# Homological stability for
general linear groups over Dedekind domains

Oscar Randal-Williams<EMAIL_ADDRESS>Centre for
Mathematical Sciences
Wilberforce Road
Cambridge CB3 0WB
UK

###### Abstract.

We prove a new kind of homological stability theorem for automorphism groups
of finitely-generated projective modules over Dedekind domains, which takes
into account all possible stabilisation maps between these, rather than only
stabilisation by the free module of rank 1. We show the same kind of stability
holds for Clausen and Jansen’s reductive Borel–Serre spaces.

##  1\. Introduction

The title of this note requires explanation, for Charney [Cha80] has already
shown that general linear groups over Dedekind domains enjoy homological
stability. Our goal is to show how these results may be obtained, and
improved, by using the machinery of cellular $E_{k}$-algebras which we
developed together with Galatius and Kupers [GKRW18a]. More precisely, in
Section 18.2 of that paper it was explained how the case of Dedekind domains
of class number 1 (i.e. PID’s) may be treated, and here we analyse the general
case. Rather than completeness, our motivation is that a new _kind_ of
homological stability theorem is appropriate when the class number is $>1$: we
will establish a generic homological stability theorem of this kind. We will
also show that it applies to Clausen and Jansen’s reductive Borel–Serre
spaces.

###  1.1. Modules over a Dedekind domain

Let $\mathcal{O}$ be a Dedekind domain, and $\mathrm{Pic}(\mathcal{O})$ denote
the abelian group of isomorphism classes of $\otimes$-invertible
$\mathcal{O}$-modules (which are the rank 1 projective modules; this group may
be identified with the ideal class group of $\mathcal{O}$). The classification
of finitely-generated modules over a Dedekind domain has the following
consequence: any finitely-generated projective $\mathcal{O}$-module has the
form $M\cong\mathcal{O}^{n-1}\oplus L$ for some
$[L]\in\mathrm{Pic}(\mathcal{O})$. In other words finitely-generated
projective $\mathcal{O}$-modules $M$ are classified up to isomorphism by the
data of

  1. (i)

their rank $n=\mathrm{rk}(M)$, and

  2. (ii)

the isomorphism class
$[\Lambda^{n}_{\mathcal{O}}M]\in\mathrm{Pic}(\mathcal{O})$ of their top
exterior power.

###  1.2. Earlier results

Unless otherwise mentioned, we consider integral homology. For an
$\mathcal{O}$-module $M$, let $\mathrm{GL}(M)$ denote the group of
$\mathcal{O}$-module automorphisms of $M$. Charney’s theorem states that for
finitely-generated projective $\mathcal{O}$-modules $M$ and $N$, the
homomorphism $-\oplus\mathrm{Id}_{N}:\mathrm{GL}(M)\to\mathrm{GL}(M\oplus N)$
induces

(1.1) $(-\oplus\mathrm{Id}_{N})_{*}:H_{d}(\mathrm{GL}(M))\longrightarrow
H_{d}(\mathrm{GL}(M\oplus N))$

which is an isomorphism for $d\leq\tfrac{\mathrm{rk}(M)-5}{4}$ and an
epimorphism for $d\leq\tfrac{\mathrm{rk}(M)-1}{4}$ (with an improvement if
$\mathcal{O}$ is a PID). Later, van der Kallen [vdK80, Theorem 4.11] has shown
that

$(-\oplus\mathrm{Id}_{\mathcal{O}})_{*}:H_{d}(\mathrm{GL}(\mathcal{O}^{n}))\longrightarrow
H_{d}(\mathrm{GL}(\mathcal{O}^{n+1}))$

is an isomorphism for $d\leq\tfrac{n-1}{2}$ and an epimorphism for
$d\leq\tfrac{n}{2}$. His results are for very general rings, but only treat
free modules. These methods were extended by Friedrich [Fri17, Theorem 2.9] to
treat _arbitrary_ (not even projective!) modules over the same class of rings:
in the case of Dedekind domains she shows111In her paper the “rank of an
$\mathcal{O}$-module $M$” denotes the largest $m$ such that $\mathcal{O}^{m}$
is a direct summand of $M$, so by the classification of finitely-generated
projective $\mathcal{O}$-modules it is usually one less than what we are
calling the rank (though they agree for free modules). that

$(-\oplus\mathrm{Id}_{\mathcal{O}})_{*}:H_{d}(\mathrm{GL}(M))\longrightarrow
H_{d}(\mathrm{GL}(M\oplus\mathcal{O}))$

is an isomorphism for $d\leq\tfrac{\mathrm{rk}(M)-4}{2}$ and an epimorphism
for $d\leq\tfrac{\mathrm{rk}(M)-3}{2}$ (with an improvement if $M$ is a free
module). One deduces the same stability range for the maps (1.1) by exploiting
the operation of tensoring with rank 1 projective modules.

###  1.3. The homological stability statement

We will prove a generic homological stability theorem in the context of
$E_{2}$-algebras, having the following explicit consequence for general linear
groups over Dedekind domains.

######  Theorem 1.1.

Let $\mathcal{O}$ be a Dedekind domain, $[L]\in\mathrm{Pic}(\mathcal{O})$, and
$M$ be a finitely-generated projective $\mathcal{O}$-module. Then

$(-\oplus\mathrm{Id}_{L})_{*}:H_{d}(\mathrm{GL}(M))\longrightarrow
H_{d}(\mathrm{GL}(M\oplus L))$

is an isomorphism for $d<\tfrac{\mathrm{rk}(M)-2}{2}$ and an epimorphism for
$d<\tfrac{\mathrm{rk}(M)}{2}$.

Given the selection of stabilisation maps, one for each element of
$\mathrm{Pic}(\mathcal{O})$, it is a little unnatural to formulate a result
such as Theorem 1.1 for one stabilisation map at a time. Instead, the
symmetric monoidal groupoid of all finitely-generated projective
$\mathcal{O}$-modules has homology

$\bigoplus_{[M]}H_{*}(\mathrm{GL}(M)),$

where the sum is over representatives of each isomorphism class. Direct sum of
modules endows this with the structure of a graded-commutative algebra, and
recording the rank of a module endows it with an additional
$\mathbb{N}$-grading. By the classification of finitely-generated projective
$\mathcal{O}$-modules, in homological degree 0 we have an isomorphism of
$\mathbb{N}$-graded algebras

$\bigoplus_{[M]}H_{0}(\mathrm{GL}(M))\cong\mathbb{Z}[\mathrm{Pic}(\mathcal{O})]/(\rho\cdot\rho^{\prime}-\sigma\cdot(\rho\otimes\rho^{\prime}))=:A_{\mathcal{O}}$

where we write $(\mathrm{Pic}(\mathcal{O}),\otimes,\sigma=[\mathcal{O}])$ for
the abelian group of $\otimes$-invertible $\mathcal{O}$-modules and
$A_{\mathcal{O}}$ is $\mathbb{N}$-graded by considering each element of
$\mathrm{Pic}(\mathcal{O})$ to have grading 1.222Here
$\mathbb{Z}[\mathrm{Pic}(\mathcal{O})]$ denotes the polynomial algebra on the
set $\mathrm{Pic}(\mathcal{O})$, not the group algebra! This discussion shows
that for each homological degree $d$,

$\bigoplus_{[M]}H_{d}(\mathrm{GL}(M))$

has the structure of an $\mathbb{N}$-graded $A_{\mathcal{O}}$-module. Our more
conceptual homological stability result in this setting is then as follows.

######  Theorem 1.2.

For each degree $d$, the $\mathbb{N}$-graded $A_{\mathcal{O}}$-module
$\bigoplus_{[M]}H_{d}(\mathrm{GL}(M))$ is generated in gradings $\leq 2d$, and
presented in gradings $\leq 2d+1$.

In Section 5 we will justify the following explicit example.

######  Example 1.3.

The Dedekind domain $\mathcal{O}:=\mathbb{Z}[\sqrt{-5}]$ has
$\mathrm{Pic}(\mathcal{O})=\\{\sigma,\lambda\\}$ with $\sigma=[\mathcal{O}]$
and nontrivial element $\lambda$ represented by the ideal
$\mathfrak{l}=(2,1+\sqrt{-5})$. The abelianisations of the automorphism groups
of the finitely-generated projective $\mathcal{O}$-modules are:

$n$ | 1 | 2 | 3 | 4
---|---|---|---|---
$\mathrm{GL}(\mathcal{O}^{n})^{ab}$ | $\mathbb{Z}/2\\{X\\}$ | $\mathbb{Z}/2\\{\sigma\cdot X,U,T,B,C\\}$ | $\mathbb{Z}/2\\{\sigma^{2}\cdot X\\}$ | $\mathbb{Z}/2\\{\sigma^{3}\cdot X\\}$
$\mathrm{GL}(\mathcal{O}^{n-1}\oplus\mathfrak{l})^{ab}$ | $\mathbb{Z}/2\\{X^{\prime}\\}$ | $\mathbb{Z}/2\\{\sigma\cdot X^{\prime},C^{\prime},D^{\prime}\\}$ | $\mathbb{Z}/2\\{\sigma^{2}\cdot X^{\prime}\\}$ | $\mathbb{Z}/2\\{\sigma^{3}\cdot X^{\prime}\\}$

where $\lambda\cdot X=\sigma\cdot X^{\prime}$, $\lambda\cdot
X^{\prime}=\sigma\cdot X$, and $U,T,B,C,C^{\prime},D^{\prime}$ are all
annihilated by $\sigma$ and $\lambda$. We see that this
$A_{\mathcal{O}}$-module is indeed generated in gradings $\leq 2$ and
presented in gradings $\leq 3$.

It is a simple exercise to deduce Theorem 1.1 from Theorem 1.2, but Theorem
1.2 is stronger. For example, it includes the statement that for each
$[L]\in\mathrm{Pic}(\mathcal{O})$ the sum of the stabilisation maps

$\bigoplus_{[L^{\prime}]\in\mathrm{Pic}(\mathcal{O})}(-\oplus
L\otimes(L^{\prime})^{-1})_{*}:\bigoplus_{[L^{\prime}]\in\mathrm{Pic}(\mathcal{O})}H_{d}(\mathrm{GL}(\mathcal{O}^{n-2}\oplus
L^{\prime}))\longrightarrow H_{d}(\mathrm{GL}(\mathcal{O}^{n-1}\oplus L))$

is surjective as long as $n>2d$ (whereas Theorem 1.1 says that the individual
stabilisation maps are surjective as long as $n+1>2d$).

###  1.4. Reductive Borel–Serre spaces

The spaces $B\mathrm{GL}(M)$ can be considered as an unstable approximation to
the algebraic $K$-theory space $\Omega^{\infty}_{0}\mathrm{K}(\mathcal{O})$,
from which point of view the homological stability results of the previous
section can be viewed as estimating the quality of this approximation.

Another unstable approximation to $\Omega^{\infty}_{0}\mathrm{K}(\mathcal{O})$
has been recently introduced by Clausen and Jansen [CJ24], in the form of the
reductive Borel–Serre spaces $|\mathrm{RBS}(M)|$; they are better
approximations in the sense that there is a factorisation

$B\mathrm{GL}(M)\longrightarrow|\mathrm{RBS}(M)|\longrightarrow\Omega^{\infty}_{0}\mathrm{K}(\mathcal{O}).$

Using recent work of Jansen [Jan24], we will explain how our general method
applies equally well to these spaces, and the exact analogues of Theorems 1.1
and 1.2 hold with $|\mathrm{RBS}(M)|$ in place of $B\mathrm{GL}(M)$.

######  Remark 1.4.

We will show elsewhere that when $\mathcal{O}$ has class number 1 then the
$|\mathrm{RBS}(M)|$ enjoy homological stability with slope 1, which is much
better than the $B\mathrm{GL}(M)$. Perhaps this is also the case for general
$\mathcal{O}$; we do not pursue that here.

###  1.5. The generic homological stability statement

To formulate our generic homological stability theorem, we assume familiarity
with the language and notation developed in [GKRW18a]. Fix a commutative ring
$\mathbbm{k}$, and work in the category
$\mathsf{sMod}_{\mathbbm{k}}^{\mathbb{N}}$ of $\mathbb{N}$-graded simplicial
$\mathbbm{k}$-modules. The statement will make use of the following
construction, which axiomatises what we did in the previous section. Let
$P=(P,\star,\sigma)$ be an abelian group, and define a commutative
$\mathbbm{k}$-algebra

$A_{P}:=\mathbbm{k}[P]/(\rho\cdot\rho^{\prime}-\sigma\cdot(\rho\star\rho^{\prime})\,|\,\rho,\rho^{\prime}\in
P).$

As the relations imposed are homogeneous with respect to word-length in the
free commutative $\mathbbm{k}$-algebra $\mathbbm{k}[P]$, we may consider
$A_{P}$ as being $\mathbb{N}$-graded. It has an augmentation
$\epsilon:A_{P}\to\mathbbm{k}$ by sending all $\rho\in P$ to 0. Following
[GKRW18a, §12.2.1], for a non-unital $E_{2}$-algebra $\mathbb{R}$ we write
$\overline{\mathbb{R}}$ for the unital associative algebra obtained by adding
a unit and strictifying the $E_{1}$-structure.

######  Theorem 1.5.

Let
$\mathbb{R}\in\mathsf{Alg}_{E_{2}}(\mathsf{sMod}_{\mathbbm{k}}^{\mathbb{N}})$
satisfy $H_{0,0}(\mathbb{R})=0$ as well as

  1. (I)

$H_{n,d}^{E_{1}}(\mathbb{R})=0$ for $d<n-1$, and

  2. (II)

there is an abelian group $(P,\star,\sigma)$ such that
$H_{*,0}(\overline{\mathbb{R}})\cong A_{P}$ as $\mathbb{N}$-graded
$\mathbbm{k}$-algebras.

Then for each degree $d$, $H_{*,d}(\overline{\mathbb{R}})$ has the structure
of an $\mathbb{N}$-graded $A_{P}$-module, and as such it is generated in
gradings $\leq 3d$ and presented in gradings $\leq 3d+1$. In particular, for
any element $\rho\in P\subset H_{1,0}(\overline{\mathbb{R}})$ the homotopy
cofibre $\overline{\mathbb{R}}/\rho$ of the map
$\rho\cdot-:S^{1,0}\otimes\overline{\mathbb{R}}\to\overline{\mathbb{R}}$
satisfies $H_{n,d}(\overline{\mathbb{R}}/\rho)=0$ for $d<\tfrac{n-1}{3}$.

Suppose that in addition to the above we have

  1. (III)

$H_{3,2}^{\overline{\mathbb{R}}}(A_{P})=0$.

Then the $\mathbb{N}$-graded $A_{P}$-module $H_{*,d}(\overline{\mathbb{R}})$
is generated in gradings $\leq 3d-1$, and is presented in gradings $\leq
3d+1$, but also $H_{n,d}(\overline{\mathbb{R}}/\rho)=0$ for
$d<\tfrac{n-1}{2}$.

Suppose that in addition to all the above we have

  1. (IV)

$H_{4,3}^{\overline{\mathbb{R}}}(A_{P})=0$.

Then the $\mathbb{N}$-graded $A_{P}$-module $H_{*,d}(\overline{\mathbb{R}})$
is generated in gradings $\leq 2d$, and is presented in gradings $\leq 2d+1$.

######  Remark 1.6.

The special case $P=\mathbb{Z}/2$ has been analysed in much more detail by
Sierra [Sie22b, Section 2]. See also [Sie22a, Theorem 2.4].

We also record here the following theorem, which resulted from early attempts
to prove a stability range of slope $\tfrac{1}{2}$ only under assumptions (I)
and (II). It is mainly notable because the stability range
$d<\tfrac{n-\sqrt{n}}{2}$ in the first part is best possible under assumptions
(I) and (II) (see Remark 4.3). We find this surprising, as we have never seen
an honestly non-linear stability range “in nature”.

######  Theorem 1.7.

Let
$\mathbb{R}\in\mathsf{Alg}_{E_{\infty}}(\mathsf{sMod}_{\mathbbm{k}}^{\mathbb{N}})$
satisfy (I) and (II) above, with $\mathbbm{k}$ a field of characteristic zero.
Then we have $H_{n,d}(\overline{\mathbb{R}}/\rho)=0$ for
$d<\tfrac{n-\sqrt{n}}{2}$.

If $P$ is finite then we also have $H_{n,d}(\overline{\mathbb{R}}/\rho)=0$ for
$d<\tfrac{n-|P|+1}{2}$.

Acknowledgements. I would like to thank Mikala Ørsnes Jansen for sharing a
draft of [Jan24], and Robin J. Sroka for a useful discussion around the
subject of Theorem 3.3. I was supported by the ERC under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 756444).

##  2\. Proof of Theorem 1.5

###  2.1. Regularity for $A_{P}$

We begin with some homological algebra over $A:=A_{P}$, considered with its
$\mathbb{N}$-grading, and with its unique augmentation
$\epsilon:A\to\mathbbm{k}$. We write $I:=I_{P}$ for its augmentation ideal.
From the relations defining $A_{P}$ we see that: $A(0)=\mathbbm{k}$,
$A(1)=\mathbbm{k}\\{P\\}$, and more generally for any $n\geq 1$ there is an
isomorphism $\mathbbm{k}\\{P\\}\overset{\sim}{\to}A(n)$ given by sending
$\rho\in P$ to $\sigma^{\cdot(n-1)}\cdot\rho$.

For an $\mathbb{N}$-graded $A$-module $M$, we will consider the (bigraded)
$\mathrm{Tor}$-groups

$H^{A}_{n,d}(M):=H_{n,d}(\mathbbm{k}\otimes^{\mathbb{L}}_{A}M)=\mathrm{Tor}^{A}_{n,d}(\mathbbm{k},M),$

and write

$h^{A}_{d}(M):=\max\\{n\in\mathbb{N}\,|\,H^{A}_{n,d}(M)\neq 0\\}.$

The module $M$ is therefore generated in gradings $\leq h_{0}^{A}(M)$, and is
presented in gradings $\leq\max(h_{0}^{A}(M),h_{1}^{A}(M))$.

######  Lemma 2.1.

We have $h^{A}_{d}(M)\leq d-1+h_{1}^{A}(M)$ for $d\geq 2$.

###### Proof.

Let $B:=\mathbbm{k}[\sigma]$, a polynomial algebra in one variable, and let
$B\to A$ send $\sigma$ to the symbol of the same name. Via this map we can
consider $M$ as a $B$-module. There is an equivalence
$\mathbbm{k}\otimes^{\mathbb{L}}_{B}M\simeq(\mathbbm{k}\otimes_{B}^{\mathbb{L}}A)\otimes^{\mathbb{L}}_{A}M$.

We may compute $B$-module homology by using the free $B$-module resolution
$0\to S^{1,0}\otimes B\overset{\sigma\cdot-}{\to}B\to\mathbbm{k}\to 0$. This
shows that $H^{B}_{*,d}(-)=0$ for $d\geq 2$, and with the explicit
presentation of $A$ gives

$H^{B}_{n,d}(A)=\begin{cases}\mathbbm{k}&(n,d)=(0,0)\\\
\mathbbm{k}\\{P\\}/\mathbbm{k}\\{\sigma\\}&(n,d)=(1,0)\\\
0&\text{else}.\end{cases}$

In particular there is a homotopy cofibre sequence of right $A$-modules of the
form

$S^{1,0}\otimes\mathbbm{k}\\{P\\}/\mathbbm{k}\\{\sigma\\}\longrightarrow\mathbbm{k}\otimes_{B}^{\mathbb{L}}A\longrightarrow
S^{0,0}\otimes\mathbbm{k}.$

Applying $-\otimes^{\mathbb{L}}_{A}M$ and using the equivalence above leads to
a long exact sequence

${\cdots(\mathbbm{k}\\{P\\}/\mathbbm{k}\\{\sigma\\})\otimes
H^{A}_{n-1,d}(M)}$${H^{B}_{n,d}(M)}$${H^{A}_{n,d}(M)}$${(\mathbbm{k}\\{P\\}/\mathbbm{k}\\{\sigma\\})\otimes
H^{A}_{n-1,d-1}(M)}$${H^{B}_{n,d-1}(M)}$${H^{A}_{n,d-1}(M)\cdots.}$$\scriptstyle{\partial}$

For $d\geq 2$, as $H^{B}_{*,d}(M)=0$ this gives an injection

$\partial:H^{A}_{n,d}(M)\longrightarrow(\mathbbm{k}\\{P\\}/\mathbbm{k}\\{\sigma\\})\otimes
H^{A}_{n-1,d-1}(M)$

so that $h^{A}_{d}(M)\leq h^{A}_{d-1}(M)+1$ as required. ∎

######  Corollary 2.2.

The algebra $A$ is Koszul, i.e.
$\mathrm{Tor}^{A}_{n,d}(\mathbbm{k},\mathbbm{k})=0$ for $n\neq d$.

###### Proof.

Applying Lemma 2.1 with $M=\mathbbm{k}$, which has $h_{1}^{A}(\mathbbm{k})=1$,
shows that $\mathrm{Tor}^{A}_{n,d}(\mathbbm{k},\mathbbm{k})=0$ for $d<n$. On
the other hand, using the reduced bar complex to calculate these
$\mathrm{Tor}$-groups shows that
$\mathrm{Tor}^{A}_{*,d}(\mathbbm{k},\mathbbm{k})$ is a subquotient of
$I^{\otimes d}$, for $I=\mathrm{Ker}(\epsilon:A\to\mathbbm{k})$ the
augmentation ideal. As $I$ is supported in gradings $\geq 1$, $I^{\otimes d}$
is supported in gradings $\geq d$, so
$\mathrm{Tor}^{A}_{n,d}(\mathbbm{k},\mathbbm{k})=0$ for $n<d$. ∎

###  2.2. Regularity for $\overline{\mathbb{R}}$

Write $\mathbb{A}$ for $A$ considered as a discrete unital $E_{2}$-algebra in
$\mathsf{sMod}_{\mathbbm{k}}^{\mathbb{N}}$, and $\mathbb{I}$ for its
augmentation ideal. We can consider an $A$-module $M$ as an
$\overline{\mathbb{R}}$-module $\mathbb{M}$, via the truncation map
$\overline{\mathbb{R}}\to\tau_{\leq 0}\overline{\mathbb{R}}\simeq\mathbb{A}$,
and we can therefore define $\overline{\mathbb{R}}$-module homology groups
$H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{M}):=H_{n,d}(\mathbbm{k}\otimes_{\overline{\mathbb{R}}}^{\mathbb{L}}\mathbb{M})$
as well as

$h^{\overline{\mathbb{R}}}_{d}(\mathbb{M}):=\max\\{n\in\mathbb{N}\,|\,H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{M})\neq
0\\}.$

######  Lemma 2.3.

Assumption (I) is equivalent to
“$H^{\overline{\mathbb{R}}}_{n,d}(\mathbbm{k})=0$ for $d<n$”.

###### Proof.

Use the equivalences
$B^{E_{1}}({\mathbb{R}}^{+},\epsilon)\simeq\mathbbm{k}\oplus\tilde{B}^{E_{1}}({\mathbb{R}})$
from [GKRW18a, Lemma 13.5] and $\tilde{B}^{E_{1}}({\mathbb{R}})\simeq\Sigma
Q^{E_{1}}_{\mathbb{L}}(\mathbb{R})$ [GKRW18a, Theorem 13.7], and note that
there are equivalences

$B^{E_{1}}({\mathbb{R}}^{+},\epsilon)\simeq
B^{E_{1}}(\overline{\mathbb{R}},\epsilon)\simeq
B(\mathbbm{k},\overline{\mathbb{R}},\mathbbm{k})$

where the first is induced by [GKRW18a, Proposition 12.9] and the second is
induced by the semi-simplicial map
$B^{E_{1}}_{\bullet}(\overline{\mathbb{R}},\epsilon)\to
B_{\bullet}(\mathbbm{k},\overline{\mathbb{R}},\mathbbm{k})$ induced on
$p_{1}$-simplices by the equivalence
$\mathcal{P}_{1}(p_{1})\overset{\sim}{\to}\\{*\\}$. ∎

######  Lemma 2.4.

We have $H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A})=0$ for $d<n-1$, and for
$d<2$ with the exception $H^{\overline{\mathbb{R}}}_{0,0}(A)=\mathbbm{k}$.

###### Proof.

Consider the homotopy cofibre sequence of left $\mathbb{A}$-modules, and hence
of $\overline{\mathbb{R}}$-modules, $\mathbb{A}\otimes
S^{1,0}\overset{-\cdot\sigma}{\to}\mathbb{A}\to\mathbb{A}/\sigma$. As
$\mathbb{A}$ is $E_{2}$ (and in fact commutative), the left-hand map is
nullhomotopic on $\overline{\mathbb{R}}$-module indecomposables, giving short
exact sequences

$0\longrightarrow H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A})\longrightarrow
H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A}/\sigma)\longrightarrow
H^{\overline{\mathbb{R}}}_{n-1,d-1}(\mathbb{A})\longrightarrow 0.$

As in the proof of Lemma 2.1 there is a homotopy cofibre sequence of left
$\overline{\mathbb{R}}$-modules
$S^{1,0}\otimes\mathbbm{k}\\{P\\}/\mathbbm{k}\\{\sigma\\}\to\mathbb{A}/\sigma\to
S^{0,0}\otimes\mathbbm{k}$, where the module structure on the outer terms is
via the augmentation. Together with the fact that
$H^{\overline{\mathbb{R}}}_{n,d}(\mathbbm{k})=0$ for $d<n$, which follows from
assumption (I) via Lemma 2.3, it follows that
$H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A}/\sigma)=0$ for $d<n-1$, and so the
same holds for $H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A})$.

The second claim follows from the fact that
$\overline{\mathbb{R}}\to\mathbb{A}$ is 1-connected in each grading, as it is
the 0-truncation, so the Hurewicz theorem [GKRW18a, Corollary 11.12] shows
that $H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A},\overline{\mathbb{R}})=0$ for
$d\leq 1$. ∎

######  Lemma 2.5.

Let $\mathbb{M}$ be an $\overline{\mathbb{R}}$-module which is discrete.

  1. (i)

We have
$h^{\overline{\mathbb{R}}}_{2}(\mathbb{M})\leq\max\\{3+h^{\overline{\mathbb{R}}}_{0}(\mathbb{M}),1+h^{\overline{\mathbb{R}}}_{1}(\mathbb{M})\\}$,
and for $d\geq 3$ we have $h^{\overline{\mathbb{R}}}_{d}(\mathbb{M})\leq
d+1+\max\\{h^{\overline{\mathbb{R}}}_{0}(\mathbb{M}),h^{\overline{\mathbb{R}}}_{1}(\mathbb{M})-1\\}$.

  2. (ii)

If $H^{\overline{\mathbb{R}}}_{3,2}(\mathbb{A})=0$, then we have
$h^{\overline{\mathbb{R}}}_{2}(\mathbb{M})\leq
2+\max\\{h^{\overline{\mathbb{R}}}_{0}(\mathbb{M}),h^{\overline{\mathbb{R}}}_{1}(\mathbb{M})-1\\}$.

  3. (iii)

In in addition $H^{\overline{\mathbb{R}}}_{4,3}(\mathbb{A})=0$, then we have
$h^{\overline{\mathbb{R}}}_{d}(\mathbb{M})\leq
d+\max\\{h^{\overline{\mathbb{R}}}_{0}(\mathbb{M}),h^{\overline{\mathbb{R}}}_{1}(\mathbb{M})-1\\}$
for $d\leq 3$.

###### Proof.

As $\mathbb{M}$ is discrete, the $\overline{\mathbb{R}}$-module structure on
it is restricted along $\overline{\mathbb{R}}\to\mathbb{A}$. Thus we have an
equivalence

$\mathbbm{k}\otimes_{\overline{\mathbb{R}}}^{\mathbb{L}}\mathbb{M}\simeq(\mathbbm{k}\otimes_{\overline{\mathbb{R}}}^{\mathbb{L}}\mathbb{A})\otimes^{\mathbb{L}}_{\mathbb{A}}\mathbb{M}.$

We may filter the right $\mathbb{A}$-module
$\mathbbm{k}\otimes_{\overline{\mathbb{R}}}^{\mathbb{L}}\mathbb{A}$ by its
grading: the associated graded may be identified with
$\mathbbm{k}\otimes_{\overline{\mathbb{R}}}^{\mathbb{L}}\mathbb{A}$, but the
$\mathbb{A}$-module structure is now via the augmentation
$\epsilon:\mathbb{A}\to\mathbbm{k}$. It gives a spectral sequence

$E^{1}_{n,p,q}=H_{p}((\mathbbm{k}\otimes_{\overline{\mathbb{R}}}^{\mathbb{L}}\mathbb{A})(q)\otimes^{\mathbb{L}}_{\mathbbm{k}}(\mathbbm{k}\otimes_{\mathbb{A}}^{\mathbb{L}}\mathbb{M})(n-q))\Longrightarrow
H_{n,p}(\mathbbm{k}\otimes^{\mathbb{L}}_{\overline{\mathbb{R}}}\mathbb{M})=H^{\overline{\mathbb{R}}}_{n,p}(\mathbb{M}).$

By the second part of Lemma 2.4 we have that
$H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A})$ vanishes for $d\leq 1$ except
that it is $\mathbbm{k}$ in bidgree $(0,0)$. From the Künneth theorem we then
see that

(2.1) $H^{\overline{\mathbb{R}}}_{n,0}(\mathbb{M})\cong
H^{\mathbb{A}}_{n,0}(\mathbb{M})\quad\text{and}\quad
H^{\overline{\mathbb{R}}}_{n,1}(\mathbb{M})\cong
H^{\mathbb{A}}_{n,1}(\mathbb{M}),$

so $h^{\overline{\mathbb{R}}}_{d}(\mathbb{M})=h^{\mathbb{A}}_{d}(\mathbb{M})$
for $d\leq 1$. More generally, we may approximate the $E^{1}$-page using the
Künneth spectral sequence

$F^{2}_{u,v}=\bigoplus_{v^{\prime}+v^{\prime\prime}=v}\mathrm{Tor}^{\mathbbm{k}}_{u}(H^{\overline{\mathbb{R}}}_{q,v^{\prime}}(\mathbb{A}),H^{\mathbb{A}}_{n-q,v^{\prime\prime}}(\mathbb{M}))\Longrightarrow
E^{1}_{n,u+v,q}.$

Thus we find that

$h^{\overline{\mathbb{R}}}_{d}(\mathbb{M})\leq\max_{v^{\prime}+v^{\prime\prime}\leq
d}\\{h^{\overline{\mathbb{R}}}_{v^{\prime}}(\mathbb{A})+h^{\mathbb{A}}_{v^{\prime\prime}}(\mathbb{M})\\}.$

Using that

  1. (i)

$h^{\mathbb{A}}_{v^{\prime\prime}}(\mathbb{M})\leq
v^{\prime\prime}-1+h_{1}^{\mathbb{A}}(\mathbb{M})=v^{\prime\prime}-1+h^{\overline{\mathbb{R}}}_{1}(\mathbb{M})$
for $v^{\prime\prime}\geq 2$, by Lemma 2.1, and

  2. (ii)

$h^{\overline{\mathbb{R}}}_{0}(\mathbb{A})=0$,
$h^{\overline{\mathbb{R}}}_{1}(\mathbb{A})=-\infty$, and
$h^{\overline{\mathbb{R}}}_{v^{\prime}}(\mathbb{A})\leq v^{\prime}+1$ for
$v^{\prime}\geq 2$, by Lemma 2.4,

we find the claimed estimates.

Under the additional assumption $h^{\overline{\mathbb{R}}}_{2}(\mathbb{A})\leq
2$ we find the slightly improved second estimate. Adding the further
assumption $h^{\overline{\mathbb{R}}}_{3}(\mathbb{A})\leq 3$ gives the third
estimate. ∎

The following can be used to verify the hypotheses in Lemma 2.5 (ii) and
(iii).

######  Lemma 2.6.

  1. (i)

If the map $H^{\overline{\mathbb{R}}}_{3,3}(\mathbbm{k})\to
H^{\mathbb{A}}_{3,3}(\mathbbm{k})$ is surjective then
$H_{3,2}^{\overline{\mathbb{R}}}(\mathbb{A})=0$.

  2. (ii)

If in addition the map $H^{\overline{\mathbb{R}}}_{4,4}(\mathbbm{k})\to
H^{\mathbb{A}}_{4,4}(\mathbbm{k})$ is surjective then
$H_{4,3}^{\overline{\mathbb{R}}}(\mathbb{A})=0$.

###### Proof.

Truncation gives a map of non-unital $E_{2}$-algebras
$\mathbb{R}\to\mathbb{I}$, which is 1-connected in each grading, i.e.
$\mathbb{I}$ can be obtained from $\mathbb{R}$ by attaching ($E_{1}$-)cells of
dimension $\geq 2$, and so $H^{E_{1}}_{n,d}(\mathbb{I},\mathbb{R})=0$ for
$d<2$. In addition $H^{E_{1}}_{n,d}(\mathbb{R})=0$ for $d<n-1$ by assumption.

Recall from the proof of Lemma 2.3 that
$H^{\overline{\mathbb{R}}}_{n,d}(\mathbbm{k})\cong
H_{n,d-1}^{E_{1}}(\mathbb{R})$ for $n>0$, and by the same argument
$H^{\mathbb{A}}_{n,d}(\mathbbm{k})\cong H_{n,d-1}^{E_{1}}(\mathbb{I})$. From
the latter and Corollary 2.2 it follows that $H^{E_{1}}_{n,d}(\mathbb{I})=0$
for $d<n-1$ (in fact for $d\neq n-1$). The long exact sequence for the pair
$(\mathbb{I},\mathbb{R})$ on $E_{1}$-homology therefore shows that
$H^{E_{1}}_{n,d}(\mathbb{I},\mathbb{R})=0$ for $d<n-1$.

In addition this sequence contains the portion

$H_{3,2}^{E_{1}}(\mathbb{R})\longrightarrow
H_{3,2}^{E_{1}}(\mathbb{I})\longrightarrow
H_{3,2}^{E_{1}}(\mathbb{I},\mathbb{R})\longrightarrow
H_{3,1}^{E_{1}}(\mathbb{R})=0.$

The assumption in part (i) of this lemma is that the left-hand map is
surjective, showing that $H_{3,2}^{E_{1}}(\mathbb{I},\mathbb{R})=0$. Similarly
there is a portion

$H_{4,3}^{E_{1}}(\mathbb{R})\longrightarrow
H_{4,3}^{E_{1}}(\mathbb{I})\longrightarrow
H_{4,3}^{E_{1}}(\mathbb{I},\mathbb{R})\longrightarrow
H_{4,2}^{E_{1}}(\mathbb{R})=0$

and the assumption in part (ii) of this lemma is that the left-hand map is
surjective, showing that $H_{4,3}^{E_{1}}(\mathbb{I},\mathbb{R})=0$.

As $E_{2}$-homology may be calculated from $E_{1}$-homology by a bar
construction (see [GKRW18a, Theorem 14.4]) we find that
$H^{E_{1}}_{n,d}(\mathbb{R})\to H^{E_{2}}_{n,d}(\mathbb{R})$ is surjective for
$d\leq n-1$, and similarly for $\mathbb{I}$, from which it follows that
$H^{E_{2}}_{n,d}(\mathbb{I},\mathbb{R})=0$ for $d<n-1$, and that on the line
$d=n-1$ there is an epimorphism

$H^{E_{1}}_{n,n-1}(\mathbb{I},\mathbb{R})\longrightarrow
H^{E_{2}}_{n,n-1}(\mathbb{I},\mathbb{R}).$

In particular $H_{2,1}^{E_{2}}(\mathbb{I},\mathbb{R})$ vanishes, and
$H_{3,2}^{E_{2}}(\mathbb{I},\mathbb{R})$ and
$H_{4,3}^{E_{2}}(\mathbb{I},\mathbb{R})$ also vanish under the assumptions in
parts (i) and (ii).

We apply [GKRW18a, Theorem 15.9] with $\rho(n)=n$ and

$\sigma(n)=\begin{cases}1&n=1\\\ 2&n=2\\\ 3&n=3\text{ and the hypothesis of
(i) holds}\\\ 4&n=4\text{ and the hypothesis of (ii) holds}\\\
n-1&\text{else},\end{cases}$

showing that there is a map

$H^{\overline{\mathbb{R}}}_{n,d}(\mathbb{A},\overline{\mathbb{R}})\longrightarrow
H^{E_{2}}_{n,d}(\mathbb{I},\mathbb{R})$

which is an isomorphism for $d<(\sigma*\sigma)(n)$. If the hypothesis of (i)
holds then $(\sigma*\sigma)(3)=3$ and so
$H^{\overline{\mathbb{R}}}_{3,2}(\mathbb{A})=H^{\overline{\mathbb{R}}}_{3,2}(\mathbb{A},\overline{\mathbb{R}})\overset{\sim}{\to}H^{E_{2}}_{3,2}(\mathbb{I},\mathbb{R})=0$.
If in addition the hypothesis of (ii) holds then $(\sigma*\sigma)(4)=4$ and so
$H^{\overline{\mathbb{R}}}_{4,3}(\mathbb{A})=H^{\overline{\mathbb{R}}}_{4,3}(\mathbb{A},\overline{\mathbb{R}})\overset{\sim}{\to}H^{E_{2}}_{4,3}(\mathbb{I},\mathbb{R})=0$.
∎

###  2.3. The $A$-module part of Theorem 1.5

We consider
$\mathbbm{k}\simeq\mathbbm{k}\otimes^{\mathbb{L}}_{\overline{\mathbb{R}}}\overline{\mathbb{R}}$,
and give the left $\overline{\mathbb{R}}$-module $\overline{\mathbb{R}}$ its
Postnikov filtration. This leads to a spectral sequence

$E^{2}_{n,s,t}=H_{n,s}^{\overline{\mathbb{R}}}(H_{*,t}(\overline{\mathbb{R}}))\Longrightarrow
H_{n,t+s}(\mathbbm{k})$

converging very nearly to zero, with differentials $d^{r}:E^{r}_{n,s,t}\to
E^{r}_{n,s-r,t+r-1}$. We have $H_{*,0}(\overline{\mathbb{R}})=A$, and so
$E^{1}_{*,s,0}=H^{\overline{\mathbb{R}}}_{*,s}(\mathbb{A})$, which is
supported in degrees $\leq s+1$ by Lemma 2.4. Furthermore, it vanishes for
$s=1$ and is $\mathbbm{k}$ for $s=0$. We consider the following chart
depicting (upper bounds for) the supports of the terms in this spectral
sequence.

[ name = BarSS, homological Serre grading, classes = draw = none , x range =
04, y range = 01, xscale = 2, yscale=0.9, y axis gap = 35pt]

[”0”](0,0) [”-∞”](1,0) [”≤3”](2,0) [”≤4”](3,0) [”≤5”](4,0) [”≤6”](5,0)

[”h_0^¯R(H_*,1(¯R))”](0,1) [”h_1^¯R(H_*,1(¯R))”](1,1)

2̣(2,0) 2̣(3,0)

[ name = BarSS, page = 2 ]

As the spectral sequence converges to zero in positive total degrees, the two
indicated $d^{2}$-differentials must be epimorphisms. This shows that
$h_{0}^{\overline{\mathbb{R}}}(H_{*,1}({\overline{\mathbb{R}}}))\leq 3$ and
$h_{1}^{\overline{\mathbb{R}}}(H_{*,1}({\overline{\mathbb{R}}}))\leq 4$, so by
Lemma 2.5 (i) we also have

$h_{s}^{\overline{\mathbb{R}}}(H_{*,1}({\overline{\mathbb{R}}}))\leq 4+s\text{
for }s\geq 2.$

This allows us to fill in the 1-row. Now we consider the differentials
entering the first two positions of the 2-row:

[ name = BarSS2, homological Serre grading, classes = draw = none , x range =
04, y range = 02, xscale = 2, yscale=0.9, y axis gap = 35pt]

[”0”](0,0) [”-∞”](1,0) [”≤3”](2,0) [”≤4”](3,0) [”≤5”](4,0) [”≤6”](5,0)

[”≤3”](0,1) [”≤4”](1,1) [”≤6”](2,1) [”≤7”](3,1) [”≤8”](4,1)

[”h_0^¯R(H_*,2(¯R))”](0,2) [”h_1^¯R(H_*,2(¯R))”](1,2)

[ name = BarSS2, page = 2—0 ]

By considering all the differentials that can arrive at these groups, we see
that $h_{0}^{\overline{\mathbb{R}}}(H_{*,2}({\overline{\mathbb{R}}}))\leq 6$
and $h_{1}^{\overline{\mathbb{R}}}(H_{*,2}({\overline{\mathbb{R}}}))\leq 7$,
and Lemma 2.5 (i) allows us to again complete this row by
$h_{s}^{\overline{\mathbb{R}}}(H_{*,2}({\overline{\mathbb{R}}}))\leq 7+s$ for
$s\geq 2$. In this way, we see that

${h_{0}^{\overline{\mathbb{R}}}(H_{*,t}({\overline{\mathbb{R}}}))}\leq
3t,\quad{h_{1}^{\overline{\mathbb{R}}}(H_{*,t}({\overline{\mathbb{R}}}))}\leq
3t+1,\quad{h_{s}^{\overline{\mathbb{R}}}(H_{*,t}({\overline{\mathbb{R}}}))}\leq
3t+1+s\text{ for }s\geq 2.$

Using (2.1) the same follows for
$h_{i}^{A}(H_{*,d}({\overline{\mathbb{R}}}))$. This proves the basic case.

Suppose now that $H_{3,2}^{\overline{\mathbb{R}}}(\mathbb{A})=0$. This allows
us to improve the start of the induction, as $E^{1}_{*,2,0}$ is supported in
degrees $\leq 2$, and also allows us to use Lemma 2.5 (ii). The same line of
reasoning as above then leads to

$\displaystyle{h_{0}^{A}(H_{*,t}({\overline{\mathbb{R}}}))}\leq 3t-1,$
$\displaystyle\quad\quad{h_{1}^{A}(H_{*,t}({\overline{\mathbb{R}}}))}\leq
3t+1,$ $\displaystyle{h_{2}^{A}(H_{*,t}({\overline{\mathbb{R}}}))}\leq 3t+2,$
$\displaystyle\quad\quad{h_{s}^{A}(H_{*,t}({\overline{\mathbb{R}}}))}\leq
3t+1+s\text{ for }s\geq 3,$

depicted as follows.

[ name = BarSS3, homological Serre grading, classes = draw = none , x range =
04, y range = 03, xscale = 2, yscale=0.9, y axis gap = 35pt]

[”0”](0,0) [”-∞”](1,0) [”≤2”](2,0) [”≤4”](3,0) [”≤5”](4,0) [”≤6”](5,0)

[”≤2”](0,1) [”≤4”](1,1) [”≤5”](2,1) [”≤7”](3,1) [”≤8”](4,1)

[”≤5”](0,2) [”≤7”](1,2) [”≤8”](2,2) [”≤10”](3,2) [”≤11”](4,2)

[”≤8”](0,3) [”≤10”](1,3) [”≤11”](2,3) [”≤13”](3,3) [”≤14”](4,3)

[ name = BarSS3, page = 2 ]

Finally, suppose that in addition
$H_{4,3}^{\overline{\mathbb{R}}}(\mathbb{A})=0$. This allows us to further
improve the start of the induction, as $E^{1}_{*,3,0}$ is supported in degrees
$\leq 3$, and also allows us to use Lemma 2.5 (iii). The same line of
reasoning as above then leads to

${h_{s}^{A}(H_{*,t}({\overline{\mathbb{R}}}))}\leq 2t+s\text{ for }s\geq 1$

depicted as follows.

[ name = BarSS4, homological Serre grading, classes = draw = none , x range =
04, y range = 03, xscale = 2, yscale=0.9, y axis gap = 35pt]

[”0”](0,0) [”-∞”](1,0) [”≤2”](2,0) [”≤3”](3,0) [”≤5”](4,0) [”≤6”](5,0)

[”≤2”](0,1) [”≤3”](1,1) [”≤4”](2,1) [”≤5”](3,1) [”≤7”](4,1)

[”≤4”](0,2) [”≤5”](1,2) [”≤6”](2,2) [”≤7”](3,2) [”≤9”](4,2)

[”≤6”](0,3) [”≤7”](1,3) [”≤8”](2,3) [”≤9”](3,3) [”≤11”](4,3)

[ name = BarSS4, page = 2 ]

###  2.4. The ordinary homological stability part of Theorem 1.5

The remaining part of Theorem 1.5 is the statement that
$H_{n,d}(\overline{\mathbb{R}}/\rho)=0$ for $d<\tfrac{n-1}{2}$ under
assumptions (I), (II), and (III). (This cannot, of course, be deduced from the
statement that the $A$-module $H_{*,d}(\mathbb{R})$ is generated in gradings
$\leq 3d-1$ and presented in gradings $\leq 3d+1$, which we proved in the
previous section under these assumptions.)

As in the proof of Lemma 2.6 we have
$H_{n,d}^{E_{2}}(\mathbb{I},\mathbb{R})=0$ for $d<2$, for $d<n-1$, and for
$(n,d)=(3,2)$. Thus by [GKRW18a, Theorem 11.21] we can construct a minimal
relative CW-approximation
$\mathbb{R}\to\mathbb{C}\overset{\sim}{\to}\mathbb{I}$, and thereby obtain a
spectral sequence

$E^{1}_{n,p,q}=H_{n,p+q,q}(\overline{\mathbb{R}}/\rho[0]\otimes
E_{\infty}^{+}(\bigoplus_{\alpha\in
I}S^{n_{\alpha},d_{\alpha},d_{\alpha}}))\Longrightarrow
H_{n,p+q}({\mathbb{A}}/\rho)$

with $d_{\alpha}\geq 2$, $d_{\alpha}\geq n_{\alpha}-1$, and
$(n_{\alpha},d_{\alpha})\neq(3,2)$. The abutment vanishes for $p+q>0$ as well
as for $n>1$.

[ name = cells, homological Serre grading, classes = draw = none , x range =
06, y range = 05, xscale = 1, yscale=0.7]

[”*”](1,2) [”*”](2,2)

[”*”](1,3) [”*”](2,3) [”*”](3,3) [”*”](4,3)

[”*”](1,4) [”*”](2,4) [”*”](3,4) [”*”](4,4) [”*”](5,4)

[”*”](1,5) [”*”](2,5) [”*”](3,5) [”*”](4,5) [”*”](5,5) [”*”](6,5)

[ name = cells, page = 2 ]

Figure 1. The $E_{\infty}$-cells of $E_{\infty}^{+}(\bigoplus_{\alpha\in
I}S^{n_{\alpha},d_{\alpha},d_{\alpha}})$ are supported in the indicated
bidegrees, with homological degree vertically and grading horizontally.

######  Lemma 2.7.

$H_{n^{\prime\prime},d^{\prime\prime},r}(E_{\infty}^{+}(\bigoplus_{\alpha\in
I}S^{n_{\alpha},d_{\alpha},d_{\alpha}}))$ is non-trivial only if
$(n^{\prime\prime},d^{\prime\prime},r)=(0,0,0)$ or if $d^{\prime\prime}\geq
r\geq 2$ and $d^{\prime\prime}-1\geq\tfrac{1}{2}n^{\prime\prime}$.

###### Proof.

The object $X:=\bigoplus_{\alpha\in I}S^{n_{\alpha},d_{\alpha},d_{\alpha}}$
has homology supported in tridegrees $(n^{\prime\prime},d^{\prime\prime},r)$
satisfying $d^{\prime\prime}=r$, and $n^{\prime\prime}$ and $d^{\prime}$ as in
Figure 1. In particular, it is supported in the region $R$ of tridegrees
$(n^{\prime\prime},d^{\prime\prime},r)$ satisfying $d^{\prime\prime}\geq r\geq
2$ and $d^{\prime\prime}-1\geq\tfrac{1}{2}n^{\prime\prime}$. The region of
tridegrees $R$ is closed under addition. As the homology of $X$ is free over
$\mathbbm{k}$, by the Künneth theorem it follows that the homology of
$X^{\otimes k}$ is also supported in the region of tridegrees $R$. The region
of tridegrees $R$ is also upwards-closed in the homological degree direction,
so it follows from the homotopy orbits spectral sequence that the homology of
$(X^{\otimes k})_{h\mathfrak{S}_{k}}$ is also supported in the region of
tridegrees $R$. As $E_{\infty}^{+}(X)\simeq\bigoplus_{k\geq 0}(X^{\otimes
k})_{h\mathfrak{S}_{k}}$, it follows that apart from the $k=0$ summand this
has homology supported in the region of tridegrees $R$. ∎

We show that $H_{n,d}(\overline{\mathbb{R}}/\rho)=0$ for $d<\tfrac{1}{2}(n-1)$
by induction on $d$: it holds for $d=0$, as
$H_{*,0}(\overline{\mathbb{R}}/\rho)=H_{*,0}({\mathbb{A}}/\rho)=A/\rho$ is
supported in gradings 0 and 1 by the explicit presentation of the algebra $A$.
Note that $E^{1}_{n,d,0}=H_{n,d}(\overline{\mathbb{R}}/\rho)$ and consider
differentials

$d^{r}:E^{r}_{n,d-r+1,r}\longrightarrow E^{r}_{n,d,0},$

with $r>0$. There is a Künneth spectral sequence

$\bigoplus_{s\geq
0}\bigoplus_{\begin{subarray}{c}n^{\prime}+n^{\prime\prime}=n\\\
d^{\prime}+d^{\prime\prime}=d+1-s\end{subarray}}\mathrm{Tor}^{\mathbbm{k}}_{s}(H_{n^{\prime},d^{\prime}}(\overline{\mathbb{R}}/\rho),H_{n^{\prime\prime},d^{\prime\prime},r}(E_{\infty}^{+}(\bigoplus_{\alpha\in
I}S^{n_{\alpha},d_{\alpha},d_{\alpha}})))\Rightarrow E^{1}_{n,d-r+1,r}.$

Using Lemma 2.7, as $r>0$ we see that if
$H_{n^{\prime\prime},d^{\prime\prime},r}(E_{\infty}^{+}(\bigoplus_{\alpha\in
I}S^{n_{\alpha},d_{\alpha},d_{\alpha}}))$ is non-trivial then

$d^{\prime}-\tfrac{1}{2}(n^{\prime}-1)=d+1-s-d^{\prime\prime}-\tfrac{1}{2}(n-n^{\prime\prime}-1)=\underbrace{d-\tfrac{1}{2}(n-1)}_{<0}+\underbrace{(\tfrac{1}{2}n^{\prime\prime}+1-d^{\prime\prime})}_{\leq
0}-s,$

so $d^{\prime}<\tfrac{1}{2}(n^{\prime}-1)$. But then as $d^{\prime}<d$ it
follows by induction that
$H_{n^{\prime},d^{\prime}}(\overline{\mathbb{R}}/\rho)$ is trivial. Thus
$E^{1}_{n,d-r+1,r}=0$ for $r>0$ when $d<\tfrac{1}{2}(n-1)$, so no non-zero
differentials can enter $E^{r}_{n,d,0}$. As $H_{n,d}(\mathbb{A}/\rho)=0$ for
$d<\tfrac{1}{2}(n-1)$, it follows that $E^{1}_{n,d,0}=0$.

##  3\. Application to Dedekind domains: Proof of Theorem 1.2

Theorem 1.2 will be deduced from Theorem 1.5 by taking the
$E_{\infty}$-algebra $\mathbf{BGL}$, constructed in [GKRW18a, Section 18.2],
having

$\mathbf{BGL}(n)=\bigoplus_{\begin{subarray}{c}[M]\\\ M\text{ f.g.\
projective}\\\ \mathrm{rk}(M)=n\end{subarray}}\mathbbm{k}[B\mathrm{GL}(M)].$

Axiom (I) of Theorem 1.5 is already verified in [GKRW18a, Section 18.2]: it
follows from the connectivity of Charney’s “split Tits building” [Cha80,
Theorem 1.1]. Axiom (II) of Theorem 1.5 holds with
$P=\mathrm{Pic}(\mathcal{O})$: this is simply a reformulation of the
classification of finitely-generated projective $\mathcal{O}$-modules (which
was the inspiration for this axiom). Verifying axioms (III) and (IV) is more
involved, and requires preparation.

###  3.1. The Tits and split Tits buildings

######  Definition 3.1.

Let $M$ be a projective $\mathcal{O}$-module. Let $T(M)$ denote the poset
consisting on nonzero proper direct summands $P$ of $M$, ordered by inclusion.
Let $\widetilde{T}(M)$ denote the poset consisting of pairs $(P,Q)$ of nonzero
submodules of $M$ such that $M=P\oplus Q$, with order relation
$(P,Q)\leq(P^{\prime},Q^{\prime})$ when $P\leq P^{\prime}$ and $Q^{\prime}\leq
Q$.

Writing $T(\mathrm{frac}(\mathcal{O})\otimes_{\mathcal{O}}M)$ for the poset of
nonzero proper vector subspaces of
$\mathrm{frac}(\mathcal{O})\otimes_{\mathcal{O}}M$, the map

$\mathrm{frac}(\mathcal{O})\otimes_{\mathcal{O}}-:T(M)\longrightarrow
T(\mathrm{frac}(\mathcal{O})\otimes_{\mathcal{O}}M)$

is an isomorphism of posets, with inverse given by $V\mapsto V\cap M$. The
Solomon–Tits theorem shows that
$T(\mathrm{frac}(\mathcal{O})\otimes_{\mathcal{O}}M)$ is Cohen–Macaulay of
dimension $(\mathrm{rk}(M)-2)$, and in particular is homotopy equivalent to a
wedge of $(\mathrm{rk}(M)-2)$-spheres. The _Steinberg module_ is its unique
nontrivial reduced homology group

$\mathrm{St}(M):=\tilde{H}_{n-2}(T(M);\mathbb{Z}).$

Charney has shown [Cha80, Theorem 1.1] that the same holds for
$\widetilde{T}(M)$:

######  Theorem 3.2 (Charney).

If $M$ has rank $n$ then $\widetilde{T}(M)$ is homotopy equivalent to a wedge
of $(n-2)$-spheres.

The _split Steinberg module_ is its unique nontrivial reduced homology group,

$\widetilde{\mathrm{St}}(M):=\tilde{H}_{n-2}(\widetilde{T}(M);\mathbb{Z}).$

There is a map of posets $(P,Q)\mapsto Q:\widetilde{T}(M)\to{T}(M)^{op}$.

######  Theorem 3.3.

For $\mathrm{rk}(M)\leq 4$ the induced map
$\widetilde{\mathrm{St}}(M)\to{\mathrm{St}}(M)$ is surjective.

######  Remark 3.4.

We only require this result in this range of ranks, so have not tried hard
(i.e. have not succeeded) to prove it for general $M$.

###### Proof of Theorem 3.3.

We apply the slight extension [GKRW18b, Theorem 4.1] of Quillen’s [Qui78,
Theorem 9.1] to the map of posets $f:\widetilde{T}(M)\to{T}(M)^{op}$ given by
$f(P,Q)=Q$, using the height function $t(V):=\mathrm{rk}(V)-1$. We have that
${T}(M)^{op}$ is $(\mathrm{rk}(M)-2)$-spherical, and for each $V\in T(M)$,
$[{T}(M)^{op}]_{>V}\cong T(V)^{op}$ is $(t(V)-1)$-spherical.

For each $V\in T(M)$, we have $f_{\leq
V}=\\{(P,Q)\in\widetilde{T}(M)\,|\,V\subseteq Q\\}$. Following [GKRW18b] we
write $\mathcal{S}^{E_{1}}(\cdot,V\subseteq\cdot\,|\,M)$ for this poset. In
order to apply [GKRW18b, Theorem 4.1] we require it to be
$(\mathrm{rk}(M)-\mathrm{rk}(V)-1)$-spherical. For $\mathrm{rk}(M)\leq 4$
there are three cases.

If $\mathrm{rk}(V)=\mathrm{rk}(M)-1$ then
$\mathcal{S}^{E_{1}}(\cdot,V\subseteq\cdot\,|\,M)$ is the discrete poset
consisting of pairs $(L,V)$ with $L\oplus V=M$. In particular it is
0-spherical.

If $\mathrm{rk}(V)=1$ then $\mathcal{S}^{E_{1}}(\cdot,V\subseteq\cdot\,|\,M)$
agrees with the complex called $[M\,|\,V]$ by Charney [Cha80], so is
$(\mathrm{rk}(M)-2)$-spherical by [Cha80, Theorem 1.1].

As long as $\mathrm{rk}(M)\leq 4$ the only remaining case is
$\mathrm{rk}(M)=4$ and $\mathrm{rk}(V)=2$, in which case we require
$\mathcal{S}^{E_{1}}(\cdot,V\subseteq\cdot\,|\,M)$ to be 1-spherical. It is
clearly 1-dimensional, so we must show that it is 0-connected. Every element
of this poset either has the form $(U,V)$ or the form $(L,W)$ for $W\supseteq
V$ and $L$ of rank 1. In the second case, writing $W=L^{\prime}\oplus V$, we
have $(L,W)\leq(L\oplus L^{\prime},V)$. So it suffices to show that any
$(U,V)$ can be connected by a path to some fixed $(U_{0},V)$. We will deduce
this from the results of Charney used above. Let $L_{0}\leq V$ be a rank 1
direct summand, and choose a complement $L_{1}$ in $V$ so that $U\oplus
L_{0}\oplus L_{1}=M$. There are maps of posets

${\mathcal{S}^{E_{1}}(\cdot,L_{1}\subseteq\cdot\,|\,U\oplus
L_{1})}$${\mathcal{S}^{E_{1}}(\cdot,V\subseteq\cdot\,|\,M)}$${\mathcal{S}^{E_{1}}(\cdot,V/L_{0}\subseteq\cdot\,|\,M/L_{0}).}$$\scriptstyle{(A,B)\mapsto(A,L_{0}\oplus
B)}$$\scriptstyle{\simeq}$$\scriptstyle{(P,Q)\mapsto(\tfrac{P\oplus
L_{0}}{L_{0}},\tfrac{Q}{L_{0}})}$

Now $L_{1}$ has rank 1 so by [Cha80, Theorem 1.1]
$\mathcal{S}^{E_{1}}(\cdot,V/L_{0}\subseteq\cdot\,|\,M/L_{0})$ is path-
connected. Thus we may find a path in it from $(\tfrac{U\oplus
L_{0}}{L_{0}},\tfrac{V}{L_{0}})$ to $(\tfrac{U_{0}\oplus
L_{0}}{L_{0}},\tfrac{V}{L_{0}})$. Considering this as a path in the isomorphic
poset $\mathcal{S}^{E_{1}}(\cdot,L_{1}\subseteq\cdot\,|\,U\oplus L_{1})$, it
starts at $(U,L_{1})$ and ends at some $(U^{\prime},L_{1})$ for which
$\tfrac{U^{\prime}\oplus L_{0}}{L_{0}}=\tfrac{U_{0}\oplus L_{0}}{L_{0}}$.
Applying the horizontal map, this gives a path from $(U,V)$ to a
$(U^{\prime},V)$ with $U^{\prime}\oplus L_{0}=U_{0}\oplus L_{0}$. Now the
latter is a rank 3 direct summand of $M$ containing the pair of rank 2 direct
summands $U^{\prime}$ and $U_{0}$, and so $U^{\prime}\cap U_{0}$ is a direct
summand (by [Cha80, Lemma 1.2]) of rank 1 or 2. If $U^{\prime}\cap U_{0}$ has
rank 2 then $U^{\prime}=U_{0}$ and so we have found the required path. If
$U^{\prime}\cap U_{0}$ has rank 1 then

$(U^{\prime},V)\geq(U^{\prime}\cap U_{0},X)\leq(U_{0},V)$

for $X$ any choice of complement of the line $U^{\prime}\cap U_{0}$ containing
$V$. In either case we have found a path from $(U,V)$ to $(U_{0},V)$, as
required.

Thus [GKRW18b, Theorem 4.1] applies. It shows that
$\widetilde{\mathrm{St}}(M)$ has a filtration whose first filtration quotient
is ${\mathrm{St}}(M)$, so in particular the natural map between them is
surjective. ∎

###  3.2. Reductive Borel–Serre spaces

Before continuing, we introduce the reductive Borel–Serre spaces example. By
the main theorem of [Jan24], there is an $E_{\infty}$-algebra in topological
spaces whose $\mathbbm{k}$-linearisation $\mathbf{RBS}$ has

$\mathbf{RBS}(n)\simeq\bigoplus_{\begin{subarray}{c}[M]\\\ M\text{ f.g.\
projective},\mathrm{rk}(M)=n\end{subarray}}\mathbbm{k}[|\mathrm{RBS}(M)|].$

By [Jan24, Theorem 10.11] there is an identification

$H^{E_{1}}_{n,d}(\mathbf{RBS})\cong\bigoplus_{\mathrm{rk}(M)=n}\tilde{H}_{d}(\Sigma
T(M)/\\!\\!/\mathrm{GL}(M);\mathbbm{k}),$

where $\Sigma$ denotes the unreduced suspension, and $/\\!\\!/$ the pointed
homotopy orbits. By the Solomon–Tits theorem, $\Sigma T(M)$ is homotopy
equivalent to a wedge of $(n-1)$-spheres, and yields the Steinberg module
$\mathrm{St}(M)$ as its top homology. Thus we can write this as

$H^{E_{1}}_{n,d}(\mathbf{RBS})\cong\bigoplus_{\mathrm{rk}(M)=n}{H}_{d-n+1}(\mathrm{GL}(M);\mathrm{St}(M)\otimes\mathbbm{k}).$

In particular these groups vanish for $d<n-1$. This verifies axiom (I).

There is a map of $E_{\infty}$-algebras $\mathbf{BGL}\to\mathbf{RBS}$ (by
consulting Jansen’s construction), which in each grading is induced by the
natural maps $B\mathrm{GL}(M)\to|\mathrm{RBS}(M)|$. As these spaces are both
connected, we have

$A_{\mathrm{Pic}(\mathcal{O})}\cong H_{*,0}(\overline{\mathbf{BGL}})\cong
H_{*,0}(\overline{\mathbf{RBS}}),$

so $\mathbf{RBS}$ also satisfies axiom (II).

###  3.3. Verifying axioms (III) and (IV)

We will verify these axioms for both $\mathbf{BGL}$ and $\mathbf{RBS}$. As we
have just mentioned, the $0$-truncations of $\overline{\mathbf{BGL}}$ and
$\overline{\mathbf{RBS}}$ are the same, namely the discrete
$E_{\infty}$-algebra $\mathbb{A}$ presented by the quadratic algebra
$A:=A_{\mathrm{Pic}(\mathcal{O})}$. In particular there are
$E_{\infty}$-algebra maps

$\overline{\mathbf{BGL}}\longrightarrow\overline{\mathbf{RBS}}\longrightarrow\mathbb{A}$

and so induced maps

$H^{\overline{\mathbf{BGL}}}_{n,d}(\mathbbm{k})\longrightarrow
H^{\overline{\mathbf{RBS}}}_{n,d}(\mathbbm{k})\longrightarrow
H^{\mathbb{A}}_{n,d}(\mathbbm{k}).$

All three groups vanish for $d<n$: the first two by assumption (I) and Lemma
2.3 in these two examples, and the third in fact vanishes for $d\neq n$, by
Corollary 2.2.

For $d=n$ these maps takes the form

(3.1)
$\bigoplus_{\mathclap{\mathrm{rk}(M)=n}}H_{0}(\mathrm{GL}(M);\widetilde{\mathrm{St}}(M)\otimes\mathbbm{k})\to\bigoplus_{\mathclap{\mathrm{rk}(M)=n}}H_{0}(\mathrm{GL}(M);{\mathrm{St}}(M)\otimes\mathbbm{k})\to\mathrm{Tor}^{A}_{n,n}(\mathbbm{k},\mathbbm{k}).$

######  Theorem 3.5.

The right-hand map in (3.1) is surjective.

###### Proof.

Church, Farb, and Putman [CFP19, §5.2] have constructed for each finitely-
generated projective $\mathcal{O}$-module $M$ of rank $n$ a certain map

$H_{0}(\mathrm{GL}(M);\mathrm{St}(M))\longrightarrow\widetilde{H}_{n-2}(|X_{n-1}(\mathrm{Pic}(\mathcal{O}))|;\mathbb{Z})$

and have shown [CFP19, Proposition 5.5] that it is surjective. We will explain
why the right-hand map above is simply the sum, over isomorphism classes of
rank $n$ projective $\mathcal{O}$-modules, of these maps (tensored with
$\mathbbm{k}$): then the claim follows.

To do so it is simplest to put $\mathbb{A}$ in the framework of “categories
with filtrations” of [CJ24, §7.1] and rely on the functoriality of that
framework. In fact it is awkward to make $\mathbb{A}$ fit exactly into that
setting, so we make the following observation. It is not actually necessary to
have an underlying category to make many of the constructions involving
“categories with filtrations”: rather, it suffices to have a collection of
objects as well as a class of “short exact sequences”, satisfying certain
properties. Among these properties one must be able to compose admissible
monomorphisms, and also compose admissible epimorphisms, but for example one
never needs to compose morphisms of the two kinds. Formally, one can view this
as as a pair $\mathcal{C}=(\mathcal{C}^{mono},\mathcal{C}^{epi})$ of
categories with the same objects, and with certain pairs of arrows
$(a\hookrightarrow b,b\twoheadrightarrow c)$ distinguished as “short exact
sequences”, again satisfying certain properties. This data suffices to define
$\mathscr{M}_{RBS}(\mathcal{C})$ as in [Jan24, Definition 6.1].

Now define $\mathcal{A}=(\mathcal{A}^{mono},\mathcal{A}^{epi})$ to have common
set of objects the isomorphism classes of finitely-generated projective
$\mathcal{O}$-modules. Let

$\mathcal{A}^{mono}([P],[Q]):=\begin{cases}*&\mathrm{rk}(P)<\mathrm{rk}(Q)\text{
or }[P]=[Q]\\\
\emptyset&\text{otherwise}\end{cases}=:\mathcal{A}^{epi}([Q],[P]),$

so both categories are posets. The short exact sequences are those pairs
$([P]\hookrightarrow[Q],[Q]\twoheadrightarrow[R])$ such that $[Q]=[P\oplus
R]$.

The category $\mathcal{P}(\mathcal{O})$ of finitely-generated projective
$\mathcal{O}$-modules is a category with filtrations as in [CJ24, Example 7.3
(3)], and can be considered in the form
$\mathcal{P}(\mathcal{O})=(\mathcal{P}(\mathcal{O})^{mono},\mathcal{P}(\mathcal{O})^{epi})$
as above. Sending a module to its isomorphism class then gives a morphism

$\phi:(\mathcal{P}(\mathcal{O})^{mono},\mathcal{P}(\mathcal{O})^{epi})\longrightarrow(\mathcal{A}^{mono},\mathcal{A}^{epi}).$

This induces a map of $E_{1}$-algebras

(3.2)
$\mathbf{RBS}:=\mathscr{M}_{RBS}(\mathcal{P}(\mathcal{O}))\longrightarrow\mathscr{M}_{RBS}(\mathcal{A}).$

We can identify the latter as (see [Jan24, Definition 6.1, Notation 6.4])

$\mathrm{RBS}([P])\simeq\begin{cases}\text{Objects are lists
}([Q_{1}],\ldots,[Q_{n}])\text{ of non-zero isomorphism classes}\\\ \text{such
that }[P]=[\bigoplus_{i}Q_{i}].\\\ \text{A morphism
}([Q_{1}],\ldots,[Q_{n}])\to([R_{1}],\ldots,[R_{m}])\text{ is an order
preserving }\\\
\theta:\\{1,\ldots,n\\}\twoheadrightarrow\\{1,\ldots,m\\}\text{ such that
}[R_{j}]=[\bigoplus_{i\in\theta^{-1}(j)}Q_{i}].\end{cases}$

Observe that this is a poset, and the one-element list $([P])$ is terminal so
$|\mathrm{RBS}([P])|\simeq*$. Thus $\mathscr{M}_{RBS}(\mathcal{A})$ has one
contractible path component for each isomorphism class of finitely-generated
projective $\mathcal{O}$-module, i.e. it is $\mathbb{A}$. The map (3.2) is
thus identified with the truncation map $\mathbf{RBS}\to\mathbb{A}$.

Now $\partial\mathrm{RBS}([P])$ is the full subposet of $\mathrm{RBS}([P])$ on
all objects other than $[P]$. We will explain how to identify it with the
poset of simplices of the the realisation of the poset
$X_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$ defined in [CFP19, Example 5.3], i.e.
with the barycentric subdivision $sdX_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$ of
this poset.

Recall that the poset $X_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$ is given by
$\\{1,2,\ldots,rk(P)-1\\}\times\mathrm{Pic}(\mathcal{O})$, with order relation
$(r,[L])<(r^{\prime},[L^{\prime}])$ if and only if $r<r^{\prime}$. We assign
to a list $([Q_{1}],\ldots,[Q_{n}])$ in the poset $\mathrm{RBS}([P])$ the
chain

$(\mathrm{rk}(Q_{1}),[\det(Q_{1})])<(\mathrm{rk}(Q_{1}\oplus
Q_{2}),[\det(Q_{1}\oplus
Q_{2})])<\cdots<(\mathrm{rk}(\oplus_{i}Q_{i}),[\det(\oplus_{i}Q_{i})])$

in $X_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$, i.e. an element of
$sdX_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$. Conversely, we assign to a chain

$(r_{1},[L_{1}])<(r_{2},[L_{2}])<\ldots<(r_{n},[L_{n}])$

in $X_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$ the list

$([\mathcal{O}^{r_{1}-1}\oplus
L_{1}],[\mathcal{O}^{r_{2}-r_{1}-1}\oplus(L_{2}\otimes
L_{1}^{-1})],\ldots,[\mathcal{O}^{r_{n}-r_{n-1}-1}\oplus(L_{n}\otimes
L_{n-1}^{-1})]).$

This defines a bijection between the poset $\partial\mathrm{RBS}([P])$ and the
barycentrically subdivided poset $sdX_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$,
and one checks that it preserves the order relation.

We can form the diagram

${{|\mathcal{F}(P)\setminus\emptyset|/\\!\\!/\mathrm{GL}(P)}}$${{|\mathcal{F}(P)|/\\!\\!/\mathrm{GL}(P)}}$${{|\partial\mathrm{RBS}(P)|}}$${{|\mathrm{RBS}(P)|}}$${{|\partial\mathrm{RBS}([P])|}}$${{|\mathrm{RBS}([P])|}}$

where the top square is [CJ24, Corollary 5.8] and is a homotopy pushout, and
the bottom square is induced by $\phi$. As in [Jan24, Observation 9.15], the
poset $\mathcal{F}(P)\setminus\emptyset$ is the barycentric subdivision of the
Tits poset $\mathcal{T}(P)$, and with the discussion above the composition

$|\mathcal{F}(P)\setminus\emptyset|\longrightarrow|\mathcal{F}(P)\setminus\emptyset|/\\!\\!/\mathrm{GL}(P)\longrightarrow|\partial\mathrm{RBS}(P)|\longrightarrow|\partial\mathrm{RBS}([P])|$

is identified with the map induced on barycentric subdivisions by the map
$\psi:\mathcal{T}(P)\to X_{rk(P)-1}(\mathrm{Pic}(\mathcal{O}))$ of [CFP19,
Section 5.2]. Using [Jan24, Theorem 10.10] we have

$\displaystyle H^{E_{1}}_{n,n-1}(\mathbf{RBS})$
$\displaystyle=\bigoplus_{[P],\mathrm{rk}(P)=n}H_{n-1}(|\mathrm{RBS}(P)|,|\partial\mathrm{RBS}(P)|)$
$\displaystyle\cong\bigoplus_{[P],\mathrm{rk}(P)=n}H_{0}(\mathrm{GL}(P);\tilde{H}_{n-1}(\mathcal{T}(P)))$
$\displaystyle H^{E_{1}}_{n,n-1}(\mathbb{A})$
$\displaystyle=\bigoplus_{[P],\mathrm{rk}(P)=n}H_{n-1}(|\mathrm{RBS}([P])|,|\partial\mathrm{RBS}([P])|)$
$\displaystyle\cong\bigoplus_{[P],\mathrm{rk}(P)=n}\tilde{H}_{n-1}(X_{rk(P)-1}(\mathrm{Pic}(\mathcal{O})))$

and the map between them is that induced by $\psi$, as required. ∎

######  Corollary 3.6.

The composition of the two maps in (3.1) is surjective for $n\leq 4$.

###### Proof.

Combine Theorems 3.5 and 3.3. ∎

######  Corollary 3.7.

Axioms (III) and (IV) hold for $\mathbf{BGL}$ and for $\mathbf{RBS}$.

###### Proof.

Apply Lemma 2.6 to the above, to see that all of
$H_{3,2}^{\overline{\mathbf{BGL}}}(\mathbb{A})$,
$H_{4,3}^{\overline{\mathbf{BGL}}}(\mathbb{A})$,
$H_{3,2}^{\overline{\mathbf{RBS}}}(\mathbb{A})$, and
$H_{4,3}^{\overline{\mathbf{RBS}}}(\mathbb{A})$ vanish. ∎

The axioms having been verified, Theorems 1.1 and 1.2 follow, as do their
analogues for $|\mathrm{RBS}(M)|$.

##  4\. Proof of Theorem 1.7

We require the following algebraic lemma, a consequence of a theorem of
Józefiak and Weyman [JW88].

######  Lemma 4.1.

Let $\mathbb{K}$ be a field of characteristic zero, $V$ be a
$\mathbb{K}$-module, and consider the cdga

$C_{\bullet,*}:=\left(\mathrm{Sym}^{*}(V[1,0])\otimes\Lambda^{*}(\mathrm{Sym}^{2}(V)[2,1]),\partial\right)$

with additional $\mathbb{N}$-grading denoted $\bullet$, where $\partial$ is
determined by the identity map $C_{2,1}=\mathrm{Sym}^{2}(V)\to
C_{2,0}=\mathrm{Sym}^{2}(V)$ and the graded Leibniz rule. Then

$H_{n,d}(C_{\bullet,*})=0\text{ for }d<\tfrac{n-\sqrt{n}}{2}.$

If $V$ is finite-dimensional then $H_{n,d}(C_{\bullet,*})=0$ for
$d<\tfrac{n-\dim(V)}{2}$. If $n$ is a square, $d=\tfrac{n-\sqrt{n}}{2}$, and
$\dim(V)\geq n-2d$, then $H_{n,d}(C_{\bullet,*})\neq 0$.

###### Proof.

Józefiak and Weyman [JW88] (see [KRW01, Theorem 2.5] for the following
formulation) identify the homology of $C_{\bullet,*}$ in terms of Schur
functors of $V$, as

$H_{n,d}(C_{\bullet,*})\cong\bigoplus_{\begin{subarray}{c}\lambda\\\
|\lambda|=n,\lambda=\lambda^{t},d(\lambda)=n-2d\end{subarray}}S_{\lambda}(V)$

where $d(\lambda)$ denotes the length of the diagonal when $\lambda$ is viewed
as a Young diagram. In order for there to be any summands at all, we must have
that $n\geq(n-2d)^{2}$ (as if $\lambda$ has diagonal of length $n-2d$ then it
contains the $(n-2d)\times(n-2d)$ square), i.e. that
$d\geq\tfrac{n-\sqrt{n}}{2}$. Thus it vanishes if $d<\tfrac{n-\sqrt{n}}{2}$.

If $V$ is finite-dimensional then $S_{\lambda}(V)=0$ if and only if the number
of rows of $\lambda$ is greater than $\dim(V)$. If $d(\lambda)=n-2d$ then it
must have at least $n-2d$ rows, so $S_{\lambda}(V)=0$ when $n-2d>\dim(V)$,
i.e. when $d<\tfrac{n-\dim(V)}{2}$.

If $n$ is a square and $d=\tfrac{n-\sqrt{n}}{2}$, so $n=(n-2d)^{2}$, then
taking $\lambda$ to be the $(n-2d)\times(n-2d)$ square we see that
$S_{\lambda}(V)\neq 0$ when $\dim(V)\geq n-2d$, which contributes to
$H_{n,d}(C_{\bullet,*})$. ∎

######  Remark 4.2.

Over a field $\mathbb{K}$ of positive characteristic, choosing a basis
$\\{b_{\alpha}\\}_{\alpha\in I}$ of $V$, a basis for $C_{\bullet,*}$ may be
given by symbols

$b_{\alpha_{1}}\cdots b_{\alpha_{i}}\otimes(b_{\alpha_{i+1}}\cdot
b_{\alpha_{i+2}})\wedge\cdots\wedge(b_{\alpha_{i+2k+1}}\cdot
b_{\alpha_{i+2k+2}}).$

The differential is given by an alternating sum of moving each wedge-summand
across the tensor: in particular the multiset of subscripts $\alpha_{i}$ is
unchanged by the differential. Thus $C_{n,*}$ may be described as a direct sum
of chain complexes, one for each $n$-element multiset in $I$. If $n\leq|I|$
then there is a summand given by an $n$-element multiset of $I$ with no
repeated elements, i.e. an injection $\\{1,2,\ldots,n\\}\to I$. The
corresponding direct summand of $C_{n,*}$ may be identified with the 1-fold
suspension of the augmented simplicial chain complex of the _matching complex_
$M(\\{1,2,\ldots,n\\})$ on the set $\\{1,2,\ldots,n\\}$, see e.g. [KRW01]. So
as long as $\dim(V)\geq n$ then

$\widetilde{H}_{d-1}(M(\\{1,2,\ldots,n\\});\mathbb{K})\subset
H_{n,d}(C_{\bullet,*}).$

Now Bouc [Bou92, Proposition 7] has shown that

$\widetilde{H}_{k}(M(\\{1,2,\ldots,3k+4\\});\mathbb{Z})\cong\mathbb{Z}/3\text{
for all }k\geq 1,$

so $H_{3d+1,d}(C_{\bullet,*};\mathbb{K})\neq 0$ if $\mathbb{K}$ has
characteristic 3. Thus Lemma 4.1 cannot hold in positive characteristic (or at
least not in characteristic 3).

###### Proof of Theorem 1.7.

Recall that $\mathbbm{k}$ is a field of characteristic zero. We construct a
CW-approximation to $\mathbb{R}$ by

$\mathbb{C}:=\mathbb{E}_{\infty}(S^{1,0}\otimes\mathbbm{k}\\{P\\})\cup^{E_{\infty}}_{\rho\cdot\rho^{\prime}-\sigma\cdot(\rho\star\rho^{\prime})}D^{2,1}\longrightarrow\mathbb{R},$

where we attach a $(2,1)$-cell for each pair $(\rho,\rho^{\prime})\in P^{2}$:
the map to $\mathbb{R}$ extends over these cells as
$[\rho]\cdot[\rho^{\prime}]=[\sigma]\cdot[\rho\star\rho^{\prime}]\in
H_{2,1}(\mathbb{R})$ by assumption (II). The map
$H_{*,0}(\overline{\mathbb{C}})\to H_{*,0}(\overline{\mathbb{R}})=A_{P}$ is an
isomorphism, by construction. As both $\mathbb{R}$ and $\mathbb{C}$ satisfy
assumption (I), it follows from [GKRW18a, Theorem 14.4] and the long exact
sequence on $E_{\infty}$-homology for this map that
$H_{n,d}^{E_{\infty}}(\mathbb{R},\mathbb{C})=0$ for $d<\tfrac{1}{2}n$. We may
therefore attach $(n,d)$-cells with $d\geq\tfrac{1}{2}n$ to $\mathbb{C}$ to
obtain a CW-approximation
$\mathbb{C}\to\mathbb{C}^{\prime}\overset{\sim}{\to}\mathbb{R}$, equipped with
a relative skeletal filtration. We will treat stability with respect to
$\sigma\in P$, but the argument can be easily adapted to any other $\rho\in
P$: the only change is notation.

As, for example, in the proof of [GKRW18a, Theorem 18.3], this leads to a
spectral sequence

$E^{1}_{n,p,q}=H_{n,p+q,q}(\overline{\mathbb{C}}/\sigma[0]\otimes
E_{\infty}^{+}(\bigoplus_{\alpha\in
I}S^{n_{\alpha},d_{\alpha},d_{\alpha}}))\Longrightarrow
H_{n,p+q}(\overline{\mathbb{R}}/\sigma),$

with $d_{\alpha}\geq\tfrac{1}{2}n_{\alpha}$. As the homology of
$E_{\infty}^{+}(\bigoplus_{\alpha\in I}S^{n_{\alpha},d_{\alpha},d_{\alpha}})$
is supported in slopes $\geq\tfrac{1}{2}$, whatever vanishing is enjoyed by
$\overline{\mathbb{C}}/\sigma$ will also hold for
$\overline{\mathbb{R}}/\sigma$, up to slope $\tfrac{1}{2}$.

We now analyse $\overline{\mathbb{C}}/\sigma$, using Lemma 4.1. Letting
$W:=\mathbbm{k}\\{P\\}$, the homology of $\overline{\mathbb{C}}$ is calculated
by the bigraded cdga

$D_{\bullet,*}:=(\mathrm{Sym}^{*}(W[1,0])\otimes\Lambda^{*}(W\otimes
W[2,1]),\partial)$

with differential determined by

$\rho\otimes\rho^{\prime}\mapsto\rho\cdot\rho^{\prime}-\sigma\cdot(\rho\star\rho^{\prime}):D_{2,1}=W\otimes
W\longrightarrow D_{2,0}=\mathrm{Sym}^{2}(W)$

and the Leibniz rule. As in the first few lines of the proof of [GKRW20,
Theorem 9.5], as $\mathbbm{k}$ has characteristic zero there is an equivalence
$\overline{\mathbb{C}}/\sigma\simeq\overline{\mathbb{C}\cup^{E_{\infty}}_{\sigma}D^{1,1}}$.
Letting $V:=\mathbbm{k}\\{P\setminus\\{\sigma\\}\\}$, the homology of
$\overline{\mathbb{C}\cup^{E_{\infty}}_{\sigma}D^{1,1}}$ may be calculated by
the bigraded cdga

$D^{\prime}_{\bullet,*}:=(\mathrm{Sym}^{*}(V[1,0])\otimes\Lambda^{*}(W\otimes
W[2,1]),\partial)$

with differential determined by

$\rho\otimes\rho^{\prime}\mapsto\begin{cases}\rho\cdot\rho^{\prime}&\rho\neq\sigma\text{
and }\rho^{\prime}\neq\sigma\\\ 0&\rho=\sigma\text{ or
}\rho^{\prime}=\sigma\end{cases}$

and the Leibniz rule. Now in terms of the cdga $C_{\bullet,*}$ introduced in
Lemma 4.1 we have
$D^{\prime}_{\bullet,*}=C_{\bullet,*}\otimes\\{\begin{subarray}{c}\text{a free
graded commutative algebra with}\\\ \text{generators of slope $\tfrac{1}{2}$
and trivial differential}\end{subarray}\\}$, and so

$H_{*,*}(\overline{\mathbb{C}}/\sigma)\cong
H_{*,*}(C_{\bullet,*})\otimes\\{\begin{subarray}{c}\text{a free graded
commutative algebra}\\\ \text{with generators of slope
$\tfrac{1}{2}$}\end{subarray}\\}.$

It then follows from Lemma 4.1 that

  1. (i)

$H_{n,d}(\overline{\mathbb{C}}/\sigma)=0$ for $d<\tfrac{n-\sqrt{n}}{2}$, and

  2. (ii)

if $P$ is finite then $H_{n,d}(\overline{\mathbb{C}}/\sigma)=0$ for
$d<\tfrac{n-(|P|-1)}{2}$.

With the discussion above, this finishes the argument. ∎

######  Remark 4.3.

The $E_{\infty}$-algebra $\mathbb{C}$ constructed in this argument satisfies
assumptions (I) and (II) of Theorem 1.5, but the argument shows that
$H_{*,*}(\overline{\mathbb{C}}/\sigma)$ contains the homology of the cdga of
Lemma 4.1 with $V=\mathbbm{k}\\{P\setminus\\{\sigma\\}\\}$. By the last part
of that lemma, the homology of this cdga does not vanish in bidegrees $(n,d)$
when $n$ is a square, $d=\tfrac{n-\sqrt{n}}{2}$, and $|P|-1\geq n-2d$. In
particular if $P$ is infinite then $H_{*,*}(\overline{\mathbb{C}}/\sigma)$
does not admit a vanishing line of slope $\tfrac{1}{2}$.

##  5\. An example

Consider the ring of integers $\mathcal{O}=\mathbb{Z}[\sqrt{-5}]$ of the
number field $\mathbb{Q}(\sqrt{-5})$, and write $\omega:=\sqrt{-5}$. This
field has class number 2, and so
$\mathrm{Pic}(\mathcal{O})=\\{\sigma,\lambda\\}$ is the group of order 2: its
nontrivial element $\lambda$ may be represented by the (fractional) ideal
$\mathfrak{l}=(2,1+\omega)$.

Rank 1. Both $\mathrm{GL}(\mathcal{O})$ and $\mathrm{GL}(\mathfrak{l})$ are
given by $\mathcal{O}^{\times}=\\{\pm 1\\}$.

Rank 2. A presentation for the group $\mathrm{SL}_{2}(\mathcal{O})$ has been
given by Swan [Swa71, Theorem 11.1]: it is generated by the matrices

$J=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix},T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},U=\begin{pmatrix}1&\omega\\\ 0&1\end{pmatrix}$
$A=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},B=\begin{pmatrix}-\omega&2\\\
2&\omega\end{pmatrix},C=\begin{pmatrix}-\omega-4&-2\omega\\\
2\omega&\omega-4\end{pmatrix}$

subject to the relations

$J^{2}=1,J\text{ central},TU=UT,A^{2}=J,B^{2}=J,(TA)^{3}=J,(AB)^{2}=J$
$(AUBU^{-1})^{2}=J,ACA=JTCT^{-1},UBU^{-1}CB=JTCT^{-1}.$

This abelianises to
$\mathbb{Z}/2\oplus\mathbb{Z}/6\oplus\mathbb{Z}\oplus\mathbb{Z}$, generated
for example by $B,T,U,C$ ([Swa71, Corollary 11.2]). Furthermore, $J\sim 0$ in
this group. The matrix $E:=\left(\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}\right)$ acts on $\mathrm{SL}_{2}(\mathcal{O})$ by
conjugation, as $B\mapsto JUBU^{-1}\sim B$, $T\mapsto T^{-1}\sim-T$, $U\mapsto
U^{-1}\sim-U$, and $C\mapsto TC^{-1}T^{-1}\sim-C$. This shows that

$\mathrm{GL}(\mathcal{O}\oplus\mathcal{O})^{ab}=(\mathbb{Z}/2)^{5}$

generated by $E,U,T,B,C$ ([Swa71, Corollary 11.3]).

It remains to describe the abelianisation of
$\mathrm{GL}(\mathcal{O}\oplus\mathfrak{l})$. Using that
$\mathfrak{l}^{-1}=\tfrac{1}{2}\mathfrak{l}$, we may describe
$\mathrm{GL}(\mathcal{O}\oplus\mathfrak{l})$ as the subgroup:

$\left\\{\begin{pmatrix}a&\tfrac{1}{2}b\\\
c&d\end{pmatrix}\in\mathrm{GL}_{2}(\mathbb{Q}(\omega))\,|\,a,d\in\mathcal{O},b,c\in\mathfrak{l}\right\\}\leq\mathrm{GL}_{2}(\mathbb{Q}(\omega)).$

In [FGT10, Section 2.2] we find a presentation for
$\mathrm{PSL}(\mathcal{O}\oplus\mathfrak{l})$ which is easily lifted to the
following presentation of $\mathrm{SL}(\mathcal{O}\oplus\mathfrak{l})$: it is
generated by the matrices

$J=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix},A=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},V=\begin{pmatrix}1&\tfrac{1+\omega}{2}\\\
0&1\end{pmatrix},C=\begin{pmatrix}1&0\\\
2&1\end{pmatrix},D=\begin{pmatrix}1&0\\\ 1-\omega&1\end{pmatrix}$

subject to the relations

$J^{2}=1,J\text{
central},AV=VA,CD=DC,(AC^{-1})^{2}=J,(DV^{-1})^{3}=1,(CD^{-1}VA^{-1})^{3}=1.$

This presentation abelianises to
$\mathbb{Z}/3\oplus\mathbb{Z}\oplus\mathbb{Z}$, generated by $DV^{-1}$, $C$,
and $D$. The matrix $E$ above acts by conjugation, as $DV^{-1}\mapsto
D^{-1}V\sim-(DV^{-1})$, $C\mapsto C^{-1}\sim-C$, and $D\mapsto D^{-1}\sim-D$.
This gives

$\mathrm{GL}(\mathcal{O}\oplus\mathfrak{l})^{ab}=(\mathbb{Z}/2)^{3}$

generated by $E$, $C$, and $D$. Furthermore, $J\sim 0$ in this group.

Rank 3. By Bass–Milnor–Serre [BMS67, Corollary 4.3 a)] the group
$\mathrm{SL}_{3}(\mathcal{O})$ is perfect, so the map
$\det:\mathrm{GL}_{3}(\mathcal{O})\to\mathcal{O}^{\times}=\\{\pm 1\\}$ is the
abelianisation. As $[\mathfrak{l}]\in\mathrm{Pic}(\mathcal{O})$ has order 2,
tensoring with it gives an isomorphism

$\mathfrak{l}\otimes-:\mathrm{GL}(\mathcal{O}\oplus\mathcal{O}\oplus\mathfrak{l})\overset{\sim}{\longrightarrow}\mathrm{GL}(\mathfrak{l}\oplus\mathfrak{l}\oplus\mathfrak{l}^{\otimes
2})\cong\mathrm{GL}(\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O})=\mathrm{GL}_{3}(\mathcal{O}),$

and so
$\det:\mathrm{GL}(\mathcal{O}\oplus\mathcal{O}\oplus\mathfrak{l})\to\mathcal{O}^{\times}=\\{\pm
1\\}$ is the abelianisation too.

## References

  * [BMS67] H. Bass, J. Milnor, and J.-P. Serre, _Solution of the congruence subgroup problem for ${\rm SL}_{n}\,(n\geq 3)$ and ${\rm Sp}_{2n}\,(n\geq 2)$_, Inst. Hautes Études Sci. Publ. Math. (1967), no. 33, 59–137.
  * [Bou92] S. Bouc, _Homologie de certains ensembles de $2$-sous-groupes des groupes symétriques_, J. Algebra 150 (1992), no. 1, 158–186.
  * [CFP19] T. Church, B. Farb, and A. Putman, _Integrality in the Steinberg module and the top-dimensional cohomology of ${\rm SL}_{n}\mathcal{O}_{K}$_, Amer. J. Math. 141 (2019), no. 5, 1375–1419.
  * [Cha80] R. M. Charney, _Homology stability for ${\rm GL}_{n}$ of a Dedekind domain_, Invent. Math. 56 (1980), no. 1, 1–17.
  * [CJ24] D. Clausen and M. Ø. Jansen, _The reductive Borel-Serre compactification as a model for unstable algebraic K-theory_ , Selecta Math. (N.S.) 30 (2024), no. 1, Paper No. 10, 93.
  * [FGT10] T. Finis, F. Grunewald, and P. Tirao, _The cohomology of lattices in ${\rm SL}(2,\mathbb{C})$_, Experiment. Math. 19 (2010), no. 1, 29–63.
  * [Fri17] N. Friedrich, _Homological stability of automorphism groups of quadratic modules and manifolds_ , Doc. Math. 22 (2017), 1729–1774.
  * [GKRW18a] S. Galatius, A. Kupers, and O. Randal-Williams, _Cellular $E_{k}$-algebras_, https://arxiv.org/abs/1805.07184. Astérisque, to appear, 2018.
  * [GKRW18b] by same author, _$E_{\infty}$ -cells and general linear groups of finite fields_, https://arxiv.org/abs/1810.11931. Annales scientifiques de l’École normale supérieure, to appear, 2018.
  * [GKRW20] by same author, _$E_{\infty}$ -cells and general linear groups of infinite fields_, https://arxiv.org/abs/2005.05620, 2020.
  * [Jan24] M. Ø. Jansen, _Unstable algebraic K-theory: homological stability and other observations_ , https://arxiv.org/abs/2405.02065, 2024.
  * [JW88] T. Józefiak and J. Weyman, _Representation-theoretic interpretation of a formula of D. E. Littlewood_ , Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 193–196.
  * [KRW01] D. B. Karaguezian, V. Reiner, and M. L. Wachs, _Matching complexes, bounded degree graph complexes, and weight spaces of ${\rm GL}_{n}$-complexes_, J. Algebra 239 (2001), no. 1, 77–92.
  * [Qui78] D. Quillen, _Homotopy properties of the poset of nontrivial $p$-subgroups of a group_, Adv. in Math. 28 (1978), no. 2, 101–128.
  * [Sie22a] I. Sierra, _Homological stability of diffeomorphism groups of high dimensional manifolds via $E_{k}$-algebras_, https://arxiv.org/abs/2209.10243, 2022.
  * [Sie22b] by same author, _Homological stability of spin mapping class groups and quadratic symplectic groups_ , https://arxiv.org/abs/2211.06219, 2022.
  * [Swa71] R. G. Swan, _Generators and relations for certain special linear groups_ , Advances in Math. 6 (1971), 1–77.
  * [vdK80] W. van der Kallen, _Homology stability for linear groups_ , Invent. Math. 60 (1980), no. 3, 269–295.