvearrowright}A$ denote the action of $G$ on $A$. Each compact
slice $U\subseteq G$ induces a $*$-homomorphism $\alpha_{U}\colon s(U)A\to
r(U)A$ defined at $a\in s(U)A$ by

$\alpha_{U}(a)\colon x\mapsto\sum_{g\in
G_{x}}\chi_{U}(g)(g\boldsymbol{\cdot}a_{x}).$

Through restriction and inclusion this extends to a $*$-homomorphism $A\to A$
which we also call $\alpha_{U}$. It is straightforward to check by [Mil24,
Lemma 2.4] that this induces the structure of a $G$-module on $K_{i}(A)$ for
$i=0,1$ with

$\chi_{U}\cdot m=\mathrm{K}_{i}(\alpha_{U})(m)$

for each compact slice $U\subseteq G$ and $m\in\mathrm{K}_{i}(A)$. For second
countable Hausdorff $G$, separable $G$-$\mathrm{C}^{\ast}$-algebras $A$ and
$B$ and $f\in\operatorname{KK}^{G}(A,B)$, the induced map
$\mathrm{K}_{i}(f)\colon\mathrm{K}_{i}(A)\to\mathrm{K}_{i}(B)$ is
$G$-equivariant, so we obtain a pair of functors

$\mathrm{K}_{0},\mathrm{K}_{1}\colon\operatorname{KK}^{G}\to
G\text{-}\mathbf{Mod}.$

The KK-theoretic induction functor of an étale correspondence has an module-
theoretic analogue constructed by the author in [Mil24, Section 3]. For an
étale correspondence $\Omega\colon G\to H$ of ample groupoids the induction
functor

$\operatorname{Ind}_{\Omega}\colon H\text{-}\mathbf{Mod}\to
G\text{-}\mathbf{Mod}$

is given by the tensor product $\mathbb{Z}[\Omega]\otimes_{H}-$ with the
$G$-$H$-bimodule $\mathbb{Z}[\Omega]$. To prepare ourselves to compare the two
induction functors, suppose $G$, $H$ and $\Omega$ are Hausdorff, let $B$ be an
$H$-$\mathrm{C}^{\ast}$-algebra and let $U\subseteq\Omega$ be a compact slice.
Let $\iota_{U}\colon s(U)B\to B$ denote the inclusion. There is also an
inclusion $\epsilon_{U}\colon\sigma(U)B\to\operatorname{Ind}_{\Omega}B$ given
at $b\in\sigma(U)B$ by the element
$\epsilon_{U}(b)\in\operatorname{Ind}_{\Omega}B\subseteq\Gamma_{b}(\Omega,\sigma^{*}\cal
B)$ as follows for $b\in\sigma(U)B$ and $\omega\in\Omega$:

$\displaystyle\epsilon_{U}\colon\sigma(U)B$
$\displaystyle\to\operatorname{Ind}_{\Omega}B$ $\displaystyle b$
$\displaystyle\mapsto\epsilon_{U}(b)$
$\displaystyle\quad\;\;\omega\mapsto\sum_{h\in
H^{\sigma(\omega)}}\chi_{U}(\omega\cdot h)(h\boldsymbol{\cdot}b_{s(h)})$

This is the equivariant extension (see [Mil23, Proposition 4.3]) of the
section in $\Gamma_{0}(U,\sigma^{*}\mathcal{B})$ determined by $b$ and the
homeomorphism $\sigma\colon U\to\sigma(U)$. The image of $\epsilon_{U}$ is the
subalgebra of functions supported on $U\cdot H\subseteq\Omega$, which is
$q(U)\operatorname{Ind}_{\Omega}B$ with respect to the
$C_{0}(\Omega/H)$-algebra structure on $\operatorname{Ind}_{\Omega}B$. We
obtain inclusions

$\displaystyle\mathrm{K}_{*}(\iota_{U})\colon\mathrm{K}_{*}(s(U)B)$
$\displaystyle\to\mathrm{K}_{*}(B)$
$\displaystyle\mathrm{K}_{*}(\epsilon_{U})\colon\mathrm{K}_{*}(s(U)B)$
$\displaystyle\to\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)$

whose images generate $\mathrm{K}_{*}(B)$ and
$\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)$ across all compact slices
$U\subseteq\Omega$.

######  Proposition 4.2 (K-theory intertwines the induction functors).

Let $\Omega\colon G\to H$ be a second countable Hausdorff étale correspondence
of second countable Hausdorff ample groupoids. Then for
$B\in\operatorname{KK}^{H}$ there is a natural isomorphism of $G$-modules

$\zeta_{\Omega,B}\colon\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B)\to\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)$

such that
$\zeta_{\Omega,B}(\chi_{U}\otimes\mathrm{K}_{*}(\iota_{U})(y))=\mathrm{K}_{*}(\epsilon_{U})(y)$
for each compact slice $U\subseteq\Omega$ and
$y\in\mathrm{K}_{*}(\sigma(U)B)$.

###### Proof.

Let $\alpha\colon G\mathbin{\curvearrowright}\operatorname{Ind}_{\Omega}B$ and
$\beta\colon H\mathbin{\curvearrowright}B$ denote the respective groupoid
actions on the $\mathrm{C}^{\ast}$-algebras. For each compact slice
$U\subseteq\Omega$ let $\eta_{U}\colon B\to\sigma(U)B$ denote the restriction.
To construct $\zeta_{\Omega,B}$ we first construct a bilinear map

$L\colon\mathbb{Z}[\Omega]\times\mathrm{K}_{*}(B)\to\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)$

such that $L(\chi_{U},x)=\mathrm{K}_{*}(\epsilon_{U}\circ\eta_{U})(x)$ for
each compact slice $U\subseteq\Omega$ and $x\in\mathrm{K}_{*}(B)$. The
assignment $(U,x)\mapsto\mathrm{K}_{*}(\epsilon_{U}\circ\eta_{U})(x)$ respects
disjoint unions of compact slices, so extends to a bilinear map $L$ by [Mil24,
Lemma 2.4]. Let $V\subseteq H$ be a compact slice. For each compact slice
$U\subseteq\Omega$ the diagram

${B}$${B}$${\operatorname{Ind}_{\Omega}B}$$\scriptstyle{\beta_{V}}$$\scriptstyle{\epsilon_{U\boldsymbol{\cdot}V}\circ\eta_{U\boldsymbol{\cdot}V}}$$\scriptstyle{\epsilon_{U}\circ\eta_{U}}$

commutes. It follows that $L$ is balanced and so extends to a homomorphism
$\zeta_{\Omega,B}\colon\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B)\to\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)$
of abelian groups. Moreover, the map $\zeta_{\Omega,B}$ is $G$-equivariant
because for each pair of compact slices $U\subseteq\Omega$ and $W\subseteq G$
the diagram

${B}$${\operatorname{Ind}_{\Omega}B}$${\operatorname{Ind}_{\Omega}B}$$\scriptstyle{\epsilon_{U}\circ\eta_{U}}$$\scriptstyle{\epsilon_{W\boldsymbol{\cdot}U}\circ\eta_{W\boldsymbol{\cdot}U}}$$\scriptstyle{\alpha_{W}}$

commutes. Note that by the same argument $\zeta_{\Omega,B}$ is
$\Omega/H$-equivariant, and thus it suffices to check bijectivity on the
restrictions to $q(U)\subseteq\Omega/H$ for each compact slice
$U\subseteq\Omega$. These restrictions are given by

$\displaystyle\chi_{q(U)}\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B)$
$\displaystyle=\\{\chi_{U}\otimes\mathrm{K}_{*}(\iota_{U})(y)\;|\;y\in\mathrm{K}_{*}(\sigma(U)B)\\}$
$\displaystyle\chi_{q(U)}\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)$
$\displaystyle=\\{\mathrm{K}_{*}(\epsilon_{U})(y)\;|\;y\in\mathrm{K}_{*}(\sigma(U)B)\\},$

upon which $\zeta_{\Omega,B}$ is bijective by construction.

Let $C\in\operatorname{KK}^{H}$ and consider a morphism
$f\in\operatorname{KK}^{H}(B,C)$. For each compact slice $U\subseteq\Omega$ we
may restrict $f$ and $\operatorname{Ind}_{\Omega}f$ to morphisms
$\operatorname{Res}_{\sigma(U)}f\in\operatorname{KK}(\sigma(U)B,\sigma(U)C)$
and
$\operatorname{Res}_{q(U)}\operatorname{Ind}_{\Omega}f\in\operatorname{KK}(q(U)\operatorname{Ind}_{\Omega}B,q(U)\operatorname{Ind}_{\Omega}C)$.
To show naturality of $\zeta_{\Omega,-}$ we show commutativity of the
following diagram:

(4.1)
${{\sigma(U)B}}$${{q(U)\operatorname{Ind}_{\Omega}B}}$${{\sigma(U)C}}$${{q(U)\operatorname{Ind}_{\Omega}C}}$$\scriptstyle{\operatorname{KK}(\epsilon_{U})}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{Res}_{q(U)}\operatorname{Ind}_{\Omega}f}$$\scriptstyle{\operatorname{Res}_{\sigma(U)}f}$$\scriptstyle{\operatorname{KK}(\epsilon_{U})}$$\scriptstyle{\cong}$

Let $(E,T)$ be an $H$-equivariant Kasparov cycle representing $f$. Consider
the étale correspondence $U\boldsymbol{\cdot}H\colon q(U)\to H$. We may
identify $q(U)\operatorname{Ind}_{\Omega}B$ with
$\operatorname{Ind}_{U\boldsymbol{\cdot}H}B\mathord{\upharpoonright}_{\sigma(U)}$
through the inclusion $\Gamma_{b}(U\cdot
H,\sigma^{*}\mathcal{B})\subseteq\Gamma_{b}(\Omega,\sigma^{*}\mathcal{B})$.
Similarly, $\operatorname{Res}_{q(U)}\operatorname{Ind}_{\Omega}f$ may be
identified with $\operatorname{Ind}_{U\boldsymbol{\cdot}H}f$ (after the
forgetful functor $\operatorname{KK}^{q(U)}\to\operatorname{KK}$) since each
cutoff function for $\Omega$ restricts to a cutoff function for
$U\boldsymbol{\cdot}H$. The correspondence $U\boldsymbol{\cdot}H\colon q(U)\to
H\mathord{\upharpoonright}_{\sigma(U)}$ has a cutoff function given by
$\chi_{U}\colon U\boldsymbol{\cdot}H\to\mathbb{R}$, and so

$\operatorname{Ind}_{U\boldsymbol{\cdot}H}f=[\operatorname{Ind}_{U\boldsymbol{\cdot}H}E,\operatorname{Ind}_{U\boldsymbol{\cdot}H,\chi_{U}}T].$

The $*$-isomorphisms labelled $\epsilon_{U}$ in (4.1) are compatible with the
analogous isomorphism
$\epsilon_{U}\colon\sigma(U)E\to\operatorname{Ind}_{U\boldsymbol{\cdot}H}E$,
through which we may identify the correspondence
$\sigma(U)E\colon\sigma(U)B\to\sigma(U)C$ with
$\operatorname{Ind}_{U\boldsymbol{\cdot}H}E\colon\operatorname{Ind}_{U\boldsymbol{\cdot}H}B\to\operatorname{Ind}_{U\boldsymbol{\cdot}H}C$.
This isomorphism conjugates the Fredholm operator $\sigma(U)T$ to
$\operatorname{Ind}_{U\boldsymbol{\cdot}H,\chi_{U}}T$, from which it follows
that (4.1) commutes. ∎

For an ample groupoid $G$ and a $G$-module $M$ the coinvariants $M_{G}$ is the
abelian group $\mathbb{Z}[G^{0}]\otimes_{G}M$. This may be considered as the
quotient of $M$ by the relations
$\chi_{U}\boldsymbol{\cdot}m\sim\chi_{s(U)}\boldsymbol{\cdot}m$ for $m\in M$
and compact slices $U\subseteq G$. The quotient map $\pi_{G}\colon M\to M_{G}$
satisfies $\pi_{G}(m)=\chi_{V}\otimes m$ for each $m\in M$ and each compact
open subset $V\subseteq G^{0}$ with $\chi_{V}\boldsymbol{\cdot}m=m$. Now
consider an action $\alpha\colon G\mathbin{\curvearrowright}A$ on a
$\mathrm{C}^{\ast}$-algebra $A$ and the inclusion map $\eta_{A}\colon A\to
G\ltimes A$. Given an element $x\in\mathrm{K}_{*}(A)$ and a compact slice
$U\subseteq G$, we have an equality
$\mathrm{K}_{*}(\eta_{A})(\chi_{U}\boldsymbol{\cdot}x)=\mathrm{K}_{*}(\eta_{A})(\chi_{s(U)}\boldsymbol{\cdot}x)$.
This is because the partial isometry $\chi_{U}\in M(G\ltimes A)$ conjugates
$\eta_{A}\circ\alpha_{U}\colon A\to G\ltimes A$ to
$\eta_{A}\circ\alpha_{s(U)}\colon A\to G\ltimes A$ in the sense that for each
$a\in A$ we have

$\displaystyle\chi_{U}^{*}\eta_{A}(\alpha_{U}(a))\chi_{U}$
$\displaystyle=\eta_{A}(\alpha_{s(U)}(a)),$
$\displaystyle\chi_{U}\eta_{A}(\alpha_{s(U)}(a))\chi_{U}^{*}$
$\displaystyle=\eta_{A}(\alpha_{U}(a)).$

It follows that
$\mathrm{K}_{*}(\eta_{A})\colon\mathrm{K}_{*}(A)\to\mathrm{K}_{*}(G\ltimes A)$
factors through the quotient map
$\pi_{G}\colon\mathrm{K}_{*}(A)\to\mathrm{K}_{*}(A)_{G}$.

######  Definition 4.3 (The comparison map).

Let $G$ be an ample groupoid and let $A$ be a $G$-$\mathrm{C}^{\ast}$-algebra.
The comparison map
$\gamma_{A}\colon\mathrm{K}_{*}(A)_{G}\to\mathrm{K}_{*}(G\ltimes A)$ is the
homomorphism induced by the inclusion $A\subseteq G\ltimes A$.

For second countable Hausdorff $G$ the comparison map $\gamma_{A}$ is natural
in $A\in\operatorname{KK}^{G}$ by naturality of the inclusion $A\subseteq
G\ltimes A$ at the level of $\operatorname{KK}$.

For an étale correspondence $\Omega\colon G\to H$ the author introduced
[Mil23, Definition 6.10] a natural transformation

$\alpha_{\Omega}\colon\mathrm{K}_{*}(G\ltimes\operatorname{Ind}_{\Omega}-)\Rightarrow\mathrm{K}_{*}(H\ltimes-)$

which relates the crossed product K-theory functors on $\operatorname{KK}^{G}$
and $\operatorname{KK}^{H}$. For modules of ample groupoids, the author also
constructed [Mil24, Proposition 3.3] an $H$-equivariant homomorphism
$\delta_{\Omega}\colon\mathbb{Z}[G^{0}]\otimes_{G}\mathbb{Z}[\Omega]\to\mathbb{Z}[H^{0}]$
which induces a natural transformation

$\delta_{\Omega}\otimes-\colon(\operatorname{Ind}_{\Omega}-)_{G}\Rightarrow(-)_{H}$

relating the coinvariants on $G\text{-}\mathbf{Mod}$ and
$H\text{-}\mathbf{Mod}$.

######  Proposition 4.4.

Let $G$ and $H$ be second countable Hausdorff ample groupoids, let
$\Omega\colon G\to H$ be a second countable Hausdorff étale correspondence and
let $B\in\operatorname{KK}^{H}$ be an $H$-C*-algebra. Then the following
diagram commutes:

${{(\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B))_{G}}}$${{\mathrm{K}_{*}(B)_{H}}}$${{\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)_{G}}}$${{\mathrm{K}_{*}(G\ltimes\operatorname{Ind}_{\Omega}B)}}$${{\mathrm{K}_{*}(H\ltimes
B)}}$$\scriptstyle{(\zeta_{\Omega,B})_{G}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{\operatorname{Ind}_{\Omega}B}}$$\scriptstyle{\alpha_{\Omega}(B)}$$\scriptstyle{\gamma_{B}}$$\scriptstyle{\delta_{\Omega}\otimes\operatorname{id}}$

###### Proof.

Fix a compact slice $U\subseteq\Omega$ and an element
$y\in\mathrm{K}_{*}(\sigma(U)B)$. We need only check commutativity at the
element
$[\chi_{U}\otimes\mathrm{K}_{*}(\iota_{U})(y)]_{G}\in(\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B))_{G}$.
The homomorphism $\alpha_{\Omega}(B)$ is induced by a proper
$\mathrm{C}^{\ast}$-correspondence

$\Omega\ltimes B\colon G\ltimes\operatorname{Ind}_{\Omega}B\to H\ltimes B$

introduced in [Mil23, Section 5]. Consider this within the following diagram
of proper $\mathrm{C}^{\ast}$-correspondences with all other maps induced by
$*$-homomorphisms:

${{\sigma(U)B}}$${B}$${{\operatorname{Ind}_{\Omega}B}}$${{G\ltimes\operatorname{Ind}_{\Omega}B}}$${{H\ltimes
B}}$$\scriptstyle{\epsilon_{U}}$$\scriptstyle{\Omega\ltimes
B}$$\scriptstyle{\iota_{U}}$

By construction, the resulting diagram in K-theory applied to $y$ computes the
values of the initial diagram applied to
$[\chi_{U}\otimes\mathrm{K}_{*}(\iota_{U})(y)]_{G}$. It therefore suffices to
show that the above diagram commutes in the category of
$\mathrm{C}^{\ast}$-correspondences. The  composition of
$\mathrm{C}^{\ast}$-correspondences is given by the Hilbert $H\ltimes
B$-submodule
$\overline{\Gamma_{c}(U\boldsymbol{\cdot}H,\sigma^{*}\mathcal{B})}$ of
$\Omega\ltimes B=\overline{\Gamma_{c}(\Omega,\sigma^{*}\mathcal{B})}$ with the
action of $b\in\sigma(U)B$ as follows on
$\xi\in\Gamma_{c}(U\boldsymbol{\cdot}H,\sigma^{*}\mathcal{B})$, for
$\omega\in\Omega$:

$\displaystyle\sigma(U)B\times\Gamma_{c}(U\boldsymbol{\cdot}H,\sigma^{*}\mathcal{B})$
$\displaystyle\to\Gamma_{c}(U\boldsymbol{\cdot}H,\sigma^{*}\mathcal{B})$
$\displaystyle(b,\xi)$
$\displaystyle\mapsto\epsilon_{U}(b)\boldsymbol{\cdot}\xi$
$\displaystyle\quad\;\;\omega\mapsto\sum_{h\in
H^{\sigma(\omega)}}\chi_{U}(\omega\cdot
h)(h\boldsymbol{\cdot}b_{s(h)})\xi(\omega)$

On the other hand, the  composition is simply the inclusion
$\sigma(U)B\subseteq H\ltimes B$, which as a
$\mathrm{C}^{\ast}$-correspondence has underlying Hilbert $H\ltimes B$-module
$\overline{\Gamma_{c}(H^{\sigma(U)},s^{*}\mathcal{B})}$, as a submodule of
$H\ltimes B=\overline{\Gamma_{c}(H,s^{*}\mathcal{B})}$. The homeomorphism
$U\boldsymbol{\cdot}H\to H^{\sigma(U)}$ satisfying
$u\boldsymbol{\cdot}h\mapsto h$ for $u\in U$ and $h\in H$ with
$\sigma(u)=r(h)$ induces an isomorphism of these
$\mathrm{C}^{\ast}$-correspondences. ∎

######  Proposition 4.5.

Let $G$ be a second countable Hausdorff ample groupoid, let
$B\in\operatorname{KK}^{G^{0}}$ be a separable $C_{0}(G^{0})$-algebra and
consider the induced algebra
$A=\operatorname{Ind}^{G}_{G^{0}}B\in\operatorname{KK}^{G}$. Then the
comparison map
$\gamma_{A}\colon\mathrm{K}_{*}(A)_{G}\to\mathrm{K}_{*}(G\ltimes A)$ is an
isomorphism.

###### Proof.

We may consider $A$ as the induced algebra $\operatorname{Ind}_{\Omega}B$ for
the étale correspondence $\Omega=G\colon G\to G^{0}$. Applying 4.4 we obtain
the following commutative diagram.

${{(\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B))_{G}}}$${{\mathrm{K}_{*}(B)}}$${{\mathrm{K}_{*}(A)_{G}}}$${{\mathrm{K}_{*}(G\ltimes
A)}}$${{\mathrm{K}_{*}(B)}}$$\scriptstyle{(\zeta_{\Omega,B})_{G}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{A}}$$\scriptstyle{\alpha_{\Omega}(B)}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\delta_{\Omega}\otimes\operatorname{id}}$

The map
$\delta_{\Omega}\colon\mathbb{Z}[G^{0}]\otimes_{G}\mathbb{Z}[G]\to\mathbb{Z}[G^{0}]$
is given by
$\delta_{\Omega}(\eta\otimes\xi)=s_{*}(\eta\vartriangleright\xi)=\eta\vartriangleleft\xi$
for $\eta\in\mathbb{Z}[G^{0}]$ and $\xi\in\mathbb{Z}[G]$, where
$\vartriangleright$ denotes the action of $\mathbb{Z}[G^{0}]$ on
$\mathbb{Z}[G]$ and $\vartriangleleft$ vice versa. It follows that
$\delta_{\Omega}$ and therefore $\delta_{\Omega}\otimes\operatorname{id}$ is
an isomorphism. The map $\alpha_{\Omega}(B)$ is induced by the proper
correspondence $\Omega\ltimes B\colon G\ltimes A\to B$. The action of $G$ on
$A$ factors through an action of $G\ltimes G$, so we may identify $G\ltimes A$
with the crossed product $(G\ltimes G)\ltimes A$. Consider the invertible
étale correspondence $\Lambda=G\colon G\ltimes G\to G^{0}$ and note that
$\operatorname{Ind}_{\Lambda}B=A$. We may identify $\Omega\ltimes B\colon
G\ltimes A\to B$ with $\Lambda\ltimes B\colon(G\ltimes G)\ltimes A\to B$,
which inherits invertibility from $\Lambda$ by [Mil23, Proposition 7.4]. Since
all other maps in the diagram are isomorphisms, $\gamma_{A}$ must also be. ∎

######  Remark 4.6.

Let $G$ be a second countable Hausdorff ample groupoid, let
$A\in\operatorname{KK}^{G}$ be a $G$-$\mathrm{C}^{\ast}$-algebra and consider
the ideal $\mathfrak{I}_{0}$ as in Example 2.18. Then the comparison map for
induced algebras induces an isomorphism

$\mathrm{H}_{p}(G;\mathrm{K}_{q}(A))\cong\mathbb{L}_{p}\mathrm{K}_{q}(G\ltimes
A)$

as follows. For $n\geq 0$ let
$P_{n}=(\operatorname{Ind}^{G}_{G^{0}}\operatorname{Res}^{G^{0}}_{G})^{n+1}A$
and consider the $\mathfrak{I}_{0}$-projective resolution $P_{\bullet}\to A$
induced by the induction-restriction adjunction
$\operatorname{Ind}^{G}_{G^{0}}\dashv\operatorname{Res}^{G^{0}}_{G}$ as in
2.13. The homology groups for the chain complex $\mathrm{K}_{q}(G\ltimes
P_{\bullet})$ are $\mathbb{L}_{*}\mathrm{K}_{q}(G\ltimes A)$ as
$P_{\bullet}\to A$ is $\mathfrak{I}_{0}$-projective. Furthermore, by 4.2 the
$G$-modules $\mathrm{K}_{q}(P_{n})$ are isomorphic to $G^{0}$-induced modules,
and so by [Mil24, Corollary 2.20] are $G$-acyclic. The homology of the chain
complex $\mathrm{K}_{q}(P_{\bullet})_{G}$ therefore produces
$\mathrm{H}_{*}(G;\mathrm{K}_{q}(A))$. By naturality the comparison maps

$\gamma_{P_{\bullet}}\colon\mathrm{K}_{q}(P_{\bullet})_{G}\to\mathrm{K}_{q}(G\ltimes
P_{\bullet})$

form a chain map. Each $\gamma_{P_{n}}$ is an isomorphism by 4.5, so they in
particular induce an isomorphism of the homology groups.

##  5\. The ABC category

We are now ready to set up a category of ABC tuples
$(\mathfrak{T},\mathfrak{I},F,A)$ so that we can study functoriality
properties of the localisation $\mathbb{L}^{\mathfrak{I}}F(A)$, the derived
functors $\mathbb{L}^{\mathfrak{I}}_{n}F(A)$ and ultimately the associated ABC
spectral sequence. Our starting point is to consider a graded homomorphism
$\varphi\colon F(A)\to F^{\prime}(A^{\prime})$ relating to a second ABC tuple
$(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$.
Morphisms of ABC tuples should be triples that lift $\varphi$ to the level of
triangulated categories, that is

  * •

a triangulated functor $\Phi\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$,

  * •

a natural transformation $\alpha\colon F_{*}\circ\Phi\Rightarrow
F^{\prime}_{*}$,

  * •

and a morphism $f\colon A\to\Phi(A^{\prime})$,

such that $\varphi=\alpha_{A^{\prime}}\circ F_{*}(f)$. We write
$\operatorname{App}(\Phi,\alpha,f)$ for the composition
$\alpha_{A^{\prime}}\circ F_{*}(f)$.

${{F(A)}}$${{F^{\prime}(A^{\prime})}}$${{F(\Phi(A^{\prime})}}$$\scriptstyle{\alpha_{A^{\prime}}}$$\scriptstyle{F(f)}$$\scriptstyle{\operatorname{App}(\Phi,\alpha,f)}$

######  Example 5.1.

Let $G$ and $H$ be second countable Hausdorff étale groupoids and let
$\Omega\colon G\to H$ be a proper second countable Hausdorff étale
correspondence. The $\mathrm{C}^{\ast}$-correspondence
$\mathrm{C}^{\ast}(\Omega)\colon\mathrm{C}^{\ast}(G)\to\mathrm{C}^{\ast}(H)$
induces a map in K-theory

$\mathrm{K}_{*}(\mathrm{C}^{\ast}(\Omega))\colon\mathrm{K}_{*}(\mathrm{C}^{\ast}(G))\to\mathrm{K}_{*}(\mathrm{C}^{\ast}(H)).$

Consider the categories $\operatorname{KK}^{G}$ and $\operatorname{KK}^{H}$,
the functors
$\mathrm{K}_{*}(G\ltimes-)\colon\operatorname{KK}^{G}\to\operatorname{\mathbf{Ab}}_{*}$
and
$\mathrm{K}_{*}(H\ltimes-)\colon\operatorname{KK}^{H}\to\operatorname{\mathbf{Ab}}_{*}$
and the objects $C_{0}(G^{0})$ and $C_{0}(H^{0})$. In [Mil23] the author
constructed a triple consisting of

  * •

the induction functor
$\operatorname{Ind}_{\Omega}\colon\operatorname{KK}^{H}\to\operatorname{KK}^{G}$,

  * •

the induction natural transformation
$\alpha_{\Omega}\colon\mathrm{K}_{*}(G\ltimes\operatorname{Ind}_{\Omega}-)\Rightarrow\mathrm{K}_{*}(H\ltimes-)$,

  * •

and the morphism $f_{\Omega}\colon
C_{0}(G^{0})\to\operatorname{Ind}_{\Omega}C_{0}(H^{0})$ in
$\operatorname{KK}^{G}$ induced by the $G$-equivariant proper map
$\overline{\rho}\colon\Omega/H\to G^{0}$,

such that
$\operatorname{App}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f_{\Omega})=\mathrm{K}_{*}(\mathrm{C}^{\ast}(\Omega))$.
More generally we can drop the properness requirement on $\Omega$ and consider
coefficient $\mathrm{C}^{\ast}$-algebras $A\in\operatorname{KK}^{G}$ and
$B\in\operatorname{KK}^{H}$ so long as we have a morphism
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$. We then set

$\mathrm{K}_{*}(\Omega;f):=\operatorname{App}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)\colon\mathrm{K}_{*}(G\ltimes
A)\to\mathrm{K}_{*}(H\ltimes B).$

Triples of this form make sense and appear outside of the triangulated setting
too:

######  Example 5.2.

Let $G$ and $H$ be ample groupoids and let $\Omega\colon G\to H$ be a proper
étale correspondence. In [Mil24] the author constructed a graded homomorphism

$\mathrm{H}_{*}(\Omega)\colon\mathrm{H}_{*}(G)\to\mathrm{H}_{*}(H)$

as the application
$\operatorname{App}(\operatorname{Ind}_{\Omega},\delta_{\Omega}\otimes\operatorname{id},\overline{\rho}^{*})$
of a triple on the level of the module categories $G\text{-}\mathbf{Mod}$ and
$H\text{-}\mathbf{Mod}$:

  * •

the induction functor $\operatorname{Ind}_{\Omega}\colon
H\text{-}\mathbf{Mod}\to G\text{-}\mathbf{Mod}$,

  * •

the natural transformation
$\delta_{\Omega}\otimes\operatorname{id}\colon(\operatorname{Ind}_{\Omega}-)_{G}\Rightarrow(-)_{H}$,

  * •

and the morphism
$\overline{\rho}^{*}\colon\mathbb{Z}[G^{0}]\to\operatorname{Ind}_{\Omega}\mathbb{Z}[H^{0}]$
induced by the $G$-equivariant proper map $\overline{\rho}\colon\Omega/H\to
G^{0}$.

######  Remark 5.3.

It is possible to place the classical homological algebra of abelian
categories such as $G\text{-}\mathbf{Mod}$ within the framework of
triangulated categories with homological ideals by considering the
(triangulated) homotopy category $\mathrm{Ho}(G\text{-}\mathbf{Mod})$ with the
kernel of the homology functor
$\mathrm{H}_{*}\colon\mathrm{Ho}(G\text{-}\mathbf{Mod})\to(G\text{-}\mathbf{Mod})_{*}$.
The above example can then be extended to a triple at the triangulated
categories level. Furthermore, the K-theory groups induce an additive (but not
triangulated!) functor
$\operatorname{KK}^{G}\to\mathrm{Ho}(G\text{-}\mathbf{Mod})$ which relates the
above two examples.

######  Definition 5.4 (Composition of triples).

Let $\mathfrak{T}$, $\mathfrak{T}^{\prime}$ and $\mathfrak{T}^{\prime\prime}$
be categories with functors $F$, $F^{\prime}$ and $F^{\prime\prime}$ to a
common target category $\cal C$ and objects $A$, $A^{\prime}$ and
$A^{\prime\prime}$. Let $\Phi\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ and
$\Psi\colon\mathfrak{T}^{\prime\prime}\to\mathfrak{T}^{\prime}$ be functors,
with natural transformations $\alpha\colon F\circ\Phi\Rightarrow F^{\prime}$
and $\beta\colon F^{\prime}\circ\Psi\Rightarrow F^{\prime\prime}$ and
morphisms $f\colon A\to\Phi A^{\prime}$ and $g\colon A^{\prime}\to\Psi
A^{\prime\prime}$. The composition $(\Psi,\beta,g)\circ(\Phi,\alpha,f)$
consists of:

  * •

the functor
$\Phi\circ\Psi\colon\mathfrak{T}^{\prime\prime}\rightarrow\mathfrak{T}$,

  * •

the natural transformation $\beta\circ(\alpha\Psi)\colon
F\circ\Phi\circ\Psi\Rightarrow F^{\prime\prime}$,

  * •

and the morphism $\Phi(g)\circ f\colon A\to\Phi\Psi A^{\prime\prime}$.

The assignment
$(\Phi,\alpha,f)\mapsto\operatorname{App}(\Phi,\alpha,f)=\alpha_{A^{\prime}}\circ
F(f)$ is functorial with respect to this composition.

Given proper étale correspondences $\Omega\colon G\to H$ and $\Lambda\colon
H\to K$ as in Example 5.1, the composition
$(\operatorname{Ind}_{\Lambda},\alpha_{\Lambda},f_{\Lambda})\circ(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f_{\Omega})$
is equivalent to the triple
$(\operatorname{Ind}_{\Lambda\circ\Omega},\alpha_{\Lambda\circ\Omega},f_{\Lambda\circ\Omega})$
in the following sense. There is a natural isomorphism

$\operatorname{Ind}_{\Omega}\circ\operatorname{Ind}_{\Lambda}\cong\operatorname{Ind}_{\Lambda\circ\Omega}$

which identifies
$\alpha_{\Lambda}\circ(\alpha_{\Omega}\operatorname{Ind}_{\Lambda})$ with
$\alpha_{\Lambda\circ\Omega}$ and
$\operatorname{Ind}_{\Omega}(f_{\Lambda})\circ f_{\Omega}$ with
$f_{\Lambda\circ\Omega}$ [Mil23, Section 7].

######  Definition 5.5 (Equivalence of triples).

Consider categories $\mathfrak{T}$ and $\mathfrak{T}^{\prime}$ with functors
$F$ and $F^{\prime}$ to a common target category $\cal C$ and objects $A$ and
$A^{\prime}$. For $i=1,2$ let
$\Phi_{i}\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ be a functor,
$\alpha_{i}\colon F\circ\Phi_{i}\Rightarrow F^{\prime}$ a natural
transformation and $f_{i}\colon A\to\Phi_{i}A^{\prime}$ a morphism. We say the
triples $(\Phi_{1},\alpha_{1},f_{1})$ and $(\Phi_{2},\alpha_{2},f_{2})$ are
equivalent if there is a natural isomorphism $\eta\colon\Phi_{1}\cong\Phi_{2}$
such that $\alpha_{1}=\alpha_{2}\circ F\eta$ and
$\eta_{A^{\prime}}f_{1}=f_{2}$.

When we work with triangulated categories $\mathfrak{T}$ and
$\mathfrak{T}^{\prime}$ with distinguished homological ideals
$\mathfrak{I}\vartriangleleft\mathfrak{T}$ and
$\mathfrak{I}^{\prime}\vartriangleleft\mathfrak{T}^{\prime}$, we typically
want a functor $\Phi\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ to be
triangulated and map $\mathfrak{I}^{\prime}$ into $\mathfrak{I}$. It then
automatically sends $\mathfrak{I}^{\prime}$-exact sequences to
$\mathfrak{I}$-exact sequences and $\mathfrak{I}^{\prime}$-equivalences to
$\mathfrak{I}$-equivalences. For an étale correspondence $\Omega\colon G\to H$
with ideals $\mathfrak{I}_{\cal E}\vartriangleleft\operatorname{KK}^{G}$ and
$\mathfrak{I}_{\cal F}\vartriangleleft\operatorname{KK}^{H}$ defined by
families $\cal E$ and $\cal F$ of open Morita embeddings into $G$ and $H$,
$\operatorname{Ind}_{\Omega}(\mathfrak{I}_{\cal F})\subseteq\mathfrak{I}_{\cal
E}$ if $\cal E$ and $\cal F$ are compatible under $\Omega$. For details on the
construction of compatible families see Example 3.21, 3.22 and 3.24.

######  Definition 5.6 (Morphism of ABC tuples).

Given ABC tuples $\mathfrak{M}=(\mathfrak{T},\mathfrak{I},F,A)$ and
$\mathfrak{M}^{\prime}=(\mathfrak{T}^{\prime},\mathfrak{I}^{\prime},F^{\prime},A^{\prime})$,
an ABC cycle from $\mathfrak{M}$ to $\mathfrak{M}^{\prime}$ is given by a
triple $(\Phi,\alpha,f)$:

  * •

a triangulated functor
$\Phi\colon\mathfrak{T}^{\prime}\rightarrow\mathfrak{T}$ mapping
$\mathfrak{I}^{\prime}$ into $\mathfrak{I}$,

  * •

a natural transformation $\alpha\colon F\circ\Phi\Rightarrow F^{\prime}$,

  * •

and a morphism $f\colon A\rightarrow\Phi(A^{\prime})$ in $\mathfrak{T}$.

An ABC morphism $\mathfrak{m}\colon\mathfrak{M}\to\mathfrak{M}^{\prime}$ is an
equivalence class $\mathfrak{m}=[\Phi,\alpha,f]$ of ABC cycles under
equivalence of triples. We will usually work directly with ABC cycles with the
understanding that our constructions depend only on the equivalence class.

######  Example 5.7.

Let $G$ and $H$ be second countable Hausdorff étale groupoids with coefficient
$\mathrm{C}^{\ast}$-algebras $A\in\operatorname{KK}^{G}$ and
$B\in\operatorname{KK}^{H}$ and countable families $\cal E$ and $\cal F$ of
open Morita embeddings into $G$ and $H$ respectively. Let $\Omega\colon G\to
H$ be a second countable Hausdorff étale correspondence under which $\cal E$
and $\cal F$ are compatible and let
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$ be a morphism.
Then

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

is an ABC morphism. This has application
$\mathrm{K}_{*}(\Omega;f)\colon\mathrm{K}_{*}(G\ltimes
A)\to\mathrm{K}_{*}(H\ltimes B)$.

We now consider the functoriality of the localisation functors. Recall that
localisation makes sense for any complementary pair
$(\mathfrak{P},\mathfrak{N})$ of subcategories of a triangulated category
$\mathfrak{T}$. A morphism is a weak equivalence if its cone is in
$\mathfrak{N}$.

######  Lemma 5.8.

Let $\mathfrak{T}$ and $\mathfrak{T}^{\prime}$ be triangulated categories with
complementary pairs of subcategories with associated localisation functors
$L\colon\mathfrak{T}\to\mathfrak{T}$ and
$L^{\prime}\colon\mathfrak{T}^{\prime}\to\mathfrak{T}^{\prime}$ and
localisation natural transformations $\mu\colon
L\Rightarrow\operatorname{id}_{\mathfrak{T}}$ and $\mu^{\prime}\colon
L^{\prime}\Rightarrow\operatorname{id}_{\mathfrak{T}^{\prime}}$. Let
$\mu\colon L\Rightarrow\operatorname{id}_{\mathfrak{T}}$ and
$\mu^{\prime}\colon
L^{\prime}\Rightarrow\operatorname{id}_{\mathfrak{T}^{\prime}}$ denote the
localisation natural transformations and let
$\Phi\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ be a functor sending weak
equivalences to weak equivalences. Then there exists a unique natural
transformation $\mathbb{L}_{\Phi}\colon L\Phi\Rightarrow\Phi L^{\prime}$ such
that the following diagram commutes.

(5.1) ${{L\Phi}}$${{\Phi
L^{\prime}}}$${\Phi}$$\scriptstyle{\Phi\mu^{\prime}}$$\scriptstyle{\mu\Phi}$$\scriptstyle{\mathbb{L}_{\Phi}}$

For $\Phi=\operatorname{id}_{\mathfrak{T}}$ and $L^{\prime}=L$, we have
$\mathbb{L}_{\operatorname{id}_{\mathfrak{T}}}=\operatorname{id}_{L}$. If
$\mathfrak{T}^{\prime\prime}$ is another triangulated category with a pair of
complementary subcategories and a functor
$\Psi\colon\mathfrak{T}^{\prime\prime}\to\mathfrak{T}^{\prime}$ maps weak
equivalences to weak equivalences, then
$\mathbb{L}_{\Phi\circ\Psi}=(\Phi\mathbb{L}_{\Psi})\circ(\mathbb{L}_{\Phi}\Psi)$.

###### Proof.

The localisation functor $L$ maps weak equivalences to isomorphisms, so
$L\Phi\mu^{\prime}$ is invertible. Applying the naturality of $\mu$ to
$\Phi\mu^{\prime}$ this fits into a commutative square:

${{L\Phi L^{\prime}}}$${{L\Phi}}$${{\Phi
L^{\prime}}}$${\Phi}$$\scriptstyle{\Phi\mu^{\prime}}$$\scriptstyle{\mu\Phi}$$\scriptstyle{\mu\Phi
L^{\prime}}$$\scriptstyle{L\Phi\mu^{\prime}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbb{L}_{\Phi}}$

We set $\mathbb{L}_{\Phi}=(\mu\Phi L^{\prime})\circ(L\Phi\mu^{\prime})^{-1}$.
For uniqueness, note first that (5.1) determines $\mathbb{L}_{\Phi}L^{\prime}$
as $(\Phi\mu^{\prime}L^{\prime})^{-1}\circ\mu\Phi L^{\prime}$. Applying
naturality of $\mathbb{L}_{\Phi}$ to $\mu^{\prime}$ we obtain the commutative
square:

${{L\Phi L^{\prime}}}$${{L\Phi}}$${{\Phi L^{\prime}L^{\prime}}}$${\Phi
L^{\prime}}$$\scriptstyle{\Phi
L^{\prime}\mu^{\prime}}$$\scriptstyle{\mathbb{L}_{\Phi}}$$\scriptstyle{\mathbb{L}_{\Phi}L^{\prime}}$$\scriptstyle{L\Phi\mu^{\prime}}$

Invertibility of $L\Phi\mu^{\prime}$ implies that $\mathbb{L}_{\Phi}$ is
determined by $\mathbb{L}_{\Phi}L^{\prime}$ and thus unique. Compatibility
with identities and with composition follow straightforwardly from uniqueness.
∎

######  Definition 5.9.

Let $\mathfrak{T}$ and $\mathfrak{T}^{\prime}$ be triangulated categories with
homological ideals $\mathfrak{I}$ and $\mathfrak{I}^{\prime}$, functors
$F\colon\mathfrak{T}\to\mathfrak{C}$ and
$F^{\prime}\colon\mathfrak{T}^{\prime}\to\mathfrak{C}$ and objects
$A\in\mathfrak{T}$ and $A^{\prime}\in\mathfrak{T}^{\prime}$. Let
$\Phi\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ be a functor sending
$\mathfrak{I}^{\prime}$-equivalences to $\mathfrak{I}$-equivalences,
$\alpha\colon F\circ\Phi\Rightarrow F^{\prime}$ a natural transformation and
$f\colon A\to\Phi(A^{\prime})$ a morphism. The localisation application
morphism $\mathbb{L}(\Phi,\alpha,f)$ is the morphism

$\mathbb{L}(\Phi,\alpha,f)=\operatorname{App}(\Phi,\alpha,\mathbb{L}_{\Phi}(A^{\prime})\circ
L(f))\colon\mathbb{L}^{\mathfrak{I}}F(A)\to\mathbb{L}^{\mathfrak{I}^{\prime}}F^{\prime}(A^{\prime}).$

By 5.8 this is compatible with composition of triples and fits into the
following diagram.

${{\mathbb{L}^{\mathfrak{I}}F(A)}}$${{\mathbb{L}^{\mathfrak{I}^{\prime}}F^{\prime}(A^{\prime})}}$${{F(A)}}$${{F^{\prime}(A^{\prime})}}$$\scriptstyle{\mathbb{L}(\Phi,\alpha,f)}$$\scriptstyle{F^{\prime}(\mu^{\prime}_{A^{\prime}})}$$\scriptstyle{F(\mu_{A})}$$\scriptstyle{\operatorname{App}(\Phi,\alpha,f)}$

######  Example 5.10.

Let $G$ and $H$ be second countable Hausdorff étale groupoids with coefficient
$\mathrm{C}^{\ast}$-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
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$. Recall that with
respect to the complementary pairs $(\mathfrak{P}_{G},\mathfrak{N}_{G})$ and
$(\mathfrak{P}_{H},\mathfrak{N}_{H})$ the localisations are given by
topological K-theory (2.5, 2.6):

$\displaystyle\mathbb{L}\mathrm{K}_{*}(G\ltimes A)$
$\displaystyle\cong\mathrm{K}^{\mathrm{top}}_{*}(G;A)$
$\displaystyle\mathbb{L}\mathrm{K}_{*}(H\ltimes B)$
$\displaystyle\cong\mathrm{K}^{\mathrm{top}}_{*}(H;B)$

The localisation application morphism for the triple
$(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)$ gives rise to functoriality
of the topological K-theory:

$\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f):=\mathbb{L}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)\colon\mathrm{K}^{\mathrm{top}}_{*}(G;A)\to\mathrm{K}^{\mathrm{top}}_{*}(H;B)$

This is compatible with the full assembly maps. To relate it to the
Baum–Connes assembly maps (i.e. the reduced assembly maps), first consider the
proper $\mathrm{C}^{\ast}$-correspondence $\Omega\ltimes B\colon
G\ltimes\operatorname{Ind}_{\Omega}B\to H\ltimes B$ which induces
$\alpha_{\Omega}$ (see [Mil23, Section 5]). The quotient $H\ltimes B\to
H\ltimes_{r}B$ induces a reduced Hilbert $H\ltimes_{r}B$-module
$\Omega\ltimes_{r}B$. Suppose that the $*$-homomorphism
$G\ltimes\operatorname{Ind}_{\Omega}B\to\cal K(\Omega\ltimes_{r}B)$ factors
through the quotient map $G\ltimes\operatorname{Ind}_{\Omega}B\to
G\ltimes_{r}\operatorname{Ind}_{\Omega}B$. Through the resulting proper
$\mathrm{C}^{\ast}$-correspondence $\Omega\ltimes_{r}B\colon
G\ltimes_{r}\operatorname{Ind}_{\Omega}B\to H\ltimes_{r}B$ we may define a map
$\mathrm{K}^{\mathrm{red}}_{*}(\Omega;f):=\mathrm{K}_{*}(\Omega\ltimes_{r}B)\circ\mathrm{K}_{*}(G\ltimes_{r}f)$
in K-theory at the reduced level making the following diagram commute:

${{\mathrm{K}^{\mathrm{top}}_{*}(G;A)}}$${{\mathrm{K}_{*}(G\ltimes
A)}}$${{\mathrm{K}_{*}(G\ltimes_{r}A)}}$${{\mathrm{K}^{\mathrm{top}}_{*}(H;B)}}$${{\mathrm{K}_{*}(H\ltimes
B)}}$${{\mathrm{K}_{*}(H\ltimes_{r}B)}}$$\scriptstyle{\mu_{G,A}}$$\scriptstyle{\mathrm{K}_{*}(\Omega;f)}$$\scriptstyle{\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)}$$\scriptstyle{\mu_{H,B}}$$\scriptstyle{\mathrm{K}^{\mathrm{red}}_{*}(\Omega;f)}$

In particular, if $(G,A)$ and $(H,B)$ satisfy the Baum–Connes conjecture with
coefficients then $\mathrm{K}^{\mathrm{top}}_{*}(\Omega;f)$ may be identified
with $\mathrm{K}^{\mathrm{red}}_{*}(\Omega;f)$.

######  Proposition 5.11 (Derived functor maps of an ABC morphism).

Let $\mathfrak{T}$ and $\mathfrak{T}^{\prime}$ be triangulated categories with
homological ideals $\mathfrak{I}$ and $\mathfrak{I}^{\prime}$, stable
homological functors $F$ and $F^{\prime}$ into
$\operatorname{\mathbf{Ab}}_{*}$ and objects $A$ and $A^{\prime}$. Let
$\Phi\colon\mathfrak{T}^{\prime}\to\mathfrak{T}$ be an additive functor
sending $\mathfrak{I}^{\prime}$-exact sequences to $\mathfrak{I}$-exact
sequences, let $\alpha\colon F\circ\Phi\Rightarrow F^{\prime}$ be a natural
transformation and let $f\colon A\to F(A^{\prime})$ be a morphism. Then for
each $n\geq 0$ there is a homomorphism

$\mathbb{L}_{n}(\Phi,\alpha,f)\colon\mathbb{L}_{n}F(A)\to\mathbb{L}_{n}F^{\prime}(A^{\prime})$

with the following property. Let $P_{\bullet}\to A$ and
$P^{\prime}_{\bullet}\to A^{\prime}$ be projective resolutions with respect to
$\mathfrak{I}$ and $\mathfrak{I}^{\prime}$ respectively and let
$\tilde{f}\colon P_{\bullet}\to\Phi(P^{\prime}_{\bullet})$ be a chain map over
$f\colon A\to\Phi(A^{\prime})$. Let $\alpha_{P^{\prime}}\colon
F\Phi(P^{\prime}_{\bullet})\to F^{\prime}(P^{\prime}_{\bullet})$ be the chain
map induced by $\alpha$. Then the chain map $\alpha_{P^{\prime}}\circ
F(\tilde{f})\colon F(P_{\bullet})\to F^{\prime}(P^{\prime}_{\bullet})$ over
$\operatorname{App}(\Phi,\alpha,f)\colon F(A)\to F^{\prime}(A^{\prime})$
induces
$\mathbb{L}_{n}(\Phi,\alpha,f)\colon\mathbb{L}_{n}F(A)\to\mathbb{L}_{n}F^{\prime}(A^{\prime})$
by taking homology.

${\cdots}$${F(P_{1})}$${F(P_{0})}$${F(A)}$${0}$${\cdots}$${F\Phi(P^{\prime}_{1})}$${F\Phi(P^{\prime}_{0})}$${F\Phi(A^{\prime})}$${0}$${\cdots}$${F^{\prime}(P^{\prime}_{1})}$${F^{\prime}(P^{\prime}_{0})}$${F^{\prime}(A^{\prime})}$${0}$$\scriptstyle{F(\tilde{f}_{1})}$$\scriptstyle{F(\tilde{f}_{0})}$$\scriptstyle{F(f)}$$\scriptstyle{\alpha_{P^{\prime}_{1}}}$$\scriptstyle{\alpha_{P^{\prime}_{0}}}$$\scriptstyle{\alpha_{A^{\prime}}}$

Furthermore, the assignment
$(\Phi,\alpha,f)\mapsto\mathbb{L}_{n}(\Phi,\alpha,f)$ respects identities,
composition and equivalence of triples.

###### Proof.

To construct $\mathbb{L}_{n}(\Phi,\alpha,f)$, we consider arbitrary projective
resolutions $P_{\bullet}\to A$ and $P^{\prime}_{\bullet}\to A^{\prime}$ with
respect to $\mathfrak{I}$ and $\mathfrak{I}^{\prime}$ respectively. By 2.17,
there is a lift $\tilde{f}\colon P_{\bullet}\to\Phi(P^{\prime}_{\bullet})$ of
$f\colon A\to\Phi(A^{\prime})$ which is unique up to chain homotopy. We may
then define $\mathbb{L}_{n}(\Phi,\alpha,f)$ as the map in homology induced by
the chain map $\alpha_{P^{\prime}}\circ F(\tilde{f})$, which is a lift of
$\operatorname{App}(\Phi,\alpha,f)=\alpha_{A^{\prime}}\circ F(f)$.

To check that this is well-defined, suppose we have some other projective
resolutions $Q_{\bullet}\to A$ and $Q^{\prime}_{\bullet}\to A^{\prime}$ and a
lift $\hat{f}\colon Q_{\bullet}\to\Phi(Q^{\prime}_{\bullet})$ of $f$. Pick
lifts $\tilde{\operatorname{id}}\colon P_{\bullet}\to Q_{\bullet}$ and
$\tilde{\operatorname{id}}^{\prime}\colon P^{\prime}_{\bullet}\to
Q^{\prime}_{\bullet}$ of $\operatorname{id}\colon A\to A$ and
$\operatorname{id}^{\prime}\colon A^{\prime}\to A^{\prime}$. Then by 2.17,
$\tilde{\operatorname{id}}^{\prime}\circ\tilde{f}$ is chain homotopic to
$\hat{f}\circ\tilde{\operatorname{id}}$, and therefore the following diagram
commutes.

${H_{n}(F(P_{\bullet}))}$${H_{n}(F^{\prime}(P^{\prime}_{\bullet}))}$${H_{n}(F(Q_{\bullet}))}$${H_{n}(F^{\prime}(Q^{\prime}_{\bullet}))}$$\scriptstyle{H_{n}\left(\alpha_{P^{\prime}}\circ
F(\tilde{f})\right)}$$\scriptstyle{H_{n}(F(\tilde{\operatorname{id}}))}$$\scriptstyle{\cong}$$\scriptstyle{H_{n}(F^{\prime}(\tilde{\operatorname{id}}^{\prime}))}$$\scriptstyle{\cong}$$\scriptstyle{H_{n}\left(\alpha_{Q^{\prime}}\circ
F(\hat{f})\right)}$

The compatibility of $(\Phi,\alpha,f)\mapsto\mathbb{L}_{n}(\Phi,\alpha,f)$
with identities, composition and equivalence each follow straightforwardly
from well-definition. ∎

######  Remark 5.12.

If we keep the triangulated category $\mathfrak{T}$, homological ideal
$\mathfrak{I}$ and stable homological functor $F$ constant, an ABC morphism
specialises to a morphism $f\colon A\to B$ in $\mathfrak{T}$. It is
straightforward to check that
$\mathbb{L}(\operatorname{id}_{\mathfrak{T}},\operatorname{id}_{F},f)=\mathbb{L}F(f)\colon\mathbb{L}F(A)\to\mathbb{L}F(B)$
and that
$\mathbb{L}_{n}(\operatorname{id}_{\mathfrak{T}},\operatorname{id}_{F},f)=\mathbb{L}_{n}F(f)\colon\mathbb{L}_{n}F(A)\to\mathbb{L}_{n}F(B)$
for each $n\geq 0$.

A more concrete description of $\mathbb{L}_{n}(\mathfrak{m})$ depends upon a
more concrete description of the derived functors. In Section 4 we analysed
the description of derived functors as groupoid homology in the setting of an
ample groupoid (see in particular 4.6). The work in that section on
compatibility with an étale correspondence $\Omega\colon G\to H$ allows us to
describe the derived functor maps $\mathbb{L}_{n}(\mathfrak{m})$ as the
induced map in homology $\mathrm{H}_{n}(\Omega;-)$ constructed by the author
in [Mil24, Theorem 3.5]:

######  Theorem 5.13.

Let $\Omega\colon G\to H$ be a second countable Hausdorff étale correspondence
of second countable Hausdorff ample groupoids, let $A\in\operatorname{KK}^{G}$
and $B\in\operatorname{KK}^{H}$ and let
$f\in\operatorname{KK}^{G}(A,\operatorname{Ind}_{\Omega}B)$. Consider the
homological ideal
$\mathfrak{I}_{0}^{G}=\ker\operatorname{Res}^{G^{0}}_{G}\vartriangleleft\operatorname{KK}^{G}$
and the associated derived functors $\mathbb{L}_{n}$ for $n\geq 0$, and
similarly for $H$. Recall the identification
$\mathrm{H}_{n}(G;\mathrm{K}_{i}(-))\cong\mathbb{L}_{n}\mathrm{K}_{i}(G\ltimes-)$
in 4.6. For $i=0,1$ let
$\mathrm{K}_{i}(f)\colon\mathrm{K}_{i}(A)\to\operatorname{Ind}_{\Omega}\mathrm{K}_{i}(B)$
denote the induced map in K-theory with the identification
$\mathrm{K}_{*}(\operatorname{Ind}_{\Omega}B)\cong\operatorname{Ind}_{\Omega}\mathrm{K}_{*}(B)$.
Then the following diagram commutes:

${\mathrm{H}_{n}(G;\mathrm{K}_{i}(A))}$${\mathrm{H}_{n}(H;\mathrm{K}_{i}(B))}$${\mathbb{L}_{n}\mathrm{K}_{i}(G\ltimes
A)}$${\mathbb{L}_{n}\mathrm{K}_{i}(H\ltimes
B)}$$\scriptstyle{\mathrm{H}_{n}(\Omega;\mathrm{K}_{i}(f))}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\mathbb{L}_{n}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)_{i}}$

###### Proof.

For $n\geq 0$ let
$P_{n}=(\operatorname{Ind}^{G}_{G^{0}}\operatorname{Res}^{G^{0}}_{G})^{n+1}A$
and
$Q_{n}=(\operatorname{Ind}^{H}_{H^{0}}\operatorname{Res}^{H^{0}}_{H})^{n+1}B$
and consider the projective resolutions $P_{\bullet}\to A$ and $Q_{\bullet}\to
B$ induced by the adjunctions as in 2.13. Recall that
$\mathbb{L}_{n}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)$ arises from a
chain map $\tilde{f}\colon
P_{\bullet}\to\operatorname{Ind}_{\Omega}Q_{\bullet}$ over $f\colon
A\to\operatorname{Ind}_{\Omega}B$ as the map induced in homology by the
composite chain map
$\alpha_{\Omega}(Q)\circ\mathrm{K}_{i}(G\ltimes\tilde{f})$:

${\cdots}$${{\mathrm{K}_{i}(G\ltimes
P_{n})}}$${\cdots}$${{\mathrm{K}_{i}(G\ltimes
P_{0})}}$${\cdots}$${{\mathrm{K}_{i}(G\ltimes\operatorname{Ind}_{\Omega}Q_{n})}}$${\cdots}$${{\mathrm{K}_{i}(G\ltimes\operatorname{Ind}_{\Omega}Q_{0})}}$${\cdots}$${{\mathrm{K}_{i}(H\ltimes
Q_{n})}}$${\cdots}$${{\mathrm{K}_{i}(H\ltimes
Q_{0})}}$$\scriptstyle{\mathrm{K}_{i}(G\ltimes\tilde{f}_{n})}$$\scriptstyle{\alpha_{\Omega}(Q_{n})}$$\scriptstyle{\mathrm{K}_{i}(G\ltimes\tilde{f}_{0})}$$\scriptstyle{\alpha_{\Omega}(Q_{0})}$

On the module side, the modules $\mathrm{K}_{i}(P_{n})$ are left $G$-acyclic
and $\mathrm{K}_{i}(Q_{n})$ are left $H$-acylic. Furthermore, the chain
complexes $\mathrm{K}_{i}(P_{\bullet})$ and
$\mathrm{K}_{i}(\operatorname{Ind}_{\Omega}Q_{\bullet})$ are exact because
$\ker\mathrm{K}_{i}\supseteq\mathfrak{I}^{G}_{0}$ and similarly for $H$, and
so form acyclic resolutions $\mathrm{K}_{i}(P_{\bullet})\to\mathrm{K}_{i}(A)$
and $\mathrm{K}_{i}(Q_{\bullet})\to\mathrm{K}_{i}(B)$. The composite chain map
$\zeta_{\Omega,Q}^{-1}\circ\mathrm{K}_{i}(\tilde{f})$ shown below lifts
$\mathrm{K}_{i}(f)\colon\mathrm{K}_{i}(A)\to\operatorname{Ind}_{\Omega}\mathrm{K}_{i}(B)$,
noting that $\mathrm{K}_{i}(f)$ is really shorthand for
$\zeta_{\Omega,B}^{-1}\circ\mathrm{K}_{i}(f)$.

${\cdots}$${{\mathrm{K}_{i}(P_{n})}}$${\cdots}$${{\mathrm{K}_{i}(P_{0})}}$${\cdots}$${{\mathrm{K}_{i}(\operatorname{Ind}_{\Omega}Q_{n})}}$${\cdots}$${{\mathrm{K}_{i}(\operatorname{Ind}_{\Omega}Q_{0})}}$${\cdots}$${{\operatorname{Ind}_{\Omega}\mathrm{K}_{i}(Q_{n})}}$${\cdots}$${{\operatorname{Ind}_{\Omega}\mathrm{K}_{i}(Q_{0})}}$$\scriptstyle{\mathrm{K}_{i}(\tilde{f}_{n})}$$\scriptstyle{\mathrm{K}_{i}(\tilde{f}_{0})}$$\scriptstyle{\zeta_{\Omega,Q_{n}}^{-1}}$$\scriptstyle{\zeta_{\Omega,Q_{0}}^{-1}}$

Therefore by [Mil24, Theorem 3.5], the map
$\mathrm{H}_{*}(\Omega;\mathrm{K}_{i}(f))$ is induced by the chain map
$(\delta_{\Omega}\otimes\operatorname{id})\circ(\zeta_{\Omega,Q_{n}}^{-1}\circ\mathrm{K}_{i}(\tilde{f}_{n}))_{G}$.
By 4.4 the diagram

${\mathrm{K}_{i}(P_{n})_{G}}$${\mathrm{K}_{i}(Q_{n})_{H}}$${\mathrm{K}_{i}(G\ltimes
P_{n})}$${\mathrm{K}_{i}(H\ltimes
Q_{n})}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{P_{n}}}$$\scriptstyle{(\delta_{\Omega}\otimes\operatorname{id})\circ(\zeta_{\Omega,Q_{n}}^{-1}\circ\mathrm{K}_{i}(\tilde{f}_{n}))_{G}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{Q_{n}}}$$\scriptstyle{\alpha_{\Omega}(Q_{n})\circ\mathrm{K}_{i}(G\ltimes\tilde{f}_{n})}$

commutes for each $n\geq 0$. On the level of homology the vertical chain maps
induce the identification in 4.6, and the horizontal chain maps induce
$\mathrm{H}_{*}(\Omega;\mathrm{K}_{i}(f))$ and
$\mathbb{L}_{*}(\operatorname{Ind}_{\Omega},\alpha_{\Omega},f)_{i}$
respectively, so we are done. ∎

##  6\. Functoriality of the ABC spectral sequence

In this section we prove the following functorial version of Theorem 2.19.

######  Theorem 6.1.

An ABC morphism $\mathfrak{m}\colon\mathfrak{M}\to\mathfrak{M}^{\prime}$
induces functorially 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 sheet 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 sheet agrees with
$\mathbb{L}(\mathfrak{m})\colon\mathbb{L}(\mathfrak{M})\to\mathbb{L}(\mathfrak{M}^{\prime})$.

We first review Meyer’s construction of the ABC spectral sequence associated
to an ABC tuple $(\mathfrak{T},\mathfrak{I},F,A)$ from [Mey08, Section 4].
This starts by picking an $\mathfrak{I}$-projective resolution over $A$ and
embedding this in a phantom tower. Note the convention that a degree one map
$f\colon A\to\Sigma B$ is written $f\colon A\mathrel{\leavevmode\hbox
to10.07pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
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