PARTIAL ACTIONS OF FREE GROUPS AND GROUPOID HOMOLOGY

BENJAMIN STEINBERG

doi:10.1017/S1446788726101670

PARTIAL ACTIONS OF FREE GROUPS AND GROUPOID HOMOLOGY

Part of: Homological methods Topological and differentiable algebraic systems Connections with homological algebra and category theory

Published online by Cambridge University Press: 04 September 2026

BENJAMIN STEINBERG

Show author details

BENJAMIN STEINBERG*: Affiliation:
Department of Mathematics, City College of New York , Convent Avenue at 138th Street, New York, NY 10031, USA
*: e-mail: bsteinberg@ccny.cuny.edu

Article contents

Abstract
Introduction
Partial group actions
Groupoid algebras and homology
Cohomological dimension and semi-saturated partial actions
Global dimension
Footnotes
References

Rights & Permissions

Abstract

We give a length-one projective resolution of the trivial module for the groupoid of a semi-saturated partial action (in the sense of Exel) of a free group on a compact Hausdorff and totally disconnected space. As a consequence, we obtain an elementary computation of the homology of these groupoids, which include transformation groupoids of free group actions and Deaconu–Renault groupoids of systems $(X,T)$, where $X$ is compact Hausdorff and totally disconnected, and $T$ is a local homeomorphism with domain a clopen subset of $X$. We also show that algebra of such a partial action groupoid over a field has global dimension at most $2$ when the space is second countable.

Keywords

partial group actions groupoid homology global dimension free groups

MSC classification

Primary: 20J05: Homological methods in group theory 22A22: Topological groupoids (including differentiable and Lie groupoids)

Secondary: 16E10: Homological dimension

Information

Type: Research Article
Information: Journal of the Australian Mathematical Society , First View , pp. 1 - 16

DOI: https://doi.org/10.1017/S1446788726101670 [Opens in a new window]
Creative Commons: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Given a collection of partial homeomorphisms $\{\theta _a\colon X_{a^{-1}}\to X_a\mid a\in A\}$ of a space $X$ , Exel [Reference Exel8] defines a partial action of the free group $F_A$ on $X$ by sending a reduced word to the corresponding composition of partial homeomorphisms; such partial actions are called semi-saturated. Associated to a partial group action of a group $G$ on a space $X$ is an étale groupoid, denoted $G\ltimes X$ [Reference Abadie1]. In this paper, we give an explicit length-one projective resolution of the trivial module $\mathbb ZX$ for $F_A\ltimes X$ when $X$ is compact Hausdorff and totally disconnected, and the partial action is semi-saturated; this resolution is flat when $X$ is merely locally compact Hausdorff and totally disconnected. As a consequence, we obtain a completely elementary computation of the homology of such a groupoid. Such groupoids include all groupoids of Deaconu–Renault systems $(X,T)$ with $X$ compact Hausdorff and totally disconnected, and with the domain of $T$ clopen by an argument of G. de Castro.

More precisely, our main results are the following three theorems.

Theorem A. Let the free group $F_A$ have a semi-saturated partial action on a compact Hausdorff totally disconnected space $X$ . Let $\mathcal G=F_A\ltimes X$ . Then, there is a projective resolution

$$ \begin{align*}0\longrightarrow \bigoplus_{a\in A}\mathbb Z\mathcal G1_{X_a}\xrightarrow{\,\partial\,} \mathbb Z\mathcal G\xrightarrow{{\boldsymbol r}_\ast}\mathbb ZX\longrightarrow 0,\end{align*} $$ 0 long right arrow circled plus Underscript a element of upper A Endscripts double struck upper Z script upper G 1 Subscript upper X Sub Subscript a Subscript Baseline ModifyingAbove right arrow With partial differential double struck upper Z script upper G right arrow Overscript bold italic r Subscript asterisk Baseline Endscripts double struck upper Z upper X long right arrow 0 comma

where $\partial $ is defined on the $a$ -component by $\partial (f) = f\delta _{a}-f$ , where $\delta _a$ is the characteristic function of $\{a\}\times X_{a^{-1}}$

Theorem B. Let $\theta $ be a semi-saturated partial action of a free group $F_A$ on a compact Hausdorff totally disconnected space $X$ . Let $\iota _a\colon \mathbb ZX_a\to \mathbb ZX$ be extension by $0$ and $\theta ^*_a\colon \mathbb ZX_a\to \mathbb ZX$ be $f\mapsto f\circ \theta _{a}$ (extended by $0$ outside $X_{a^{-1}}$ ). Then, $H_n(F_A\ltimes X)=0$ for $n\geq 2$ and

$$ \begin{align*} H_0(F_A\ltimes X) &= \mathop{\mathrm{coker}}\bigg(\bigoplus_{a\in A} (\iota_a-\theta_a^*)\bigg),\\ H_1(F_A\ltimes X) &= \ker \bigg(\bigoplus_{a\in A} (\iota_a-\theta_a^*)\bigg). \end{align*} $$

StartLayout 1st Row 1st Column upper H 0 left parenthesis upper F Subscript upper A Baseline left normal factor semidirect product upper X right parenthesis 2nd Column equals coker left parenthesis circled plus Underscript a element of upper A Endscripts left parenthesis iota Subscript a Baseline minus theta Subscript a Superscript asterisk Baseline right parenthesis right parenthesis comma 2nd Row 1st Column upper H 1 left parenthesis upper F Subscript upper A Baseline left normal factor semidirect product upper X right parenthesis 2nd Column equals kernel left parenthesis circled plus Underscript a element of upper A Endscripts left parenthesis iota Subscript a Baseline minus theta Subscript a Superscript asterisk Baseline right parenthesis right parenthesis period EndLayout

This leads to a computation of the homology of Deaconu–Renault groupoids without assuming the local homeomorphism is surjective or totally defined (but with restrictions on $X$ ). See Corollary 4.8 and Remark 4.9.

Theorem C. Let the free group $F_A$ have a semi-saturated partial action on a second countable compact Hausdorff and totally disconnected space $X$ . Then, for any field $K$ , the algebra $K[F_A\ltimes X]\cong KX\rtimes F_A$ has global dimension at most $2$ .

Theorem C is derived from Theorem A using a spectral sequence connecting ${\mathrm {Ext}}$ with groupoid cohomology that may be of interest in its own right.

2 Partial group actions

2.1 Definition of a partial action

See [Reference Kellendonk and Lawson12] or [Reference Exel8] for basics on partial group actions.

Recall that an inverse semigroup is a semigroup $S$ such that, for each $s\in S$ , there is a unique $s^*\in S$ such that $ss^*s=s$ and $s^*ss^*=s^*$ . If $S,T$ are inverse monoids, a dual prehomomorphism $\theta \colon S\to T$ is a map such that:

(1) $\theta (1)=1$ ;
(2) $\theta (s^*) = \theta (s)^*$ ;
(3) $\theta (s)\theta (t)\leq \theta (st)$ .

Here, we recall that an inverse semigroup is partially ordered via $s\leq t$ if $ss^*t=s$ or, equivalently, $ts^*s=s$ .

If $X$ is a locally compact Hausdorff totally disconnected space, $I_X$ denotes the inverse semigroup of partial homeomorphisms of $X$ with clopen domains.

A partial action of a group $G$ on $X$ is a dual prehomorphism $\theta \colon G\to I_X$ . What this entails is, for each $g\in G$ , a homeomorphism $\theta _g\colon X_{g^{-1}}\to X_g$ between clopen subsets of $X$ such that $\theta _1=1_X$ , $\theta _g^{-1}=\theta _{g^{-1}}$ and, whenever $\theta _g\theta _h(x)$ is defined, it equals $\theta _{gh}(x)$ .

Let $F_A$ be a free group on $A$ . Exel [Reference Exel8] calls a partial action $\theta $ of $F_A$ on $X$ semi-saturated if $\theta _{uv}=\theta _u\circ \theta _v$ whenever $|uv|=|u|+|v|$ , that is, there is no cancellation in forming the product $uv$ . Semi-saturated actions are determined uniquely by a collection of partial homeomorphisms $\theta _a\colon X_{a^{-1}}\to X_a$ for $a\in A$ . The action for an element with reduced form $w=a_1^{\epsilon _1}\cdots a_n^{\epsilon _n}$ must be given by $\theta _w=\theta _{a_1}^{\epsilon _1}\cdots \theta _{a_n}^{\epsilon _n}$ . See [Reference Exel8, Proposition 4.10] for details.

One way to see this is to note that this collection gives us a homomorphism $\alpha \colon FIM_A\to I_X$ , where $FIM_A$ is the free inverse monoid on $A$ , sending $a$ to $\theta _a$ . However, there is a well-known dual prehomomorphism $\gamma \colon F_A\to FIM_A$ sending an element $w\in F_A$ to the element of $FIM_A$ represented by the reduced form of $w$ , that is, the element with Munn tree [Reference Lawson13] a straight-line labeled by $w$ . Then, the above partial action is $\alpha \gamma $ .

Elementary properties of semi-saturated actions that we use throughout, often without comment, are the following.

Proposition 2.1. Suppose that $\theta \colon F_A\to I_X$ is a semi-saturated action.

(1) If $w=uv$ with $|w|=|u|+|v|$ , then $X_w\subseteq X_u$ .
(2) We have $\theta _u(X_{u^{-1}} \cap X_v)\subseteq X_{uv}$ with equality if $|uv|=|u|+|v|$ .

Proof. This first item is immediate as $\theta _w = \theta _u\circ \theta _v$ . For the second, if $x\in X_{u^{-1}}\cap X_v$ , write $x=\theta _v(y)$ . Then, $\theta _u(x) = \theta _u(\theta _v(y))=\theta _{uv}(y)$ and so $\theta _u(x)\in X_{uv}$ . If $|uv|=|u|+|v|$ , then $\theta _{uv} = \theta _u\circ \theta _v$ . Thus, if $z\in X_{uv}$ , then $z=\theta _{uv}(y) = \theta _u(\theta _v(y))$ . Therefore, $\theta _v(y)\in X_{u^{-1}}\cap X_v$ and so $z\in \theta _u(X_{u^{-1}}\cap X_v)$ .

Any total action of $F_A$ is, of course, a semi-saturated partial action.

Important examples of semi-saturated partial actions come from graphs. We follow here the southern-hemisphere convention on graphs: so an edge is drawn ${\boldsymbol r}(e)\xleftarrow {\,e\,}\mathop {\boldsymbol s}(e)$ . Let $E=(E^0,E^1)$ be a graph and let $\partial E$ be the boundary path space. For each edge $e$ , put $X_e = e\partial E$ and $X_{e^{-1}} =\mathop {\boldsymbol s}(e)\partial E$ . Then, $\theta _e(z) = ez$ . For the corresponding semi-saturated action of $F_{E^1}$ , one can check that only reduced words of the form $pq^{-1}$ , where $p,q$ are paths with $\mathop {\boldsymbol s}(p)=\mathop {\boldsymbol s}(q)$ have nonempty domain. This element has domain $q\partial E$ and range $p\partial E$ .

2.2 The groupoid of a partial group action

A topological groupoid $\mathcal G$ is étale if the source map is a local homeomorphism and the unit (or object) space $\mathcal G^0$ is locally compact Hausdorff. If, in addition, $\mathcal G^0$ is totally disconnected, the groupoid is said to be ample. Recall that a bisection of $\mathcal { G}$ is an open subset $U$ on which the source and range maps are both injective. The compact open bisections of an ample groupoid form a basis for the topology and an inverse semigroup under multiplication.

Abadie [Reference Abadie1] associated a Hausdorff étale groupoid $G\ltimes X$ to a partial group action of $G$ on $X$ . The object space is $X$ and the arrow space is $\{(g,x)\mid x\in X_{g^{-1}}\}\subseteq G\times X$ with the subspace topology coming from the product topology. One has $\mathop {\boldsymbol s}(g,x)=x$ , ${\boldsymbol r}(g,x)=\theta _g(x)$ , $(g,\theta _h(x))(h,x) = (gh,x)$ and $(g,x)^{-1} = (g^{-1},\theta _g(x))$ . The identities are $(1,x)$ with $x\in X$ .

If $E$ is a graph and we consider the semi-saturated partial action of $F_{E^1}$ on $\partial E$ , then $F_{E^1}\ltimes \partial E$ is well known to be isomorphic to the usual graph groupoid $\mathcal G_E$ via $(pq^{-1}, qz)\mapsto (pz, |p|-|q|, qz)$ .

More generally, the groupoid of any Deaconu–Renault system $(X,T)$ with $X$ compact Hausdorff and totally disconnected, and with the domain of $T$ clopen is the groupoid of a semi-saturated partial action of a free group. Recall that a Deaconu–Renault system consists of a locally compact Hausdorff space $X$ and a local homeomorphism $T\colon {\mathrm {dom}}(T)\to X$ , where ${\mathrm {dom}}(T)$ is an open subset of $X$ . The Deaconu–Renault groupoid $\mathcal G_{(X,T)}$ has object space $X$ and arrows

$$ \begin{align*}\{(x,m-n,y)\in {\mathrm{dom}}(T^m)\times \mathbb Z\times {\mathrm{dom}}(T^n)\mid T^m(x)=T^n(y)\}\end{align*} $$

with $\mathop {\boldsymbol s}(x,k,y)=y$ , ${\boldsymbol r}(x,k,y)=x$ . The topology has basis the bisections of the form

$$ \begin{align*}(U,(m,n),V) = \{(x,m-n,y)\in U\times \mathbb Z\times V\mid T^m(x)=T^n(y)\}\end{align*} $$

with $U\subseteq {\mathrm {dom}}(T^m)$ , $V\subseteq {\mathrm {dom}}(T^n)$ open, $T|_U,T|_V$ injective, and $T^m(U)=T^n(V)$ . The graph groupoid is the Deaconu–Renault system of the shift map on the boundary path space $\partial E$ .

A partial action of a free group $F_A$ on a space $X$ is called orthogonal if $X_a\cap X_b=\emptyset $ for $a\neq b\in A$ [Reference Exel and Steinberg9]. The following argument was given to me by Gilles de Castro (private communication).

Theorem 2.2. The groupoids of Deaconu–Renault systems $(X,T)$ with $X$ compact Hausdorff and totally disconnected, and ${\mathrm {dom}}(T)$ clopen are precisely the groupoids of semi-saturated orthogonal partial actions of finitely generated free groups on compact Hausdorff and totally disconnected spaces with clopen domains.

Proof. If $A$ is finite, and $\theta $ is an orthogonal and semi-saturated partial action of $F_A$ on a compact Hausdorff and totally disconnected $X$ , then we can define a local homeomorphism with clopen domain $T\colon \bigcup _{a\in A} X_a\to X$ by $T|_{X_a} = \theta _{a^{-1}}$ , using orthogonality of the action. Then, [Reference Exel and Steinberg9, Theorem 9.6] shows that $F_A\ltimes X\cong \mathcal G_{(X,T)}$ (see also [Reference de Castro and Kang4, Theorem 4.1]).

Conversely, if $(X,T)$ is a Deaconu–Renault system with the domain ${\mathrm {dom}}(T)$ of $T$ clopen (hence, compact), then we can partition ${\mathrm {dom}}(T)$ into finitely many clopen sets $U_1\sqcup \cdots \sqcup U_n$ with $T|_{U_i}$ a homeomorphism with its image $V_i$ . We can then define a semi-saturated orthogonal partial action of the free group on $a_1,\ldots , a_n$ on $X$ by putting $X_{a_i}=U_i$ , $X_{a_i^{-1}} = V_i$ , and $\theta _{a_i} = (T|_{U_i})^{-1}$ using [Reference Exel8, Proposition 4.10]. Note then that $T=\bigcup _{i=1}^n\theta _{a_i^{-1}}$ and so $F_A\ltimes X\cong \mathcal G_{(X,T)}$ by the previous paragraph.

We now show that groupoids $F_A\ltimes X$ of semi-saturated partial actions are exactly the groupoids of germs $FIM_A\ltimes X$ for actions of the free inverse monoid $FIM_A$ on $A$ . See [Reference Lawson13] for more on the free inverse monoid and [Reference Exel7] for groupoids of germs.

Proposition 2.3. Let $\{\theta _a\colon X_{a^{-1}}\to X_A\colon a\in A\}$ be a collection of homeomorphisms of a locally compact Hausdorff totally disconnected space $X$ with the $X_a$ clopen. Then, $F_A\ltimes X\cong FIM_A\ltimes X$ , where the groupoid of germs $FIM_A\ltimes X$ is taken with respect to the action of $FIM_A$ on $X$ given by $a\mapsto \theta _a$ .

Proof. The key fact that we use is that $FIM_A$ is an $F$ -inverse monoid. More precisely, if $\sigma \colon FIM_A\to F_A$ is the canonical projection and if $w\in F_A$ is viewed as a reduced word over $A$ , then $w$ evaluated in $FIM_A$ is the maximum element of $\sigma ^{-1}(w)$ . Thus, each germ of $FIM_A\ltimes X$ is uniquely represented by a germ $[w,x]$ with $w$ reduced since if $[u,x]=[v,x]$ , then $\sigma (u)=\sigma (v)$ . Moreover, basic neighborhoods in $FIM_A\ltimes X$ can be taken to be of the form $D(w,U) =\{[w,u]\mid u\in U\}$ , where $w$ is a reduced word and $U$ is an open subset of $X_{w^{-1}}$ . The actions of a reduced word viewed as an element of $F_A$ and as an element of $FIM_A$ are the same. It follows directly that $(w,x)\mapsto [w,x]$ is an isomorphism of topological groupoids $F_A\ltimes X\to FIM_A\ltimes X$ .

3 Groupoid algebras and homology

3.1 Groupoid algebras

If $K$ is a commutative ring, and $X$ is a locally compact Hausdorff and totally disconnected space, then $KX$ denotes the $K$ -module of locally constant functions $f\colon X\to K$ with compact support. If $\mathcal G$ is a Hausdorff ample groupoid, then $K\mathcal G$ becomes a $K$ -algebra [Reference Steinberg19] with respect to the convolution product

$$ \begin{align*}f\ast g(\gamma) = \sum_{{\boldsymbol r}(\alpha)={\boldsymbol r}(\gamma)}f(\alpha)g(\alpha^{-1} \gamma).\end{align*} $$ StartLayout 1st Row f asterisk g left parenthesis gamma right parenthesis equals sigma summation Underscript bold italic r left parenthesis alpha right parenthesis equals bold italic r left parenthesis gamma right parenthesis Endscripts f left parenthesis alpha right parenthesis g left parenthesis alpha Superscript negative 1 Baseline gamma right parenthesis period EndLayout

Often, we write the convolution product as juxtaposition.

In the case of $G\ltimes X$ , one has that $K[G\ltimes X]$ is isomorphic to the partial skew group ring $KX\rtimes G$ , where $KX$ is given the pointwise product; see [Reference Dokuchaev and Exel5] for more on partial skew group rings.

To simplify the discussion, we assume from now on that $X$ is compact and, hence, each $X_g$ is compact. Let $\delta _g$ be the characteristic function of the compact open bisection $\{g\}\times X_{g^{-1}}$ . Then, the clopen partition $G\ltimes X=\bigsqcup _{g\in G} \{g\}\times X_{g^{-1}}$ leads to a $K$ -module direct sum decomposition

$$ \begin{align*}K[G\ltimes X] = \bigoplus KX_g\delta_g.\end{align*} $$

That is, if $f$ is supported on $\{g\}\times X_{g^{-1}}$ , we can write $f= (f\delta _{g^{-1}})\delta _g$ and $f\delta _{g^{-1}}$ is in $KX_g$ . In particular, in the direct sum decomposition, $\delta _g = 1_{X_g}\delta _g$ .

The reader can check that if $\alpha _g\colon KX_{g^{-1}}\to KX_g$ is defined by $\alpha _g(f) = f\circ \theta _{g^{-1}}$ , then

$$ \begin{align*}f\delta_g\ast f'\delta_g' = \alpha_g(\alpha_{g^{-1}}(f)f')\delta_{gg'}, \end{align*} $$

although we do not use this formula much.

Proposition 3.1. Let $F_A$ have a semi-saturated partial action on $X$ . Let $u,v\in F_A$ . Then,

$$ \begin{align*}(\{u\}\times X_{u^{-1}})(\{v\}\times X_{v^{-1}})\subseteq \{uv\}\times X_{(uv)^{-1}}\end{align*} $$

with equality if $|uv|=|u|+|v|$ . Consequently, if $|uv|=|u|+|v|$ , then $\delta _u\delta _v = \delta _{uv}$ .

Proof. The inclusion is immediate from the definition of a partial action. If ${|u|+|v|=|uv|}$ and $x\in X_{(uv)^{-1}}$ , then $\theta _{uv}(x) = \theta _u(\theta _v(x))$ by the semi-saturated property and so $(uv,x) = (u,\theta _v(x))(v,x)\in (\{u\}\times X_{u^{-1}})(\{v\}\times X_{v^{-1}})$ .

3.2 Homology and cohomology of ample groupoids

Let $\mathcal G$ be a Hausdorff ample groupoid. Then, $K\mathcal G^0$ is a left $K\mathcal G$ -module via

$$ \begin{align*}f\cdot g(x) = \sum_{{\boldsymbol r}(\gamma)=x}f(\gamma)g(\mathop{\boldsymbol s}(\gamma)).\end{align*} $$ StartLayout 1st Row f dot g left parenthesis x right parenthesis equals sigma summation Underscript bold italic r left parenthesis gamma right parenthesis equals x Endscripts f left parenthesis gamma right parenthesis g left parenthesis bold italic s left parenthesis gamma right parenthesis right parenthesis period EndLayout

We call this the trivial left $K\mathcal G$ -module. If $U$ is a compact open bisection, then ${1_U\cdot g=1_U\ast g\ast 1_{U^{-1}}}$ . The action restricts to $K\mathcal G_0$ as left multiplication. The trivial right module is defined dually.

The homology of $\mathcal G$ with coefficients in a right $\mathbb Z\mathcal G$ -module $M$ is

$$ \begin{align*}H_n(\mathcal G,M) = {\mathrm{Tor}}_n^{\mathbb Z\mathcal G}(M,\mathbb Z\mathcal G^0)\end{align*} $$

and the cohomology of $\mathcal G$ in a left $\mathbb Z\mathcal G$ -module $M$ is given by

$$ \begin{align*}H^n(\mathcal G,M) = {\mathrm{Ext}}^n_{\mathbb Z\mathcal G}(\mathbb Z\mathcal G^0,M).\end{align*} $$

One puts $H_n(\mathcal G) = H_n(\mathcal G,\mathbb Z\mathcal G^0)$ and $H^n(\mathcal G) = H^n(\mathcal G,\mathbb Z\mathcal G^0)$ . A good reference for groupoid homology is [Reference Bönicke, Dell’Aiera, Gabe and Willett3].

The cohomological dimension of $\mathcal G$ is the shortest length of a projective resolution of $\mathbb Z\mathcal G^0$ or, equivalently, the least $n$ such that $H^{n+1}(\mathcal G,M)=0$ for all $\mathbb Z\mathcal G$ -modules $M$ .

The famous Stallings–Swan theorem says that free groups are precisely the groups of cohomological dimension $1$ . We show that semi-saturated actions of free groups on compact Hausdorff totally disconnected spaces lead to groupoids of cohomological dimension one.

If $f\colon X\to Y$ is a local homeomorphism of locally compact totally disconnected Hausdorff spaces, we have an induced homomorphism $f_\ast \colon \mathbb ZX\to \mathbb ZY$ given by $f_*(g)(y) = \sum _{f(x)=y}g(x)$ .

It is well known that ${\boldsymbol r}_\ast \colon \mathbb Z\mathcal G\to \mathbb Z\mathcal G^0$ is a $\mathbb Z\mathcal G$ -module homomorphism. If $U$ is a compact open bisection, then ${\boldsymbol r}_\ast (1_U) = 1_{{\boldsymbol r}(U)}$ .

4 Cohomological dimension and semi-saturated partial actions

For this section, we fix a semi-saturated partial action $\theta $ of a free group $F_A$ on a compact totally disconnected Hausdorff space $X$ . We put $\mathcal G=F_A\ltimes X$ . Our goal is to give a length-one projective resolution of $\mathbb Z \mathcal G^0=\mathbb ZX$ .

We begin our projective resolution with ${\boldsymbol r}_\ast \colon \mathbb Z\mathcal G\to \mathbb ZX$ . It is convenient to understand this map with respect to our direct sum decomposition $\mathbb Z\mathcal G$ ${=}\ \bigoplus _{w\in F_A}\mathbb ZX_w\delta _w$ .

Proposition 4.1. Suppose that $f\in \mathbb ZX_w$ . Then, ${\boldsymbol r}_\ast (f\delta _w)=f$ .

Proof. Indeed, ${\boldsymbol r}_\ast (f\delta _w) = f{\boldsymbol r}_\ast (\delta _w) = f1_{X_w}= f$ using that ${\boldsymbol r}_\ast $ is a module homomorphism and $\delta _w = 1_{\{w\}\times X_{w^{-1}}}$ .

Set $P_0=\mathbb Z\mathcal G$ . Our next projective module is $P_1=\bigoplus _{a\in A}\mathbb Z\mathcal G1_{X_a}\gamma _a$ , where $\gamma _a$ is just a symbol to indicate to which summand an element belongs.

Proposition 4.2. One has $\mathbb ZX_w\delta _w1_{X_a} = \mathbb Z[\theta _w(X_{w^{-1}} \cap X_a)]\delta _w$ . Hence, $\mathbb Z\mathcal G1_{X_a}=\bigoplus _{w\in F_X}\mathbb Z [\theta _w(X_w^{-1} \cap X_a)]\delta _w$ .

Proof. Note that $(\{w\}\times X_{w^{-1}})(\{1\}\times X_a)$ consists of all $(w,x)$ with $x\in X_{w^{-1}}\cap X_a$ , as does $(\{1\}\times \theta _w(X_{w^{-1}}\cap X_a))(\{w\}\times X_{w^{-1}})$ . Thus, $\delta _w 1_{X_a} = 1_{\theta _w(X_{w^{-1}}\cap X_a)}\delta _w$ , from which the proposition follows.

We think of $f\delta _w\gamma _a$ as the edge $w\xrightarrow {a}wa$ in the Cayley graph of $F_A$ with weight $f$ . With this intuition, we define $\partial \colon P_1\to P_0$ by

$$ \begin{align*}\partial (f\delta_w\gamma_a) = f\delta_{wa}-f\delta_w.\end{align*} $$

Note that Propositions 4.2 and 2.1(2) imply that $f\in \mathbb ZX_{wa}$ . Therefore, ${\boldsymbol r}_\ast \partial (f\delta _w\gamma _a) = f-f=0$ by Proposition 4.1.

Proposition 4.3. For $f\in \mathbb Z\mathcal G1_{X_a}$ , one has $\partial (f\gamma _a)=f\delta _a-f$ . Consequently, $\partial $ is a $\mathbb Z\mathcal G$ -module homomorphism.

Proof. One computes $\delta _w\delta _a = 1_{\theta _w(X_{w^{-1}}\cap X_a)}\delta _{wa}$ . So, if $h\delta _w\gamma _a\in \mathbb Z\mathcal G1_{X_a}$ , then $h\delta _{wa} = h1_{\theta _w(X_{w^{-1}}\cap X_a)}\delta _{wa}=h\delta _w\delta _a$ as a consequence of Proposition 4.2, and so $\partial (h\delta _w\gamma _a) = h\delta _{wa}-h\delta _w=h\delta _w\delta _a-h\delta _w$ , as required.

We now show that the chain complex $0\to P_1\xrightarrow {\partial }P_0\xrightarrow {{\boldsymbol r}_\ast }\mathbb ZX\to 0$ is acyclic by constructing a contracting homotopy over $\mathbb Z$ , based on the corresponding contracting homotopy for the special case that $X$ is a singleton, that is, for the projective resolution of the trivial module for the free group,

$$ \begin{align*}0\to \bigoplus_{a\in A} \mathbb ZF_A\gamma_a\to \mathbb ZF_A\to \mathbb Z\to 0.\end{align*} $$

In that contracting homotopy, $s_1\colon \mathbb ZF_A\to \bigoplus _{a\in A}\mathbb ZF_A\gamma _a$ sends a word $w$ to the signed sum of the edges in the Cayley graph of $F_A$ on the geodesic path from $1$ to $w$ , where we view $g\gamma _a$ as the edge $g\xrightarrow {a}ga$ . We emulate this contracting homotopy in our context. The semi-saturated condition ensures that if $f\in X_w$ , then $f\in X_u$ for any prefix $u$ of $w$ by Proposition 2.1, which allows us to perform this homotopy.

First, define $s_0\colon \mathbb ZX\to \mathbb Z\mathcal G=P_0$ by $s_0(f) = f\delta _1$ . The definition of $s_1\colon P_0\to P_1$ is more involved, as we must encode signed traversals of edges.

Let $a\in A$ and $w\in F_A$ . If $f\in X_{wa}$ and $|wa|=|w|+1$ , let us put

$$ \begin{align*}f[w,a] = f\delta_{w}\gamma_a.\end{align*} $$

If $f\in X_{wa^{-1}}$ with $|wa^{-1}| = |w|+1$ , then we put

$$ \begin{align*}f[w,a^{-1}] = -f\delta_{wa^{-1}}\gamma_a.\end{align*} $$

(Imagine traversing the edge $wa^{-1}\xrightarrow {a}w$ in the negative direction.)

Lemma 4.4. Suppose that $|wa^{\epsilon }| =|w|+1$ for $a\in A$ , $\epsilon =\pm 1$ , and $f\in X_{wa^\epsilon }$ :

(1) $f[w,a^\epsilon ]\in \mathbb Z\mathcal G1_{X_a}\gamma _a$ ;
(2) $\partial (f[w,a^\epsilon ]) = f\delta _{wa^\epsilon } - f\delta _w$ .

Proof. We first prove the two items when $\epsilon =1$ . Proposition 2.1(2) implies that ${f\in \mathbb Z[\theta _w(X_{w^{-1}}\cap X_a)]}$ . Moreover, $\partial (f[w,a]) = \partial (f\delta _w\gamma _a) = f\delta _{wa}-f\delta _w$ .

Suppose now that $\epsilon =-1$ . Then, $f\delta _{wa^{-1}} =f\delta _w\delta _{a^{-1}} = f\delta _w\delta _{a^{-1}}1_{X_a}\in \mathbb Z\mathcal G1_{X_a}$ , where the first equality is from Proposition 3.1. Furthermore,

$$ \begin{align*}\partial (f[w,a^{-1}]) =- \partial (f\delta_{wa^{-1}}\gamma_a) = -(f\delta_{w}-f\delta_{wa^{-1}}) = f\delta_{wa^{-1}} - f\delta_w,\end{align*} $$

as required.

We now define $s_1\colon P_0\to P_1$ as follows. Suppose that $w=a_1^{\epsilon _1}\cdots a_n^{\epsilon _n}$ in reduced form with $a_i\in A$ , $\epsilon _i\in \{\pm 1\}$ , and $n\geq 0$ . Set, for $f\in \mathbb ZX_w$ ,

$$ \begin{align*}s_1(f\delta_w) = \sum_{i=1}^n f[a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}},a_i^{\epsilon_i}],\end{align*} $$

where we note that by Proposition 2.1(1), we have that $f\in \mathbb ZX_{a_1^{\epsilon _1}\cdots a_i^{\epsilon _i}}$ for all $0\leq i\leq n$ since $f\in \mathbb ZX_w$ . Observe that if $w=1$ , then $s_1(f\delta _1)=0$ since the sum is empty.

Proposition 4.5. The maps $s_0,s_1$ form a contracting homotopy of the chain complex of abelian groups $0\to P_1\to P_0\to \mathbb ZX\to 0$ .

Proof. Clearly, $s_0,s_1$ are homomorphisms of abelian groups. To show that they form a contracting homotopy of abelian groups, we need to verify: ${\boldsymbol r}_\ast s_0=1$ ; $\partial s_1+s_0{\boldsymbol r}_\ast =1$ ; and $s_1\partial = 1$ . We begin with the first property.

If $f\in \mathbb ZX$ , then ${\boldsymbol r}_\ast s_0(f) = {\boldsymbol r}_\ast (f\delta _1) = f$ by Proposition 4.1.

Next, we turn to the second property: $\partial s_1+s_0{\boldsymbol r}_\ast =1$ . We apply the left-hand side to $f\delta _w$ with $f\in \mathbb ZX_w$ . Suppose that $w=a_1^{\epsilon _1}\cdots a_n^{\epsilon _n}$ in reduced form with $n\geq 0$ . Note that $s_0{\boldsymbol r}_\ast (f\delta _w) = f\delta _1$ by Proposition 4.1. Thus,

$$ \begin{align*} (\partial s_1+s_0{\boldsymbol r}_\ast)(f\delta_w) &= \sum_{i=1}^n \partial f[a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}},a_i^{\epsilon_i}]+f\delta_1\\ &= \sum_{i=1}^n (f\delta_{a_1^{\epsilon_1}\cdots a_{i}^{\epsilon_{i}}}- f\delta_{a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}}})+f\delta_1\\ &= f\delta_w \end{align*} $$

StartLayout 1st Row 1st Column left parenthesis partial differential s 1 plus s 0 bold italic r Subscript asterisk Baseline right parenthesis left parenthesis f delta Subscript w Baseline right parenthesis 2nd Column equals sigma summation Underscript i equals 1 Overscript n Endscripts partial differential f left bracket a 1 Superscript epsilon 1 Baseline midline horizontal ellipsis a Subscript i minus 1 Superscript epsilon Super Subscript i minus 1 Superscript Baseline comma a Subscript i Superscript epsilon Super Subscript i Superscript Baseline right bracket plus f delta 1 2nd Row 1st Column Blank 2nd Column equals sigma summation Underscript i equals 1 Overscript n Endscripts left parenthesis f delta Subscript a 1 Sub Superscript epsilon 1 Subscript midline horizontal ellipsis a Sub Subscript i Sub Superscript epsilon Sub Super Subscript i Sub Superscript Subscript Baseline minus f delta Subscript a 1 Sub Superscript epsilon 1 Subscript midline horizontal ellipsis a Sub Subscript i minus 1 Sub Superscript epsilon Sub Super Subscript i minus 1 Sub Superscript Subscript Baseline right parenthesis plus f delta 1 3rd Row 1st Column Blank 2nd Column equals f delta Subscript w EndLayout

since the sum is telescoping, where we have used Lemma 4.4(2) for the second equality.

Finally, we prove the third property: the equality $s_1\partial =1$ holds. We compute $s_1\partial (f\delta _w\gamma _a)$ . Let $w=a_1^{\epsilon _1}\cdots a_n^{\epsilon _n}$ in reduced form with $n\geq 0$ . There are two cases. If $a_n^{\epsilon _1}\neq a^{-1}$ , then $|wa|=|w|+1$ and we compute

$$ \begin{align*} s_1\partial(f\delta_w\gamma_a)&=s_1(f\delta_{wa}-f\delta_w) \\ &= \sum_{i=1}^n f[a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}},a_i^{\epsilon_i}]+f[w,a]- \sum_{i=1}^n f[a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}},a_i^{\epsilon_i}]\\ &= f\delta_w\gamma_a. \end{align*} $$

StartLayout 1st Row 1st Column s 1 partial differential left parenthesis f delta Subscript w Baseline gamma Subscript a Baseline right parenthesis 2nd Column equals s 1 left parenthesis f delta Subscript w a Baseline minus f delta Subscript w Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column equals sigma summation Underscript i equals 1 Overscript n Endscripts f left bracket a 1 Superscript epsilon 1 Baseline midline horizontal ellipsis a Subscript i minus 1 Superscript epsilon Super Subscript i minus 1 Superscript Baseline comma a Subscript i Superscript epsilon Super Subscript i Superscript Baseline right bracket plus f left bracket w comma a right bracket minus sigma summation Underscript i equals 1 Overscript n Endscripts f left bracket a 1 Superscript epsilon 1 Baseline midline horizontal ellipsis a Subscript i minus 1 Superscript epsilon Super Subscript i minus 1 Superscript Baseline comma a Subscript i Superscript epsilon Super Subscript i Superscript Baseline right bracket 3rd Row 1st Column Blank 2nd Column equals f delta Subscript w Baseline gamma Subscript a Baseline period EndLayout

If $a_n^{\epsilon _n}=a^{-1}$ , put $u=a_1^{\epsilon _1}\cdots a_{n-1}^{\epsilon _{n-1}}$ , so that $w=ua^{-1}$ . Then,

$$ \begin{align*} s_1\partial(f\delta_w\gamma_a)&=s_1(f\delta_{u}-f\delta_{ua^{-1}}) \\ &= \sum_{i=1}^{n-1} f[a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}},a_i^{\epsilon_i}]- \sum_{i=1}^{n-1} f[a_1^{\epsilon_1}\cdots a_{i-1}^{\epsilon_{i-1}},a_i^{\epsilon_i}]-f[u,a^{-1}]\\ &= f\delta_{ua^{-1}}\gamma_a=f\delta_w\gamma_a, \end{align*} $$

as required.

Proof of Theorem A

The exactness of the resolution follows from the existence of a contracting homotopy over $\mathbb Z$ ; see Proposition 4.5. Since $X$ is compact, $\mathbb Z\mathcal G$ is unital, and hence projective as a $\mathbb Z\mathcal G$ -module. Since $1_{X_a}$ is an idempotent, each $\mathbb Z\mathcal G1_{X_a}$ is projective. This completes the proof.

Corollary 4.6. Let $F_A$ have a semi-saturated partial action on a compact totally disconnected Hausdorff space $X$ . Then, $\mathcal G=F_A\ltimes X$ has cohomological dimension at most $1$ . Hence, for any $\mathbb Z\mathcal G$ -module $M$ , $H_n(\mathcal G,M)=0$ and $H^n(\mathcal G,M)=0$ for $n\geq 2$ .

Corollary 4.7. Let $(X,T)$ be a Deaconu–Renault system with $X$ compact Hausdorff and totally disconnected, and the domain of $T$ clopen. Then, $\mathcal G_{(X,T)}$ has cohomological dimension at most $1$ .

We are now in a position to prove Theorem B.

Proof of Theorem B

Tensoring the deleted projective resolution from Theorem A with the right module $\mathbb ZX$ yields the chain complex

$$ \begin{align*}\bigoplus_{a\in A}\mathbb ZX_a\xrightarrow{\,d\,}\mathbb ZX\end{align*} $$ circled plus Underscript a element of upper A Endscripts double struck upper Z upper X Subscript a Baseline right arrow Overscript d Endscripts double struck upper Z upper X

computing homology, where $d$ is identified with $1_{\mathbb ZX}\otimes \partial $ under the isomorphisms $\mathbb ZX\otimes _{\mathbb Z\mathcal G}\mathbb Z\mathcal {G}1_{X_A}\cong \mathbb ZX_a$ and $\mathbb ZX\otimes _{\mathbb Z\mathcal G}\mathbb Z\mathcal {G}\cong \mathbb ZX$ . If $f\in \mathbb ZX_a$ , then $d(f)$ corresponds to $(1_{\mathbb ZX}{\kern1pt}\otimes{\kern1pt} \partial )(1_{X}\otimes f) {\kern1pt}={\kern1pt} 1_{X}{\kern1pt}\otimes{\kern1pt} \partial f {\kern1pt}={\kern1pt} 1_{X}{\kern1pt}\otimes{\kern1pt} f\delta _a-1_{X}{\kern1pt}\otimes \!f = \delta _{a^{-1}}f\delta _a{\kern1pt}\otimes{\kern1pt} 1_{X_{a^{-1}}} -f{\kern1pt}\otimes{\kern1pt} 1_{X_a}=\! f\circ \theta _{a}\ {\otimes }\ 1_{X_{a^{-1}}}-f{\kern1pt}\otimes{\kern1pt} 1_{X_a}$ , which corresponds to $\theta _a^*(f)-\iota _a(f)$ under the isomorphisms. The result now follows by definition of groupoid homology as $\mathrm {Tor}_{\bullet }^{\mathbb Z\mathcal G}(\mathbb Z\mathcal {G}^0,\mathbb Z\mathcal {G}^0)$ .

Theorem B reduces to a known result for Deaconu–Renault groupoids [Reference Farsi, Kumjian, Pask and Sims10, Theorem 6.7], although that reference assumes that the local homeomorphism is fully defined and surjective, whereas we allow it to be defined on a clopen subset and not be surjective. However, we require $X$ to be compact, although that can be relaxed somewhat; see Remark 4.9.

Corollary 4.8. Let $(X,T)$ be a Deaconu–Renault groupoid with $X$ compact Hausdorff and totally disconnected, and where $T$ has clopen domain $U$ . Then, ${H_n(\mathcal G_{(X,T)})=0}$ for $n\geq 2$ and

$$ \begin{align*} H_0(\mathcal G_{(X,T)}) &= \mathop{\mathrm{coker}}(\iota-T_*),\\ H_1(\mathcal G_{(X,T)}) &= \ker (\iota- T_*), \end{align*} $$

where $\iota \colon \mathbb ZU\to \mathbb ZX$ is extension by $0$ and $T_\ast \colon \mathbb ZU\to \mathbb ZX$ is the induced homomorphism.

Proof. The proof of Theorem 2.2 shows that there is an orthogonal semi-saturated partial action $\theta $ of a free group $F_A$ on a finite set $A$ such that $U = \bigsqcup _{a\in A} X_a$ , $T=\bigcup _{a\in A}\theta _{a^{-1}}$ , and $\mathcal G_{(X,T)} \cong F_A\ltimes X$ . We can identify $\bigoplus _{a\in A}\mathbb ZX_a$ with $\mathbb ZU$ via the inclusions, and under this identification, using $T=\bigcup _{a\in A}\theta _{a^{-1}}$ , the map $\bigoplus _{a\in A}(\iota _a-\theta _a^*)$ becomes $\iota -T_*$ . Indeed, $T_*(f)(x)=\sum _{T(y)=x}f(y) = \sum _{a\in A}f(\theta _a(x))=\sum _{a\in A}\theta _a^*(f)(x)$ . The result follows from Theorem B.

Notice that the isotropy groups of $F_A\ltimes X$ are subgroups of $F_A$ and, hence, free. Thus, they are torsion-free and satisfy the Baum–Connes conjecture. Since ${H_n(F_A\ltimes X)=0}$ for $n\geq 2$ , the Matui HK conjecture holds for $F_A\ltimes X$ and we have $K_i(C^*_r(F_A\ltimes X))\cong H_i(F_A\ltimes X)$ for $i=0,1$ by [Reference Proietti and Yamashita16, Remark 3.5]. By Abadie’s results [Reference Abadie1], $C^*_r(F_A\ltimes X)$ is the reduced partial crossed product $C(X)\rtimes _r F_A$ .

Remark 4.9. Most of the results of this paper can be adapted to the case where $X$ is merely assumed to be locally compact Hausdorff and totally disconnected, except that one obtains a flat resolution instead of a projective resolution in Theorem A (as $\mathbb Z\mathcal G$ and its direct summands are merely flat), and so one can only make conclusions about homology and not cohomology. In this context, $\delta _g$ is just the characteristic function of a clopen bisection, but convolving it with a compactly supported locally constant function still gives a compactly supported locally constant function, and so our products with it make sense when adapting the arguments. The proof that Deaconu–Renault groupoids come from orthogonal semi-saturated partial actions of free groups in the locally compact Hausdorff and totally disconnected case requires ${\mathrm {dom}}\ T$ to be paracompact to partition it into disjoint compact open sets on which $T$ is injective. Otherwise, there are no obstacles and, in particular, Theorem B and Corollary 4.8 still go through under these hypotheses.

5 Global dimension

In this section, we show that for semi-saturated partial group actions of a free group $F_A$ , the algebra $K[F_A\ltimes X]$ has global dimension at most $2$ if $K$ is a field and $X$ is compact Hausdorff totally disconnected and second countable. I conjecture such algebras are, in fact, hereditary. This would generalize the well-known fact that Leavitt path algebras of finite graphs are hereditary [Reference Abrams, Ara and Siles Molina2] and free group algebras are hereditary.

Recall that a ring is (left) hereditary if each left ideal is projective. It is well known that a ring is hereditary if and only if any submodule of a projective module is projective.

The (left) global dimension of a ring $R$ is the supremum of the projective dimensions of its (left) modules. A ring is hereditary if and only if it has global dimension at most one. A result of Auslander [Reference Rotman17, Theorem 8.16] says that the global dimension of $R$ is the supremum of the projective dimensions of cyclic modules $R/L$ with $L$ a left ideal. In particular, by considering the exact sequence $0\to L\to R\to R/L\to 0$ , it follows that if each left ideal has projective dimension at most $n$ , then $R$ has global dimension at most $n+1$ .

We develop some preliminaries. First, we assume that $\mathcal G$ is a Hausdorff ample groupoid with compact unit space in what follows. If $X$ is a locally compact and totally disconnected Hausdorff space, and $K$ is a commutative ring, then $K\otimes _{\mathbb Z} \mathbb ZX\cong KX$ by [Reference Li14, Corollary 2.3]. We use $\mathop {\mathrm {pd}}_R M$ for the projective dimension of a module over a ring $R$ .

Lemma 5.1. Let $K$ be a commutative ring. Then, $\mathop {\mathrm {pd}}_{K\mathcal G}K\mathcal G^0\leq \mathop {\mathrm {pd}}_{\mathbb Z\mathcal G}\mathbb Z\mathcal G^0$ .

Proof. First, note that since $K\mathcal G\cong K\otimes _{\mathbb Z}\mathbb Z\mathcal G$ , extension of scalars preserves projectivity. Let $P_{\bullet }\to \mathbb Z\mathcal G^0$ be a projective resolution of length $\mathop {\mathrm {pd}}_{\mathbb Z\mathcal G}\mathbb Z\mathcal G^0$ . Then, ${K\otimes _{\mathbb Z}P_{\bullet }\to K\mathcal G^0}$ is a chain complex of projective $K\mathcal G$ -modules. Since $\mathbb Z\mathcal G$ is torsion-free as an abelian group, $P_{\bullet }\to \mathbb Z\mathcal G^0$ is a flat resolution over $\mathbb Z$ of the torsion-free, and hence flat, abelian group $\mathbb Z\mathcal G^0$ . Thus, ${H_n(K\otimes _{\mathbb Z}P_{\bullet }){\kern-1pt}\cong{\kern-1pt} {\mathrm {Tor}}_n^{\mathbb Z}(K,\mathbb Z\mathcal G^0){\kern-1pt}={\kern-1pt}0}$ . We conclude that $K\otimes _{\mathbb Z}P_{\bullet }\to K\mathcal G^0$ is a projective resolution of length $\mathop {\mathrm {pd}}_{\mathbb Z\mathcal G}\mathbb Z\mathcal G^0$ , establishing the result.

Let $S$ be the inverse monoid of compact open bisections of $\mathcal G$ . Then, $K\mathcal G\cong KS/I$ via $1_U\unicode{x21A4} U$ , where $I$ is generated by all differences $U+V - (U\cup V)$ , where $U,V\subseteq \mathcal G^0$ are disjoint and compact open [Reference Steinberg18, Theorem 3.11]. Moreover, any idempotent of $S$ , such as $ss^*$ , maps into $K\mathcal {G}^0$ .

Let $M,N$ be left $K\mathcal G$ -modules. We wish to show that $M\otimes _{K\mathcal G^0} N$ is a $K\mathcal G$ -module. First, note that $M\otimes _K N$ is a $KS$ -module in the standard way: $s(m\otimes n) = sm\otimes sn$ . Note that $M\otimes _{K\mathcal G^0}N$ is the quotient of $M\otimes _K N$ by the $K$ -module $A$ generated by the elements $em\otimes n -m\otimes en$ with $e=1_U$ , where $U\subseteq \mathcal G^0$ is compact open. If $s=V$ is a compact open bisection, then $ses^* = 1_{VUV^{-1}}$ . Therefore, $s(em\otimes n -m\otimes en) = ses^*sm\otimes sn-sm\otimes ses^*sn$ belongs to $A$ . It follows that $M\otimes _{K\mathcal G^0} N$ is a $KS$ -module. To show that it is a $K\mathcal G$ -module, we must show that if $m\otimes n\in M\otimes _{K\mathcal G^0} N$ is a basic tensor and $U,V\subseteq G^0$ are disjoint compact open sets, then $(1_U+1_V)(m\otimes n) = 1_{U\cup V}(m\otimes n)$ . Indeed, we have that

$$ \begin{align*} 1_{U\cup V}(m\otimes n) &=1_{U\cup V}m\otimes 1_{U\cup V}n =m\otimes 1_{U\cup V}n =m\otimes (1_U+1_V)n \\ &= m\otimes 1_Un+m\otimes 1_Vn= 1_Um\otimes 1_Un+1_Vm\otimes 1_Vn\\ &= 1_U(m\otimes n)+1_V(m\otimes n)= (1_U+1_V)(m\otimes n) \end{align*} $$

StartLayout 1st Row 1st Column 1 Subscript upper U union upper V Baseline left parenthesis m circled times n right parenthesis 2nd Column equals 1 Subscript upper U union upper V Baseline m circled times 1 Subscript upper U union upper V Baseline n equals m circled times 1 Subscript upper U union upper V Baseline n equals m circled times left parenthesis 1 Subscript upper U Baseline plus 1 Subscript upper V Baseline right parenthesis n 2nd Row 1st Column Blank 2nd Column equals m circled times 1 Subscript upper U Baseline n plus m circled times 1 Subscript upper V Baseline n equals 1 Subscript upper U Baseline m circled times 1 Subscript upper U Baseline n plus 1 Subscript upper V Baseline m circled times 1 Subscript upper V Baseline n 3rd Row 1st Column Blank 2nd Column equals 1 Subscript upper U Baseline left parenthesis m circled times n right parenthesis plus 1 Subscript upper V Baseline left parenthesis m circled times n right parenthesis equals left parenthesis 1 Subscript upper U Baseline plus 1 Subscript upper V Baseline right parenthesis left parenthesis m circled times n right parenthesis EndLayout

since the tensor product is over $K\mathcal G^0$ .

It is trivial to verify that the natural isomorphism $(L\otimes _{K\mathcal G^0} M)\otimes _{K\mathcal G^0} N\cong L\otimes _{K\mathcal G^0} (M\otimes _{K\mathcal G^0} N)$ is a $K\mathcal G$ -module isomorphism.

Note that the action of $S$ on the trivial module $K\mathcal G^0$ is $(s,f)\mapsto sfs^*$ .

Proposition 5.2. There is a natural isomorphism $K\mathcal{G}^0\otimes _{K\mathcal G^0} M\to M$ for any $K\mathcal G$ -module $M$ .

Proof. The action map induces a $K\mathcal G^0$ -module isomorphism $f\otimes m\mapsto fm$ . We check this is a $K\mathcal G$ -module isomorphism. Indeed, if $s\in S$ and $e=1_U$ with $U\subseteq \mathcal G^0$ compact open, then $s(e\otimes m) = ses^*\otimes sm\mapsto ses^*sm=ss^*sem=sem$ .

We can similarly make ${\mathrm {Hom}}_{K\mathcal G^0}(M,N)$ into a $K\mathcal G$ -module for $K\mathcal G$ -modules $M,N$ . The action is $(sf)(m) = sf(s^*m)$ . To see that this is $K\mathcal G^0$ -linear, let $e=1_U$ with $U\subseteq \mathcal G^0$ compact open and let $s\in S$ . Then, $(sf)(em) = sf(s^*em) = sf(s^*ess^*m) = ss^*esf(s^*m) = esf(s^*m)=e(sf)(m)$ . This makes ${\mathrm {Hom}}_{K\mathcal G^0}(M,N)$ into a $KS$ -module. To check that it is a $K\mathcal G$ -module, we take disjoint compact open $U,V\subseteq \mathcal G^0$ and $f\colon M\to N$ , a $K\mathcal G^0$ -module homomorphism, and check that $1_{U\cup V}f = 1_Uf+ 1_Vf$ . Indeed,

$$ \begin{align*} (1_{U\cup V}f)(m) &= 1_{U\cup V}f(1_{U\cup V}m) = f(1_{U\cup V}m)=f((1_U+1_V)m) \\ &= f(1_Um)+f(1_Vm) = 1_Uf(1_Um)+1_Vf(1_Vm) \\ &= (1_Uf)(m)+(1_Vf)(m) = ((1_U+1_V)f)(m). \end{align*} $$

StartLayout 1st Row 1st Column left parenthesis 1 Subscript upper U union upper V Baseline f right parenthesis left parenthesis m right parenthesis 2nd Column equals 1 Subscript upper U union upper V Baseline f left parenthesis 1 Subscript upper U union upper V Baseline m right parenthesis equals f left parenthesis 1 Subscript upper U union upper V Baseline m right parenthesis equals f left parenthesis left parenthesis 1 Subscript upper U Baseline plus 1 Subscript upper V Baseline right parenthesis m right parenthesis 2nd Row 1st Column Blank 2nd Column equals f left parenthesis 1 Subscript upper U Baseline m right parenthesis plus f left parenthesis 1 Subscript upper V Baseline m right parenthesis equals 1 Subscript upper U Baseline f left parenthesis 1 Subscript upper U Baseline m right parenthesis plus 1 Subscript upper V Baseline f left parenthesis 1 Subscript upper V Baseline m right parenthesis 3rd Row 1st Column Blank 2nd Column equals left parenthesis 1 Subscript upper U Baseline f right parenthesis left parenthesis m right parenthesis plus left parenthesis 1 Subscript upper V Baseline f right parenthesis left parenthesis m right parenthesis equals left parenthesis left parenthesis 1 Subscript upper U Baseline plus 1 Subscript upper V Baseline right parenthesis f right parenthesis left parenthesis m right parenthesis period EndLayout

Proposition 5.3. If $M$ is a $K\mathcal G$ -module, then ${\mathrm {Hom}}_{K\mathcal G^0}(K\mathcal G^0, M)\cong M$ naturally in $M$ .

Proof. There is a $K\mathcal G^0$ -module isomorphism taking $f\colon \mathcal G^0\to M$ to $f(1)$ . Let us show it is a $K\mathcal G$ -module isomorphism. Indeed, if $s\in S$ , then $(sf)(1) = sf(s^*1s) = sf(s^*s) = ss^*sf(1) = sf(1)$ .

We now show that the hom-tensor adjunction holds at the level of $K\mathcal G$ -modules.

Proposition 5.4. Let $L,M,N$ be $K\mathcal G$ -modules. Then, there are natural isomorphisms of $K$ -modules ${\mathrm {Hom}}_{K\mathcal G}(L,{\mathrm {Hom}}_{K\mathcal G^0}(M,N))\cong {\mathrm {Hom}}_{K\mathcal G}(L\otimes _{K\mathcal G^0} M,N)\cong {\mathrm {Hom}}_{K\mathcal G}(M,{\mathrm {Hom}}_{K\mathcal G^0}(L,N))$ .

Proof. Indeed, all three $K$ -modules are isomorphic to the $K$ -module of $K\mathcal G^0$ -bilinear maps $f\colon L\times M\to N$ satisfying $f(s\ell ,sm) = sf(\ell ,m)$ for $s\in S$ .

Since this is clear for the middle hom set and the third hom set is dual to the first, we just handle the first hom set. If $f\colon L\times M\to N$ satisfies the above property, define $F\colon L\to {\mathrm {Hom}}_{K\mathcal G}(M,N)$ by $F(\ell )(m) = f(\ell ,m)$ . Then, $F(s\ell ) = f(s\ell , m)= f(ss^*s\ell ,m) = f(s\ell ,ss^*m) = sf(\ell , s^*m) =sF(\ell )(s^*m) = (sF(\ell ))(m)$ , where we have used that $f$ is $K\mathcal G^0$ -bilinear in the third equality.

Conversely, if $F\colon L\to {\mathrm {Hom}}_{K\mathcal G^0}(M,N)$ is a homomorphism and $f(\ell ,m)=F(\ell )(m)$ , then $f(s\ell ,sm) = F(s\ell )(sm) = (sF(\ell ))(sm) = sF(\ell )(s^*sm) = ss^*sF(\ell )(m) = sF(\ell )(m) = sf(\ell ,m)$ .

Our first corollary of the hom-tensor adjunction is the following.

Corollary 5.5. If $M,N$ are $K\mathcal G$ -modules, then there is a natural isomorphism ${\mathrm {Hom}}_{K\mathcal G}(M,N)\cong {\mathrm {Hom}}_{K\mathcal G}(K\mathcal G^0,{\mathrm {Hom}}_{K\mathcal G^0}(M,N))$ .

Proof. By Propositions 5.2 and 5.4, we have isomorphisms ${\mathrm {Hom}}_{K\mathcal G}(M,N)\cong {\mathrm {Hom}}_{K\mathcal G}(K\mathcal G^0\otimes _{K\mathcal G^0}M,N)\cong {\mathrm {Hom}}_{K\mathcal G}(K\mathcal G^0,{\mathrm {Hom}}_{K\mathcal G^0}(M,N))$ .

Another consequence of the hom-tensor adjunction is that the tensor product of projective modules is projective when $\mathcal G$ is paracompact (and Hausdorff). The following is a slight variation of [Reference Bönicke, Dell’Aiera, Gabe and Willett3, Lemma 2.13].

Lemma 5.6. Let $\mathcal G$ be paracompact ample Hausdorff groupoid with compact unit space. Then, $K\mathcal G$ is a projective $K\mathcal G^0$ -module. Consequently, any projective $K\mathcal G$ -module is also a projective $K\mathcal G^0$ -module.

Proof. Since $\mathcal G$ is a paracompact locally compact Hausdorff and totally disconnected space, it has a covering by pairwise disjoint compact open bisections $\{U_{\alpha }\}_{\alpha \in A}$ . Then, $K\mathcal G\cong \bigoplus _{\alpha \in A} K\mathcal G^01_{U_\alpha }\cong \bigoplus _{\alpha \in A} K\mathcal G^01_{{\boldsymbol r}(U_\alpha )}$ as a $K\mathcal G^0$ -module.

Corollary 5.7. Let $\mathcal G$ be an ample groupoid with compact unit space. Let $P,Q$ be $K\mathcal G$ -modules with $P$ projective over $K\mathcal G$ and $Q$ projective over $K\mathcal G^0$ . Then, $P\otimes _{K\mathcal G^0}Q$ is projective over $K\mathcal G$ .

Proof. We have by Proposition 5.4 that

$$ \begin{align*}{\mathrm{Hom}}_{K\mathcal G}(P\otimes_{K\mathcal G^0}Q,-)\cong {\mathrm{Hom}}_{K\mathcal G}(P,{\mathrm{Hom}}_{K\mathcal G^0}(Q,-)),\end{align*} $$

and the right-hand side is exact as $P$ is projective over $K\mathcal G$ and $Q$ is projective over $K\mathcal G^0$ .

Assume now that $\mathcal G$ is a paracompact Hausdorff ample groupoid with compact unit space. Notice that if $M,N$ are $K\mathcal G$ -modules and $P_{\bullet }\to M$ is a projective resolution over $K\mathcal G$ , then it is a projective resolution over $K\mathcal G^0$ by Lemma 5.6. Therefore, $H^n({\mathrm {Hom}}_{K\mathcal G^0}(P_{\bullet },N))={\mathrm {Ext}}^n_{K\mathcal G^0}(M,N)$ , and it has a $K\mathcal G$ -module structure, which is easily seen to be independent of the projective resolution by a standard argument.

We now construct a spectral sequence connecting $\mathrm {Ext}$ with groupoid cohomology. A similar spectral sequence for computing Hochschild cohomology appears in [Reference Dokuchaev, Khrypchenko and Jacobo Simón6], but they work with the cohomology of the inverse semigroup of compact open bisections.

Theorem 5.8. Let $\mathcal G$ be a paracompact Hausdorff ample groupoid with compact unit space. Then, there is a convergent spectral sequence

$$ \begin{align*}{\mathrm{Ext}}^p_{K\mathcal G}(K\mathcal G^0,{\mathrm{Ext}}^q_{K\mathcal G^0}(M,N))\Rightarrow {\mathrm{Ext}}_{K\mathcal G}^{p+q}(M,N)\end{align*} $$

for $K\mathcal G$ -modules $M,N$ .

Proof. Let $P_{\bullet }\to K\mathcal G^0$ and $Q_{\bullet }\to M$ be projective resolutions over $K\mathcal G$ . Consider the double cochain complex $C^{p,q} = {\mathrm {Hom}}_{K\mathcal G}(P_p\otimes _{K\mathcal G^0}Q_q,N)$ . Let us compute the spectral sequence of the first filtration. We have

$$ \begin{align*}C^{p,\bullet} = {\mathrm{Hom}}_{K\mathcal G}(P_p\otimes_{K\mathcal G^0} Q_{\bullet},N)\cong {\mathrm{Hom}}_{K\mathcal G}(P_p,{\mathrm{Hom}}_{K\mathcal G^0}(Q_{\bullet},N)),\end{align*} $$

which has cohomology ${\mathrm {Hom}}_{K\mathcal G}(P_p,{\mathrm {Ext}}^q_{K\mathcal G^0}(M,N))$ since $P_p$ is projective for all $p$ . This complex in turn has cohomology ${\mathrm {Ext}}^p_{K\mathcal G}(K\mathcal G^0,{\mathrm {Ext}}^q_{K\mathcal G^0}(M,N))$ .

For the second filtration, we compute

$$ \begin{align*}C^{\bullet,p}= {\mathrm{Hom}}_{K\mathcal G}(P_{\bullet}\otimes_{K\mathcal G^0} Q_p,N)\cong {\mathrm{Hom}}_{K\mathcal G}(Q_p,{\mathrm{Hom}}_{K\mathcal G^0}(P_{\bullet},N)).\end{align*} $$

Since $Q_p$ is projective, the cohomology of this is ${\mathrm {Hom}}_{K\mathcal G}(Q_p,{\mathrm {Ext}}^q_{K\mathcal G^0}(K\mathcal G^0,N))$ , which then has cohomology ${\mathrm {Ext}}^p_{K\mathcal G}(M,{\mathrm {Ext}}^q_{K\mathcal G^0}(K\mathcal G^0,N))$ . However,

$$ \begin{align*}{\mathrm{Ext}}^q_{K\mathcal G^0}(K\mathcal G^0,N) \cong\begin{cases} N & \text{if}\ q=0,\\ 0 & \text{if}\ q>0, \end{cases}\end{align*} $$ upper E x t Subscript upper K script upper G Sub Superscript 0 Superscript q Baseline left parenthesis upper K script upper G Superscript 0 Baseline comma upper N right parenthesis approximately equals StartLayout Enlarged left brace 1st Row 1st Column upper N 2nd Column if q equals 0 comma 2nd Row 1st Column 0 2nd Column if q greater than 0 comma EndLayout

upper E x t Subscript upper K script upper G Sub Superscript 0 Superscript q Baseline left parenthesis upper K script upper G Superscript 0 Baseline comma upper N right parenthesis approximately equals StartLayout Enlarged left brace 1st Row 1st Column upper N 2nd Column if q equals 0 comma 2nd Row 1st Column 0 2nd Column if q greater than 0 comma EndLayout

by Proposition 5.3. Thus, this spectral sequence collapses on the axis $q=0$ and so we deduce the total complex has homology ${\mathrm {Ext}}^n_{K\mathcal G}(M,N)$ . This completes the proof of the theorem.

Next, we need an observation about $KX$ when $X$ is a second countable compact Hausdorff totally disconnected space. Recall that a ring $R$ is (von Neumann) regular [Reference Goodearl11] if, for all $a\in R$ , there exists $b\in R$ with $aba=a$ .

Theorem 5.9. Let $X$ be a second countable compact Hausdorff totally disconnected space and $K$ a field. Then, $KX$ is hereditary (as a ring with pointwise product).

Proof. Note first that $KX$ is regular. Indeed, if $f\in KX$ and one defines $g\in KX$ by

$$ \begin{align*}g(x) = \begin{cases}f(x)^{-1} & \text{if}\ f(x)\neq 0,\\ 0 & \text{otherwise,}\end{cases}\end{align*} $$ StartLayout 1st Row g left parenthesis x right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column f left parenthesis x right parenthesis Superscript negative 1 Baseline 2nd Column if f left parenthesis x right parenthesis not equals 0 comma 2nd Row 1st Column 0 2nd Column otherwise comma EndLayout EndLayout

StartLayout 1st Row g left parenthesis x right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column f left parenthesis x right parenthesis Superscript negative 1 Baseline 2nd Column if f left parenthesis x right parenthesis not equals 0 comma 2nd Row 1st Column 0 2nd Column otherwise comma EndLayout EndLayout

then $fgf=f$ . Since $X$ is second countable, $KX$ has countable dimension over $K$ (being spanned by the characteristic functions of the countably many compact open sets). Hence, all ideals in $KX$ are countably generated. A theorem of Kaplansky says that any countably generated left ideal in a regular ring is projective [Reference Goodearl11, Corollary 2.15]. We conclude that $KX$ is hereditary.

We remark that second countability is needed here. There are examples of Boolean rings of arbtirary global dimension [Reference Pierce15]. Every Boolean ring is of the form $\mathbb F_2X$ for a compact Hausdorff and totally disconnected space $X$ , where $\mathbb F_2$ is the two-element field.

We can now prove Theorem C.

Proof of Theorem C

Let $\mathcal G=F_A\ltimes X$ . Note that $\mathcal G=\bigcup _{w\in F_A}\{w\}\times X_{w^{-1}}$ is a covering of $\mathcal G$ by pairwise disjoint compact open bisections, from which it follows that $\mathcal G$ is paracompact. Let $L$ be a left ideal of $K\mathcal G$ . We show that $L$ has projective dimension at most $1$ ; it then follows $K\mathcal G$ has global dimension at most $2$ . First, note that $L$ is projective as a $KX$ -module since $KX$ is hereditary by Theorem 5.9 and $L$ is a submodule of a projective $KX$ -module by Lemma 5.6. Therefore, the spectral sequence in Theorem 5.8 (with $M=L$ ) collapses on the axis $q=0$ , from which we conclude that ${\mathrm {Ext}}^n_{K\mathcal G}(L,N)\cong {\mathrm {Ext}}^n_{K\mathcal G}(K\mathcal G^0,{\mathrm {Hom}}_{K\mathcal G^0}(L,N))$ . By Corollary 4.6 and Lemma 5.1, we have that $\mathop {\mathrm {pd}}_{K\mathcal G} K\mathcal G^0\leq 1$ . It follows that ${\mathrm {Ext}}^2_{K\mathcal G}(L,N)=0$ for all modules $N$ . Thus, $L$ has projective dimension at most $1$ .

I conjecture that under the hypothesis of Theorem C, the global dimension is at most one, that is, $K[F_A\ltimes X]$ is hereditary.

Corollary 5.10. Let $(X,T)$ be a Deaconu–Renault system with $X$ second countable compact Hausdorff and totally disconnected, and ${\mathrm {dom}}\ T$ clopen. Then, $K\mathcal G_{(X,T)}$ has global dimension at most $2$ for any field $K$ .

The following result about the algebra of a free inverse monoid seems to be new.

Corollary 5.11. The algebra of a countably generated free inverse monoid over a field has global dimension at most $2$ .

Proof. Let $K$ be a field. For any countable inverse monoid $M$ , there is an action of $M$ on the spectrum $\widehat {E(M)}$ of its idempotents, which is a second countable compact totally disconnected Hausdorff space, such that $KM\cong K[M\ltimes \widehat {E(M)}]$ [Reference Steinberg19, Theorem 6.3]. Applying this to the free inverse monoid and using Proposition 2.3, we see that $KFIM_A\cong K[F_A\ltimes \widehat {E(FIM_A)}]$ for a semi-saturated partial action of $F_A$ on the second countable compact Hausdorff and totally disconnected space $\widehat {E(FIM_A)}$ . Thus, the global dimension of $KFIM_A$ is at most $2$ by Theorem C.

I conjecture $KFIM_A$ is hereditary.

Acknowledgements

The author thanks Gilles Gonçalves de Castro for permission to use his proof of Theorem 2.2.

Footnotes

Communicated by Aidan Sims

The author was supported by the NSF grant DMS-2452324, a Simons Foundation Collaboration Grant, award number 849561, the Australian Research Council Grant DP230103184, and Marsden Fund Grant MFP-VUW2411.

References

Abadie, F., ‘Enveloping actions and Takai duality for partial actions’, J. Funct. Anal. 197(1) (2003), 14–67.10.1016/S0022-1236(02)00032-0CrossRef Google Scholar

Abrams, G., Ara, P. and Siles Molina, M., Leavitt Path Algebras, Lecture Notes in Mathematics, 2191 (Springer, London, 2017).10.1007/978-1-4471-7344-1CrossRef Google Scholar

Bönicke, C., Dell’Aiera, C., Gabe, J. and Willett, R., ‘Dynamic asymptotic dimension and Matui’s HK conjecture’, Proc. Lond. Math. Soc. (3) 126(4) (2023), 1182–1253.10.1112/plms.12510CrossRef Google Scholar

de Castro, G. G. and Kang, E. J., ‘A categorical interpretation of continuous orbit equivalence for partial dynamical systems’, Bull. Lond. Math. Soc. 57(12) (2025), 3637–3656.10.1112/blms.70207CrossRef Google Scholar

Dokuchaev, M. and Exel, R., ‘The ideal structure of algebraic partial crossed products’, Proc. Lond. Math. Soc. (3) 115(1) (2017), 91–134.10.1112/plms.12036CrossRef Google Scholar

Dokuchaev, M., Khrypchenko, M. and Jacobo Simón, J., ‘Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras’, Preprint, 2025, arXiv:2506.20321.Google Scholar

Exel, R., ‘Inverse semigroups and combinatorial ${C}^{\ast }$

-algebras’, Bull. Braz. Math. Soc. (N.S.) 39(2) (2008), 191–313.10.1007/s00574-008-0080-7CrossRef Google Scholar

Exel, R., Partial Dynamical Systems, Fell Bundles and Applications, Mathematical Surveys and Monographs, 224 (American Mathematical Society, Providence, RI, 2017).10.1090/surv/224CrossRef Google Scholar

Exel, R. and Steinberg, B., ‘Subshift semigroups’, Preprint, 2019, arXiv:1908.08315 [math.OA].Google Scholar

Farsi, C., Kumjian, A., Pask, D. and Sims, A., ‘Ample groupoids: equivalence, homology, and Matui’s HK conjecture’, Münster J. Math. 12(2) (2019), 411–451.Google Scholar

Goodearl, K. R., von Neumann Regular Rings, 2nd edn (Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991).Google Scholar

Kellendonk, J. and Lawson, M. V., ‘Partial actions of groups’, Internat. J. Algebra Comput. 14(1) (2004), 87–114.10.1142/S0218196704001657CrossRef Google Scholar

Lawson, M. V., Inverse Semigroups: The Theory of Partial Symmetries (World Scientific Publishing Co., Inc., River Edge, NJ, 1998).10.1142/3645CrossRef Google Scholar

Li, X., ‘Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces’, Forum Math. Pi 13 (2025), e9.10.1017/fmp.2024.31CrossRef Google Scholar

Pierce, R., ‘The global dimension of Boolean rings’, J. Algebra 7(1) (1967), 91–99.10.1016/0021-8693(67)90069-5CrossRef Google Scholar

Proietti, V. and Yamashita, M., ‘Homology and $K$

-theory of dynamical systems I. Torsion-free ample groupoids’, Ergod Th. & Dynam. Sys. 42(8) (2022), 2630–2660.10.1017/etds.2021.50CrossRef Google Scholar

Rotman, J. J., An Introduction to Homological Algebra, 2nd edn, Universitext, 223 (Springer, New York, 2009).10.1007/b98977CrossRef Google Scholar

Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Preprint, 2009, arXiv:0903.3456.Google Scholar

Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Adv. Math. 223(2) (2010), 689–727.10.1016/j.aim.2009.09.001CrossRef Google Scholar

Article contents

PARTIAL ACTIONS OF FREE GROUPS AND GROUPOID HOMOLOGY

Abstract

Keywords

MSC classification

Information

1 Introduction

2 Partial group actions

2.1 Definition of a partial action

2.2 The groupoid of a partial group action

3 Groupoid algebras and homology

3.1 Groupoid algebras

3.2 Homology and cohomology of ample groupoids

4 Cohomological dimension and semi-saturated partial actions

Proof of Theorem A

Proof of Theorem B

5 Global dimension

Proof of Theorem C

Acknowledgements

Footnotes

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests