Central stability homology

TL;DR

This paper introduces a new categorical framework for central stability homology, extending previous work to categories with infinite automorphism groups and linking it to broader stability concepts.

Contribution

It develops a novel categorical approach to central stability homology, applicable to categories with infinite automorphisms, and connects it to existing stability theories.

Findings

Provides a new categorical construction for central stability homology.

Establishes a connection between central stability and homological stability.

Develops a criterion for polynomial functors to be centrally stable.

Abstract

We give a new categorical way to construct the central stability homology of Putman and Sam and explain how it can be used in the context of representation stability and homological stability. In contrast to them, we cover categories with infinite automorphism groups. We also connect central stability homology to Randal-Williams and Wahl's work on homological stability. We also develop a criterion that implies that functors that are polynomial in the sense of Randal-Williams and Wahl are centrally stable in the sense of Putman.

Equations

$$H_i({\mathfrak{S}}_{n-1})⟶H_i({\mathfrak{S}}_n)$$

$${\mathfrak{S}}_n/{\mathfrak{S}}_{n-p-1}$$

$$\{0,1,\dots ,p\}⟼{\mathfrak{S}}_n/{\mathfrak{S}}_{n-p-1}.$$

$$\{0,1,\dots ,p\}\text{lhook}\text{joinrel}⟶\{1,2,\dots ,n\},$$

$$({\Delta }_{+}^{\prime }{)}^{\mathrm{op}}⟶{\mathsf{FI}}^{\mathrm{op}}\stackrel{{\mathrm{Hom}}_{\mathsf{FI}}(-,\{1,\dots ,n\})}{⟶}\mathsf{Set}$$

$$\operatorname{coeq}\big{(}\operatorname{Ind}^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-2}}V_{n-2}\rightrightarrows\operatorname{Ind}^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-1}}V_{n-1}\big{)}.$$

$${\mathrm{inc}}_N:{\mathsf{FI}}^{\le N}\text{lhook}\text{joinrel}⟶\mathsf{FI}.$$

$$V\cong {\mathrm{Lan}}_{{\mathrm{inc}}_N}(V\circ {\mathrm{inc}}_N)$$

$$f:V\circ {\mathrm{inc}}_N⟶W\circ {\mathrm{inc}}_N$$

$$\cdots \to {\mathrm{Ind}}_{{\mathfrak{S}}_{n-p-1}\times {\mathfrak{S}}_{p+1}}^{{\mathfrak{S}}_n}{V}_{n-p-1}\otimes {\mathcal{A}}_{p+1}\to {\mathrm{Ind}}_{{\mathfrak{S}}_{n-p}\times {\mathfrak{S}}_p}^{{\mathfrak{S}}_n}{V}_{n-p}\otimes {\mathcal{A}}_p\to \dots$$

$$\cdots \to \underset{\begin{array}{c}S\subset \{1,\dots ,n\}\\ |S|=p+1\end{array}}{⨁}V(\{1,\dots ,n\}\setminus S)\to \underset{\begin{array}{c}S\subset \{1,\dots ,n\}\\ |S|=p\end{array}}{⨁}V(\{1,\dots ,n\}\setminus S)\to \dots$$

$$d_{i}\colon V\big{(}\{1,\dots,n\}\setminus\{j_{0}<\dots<j_{p}\}\big{)}\longrightarrow V\big{(}\{1,\dots,n\}\setminus\{j_{0}<\dots<\hat{j_{i}}<\dots<j_{p}\}\big{)}$$

$$\cdots \to \underset{f:\{0,1,\dots ,p\}\text{lhook}\text{joinrel}⟶\{1,\dots ,n\}}{⨁}V(\{1,\dots ,n\}\setminus \mathrm{im}f)\to \underset{f:\{0,1,\dots ,p-1\}\text{lhook}\text{joinrel}⟶\{1,\dots ,n\}}{⨁}V(\{1,\dots ,n\}\setminus \mathrm{im}f)\to \dots$$

$$\underset{f:\{0,1,\dots ,p\}\text{lhook}\text{joinrel}⟶\{1,\dots ,n\}}{⨁}V(\{1,\dots ,n\}\setminus \mathrm{im}f)\cong {\mathrm{Ind}}_{{\mathfrak{S}}_{n-p}}^{{\mathfrak{S}}_n}V(\{1,\dots ,n-p\}).$$

$${\mathrm{Ind}}_{{\mathfrak{S}}_{n-1}}^{{\mathfrak{S}}_n}{V}_{n-1}⟶V_n$$

$$V_n\cong \mathrm{coeq}({\mathrm{Ind}}_{{\mathfrak{S}}_{n-2}}^{{\mathfrak{S}}_n}{V}_{n-2}\rightrightarrows {\mathrm{Ind}}_{{\mathfrak{S}}_{n-1}}^{{\mathfrak{S}}_n}{V}_{n-1})$$

$${\mathrm{Ind}}_{{\mathfrak{S}}_{n-2}}^{{\mathfrak{S}}_n}{V}_{n-2}\to {\mathrm{Ind}}_{{\mathfrak{S}}_{n-1}}^{{\mathfrak{S}}_n}{V}_{n-1}\to V_n\to 0$$

$${\mathrm{Ind}}_{{\mathfrak{S}}_{n-2}\times {\mathfrak{S}}_2}^{{\mathfrak{S}}_n}{V}_{n-2}\otimes {\mathcal{A}}_2\to {\mathrm{Ind}}_{{\mathfrak{S}}_{n-1}\times {\mathfrak{S}}_1}^{{\mathfrak{S}}_n}{V}_{n-1}\otimes {\mathcal{A}}_1\to V_n⟶0$$

$$\mathrm{Fun}(\mathcal{C},\mathcal{D})⟶\mathrm{Fun}(\mathcal{C},\mathrm{Fun}(({\Delta }_{+}^{\prime }{)}^{\mathrm{op}},\mathcal{D}))$$

$${\mathrm{Hom}}_{\mathsf{FI}}(\varnothing ,-):\mathsf{FI}⟶\mathsf{Set},$$

$$\{1,\dots ,n\}⟼W_n.$$

$${\tilde{C}}_p(V)(S)\cong \underset{f:\{0,1,\dots ,p\}\text{lhook}\text{joinrel}⟶S}{⨁}V(S\setminus \mathrm{im}f)$$

$$\oplus :G_m\times G_n\text{lhook}\text{joinrel}⟶G_{m+n}$$

$${\mathrm{Hom}}_{U\mathcal{G}}(m,n)\cong \{\begin{array}{ll}{G}_n/G_{n-m}& m\le n\\ \varnothing & m>n.\end{array}$$

$${\tilde{C}}_p(V{)}_n\cong \underset{f\in {\mathrm{Hom}}_{U\mathcal{G}}(p+1,n)}{⨁}{V}_{n-p-1}\cong {\mathrm{Ind}}_{G_{n-p-1}}^{G_n}{V}_{n-p-1}.$$

$${\mathrm{Hom}}_{U\mathcal{G}}(0,n)\cong G_n/G_{n-0}\cong \{\ast \}.$$

$$R{\mathrm{Hom}}_{U\mathcal{G}}(0,-):U\mathcal{G}⟶R-\mathsf{mod}$$

$$\cdots \to P_1\to P_0\to V\to 0$$

$${\tilde{H}}_i(V{)}_n=0$$

$$V(0\oplus n)⟶V(1\oplus n)$$

Full text

1. 1 Introduction
2. 2 Central stability homology
3. 3 Stability categories
4. 4 Central stability and stability categories
5. 5 Central stability and finiteness properties
6. 6 Notions of central stability
7. 7 Polynomial degree
8. 8 Short exact sequences
9. 9 Quillen’s argument revisited

1. Introduction

In 1960 Nakaoka [Nak60] proved the pioneering result that for the symmetric groups

[TABLE]

is an isomorphism for all $n$ large enough in comparison to $i$. Quillen [Qui70] was able to prove this result, using a new method that has since been generalized in many ways. He was considering a highly connected simplicial complex $X_{n}$ on which $\mathfrak{S}_{n}$ acts and the groups that stabilizes simplices pointwise are isomorphic to $\mathfrak{S}_{m}$ for large enough $m\leq n$. For the symmetric groups $X_{n}=\Delta^{n-1}$ the $(n-1)$–dimensional standard simplex works. Randal-Williams and Wahl [RWW15] turned this game around: They construct a semisimplicial set $W_{n}$ whose $p$–simplices are given by

[TABLE]

such that it is a transitive $\mathfrak{S}_{n}$–set whose stabilizers are $\mathfrak{S}_{n-p-1}$. Then they prove that $W_{n}$ is highly connected. More precisely, let $\Delta^{\prime}_{+}$ be the category of finite ordered sets (including the empty set) and strictly monotone maps and $\mathsf{FI}$ the category of finite sets and injection. Being a semisimplicial set, $W_{n}$ is a functor from $(\Delta^{\prime}_{+})^{\operatorname{op}}$ to the category of sets with

[TABLE]

Because $\mathfrak{S}_{n}/\mathfrak{S}_{n-p-1}$ is as an $\mathfrak{S}_{n}$–set the set of injective maps

[TABLE]

this functor factors through $\mathsf{FI}^{\operatorname{op}}$ the opposite category of $\mathsf{FI}$:

[TABLE]

The category $\mathsf{FI}$ has taken up a prominent role in the theory of representation stability. Established by Church and Farb in [CF13], representation stability characterizes sequences $(V_{n})_{n\in\mathbb{N}}$ of representations of $\mathfrak{S}_{n}$ or other sequences of groups, that admit a uniform description as a direct sum of irreducible representation once $n$ is large enough. For example, the permutation representation $\mathbb{Q}^{n}$ is the direct sum of the trivial representation and the standard representation for $n\geq 2$. The permutation representation can also be realized as an $\mathsf{FI}$–module, by which we mean a functor from $\mathsf{FI}$ to $\mathbb{Q}$–vector spaces (and later more generally a functor from $\mathsf{FI}$ to $R$–modules). This can be done by sending a finite set $S$ to the finite dimensional vector space with basis $S$. For an $\mathsf{FI}$–module $V$, we will denote the images of $\{1,\dots,n\}$ by $V_{n}$.

Notions of representation stability

Church and Farb’s original definition [CF13, Def 2.3] was given in terms of irreducible representations. Later Church, Ellenberg and Farb [CEF15] proved that a finitely generated $\mathsf{FI}$–module is representation stable. Putman [Put15] noticed that for an $\mathsf{FI}$–module $V$ one cannot expect that $V_{n}\cong\operatorname{Ind}^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-1}}V_{n-1}$, because the transposition of $(n-1\ n)$ acts trivially on the image of $V_{n-2}$ in $V_{n}$. But he calls an $\mathsf{FI}$–module $V$ centrally stable if $V_{n}$ is isomorphic to the largest quotient of $\operatorname{Ind}^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-1}}V_{n-1}$ such that $(n-1\ n)$ acts trivially on the image of $V_{n-2}$ for all $n$ large enough. This quotient can be expressed as a coequalizer

[TABLE]

Let $\mathsf{FI}^{\leq N}$ be the full subcategory of $\mathsf{FI}$ whose objects all have cardinality at most $N$ and denote the inclusion by

[TABLE]

Church, Ellenberg, Farb and Nagpal [CEFN14] then proved that an $\mathsf{FI}$–module is centrally stable if and only if it is presented in $\mathsf{FI}^{\leq N}$ (having a presentation by sums of representable functors from $\mathsf{FI}^{\leq N}$, also see Section 6(a)) for some sufficiently large $N$. Later Putman and Sam [PS14] also gave a categorical concept the name central stability. They call an $\mathsf{FI}$–module $V$ is centrally stable if there is an $N$ large enough such that $V$ is the left Kan extension of its restriction to $\mathsf{FI}^{\leq N}$:

[TABLE]

In other words, for any $\mathsf{FI}$–module $W$ and any morphism (ie natural transformation)

[TABLE]

between the restrictions, there is a unique morphism $V\to W$ which induces $f$. In a more general setup, Djament [Dja16] proves that an $\mathsf{FI}$–module fulfills this condition if and only if it is presented in $\mathsf{FI}^{\leq N}$ and is thereby equivalent to the original definition from [Put15]. Unfortunately the two notions of central stability differ in natural generalizations of the category $\mathsf{FI}$, we consider later. We compare many different notions in Section 6.

The central stability chain complex

The term central stability chain complex was first coined by Putman [Put15] for a sequence $(V_{n})_{n\in\mathbb{N}}$ of $\mathfrak{S}_{n}$–representations. He defined it by

[TABLE]

where $\mathcal{A}_{p}$ is the sign representation of $\mathfrak{S}_{p}$. It turns out, that for an $\mathsf{FI}$–module $V$ this chain complex is given by

[TABLE]

where the differential is the alternated sum of face maps

[TABLE]

that is induced by the inclusion of sets. This complex also computes $\mathsf{FI}$–homology – the derived functor analyzed by Church and Ellenberg [CE16].

There is a related chain complex

[TABLE]

for $\mathsf{FI}$–modules $V$. It was first mentioned by Church, Ellenberg, Farb and Nagpal [CEFN14] and considered in more detail by Putman and Sam [PS14]. One notes

[TABLE]

To recover Putman’s chain complex (1.1), one factors out the $\mathfrak{S}_{p+1}$–action. The second complex (1.2) computes what we will call central stability homology of an $\mathsf{FI}$–module in this paper.

Both complexes are related to the notions of representation stability. For example, the augmentation map

[TABLE]

is surjective for all $n>N$ if and only if $V$ is generated in $\mathsf{FI}^{\leq N}$.

A similar connection can be found for central stability. By definition

[TABLE]

if and only if the part

[TABLE]

of (1.2) is exact. Coincidently this is true if and only if the part

[TABLE]

of (1.1) is exact. The remaining parts of the two complexes may very well differ.

Such a generalization was done for both complexes by Putman and Sam [PS14] for modules over complemented categories (see Section 3). $\mathsf{FI}$ is an example of a complemented category.

Let us plug in the constant $\mathsf{FI}$–module $V(S)=\mathbb{Q}$ that sends all maps to the identity. We find that (1.2) is the chain complex associated to $W_{n}$ from above. Whereas (1.1) computes to be the simplicial complex of the standard simplex $\Delta^{n-1}$. Randal-Williams and Wahl [RWW15] gave a general construction of the semisimplicial set $W_{n}$ for a homogeneous category (see Section 3). $\mathsf{FI}$ is also an example of a homogeneous category. Moreover, they have a general construction of a simplicial complex $S_{n}$ whose simplicial chain complex is a generalization of (1.1).

Let $\mathcal{C}$ be a small monoidal category with initial unit and let $\mathcal{D}$ be a cocomplete category. In Section 2, we give a construction of a functor

[TABLE]

that turns a functor from $\mathcal{C}$ to $\mathcal{D}$ into a functor from $\mathcal{C}$ to augmented semisimplicial objects in $\mathcal{D}$. This generalizes Randal-Williams and Wahl’s construction of the augmented semisimplicial set $W_{n}$ and the augmented semisimplicial module in (1.2). For example plugging in the constant functor

[TABLE]

we get a functor from $\mathsf{FI}$ to augmented semisimplicial sets with

[TABLE]

In our construction functorial dependencies become more visible than in previously existing literature. If $\mathcal{D}$ is abelian, we denote the associated chain complex for a functor $F\colon\mathcal{C}\to\mathcal{D}$ by $\widetilde{C}_{*}(F)$ and its homology by $\widetilde{H}_{*}(F)$. This is what we call the central stability chain complex and the central stability homology in this paper. When $V$ is an $\mathsf{FI}$–modules it computes to be

[TABLE]

as in (1.2).

Stability categories

[PS14] and [RWW15] generalize $\mathsf{FI}$ by weakly complemented categories (see Section 3) and homogeneous categories (see Section 3), respectively. We will work with a different generalization of $\mathsf{FI}$ that we call stability categories which are both weakly complemented and homogeneous. In Section 3, we define stability groupoids, that are sequences of groups $\mathcal{G}=(G_{n})_{n\in\mathbb{N}}$ together with a monoidal structure

[TABLE]

and some mild condition (see Section 3). We then use a construction of Quillen, which is also used in [RWW15], to get a category $U\mathcal{G}$ whose objects are the natural numbers and whose homomorphisms are

[TABLE]

We call $U\mathcal{G}$ a stability category if $\mathcal{G}$ is braided. All examples considered by [PS14] and [RWW15] are covered in this setup:

Examples \theexsauto.

Here is a list of some braided stability groupoids $\mathcal{G}$ and their stability categories $U\mathcal{G}$.

1. (a)

The symmetric groups $\mathfrak{S}=(\mathfrak{S}_{n})_{n\in\mathbb{N}}$ is a symmetric stability groupoid and its stability category $U\mathfrak{S}$ is a skeleton of $\mathsf{FI}$. 2. (b)

Fix a commutative ring $R$. The general linear groups $\operatorname{GL}(R)=(\operatorname{GL}_{n}(R))_{n\in\mathbb{N}}$ is a symmetric stability groupoid and its stability category $U\operatorname{GL}(R)$ is a skeleton of $\mathsf{VIC}_{R}$ whose objects are finitely generated free $R$–modules and its morphisms $R^{m}\to R^{n}$ are given by pairs $(f,C)$ of a monomorphism $f\colon R^{m}\to R^{n}$ and a free complement $C\oplus\operatorname{im}f=R^{n}$. 3. (c)

Fix a commutative ring $R$. The symplectic groups $\operatorname{Sp}(R)=(\operatorname{Sp}_{2n}(R))_{n\in\mathbb{N}}$ is a symmetric stability groupoid and its stability category $U\operatorname{Sp}(R)$ is a skeleton of $\mathsf{SI}_{R}$ whose objects are finitely generated free symplectic $R$–modules and its morphisms are isometries. 4. (d)

The automorphism groups of free groups $\operatorname{AutF}=(\operatorname{Aut}F_{n})_{n\in\mathbb{N}}$ is a symmetric stability groupoid and its stability category $U\operatorname{AutF}$ is given by pairs $(f,C)$ of a monomorphism $f\colon F_{m}\to F_{n}$ and a complement $C*\operatorname{im}f=F_{n}$. 5. (e)

There is a braided stability groupoid for the braid groups $\beta=(\beta_{n})_{n\in\mathbb{N}}$. Its monoidal structure is described in Section 3. 6. (f)

There is a braided stability groupoid for the mapping class groups $\operatorname{Mod}\Sigma=(\operatorname{Mod}\Sigma_{g,1})_{g\in\mathbb{N}}$ of connected, oriented surfaces of genus $g\in\mathbb{N}$ and one boundary component. Its monoidal structure is described in **[RWW15, Sec 5.6]**. A description of the corresponding stability categories of $\beta$ and $\operatorname{Mod}\Sigma$ can also be found in **[RWW15, Sec 5.6]**.

From now on, we call a functor from $U\mathcal{G}$ to the category of $R$–modules for some fixed commutative ring $R$ a $U\mathcal{G}$–module. In Section 4, we compute the central stability complex of a $U\mathcal{G}$–module $V$ very concretely to be isomorphic to

[TABLE]

This complex clearly generalizes (1.2) and (1.3).

The stability category $U\mathcal{G}$ of a braided stability groupoid $\mathcal{G}$ is monoidal (see [RWW15, Prop 2.6] or Section 3) and has an initial unit [math] as

[TABLE]

That means

[TABLE]

is the functor sending all objects to $R$ and all morphisms to the identity on $R$. The connectivity condition on $U\mathcal{G}$ used in [RWW15] can be described by the following condition.

**H3: **

The central stability homology $\widetilde{H}_{i}(R\operatorname{Hom}_{U\mathcal{G}}(0,-))_{n}=0$ for all $i\geq-1$ and all $n$ large enough in comparison to $i$.

All stability categories in Section 1 satisfy H3. Also, Putman and Sam’s [PS14, Thm 3.7] implies H3 for all stability categories $U\mathcal{G}$ that fulfill the noetherian condition that every submodule of a finitely generated $U\mathcal{G}$–module is again finitely generated. Using H3 in Section 5, we relate resolutions by $U\mathcal{G}$–modules freely generated in finite ranks (see Section 5) with the central stability homology. We prove a quantitive version of the following theorem as Section 5, where we also consider partial resolutions.

Theorem A**.**

Let $U\mathcal{G}$ be a stability category. Assume H3. Let $V$ be a $U\mathcal{G}$–module, then the following statements are equivalent.

1. (a)

There is a resolution

[TABLE]

of $V$ by $U\mathcal{G}$–modules $P_{i}$ that are freely generated in finite ranks. 2. (b)

The homology

[TABLE]

for all $i\geq-1$ and all $n$ large enough in comparison to $i$.

Let $U\mathcal{G}^{\leq N}$ denote the full subcategory of $U\mathcal{G}$ whose objects (which are natural numbers) are exactly $\{0,\dots,N\}$. Previously it has only been known that $\widetilde{H}_{-1}(V)_{n}=0$ for all $n>d$ if and only if $V$ is generated in ranks $\leq d$. For $\mathsf{FI}$, our $\widetilde{H}_{-1}$ is what is denoted by $H_{0}$ in [CEF15]. [CEFN14] and [PS14] have then used a noetherian property of the categories, they worked with, to show that all central stability homology vanishes for large enough $n$. Theorem A explains this phenomenon very clearly. It also makes the existence of free resolutions tangible, when such a noetherian condition is not true. For example for $\mathsf{VIC}_{\mathbb{Z}}$ and $\mathsf{SI}_{\mathbb{Z}}$.

We call a $U\mathcal{G}$–module $V$ centrally stable if $\widetilde{H}_{-1}(V)_{n}=\widetilde{H}_{0}(V)_{n}=0$ for all large enough $n$. This notion and terminology was introduced by Putman [Put15] for sequences of representations of the symmetric groups. Theorem A proves that under the assumption of H3, a $U\mathcal{G}$–module $V$ is centrally stable if and only if $V$ is presented in $U\mathcal{G}^{\leq N}$ for some sufficiently large $N$. A $U\mathcal{G}$–module $V$ is presented in a full subcategory with finitely many objects if and only if it is the left Kan extension of its restriction along the inclusion. Unfortunately this notion was also dubbed central stability in [PS14] and [CEFN14]. See Section 6 for more clarification on different notion used for central stability.

Polynomial functors

Many interesting examples of $U\mathcal{G}$–modules satisfy polynomiality conditions. For example the dimensions of the permutation representations grow polynomially (even linearly). In fact [CEF15] proved that, assuming $R=\mathbb{Q}$, dimensional growth is eventually polynomial for all finitely generated $\mathsf{FI}$–modules. In Section 7, we recall the definition of the polynomial degree of a $U\mathcal{G}$–module $V$ from [RWW15]. We say $V$ has polynomial degree $-\infty$ if it is zero. For $r\geq 0$ we say $V$ has polynomial degree $\leq r$ if the map

[TABLE]

is always injective and its cokernel is a $U\mathcal{G}$–module of polynomial degree $\leq r-1$. For example the constant $U\mathcal{G}$–module has polynomial degree $\leq 0$ and the permutation representations have polynomial degree $\leq 1$.

We formulate a new condition on stability categories, that involves the cokernel

[TABLE]

of the representable functors $R\operatorname{Hom}_{U\mathcal{G}}(m,-)$.

**H4: **

The central stability homology $\widetilde{H}_{i}(\operatorname{coker}R\operatorname{Hom}_{U\mathcal{G}}(m,-))_{n}=0$ for all $m\in\mathbb{N}$, all $i\geq-1$ and all $n$ large enough in comparison to $m$ and $i$.

It is easy to check that

[TABLE]

is free (eg see [CEFN14, Prop 2.2]) and thus H4 follows from H3 for $U\mathfrak{S}$ (which is equivalent to $\mathsf{FI}$). We prove together with Jeremy Miller and Jennifer Wilson [MPW17] that $U\operatorname{GL}(R)$ and $U\operatorname{Sp}(R)$ satisfy H4 if $R$ is a PID. In Section 7 we show that the stability category $U\beta$ of the braid groups does not satisfy H4. It is unknown whether $U\operatorname{AutF}$ and $U\operatorname{Mod}\Sigma$ satisfy H4. Assuming H4, we can make a statement about the central stability homology of $U\mathcal{G}$–module with finite polynomial degree.

Theorem B**.**

Let $U\mathcal{G}$ be a stability category. Assume H3 and H4. Let $V$ be a $U\mathcal{G}$–module with finite polynomial degree, then $\widetilde{H}_{i}(V)_{n}=0$ for all $i\geq-1$ and all $n$ large enough in comparison to $i$.

We prove a quantified version of this theorem with Section 7. This theorem is the main new ingredient of [MPW17], where we together with Jeremy Miller and Jennifer Wilson prove central stability for the second homology groups of Torelli subgroups of automorphism groups of free groups and mapping class groups as well as certain congruence subgroups of general linear groups.

Stability SES

In Section 8, we consider families of short exact sequences

[TABLE]

that are coming from morphisms of stability groupoids

[TABLE]

We call these families stability SES. For every $U\mathcal{G}$–module $V$, there is a $U\mathcal{Q}$–module $H_{i}(\mathcal{N};V)$ such that

[TABLE]

Examples \theexsauto.

Here is a (very incomplete) list of interesting stability SES’s.

1. (a)

The pure braid groups $P\beta=(P\beta_{n})_{n\in\mathbb{N}}$ are the kernels in the sequence

[TABLE] 2. (b)

Fix a commutative ring $R$ and an ideal $I\subset R$. The congruence subgroups $\operatorname{GL}(R,I)=(\operatorname{GL}_{n}(R,I))_{n\in\mathbb{N}}$ are the kernels in the sequence

[TABLE]

where $\overline{\operatorname{GL}(R/I)}=(\overline{\operatorname{GL}_{n}(R/I)})_{n\in\mathbb{N}}$ are the images of $\operatorname{GL}_{n}(R)$ in $\operatorname{GL}_{n}(R/I)$. 3. (c)

Fix a commutative ring $R$ and an ideal $I\subset R$. The congruence subgroups $\operatorname{Sp}(R,I)=(\operatorname{Sp}_{n}(R,I))_{n\in\mathbb{N}}$ are the kernels in the sequence

[TABLE]

where $\overline{\operatorname{Sp}(R/I)}=(\overline{\operatorname{Sp}_{n}(R/I)})_{n\in\mathbb{N}}$ are the images of $\operatorname{Sp}_{n}(R)$ in $\operatorname{Sp}_{n}(R/I)$. 4. (d)

The Torelli subgroups $\operatorname{IA}=(\operatorname{IA}_{n})_{n\in\mathbb{N}}$ of the automorphism groups $\operatorname{Aut}F_{n}$ are the kernels in the sequence

[TABLE] 5. (e)

The Torelli subgroups $\mathcal{I}=(\mathcal{I}_{g,1})_{g\in\mathbb{N}}$ of the mapping class groups $\operatorname{Mod}\Sigma_{g,1}$ are the kernels in the sequence

[TABLE] 6. (f)

If there is an inclusion of $\mathfrak{S}\subset\mathcal{Q}$, we can twist $\mathcal{N}$ to receive a braided stability groupoid $\widetilde{\mathcal{N}}=(\widetilde{N}_{n})_{n\in\mathbb{N}}$, where $\widetilde{N}_{n}\subset G_{n}$ is the preimage of $\mathfrak{S}_{n}$. Then

[TABLE]

is a stability SES. This has been done by Putman **[Put15]** for (b) and by the author together with Wu **[PW16]** for the Houghton groups. It could also be done for (c), (d), and (e).

Note that in all these examples $\mathcal{G}$ and $\mathcal{Q}$ are braided, but $\mathcal{N}$ is not. Assume for the following that $\mathcal{G}$ and $\mathcal{Q}$ both are braided. The central technical tools of this paper are the following spectral sequences. Let $V$ be a $U\mathcal{G}$–module and $n\in\mathbb{N}$, then there are two spectral sequences

[TABLE]

and

[TABLE]

that converge to the same limit. In particular when $\widetilde{H}_{q}(V)_{n}=0$ for all $n$ large enough in comparison to $q$, then both spectral sequences converge to zero in each diagonal $p+q$ for large enough $n$.

As an application of this spectral sequence, we prove a generalization of a theorem of Putman and Sam [PS14, Thm 5.13].

Theorem C**.**

Let

[TABLE]

be a stability SES. Assume that $\mathcal{G}$, $\mathcal{Q}$, and $\mathcal{G}\to\mathcal{Q}$ are braided. Let $V$ be a $U\mathcal{G}$–module. Assume furthermore:

1. (a)

All submodules of finitely generated $U{\mathcal{Q}}$–modules are finitely generated. (Noetherian condition) 2. (b)

$H_{i}(\mathcal{N};V)_{n}$* is a finitely generated $R$–module for all $i,n\in\mathbb{N}$.* 3. (c)

$V$* is a finitely generated $U\mathcal{G}$–module.* 4. (d)

$\widetilde{H}_{i}(V)_{n}=0$* for all $n$ large enough in comparison to $i$.*

Then $H_{i}(\mathcal{N};V)$ is a finitely generated $U{\mathcal{Q}}$–module for every $i\in\mathbb{N}$.

This spectral sequence is also an important tool in [MPW17], where we apply it to the stability SES’s in Section 1 (b), (d), and (e).

Finally in Section 9, we revisit Quillen’s argument for homological stability. We prove a theorem on homological stability with twisted coefficients. Randal-Williams and Wahl [RWW15, Thm A] prove a similar result but require their coefficients to have finite polynomial degree. If we additionally assume H4, our Theorem B implies that our condition is weaker, but in general our ranges are worse than theirs.

Theorem D**.**

Let $U\mathcal{G}$ be a stability category. Assume H3. Let $V$ be a $U\mathcal{G}$–module with

[TABLE]

for all $n\geq k\cdot i+a$ for some $a\in\mathbb{Z}$ and $k\geq 2$, then the stabilization map

[TABLE]

is an epimorphism for all $n\geq k\cdot i+a-1$ and an isomorphism for all $n\geq k\cdot i+a$.

Remark \theremauto.

Notice that Theorem D and Section 7 give criterions to check whether the double cosets

[TABLE]

stabilize with growing $n$ and fixed $a,b$.

Acknowledgement

The author was supported by the Berlin Mathematical School and the Dahlem Research School. The author also wants to thank Aur$\acute{\text{e}}$lien Djament, Daniela Egas Santander, Reiner Hermann, Henning Krause, Daniel Lütgehetmann, Jeremy Miller, Holger Reich, Steven Sam, Elmar Vogt, Nathalie Wahl, Jenny Wilson for helpful conversations. Special thanks to Reiner Hermann and his invitation to NTNU where the idea for this project was born.

2. Central stability homology

Let $(\mathcal{C},\oplus,0)$ be a small monoidal category with unit object [math]. In particular,

[TABLE]

is a bifunctor. Let $\mathcal{D}$ be cocomplete category and $\operatorname{Fun}(\mathcal{C},\mathcal{D})$ the category of functors from $\mathcal{C}$ to $\mathcal{D}$. Then the monoidal structure gives rise to a functor

[TABLE]

by precomposition of $\oplus$. This is the same as giving a functor

[TABLE]

from $\mathcal{C}$ to the category of endofunctors of $\operatorname{Fun}(\mathcal{C},\mathcal{D})$. Explicitly it sends an object $X\in\mathcal{C}$ to the suspension endofunctor

[TABLE]

Because $S_{X}$ is just the precomposition of $-\oplus X$, the functor $S_{X}$ has a left adjoint, namely the left Kan extension along $-\oplus X$ (see [ML98, Cor X.3.2]). We will call its left adjoint the desuspension endofunctor and denote it by $\Sigma_{X}$. There is an antiequivalence of categories between left and right adjoint endofunctors. (Cf [Kan58, Thm 3.2]) Consequently, there is a functor

[TABLE]

that sends $X$ to $\Sigma_{X}$.

Clearly $S_{X\oplus Y}=S_{Y}\circ S_{X}$. Because $\Sigma_{X}$ is the right adjoint of $S_{X}$

[TABLE]

Thus $\Sigma$ is in fact a monoidal functor, where $\operatorname{End}(\operatorname{Fun}(\mathcal{C},\mathcal{D}))$ is equipped with the monoidal structure $\circ$.

Example \theexauto.

Consider the monoidal category $\mathcal{C}=\mathsf{FI}$ (which is not small but has a small skeleton) and $\mathcal{D}=\mathbb{Z}\mathsf{-mod}$ the cocomplete category of abelian groups. Then $\Sigma_{X}$ is given by

[TABLE]

for finite sets $X$ and $A$ and an $\mathsf{FI}$–module $F\colon\mathsf{FI}\to\mathbb{Z}\mathsf{-mod}$. This $\mathbb{Z}$–module is isomorphic to

[TABLE]

when $X=\{1,\dots,m\}$ and $A=\{1,\dots,n\}$ with $m\leq n$.

Definition \theDefauto.

Let the augmented semisimplicial category $\Delta^{\prime}_{+}$ be the category with objects the sets $\{0,\dots,n\}$ for $n\geq-1$ and morphisms strictly increasing set maps.

[TABLE]

defines a monoidal structure on $\Delta^{\prime}_{+}$ with the unit $\emptyset$, which is additionally initial.

The following theorem describes how $\Delta^{\prime}_{+}$ behaves universally among all monoidal categories with initial unit object. Let for the remainder of this section $(\mathcal{C},\oplus,0)$ be a small monoidal category with initial unit object [math] and denote the initial maps by $\iota_{X}\colon 0\to X$.

Theorem \thethmauto.

Let $X$ be an object in $\mathcal{C}$. Then there exists a unique monoidal functor $\Delta^{\prime}_{+}\to\mathcal{C}$ that sends $\{0\}$ to $X$.

Proof.

Similar to [Wei94, Ex 8.1.6], a functor $\Delta^{\prime}_{+}\to\mathcal{C}$ is given by a sequence of objects $K_{-1},K_{0},\dots$ and maps $d_{i}\colon K_{p-1}\to K_{p}(i=0,\dots,p)$ such that if $i<j$ then $d_{j}d_{i}=d_{i}d_{j-1}$. To be monoidal $K_{p}$ must be $X^{\oplus p+1}$ for all $n\geq-1$. Because $0\in\mathcal{C}$ is initial, there is exactly one candidate for $d_{0}\colon K_{-1}=0\to K_{0}=X$, which is the initial map $\iota_{X}\colon 0\to X$. Further the other maps must have the following form.

[TABLE]

One easily checks that this satisfies the condition given. ∎

Definition \theDefauto.

Let $X$ an object of $\mathcal{C}$. Then we get an augmented semisimplicial object

[TABLE]

This is the same as a functor

[TABLE]

Thus for every functor $F\colon\mathcal{C}\to\mathcal{D}$ we get an augmented semisimplicial object $K_{\bullet}^{X}F$ in $\operatorname{Fun}(\mathcal{C},\mathcal{D})$ with

[TABLE]

Definition \theDefauto.

Assume $\mathcal{D}$ is an abelian category, we denote the associated chain complex of $K_{\bullet}^{X}F$ by $\widetilde{C}^{X}_{*}F$ and the homology of this chain complex by $\widetilde{H}^{X}_{*}F$. We call this the central stability homology of $F$.

A well known consequence of the Yoneda Lemma is that

[TABLE]

Example \theexauto.

Let us consider $\mathcal{C}=\mathsf{FI}$ and $\mathcal{D}=\mathsf{Set}$, and let us fix $X=\{*\}$ a singleton. Given $F=\operatorname{Hom}_{\mathsf{FI}}(\emptyset,-)$ the representable functor, $K_{\bullet}^{X}F$ is given by

[TABLE]

with face maps given by precomposition of the morphisms

[TABLE]

where the hat indicates that $i$ is omitted.

Evaluating at $\{1,\dots,n\}$, we get the (augmented) semisimplicial set

[TABLE]

from [RWW15, Def 2.1] associated to the homogeneous category $\mathsf{FI}$.

More generally we can describe $\Sigma_{X}F$ for a functor $F\colon\mathcal{C}\to\mathcal{D}$ as a colimit. Let $(-\oplus X\downarrow B)$ be the comma category whose objects are the pairs $(A,\psi\colon A\oplus X\to B)$, then there is a forgetful functor

[TABLE]

Precomposing $F$ with this functor we get

[TABLE]

For a morphism $X\to Y$ in $\mathcal{C}$, the natural transformation $\Sigma_{Y}F\to\Sigma_{X}F$ is given by the functor

[TABLE]

mapping to the composition

[TABLE]

Example \theexauto.

For $\mathcal{C}=\mathsf{FI}$, $\mathcal{D}=\mathbb{Z}\mathsf{-mod}$ and $X=\{*\}$, the central stability homology is given by the semisimplicial $\mathsf{FI}$–module $K_{\bullet}^{X}F$ with

[TABLE]

*where $C_{f}$ is the complement of the image of $f$ and the face maps given by the precomposition of the morphisms above and the inclusion of the complements. *

3. Stability categories

In this section we introduce stability categories that are a specific kind of homogeneous categories (see [RWW15, Def 1.3] or Section 3) and weakly complemented categories (see [PS14] or Section 3).

Definition \theDefauto.

Let $(\mathcal{G},\oplus,0)$ be a monoidal groupoid whose monoid of objects is the natural numbers $\mathbb{N}$. The automorphism group of the object $n\in\mathbb{N}$ is denoted $G_{n}=\operatorname{Aut}^{\mathcal{G}}(n)$. Then $\mathcal{G}$ is called a stability groupoid if it satisfies the following properties.

1. (a)

The monoidal structure

[TABLE]

is injective for all $m,n\in\mathbb{N}$. 2. (b)

The group $G_{0}$ is trivial. 3. (c)

$(G_{l+m}\times 1)\cap(1\times G_{m+n})=1\times G_{m}\times 1\subset G_{l+m+n}$* for all $l,m,n\in\mathbb{N}$.*

Definition \theDefauto.

A homomorphism $\mathcal{G}\to\mathcal{H}$ of stability groupoids is a monoidal functor sending $1$ to $1$.

Remark \theremauto.

A stability groupoid $\mathcal{G}$ is given by a sequence of groups $(G_{n})_{n\in\mathbb{N}}$ and group homomorphisms

[TABLE]

satisfying additional properties that would require spelling out what a monoidal structure is. In notation we will usually suppress the monoidal structure and just write $\mathcal{G}=(G_{n})_{n\in\mathbb{N}}$ for a stability groupoid.

Similarly a homomorphism of stability groupoids

[TABLE]

is given by a sequence of group homomorphisms

[TABLE]

for which the diagrams

[TABLE]

commute.

Recall that a braiding $b$ of a monoidal category $(\mathcal{C},\oplus,0)$ is a natural isomorphism between the functors $\oplus\colon\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ and $\oplus$ precomposed with a swap that has some additional conditions. In particular, there are isomorphisms

[TABLE]

for all objects $A,B\in\mathcal{C}$, such that for all morphisms $f\colon A\to C$ and $g\colon B\to D$ the diagram

[TABLE]

commutes. For such natural isomorphism to be a braiding one also needs two hexagon identities. (Cf [ML98, Sec XI.1] for details.) If

[TABLE]

the braiding is called a symmetry.

For a stability groupoid $\mathcal{G}=(G_{n})_{n\in\mathbb{N}}$, a braiding $b$ is given by isomorphisms

[TABLE]

such that

[TABLE]

for all $f\in G_{m}$ and $g\in G_{n}$ and

[TABLE]

which amounts to the hexagon identities.

For any monoidal groupoid $\mathcal{G}$, Randal-Williams and Wahl [RWW15, Sec 1.1] give a construction originally due to Quillen that yields the category $U\mathcal{G}$.

Definition \theDefauto.

Let $(\mathcal{G},\oplus,0)$ be a monoidal groupoid. The category $U\mathcal{G}$ has the same objects as $\mathcal{G}$ and its morphisms from $A$ to $B$ are equivalence classes of pairs $(f,C)$ where $C$ is an object in $\mathcal{G}$ and $f$ is an (iso)morphism in $\mathcal{G}$ from $C\oplus A\to B$. Two of these pairs $(f,C)$ and $(f^{\prime},C^{\prime})$ are equivalent if there is an isomorphism $g\colon C\to C^{\prime}$ (in $\mathcal{G}$) such that the diagram

[TABLE]

commutes.

Definition \theDefauto.

If $\mathcal{G}$ is a braided stability groupoid, we call $(U\mathcal{G},\oplus,0)$ the stability category of $\mathcal{G}$.

The following definition is due to Randal-Williams and Wahl [RWW15, Def 1.3].

Definition \theDefauto.

Let $(\mathcal{C},\oplus,0)$ be a monoidal category with initial [math]. $\mathcal{C}$ is called a homogeneous category if it satisfies the following conditions for all objects $A,B\in\mathcal{C}$:

**H1: **

$\operatorname{Aut}(B)$* acts transitively on $\operatorname{Hom}(A,B)$ by postcomposition.*

**H2: **

The map $\operatorname{Aut}(A)\to\operatorname{Aut}(A\oplus B)$ taking $f$ to $f\oplus\mathrm{id}_{B}$ is injective and its image is

[TABLE]

where $\iota_{A}\colon 0\to A$ denotes the initial morphism.

$\mathcal{C}$* is called prebraided if its underlying groupoid is braided with a braiding $b$, that satisfies the equation*

[TABLE]

for all objects $A,B\in\mathcal{C}$.

Remark \theremauto.

The condition H2 is not only symmetric for symmetric homogeneous category $(\mathcal{C},\oplus,0)$; prebraided suffices for this conclusion. Here is the reason. Clearly $\operatorname{Aut}(A)\to\operatorname{Aut}(B\oplus A)$ is also injective and the image lies in

[TABLE]

But assume $f\in\operatorname{Fix}(B,B\oplus A)$ then $b_{B,A}\circ f\circ b^{-1}_{B,A}=g\oplus B$ for some $g\in\operatorname{Aut}(A)$ by H2 and thus

[TABLE]

The stability category of a braided stability groupoid is always a homogeneous category.

Proposition \thepropauto.

Let $\mathcal{G}$ be a braided (symmetric) stability groupoid, then $U\mathcal{G}$ is a prebraided (symmetric) homogeneous category. And the underlying groupoid of $U\mathcal{G}$ is $\mathcal{G}$.

Proof.

In [RWW15, Prop 2.6(i)+(ii)] it is shown that [math] is initial in $U\mathcal{G}$ and that $U\mathcal{G}$ is prebraided. For H1 and H2 the statements [RWW15, Thm 1.8(c)+(d)] apply. For the second statement [RWW15, Prop 2.10] applies. ∎

In [RWW15, Rem 1.4] it is already stated that in every homogeneous category

[TABLE]

For $U\mathcal{G}$ this means

[TABLE]

Because there is cancellation in $\mathbb{N}$, the homomorphisms are given by equivalence classes of pairs $(f,m)$ for some $f\in G_{m+n}$ and the above isomorphism is given by

[TABLE]

The composition is then

[TABLE]

for $fG_{l}\colon m\oplus n\to l\oplus m\oplus n$ and $gG_{m}\colon n\to m\oplus n$.

The monoidal structure on $U\mathcal{G}$ is then given by

[TABLE]

for $f_{1}G_{m_{1}}\in\operatorname{Hom}(n_{1},m_{1}\oplus n_{1})$ and $f_{2}G_{m_{2}}\in\operatorname{Hom}(n_{2},m_{2}\oplus n_{2})$.

Example \theexauto.

$\mathfrak{S}=(\mathfrak{S}_{n})_{n\in\mathbb{N}}$* is a braided stability groupoid with the braiding $b_{m,n}\in\mathfrak{S}_{m+n}$ given by the permutation*

[TABLE]

In fact, the braiding of $\mathfrak{S}$ is a symmetry, wherefore it is also a symmetry on $U\mathfrak{S}$.

An example of a braided stability groupoid that is not symmetric is given by the braid groups $\beta=(\beta_{n})_{n\in\mathbb{N}}$. Its braiding $b_{m,n}\in\beta_{m+n}$ is given by the following diagram.

[TABLE]

A stability category of a braided stability groupoid is also a weakly complemented category as defined by Putman and Sam [PS14].

Definition \theDefauto.

Let $(\mathcal{C},\oplus,0)$ be a monoidal category with initial [math]. $\mathcal{C}$ is called a weakly complemented category if it satisfies the following conditions for all objects $A,B,C\in\mathcal{C}$:

1. (a)

All morphisms are monomorphisms. 2. (b)

The map $\operatorname{Hom}(A\oplus B,C)\to\operatorname{Hom}(A,C)\times\operatorname{Hom}(B,C)$ given by

[TABLE]

is injective. 3. (c)

For every morphism $\psi\in\operatorname{Hom}(A,B)$ there is an object $D$ and an isomorphism $f\colon D\oplus A\to B$ such that $\psi=f\circ(\iota_{D}\oplus\mathrm{id}_{A})$. The pair $(D,f)$ is uniquely determined up to isomorphisms in the comma category $(-\oplus A\downarrow B)$.

Proposition \thepropauto.

Let $\mathcal{G}=(G_{n})_{n\in\mathbb{N}}$ be a braided stability groupoid, then $U\mathcal{G}$ is a weakly complemented category.

Proof.

Let $l,m,n\in\mathbb{N}$, ie objects of $\mathcal{G}$ and $U\mathcal{G}$. To prove (a), by H1, we only have to show that $\phi=\mathrm{id}_{l\oplus m\oplus n}G_{l}$ is a monomorphism. Let $\psi=fG_{m},\psi^{\prime}=f^{\prime}G_{m}\colon n\to m\oplus n$ such that

[TABLE]

Then there is an automorphism $g\in G_{l+m}$ such that

[TABLE]

By definition

[TABLE]

Therefore $\psi$ and $\psi^{\prime}$ must coincide.

Using H1 to prove (b), it is enough to show that for every $g\in G_{l+m+n}$ and $\phi=\mathrm{id}_{l\oplus m\oplus n}G_{l}$ the two equations

[TABLE]

imply $\phi=g\phi$. Using the second equation we get that

[TABLE]

thus by H2 there is a $g^{\prime}\in G_{l+m}$ with $g=g^{\prime}\oplus\mathrm{id}_{n}$. Applying the same trick to the first equation we get

[TABLE]

and therefore the existence of a $g^{\prime\prime}\in G_{l}$ with $g^{\prime}=g^{\prime\prime}\oplus\mathrm{id}_{m}$. For $g=g^{\prime\prime}\oplus\mathrm{id}_{m}\oplus\mathrm{id}_{n}$ clearly $\phi=g\phi$.

(c) is true by definition of $U\mathcal{G}$. ∎

For the remainder of this paper we assume that $\mathcal{G}=(G_{n})_{n\in\mathbb{N}}$ is a braided stability groupoid and thus $U\mathcal{G}$ is its prebraided stability category. Given a functor $F$ from $U\mathcal{G}$ to some category $\mathcal{D}$, we denote the images $F(n)$ by $F_{n}$.

Further for the object $1\in U\mathcal{G}$, we abbreviate $S_{1},\Sigma_{1},K^{1}_{\bullet},\widetilde{C}_{*}^{1},\widetilde{H}_{*}^{1}$ by $S,\Sigma,K_{\bullet},\widetilde{C}_{*},\widetilde{H}_{*}$, respectively.

4. Central stability and stability categories

In this section, we explain some basic properties of the functor $K_{\bullet}$ from Section 2 in the context of stability categories.

Proposition \thepropauto.

Let $l,m,n$ be objects in $U\mathcal{G}$. Then the map $\operatorname{Hom}(m,l\oplus m)\to\operatorname{Hom}(m\oplus n,l\oplus m\oplus n)$ given by $\psi\mapsto\psi\oplus\mathrm{id}_{n}$ is injective and its image is

[TABLE]

(Because of Section 3, the symmetric version of this statement is also true.)

Proof.

Given two morphisms from $fG_{l},f^{\prime}G_{l}\colon m\to l\oplus m$, and assume that

[TABLE]

Then there is an automorphism $g\in G_{l}$ such that

[TABLE]

which implies that

[TABLE]

Because $G_{l+m}\to G_{l+m+n}$ is injective,

[TABLE]

Thus

[TABLE]

and the map

[TABLE]

is injective.

If $\psi\in\operatorname{Hom}(m,m\oplus n)$,

[TABLE]

Let on the other hand $\chi=hG_{l}\colon m\oplus n\to l\oplus m\oplus n$ such that

[TABLE]

Then

[TABLE]

whence there is an isomorphism $g\in G_{l+m}$ with $h=g\oplus\mathrm{id}_{n}$. Thus

[TABLE]

Proposition \thepropauto.

Let $F\colon U\mathcal{G}\to\mathsf{Set}$. Then there is an isomorphism

[TABLE]

For a map $\eta=hG_{p-q}\in\operatorname{Hom}(q,p)$ the corresponding morphism

[TABLE]

is given by

[TABLE]

with $\phi^{\prime}=F(\mathrm{id}_{n-p}\oplus\iota_{p-q})$. This morphism is independent of the choice of $h\in\eta$.

Proof.

We know from Section 2 that

[TABLE]

Note that by construction, every morphism $m\oplus p\to n$ factors through an automorphism $(n-p)\oplus p\to n$. Therefore there is a surjection

[TABLE]

There are still some relations, which are given by the precomposition by elements in $G_{n-p}$. Therefore

[TABLE]

together with the (choice-free) maps

[TABLE]

given by

[TABLE]

where $\phi=F(\iota_{n-p-m}\oplus\mathrm{id}_{m})$ and $\psi=gG_{n-p-m}$.

To understand the functoriality, we see that

[TABLE]

is given by

[TABLE]

Therefore

[TABLE]

maps to

[TABLE]

via $F_{n-p}$ corresponding to $(n-p,g\circ(\mathrm{id}_{n-p}\oplus\eta))\in(-\oplus q\downarrow n)$. Thus

[TABLE]

where $h\in G_{p}$ with $\eta=hG_{p-q}$. ∎

A similar proposition can be made for functors $U\mathcal{G}\to R\mathsf{-mod}$:

Proposition \thepropauto.

Let $V\colon U\mathcal{G}\to R\mathsf{-mod}$. Then there is an isomorphism

[TABLE]

For a map $\eta=hG_{p-q}\in\operatorname{Hom}(q,p)$ the corresponding morphism

[TABLE]

is given by

[TABLE]

with $\phi^{\prime}=V(\mathrm{id}_{n-p}\oplus\iota_{p-q})$. This morphism is independent of the choice of $h\in\eta$.

This allows us to describe the face maps of the semisimpicial set $(K_{\bullet}F)_{n}$ for a $U\mathcal{G}$–set $F$. Recall that

[TABLE]

Then

[TABLE]

is given by

[TABLE]

Example \theexauto.

The functor $F=\operatorname{Hom}(0,-)\colon\mathcal{C}\to\mathsf{Set}$ sends every object to a singleton. Then

[TABLE]

Which is the semisimplicial set $W_{n}(0,1)_{n}$ from [RWW15, Def 2.1].

Example \theexauto.

We can now pick up Section 2 again. Let $V\colon\mathsf{FI}\to\mathbb{Z}\mathsf{-mod}$. As $U\mathfrak{S}$ from Section 3 is a skeleton of $\mathsf{FI}$, we have

[TABLE]

by sending $g\otimes v$ to

[TABLE]

in the summand corresponding to

[TABLE]

Functors $F\colon U\mathcal{G}\to\mathsf{Set}$ that preserve monomorphisms, such as representable functors, send all maps in $U\mathcal{G}$ to injective set maps. For such functors we can split $K_{\bullet}F$ into a disjoint union.

Proposition \thepropauto.

Let $F\colon U\mathcal{G}\to\mathsf{Set}$ preserve monomorphisms. Then the semisimplicial set $(K_{\bullet}F)_{n}$ splits disjointly

[TABLE]

into augmented semisimplicial sets $L_{\bullet}^{x}\subset(K_{\bullet}\operatorname{Hom}(0,-))_{n}$ with

[TABLE]

where $\sigma^{\prime}=f\circ(\mathrm{id}_{n-p-1}\oplus\iota_{p+1})$ is a complement of $\sigma=fG_{n-p-1}$.

Proof.

Because $(K_{-1}F)_{n}=F_{n}$, the augmented semisimplicial set splits into

[TABLE]

with $L_{p}^{x}\subset(K_{p}F)_{n}$ the preimage of $x\in F_{n}$ which is

[TABLE]

using Section 4. Because $F$ preserves monomorphims, the set map $\phi^{\prime}=F(\mathrm{id}_{n-p-1}\oplus\iota_{p+1})$ is injective. Thus for every $g\in G_{n}$ there is at most one $y\in F_{n-p-1}$ with $g\phi^{\prime}(y)=x$. Therefore $L_{\bullet}^{x}$ can be included into $(K_{\bullet}\operatorname{Hom}(0,-))_{n}$ by sending $[g,y]$ to $gG_{n-p-1}$.

Finally, note that

[TABLE]

5. Central stability and finiteness properties

Definition \theDefauto.

We call a functor from a category $\mathcal{C}$ to the category of sets a $\mathcal{C}$–set and write $\mathcal{C}\mathsf{-Set}:=\operatorname{Fun}(\mathcal{C},\mathsf{Set})$. A $U\mathcal{G}$–set $V$ is generated in ranks $\leq m$ if there is an epimorphism $P\relbar\joinrel\twoheadrightarrow V$ of $U\mathcal{G}$–sets where $P$ is a disjoint union

[TABLE]

of representable functors such that $m_{i}\leq m$ for all $i\in I$.

Fix a ring $R$. We call a functor from a category $\mathcal{C}$ to the category of $R$–modules a $\mathcal{C}$–module and write $\mathcal{C}\mathsf{-mod}:=\operatorname{Fun}(\mathcal{C},R\mathsf{-mod})$. A $U\mathcal{G}$–module $V$ is generated in ranks $\leq m$ if there is an epimorphism $P\relbar\joinrel\twoheadrightarrow V$ of $U\mathcal{G}$–modules where $P$ is a direct sum

[TABLE]

such that $m_{i}\leq m$ for all $i\in I$.

In both situations we call such a $P$ freely generated in ranks $\leq m$.

Remark \theremauto.

For $U\mathcal{G}$–modules, a similar notion has been called “freely generated” by for example [CEF15]. The $U\mathcal{G}$–module $R\operatorname{Hom}(m,-)$ has a linear right action by $G_{m}$. Let $W_{m}$ be a not necessarily free $RG_{m}$–module, then

[TABLE]

is a $U\mathcal{G}$–module. These modules are projective over $U\mathcal{G}$ if and only if $W_{m}$ is a projective $RG_{m}$–module. Therefore we will not call these freely generated as [CEF15] do in the case of $\mathsf{FI}$–modules.

Let $V$ be a $U\mathcal{G}$–set or a $U\mathcal{G}$–module, then $V_{n}=V(n)$ is a sequence of $G_{n}$–sets or $G_{n}$–representations, respectively, and $\phi_{n}=V(\iota_{1}\oplus\mathrm{id}_{n})$ is $G_{n}$–equivariant. The following lemma is a criterion when such a sequence is actually a $U\mathcal{G}$–set or a $U\mathcal{G}$–module. This criterion has been observed by Church–Ellenberg–Farb [CEF15, Rmk 3.3.1] for the special case of $\mathsf{FI}$–modules but the argument easily generalizes as shown in [RWW15, Prop 4.2].

Lemma \thelemauto.

Let $(V_{n},\phi_{n})_{n\in\mathbb{N}}$ be a sequence of $RG_{n}$–modules or $G_{n}$–sets (resp.) $V_{n}$ and $G_{n}$–equivariant maps $\phi_{n}\colon V_{n}\to V_{n+1}$.

There is a unique $U\mathcal{G}$–module or $U\mathcal{G}$–set (resp.) $V$ with $V(n)=V_{n}$ and $V(\iota_{1}\oplus\mathrm{id}_{n})=\phi_{n}$ if and only if for all $m\leq n$ and $g\in G_{n-m}$

[TABLE]

where $\phi_{m,n}=\phi_{n-1}\circ\dots\circ\phi_{m}$.

We want to connect the central stability homology to generation properties.

Proposition \thepropauto.

Let $V\colon U\mathcal{G}\to\mathcal{D}$ be a functor to $\mathcal{D}=\mathsf{Set}$ or to $\mathcal{D}=R\mathsf{-mod}$. Then $V$ is generated in the ranks $\leq d$ if and only if the map

[TABLE]

induced by $\iota_{1}\in\operatorname{Hom}(0,1)$ is surjective for all $n>d$.

Proof.

We give the proof for $\mathcal{D}=\mathsf{Set}$. For $\mathcal{D}=R\mathsf{-mod}$ the proof is analogous after linearlizing.

Let $p\leq d<n$. Then for a representable functor

[TABLE]

is given by $f\mapsto f\circ(\mathrm{id}_{p}\oplus\iota_{1})$ is surjective because $G_{n}$ acts transitively on $\operatorname{Hom}(p,n)$ by postcomposition. Let there be an epimorphism

[TABLE]

such that all $m_{i}\leq d$. The diagram

[TABLE]

commutes because $\Sigma\to\mathrm{id}$ is a natural transformation between endofunctors. This implies the first implication.

Let $V$ be a functor for which

[TABLE]

is surjective for all $n>d$. Let $P$ be a disjoint union

[TABLE]

of representable functors with $m\leq d$ together with a morphism of $U\mathcal{G}$–sets $P\to V$ such that

[TABLE]

is surjective for all $n\leq d$. (The Yoneda Lemma lets us find such a $P$.)

Let $n>d$ and assume that $P_{n-1}\to V_{n-1}$ is surjective by induction. Then the following commutative diagram show that $P_{n}\to V_{n}$ is surjective.

[TABLE]

The previous proposition says that a $U\mathcal{G}$–module $V$ is generated in finite ranks if and only if $\widetilde{H}_{-1}V$ is stably zero, ie $\widetilde{H}_{-1}V_{n}=0$ for all $n$ large enough. We next want to generalize this concept. For this we need an additional condition.

Definition \theDefauto.

Let $a,k\in\mathbb{N}$. We define the following condition.

**H3($N$): **

$\widetilde{H}_{i}(R\operatorname{Hom}(0,-))_{n}=0$* for all $i<N$ and all $n>k\cdot i+a$.*

Remark \theremauto.

Note that the condition LH3 in [RWW15, Def 2.2] implies H3($\infty$) with $a=\operatorname{rk}A+1$. For all of Section 1, they prove or gather the following values from existing literature.

[TABLE]

Theorem \thethmauto.

Assume H3($N$). Let $V$ be a $U\mathcal{G}$–module and $d_{0},\dots,d_{N}\in\mathbb{Z}$ with $d_{i+1}-d_{i}\geq\max(k,a)$, then the following statements are equivalent.

1. (a)

There is a partial resolution

[TABLE]

with $P_{i}$ that are freely generated in ranks $\leq d_{i}$. 2. (b)

The homology

[TABLE]

for all $-1\leq i<N$ and all $n>d_{i+1}$.

To prove this theorem we first compute the central stability homology of free $U\mathcal{G}$–modules. This part is quite technical and can easily be skipped upon the first read. Assuming Section 5, the proof of Section 5 is straight forward.

To compute the central stability homology of free $U\mathcal{G}$–modules, we investigate $K_{\bullet}\operatorname{Hom}(m,-)$. We first introduce the concept of joins and links for augmented semisimplicial sets. We say $\rho$ is a join $\tau*\sigma$ of the $p$–simplex $\tau$ and the $q$–simplex $\sigma$ in a semisimplicial object if

[TABLE]

and let the left- and right-sided link be

[TABLE]

In $K_{\bullet}\operatorname{Hom}(0,-)_{n}$ the $p$–simplices are morphisms from $p+1\to n$. And for $\rho\in K_{p+q+1}\operatorname{Hom}(0,-)_{n}$

[TABLE]

Condition (b) of Section 3 implies that $\tau*\sigma$ is unique if it exists. More generally we write $\tau*\sigma=\rho\colon l\oplus m\to n$ for $\tau\colon l\to n$ and $\sigma\colon m\to n$ if

[TABLE]

Lemma \thelemauto.

Let $fG_{n-m}=\sigma\colon m\to n$, then

[TABLE]

and

[TABLE]

Proof.

Assume

[TABLE]

then

[TABLE]

Similarly if

[TABLE]

then

[TABLE]

For the opposite direction apply Section 4 to

[TABLE]

and

[TABLE]

Corollary \thecorauto.

Let $\sigma\in K_{p}\operatorname{Hom}(0,-)_{n}$, then both

[TABLE]

Proposition \thepropauto.

Assume H3($N$). Then

[TABLE]

for all $i<N$ and all $n>k\cdot i+a+m$.

Proof.

From Section 4 we know that $K_{p}\operatorname{Hom}({m},-)_{n}$ is isomorphic to

[TABLE]

with

[TABLE]

where $\sigma^{\prime}=f\circ(\mathrm{id}_{n-p-1}\oplus\iota_{p+1})$ is a complement of $\sigma=fG_{n-p-1}$. Note that

[TABLE]

that is the set of those morphisms that factor through $\sigma^{\prime}$, is independent of the choice of $f$. Because

[TABLE]

Section 5 gives that $\tau\in\operatorname{Lk}^{L}\sigma$, hence

[TABLE]

This means that for every $U\mathcal{G}$–module $P$ freely generated in ranks $\leq m$, $\widetilde{H}_{i}(P)_{n}=0$ for all $-1\leq i<N$ and $n>k\cdot i+a+m$. ∎

This proposition suffices to prove Section 5. First we will derive a corollary that the modules described in Section 5 have vanishing central stability homology in the same range as the freely generated $U\mathcal{G}$–module $R\operatorname{Hom}(m,-)$. Note that in Section 5 (a) implies (b) for any partial resolution with vanishing central stability homology in the same ranges as the $P_{i}$ that are freely generated in ranks $\leq d_{i}$.

Corollary \thecorauto.

Assume H3($N$). Let $W_{m}$ be an $RG_{m}$–module. Then

[TABLE]

for all $i<N$ and all $n>k\cdot i+a+m$.

Proof.

Let

[TABLE]

be a projective resolution of $W_{m}$ by $RG_{m}$–modules. We consider the complex

[TABLE]

and its two spectral sequences. The first spectral sequence

[TABLE]

is given by

[TABLE]

because

[TABLE]

is a free $RG_{m}$–module.

The second spectral sequence

[TABLE]

that converges to the same limit, computes to

[TABLE]

Therefore the central stability homology of $R\operatorname{Hom}(m,-)\otimes_{RG_{m}}W_{m}$ vanishes in the same range as $R\operatorname{Hom}(m,-)$. ∎

Proof of Section 5.

Assume (a). Let $K_{-1}=P_{-1}:=V$ and let

[TABLE]

for $i\geq 0$. Then the long exact sequence of $\widetilde{H}_{*}$ for the short exact sequence

[TABLE]

implies that

[TABLE]

is injective for all $i,j\leq N-1$ and all $n>kj+a+d_{i+1}$. Because $P_{i+1}\relbar\joinrel\twoheadrightarrow K_{i}$ for all $i\leq N-1$, we know from Section 5 that

[TABLE]

for all $n>d_{i+1}$. Thus

[TABLE]

is zero for all $i\leq N-1$ and $n>d_{i+1}$ because

[TABLE]

for $j=0,\dots,i+1$.

Now assume (b). Let by induction

[TABLE]

be a partial resolution such that $P_{i}$ is freely generated in ranks $\leq d_{i}$ for all $i\leq N-1$. Let as before $K_{-1}=P_{-1}:=V$ and let

[TABLE]

for $i\geq 0$. We need to prove that $K_{N-1}$ is generated in ranks $d_{N}$. Similar as before

[TABLE]

is surjective for all $i,j\leq N-1$ and all $n>k(j-1)+a+d_{i+1}$. Thus

[TABLE]

for all

[TABLE]

Definition \theDefauto.

We call a $U\mathcal{G}$–modules $V$ stably acyclic if $\widetilde{H}_{i}(V)_{n}=0$ for all $i\geq-1$ and all $n$ large enough.

We want to conclude this section by giving an example that is by definition stably acyclic, but the existence of the free resolution as in Section 5 is not a priori clear. More generally, in Section 7 we find a condition H4 on $U\mathcal{G}$ such that all $U\mathcal{G}$–modules with finite polynomial degree (see Section 7) are stably acyclic.

Corollary \thecorauto.

Assume H3($N$). Let $V$ be a $U\mathcal{G}$–module such that $V_{n}=0$ for all $n>d$. Then there is a resolution

[TABLE]

with $P_{i}$ that are freely generated in ranks $\leq\max(k,a)\cdot i+d$.

6. Notions of central stability

Different notions of central stability have been used in the past. We want to clear this up for stability categories. In the following proposition we prove the equivalence of four conditions. The condition (a) is called “presented in degree $\leq d$” in [CEF15, CE16]. In [PS14], condition (b) is called “central stability”. And conditions (c) and (d) are used in [CEFN14], although they phrase the colimit over an equivalent category. We will refer to this notion as in (a) by presented in the ranks $\leq d$.

Proposition \thepropauto.

Let $V$ be a $U\mathcal{G}$–module, then the following are equivalent.

1. (a)

$V$* is generated and presented in the ranks $\leq d$, ie there are $U\mathcal{G}$–modules $P_{0},P_{1}$ freely generated in the ranks $\leq d$ such that*

[TABLE]

is exact. 2. (b)

Let $\operatorname{inc}_{d}\colon U\mathcal{G}^{\leq d}\lhook\joinrel\longrightarrow U\mathcal{G}$ be the full subcategory with the objects $0,\dots,d$. The left Kan extension

[TABLE]

is naturally isomorphic to $V$. 3. (c)

For all $n>d$ there is a natural isomorphism

[TABLE] 4. (d)

For all $n>d$ there is a natural isomorphism

[TABLE]

Proof.

Djament [Dja16, Prop 2.14] proves that (a) and (b) are equivalent in an even more general setting. Gan–Li [GL17, Thm 3.2] gave a different proof with the viewpoint of graded modules over graded nonunital algebra.

The left Kan extension evaluated at $n\in\mathbb{N}$ can be expressed as the colimit in (c) (see [ML98, Cor X.3.4]):

[TABLE]

This proves that (c) is equivalent to (b).

Finally we want to prove that (c) and (d) are equivalent. First note that

[TABLE]

is just a different way of writing the same colimit. Further there is a natural map

[TABLE]

And because $(\operatorname{inc}_{d}\downarrow n)$ is a full subcategory of $(\operatorname{inc}_{n-1}\downarrow n)$ there is a map

[TABLE]

and similarly fixing a morphism $m\to n$ there is a map

[TABLE]

Let us prove the following claim.

Claim. Let $n\geq d$ and assume that

[TABLE]

is an isomorphism for all $m<n$, then

[TABLE]

is an isomorphism.

Proof.

For every morphism $f\colon m\to n$ which is an object in $(\operatorname{inc}_{n-1}\downarrow n)$, we get the following diagram.

[TABLE]

Here $\delta$ is the unique map such that $\alpha=\delta\beta$. One checks also checks that $\beta=\gamma\alpha$. This implies that

[TABLE]

The universal property of the colimits implies that $\gamma$ and $\delta$ are inverses. ∎

We now want to finish our proof of the equivalence of (c) and (d) by induction over $n$. For $n=d+1$ the statements are identical. Now assume that

[TABLE]

for all $d<m<n$. Then we can certainly assume

[TABLE]

is an isomorphism for all $m<n$. The claim then finishes the induction step. ∎

Putman’s [Put15] original notion of central stability for $\mathsf{FI}$–modules is reflected in condition (c) of the following proposition. In the case of symmetric stability categories, this is equivalent to (a), which we call centrally stable in the ranks $>d$.

Proposition \thepropauto.

Let $V$ be a $U\mathcal{G}$–module, then the following are equivalent.

1. (a)

$\widetilde{H}_{0}(V)_{n}=\widetilde{H}_{-1}(V)_{n}=0$* for all $n>d$.* 2. (b)

V_{n}\cong\operatorname{coeq}\big{(}\operatorname{Ind}^{G_{n}}_{G_{n-2}}V_{n-2}\rightrightarrows\operatorname{Ind}^{G_{n}}_{G_{n-1}}V_{n-1}\big{)}* for all $n>d$.*

If $\mathcal{G}$ is symmetric, then the following condition is equivalent to the first two.

1. (c)

V_{n}\cong\operatorname{coeq}\big{(}\operatorname{Ind}^{G_{n}}_{G_{n-2}\times\mathfrak{S}_{2}}V_{n-2}\otimes\mathcal{A}_{2}\rightrightarrows\operatorname{Ind}^{G_{n}}_{G_{n-1}}V_{n-1}\big{)}* for all $n>d$, where $\mathfrak{S}_{2}\subset G_{2}$ and $\mathcal{A}_{2}$ is its sign representation.*

Proof.

Conditions (a) and (b) are equivalent because (b) is equivalent to the sequence

[TABLE]

being exact, which is part of the central stability complex.

To see that (b) and (c) are equivalent, we have to check the image that the images in $\operatorname{Ind}^{G_{n}}_{G_{n-1}}V_{n-1}$ coincide. Let $t\in\mathfrak{S}_{2}$ be the transposition, then both maps are given by

[TABLE]

for $v\in V_{n-2}$. This completes the proof. ∎

With Section 5 we can connect these two notions. Note that the assumption of the theorem can be expressed as follows.

**H3($1$): **

$b_{1,1}\oplus\mathrm{id}_{n-2}$ and $1\times G_{n-1}$ generate $G_{n}$ for all $n>a$.

Corollary \thecorauto.

Assume H3($1$). Let $V$ be a $U\mathcal{G}$–module. Then the following are equivalent.

1. (a)

$V$* is generated in ranks $\leq d$ and presented in ranks $\leq d+a$.* 2. (b)

$V$* is generated in ranks $\leq d$ and centrally stable in the ranks $>d+a$.*

Corollary \thecorauto.

1. (a)

A $U\mathfrak{S}$–module is generated in ranks $\leq d$ and presented in ranks $\leq d+1$ if and only if it is generated in ranks $\leq d$ and centrally stable in the ranks $>d+1$. 2. (b)

Let $R$ be a ring with stable rank $s$. A $U\operatorname{GL}(R)$–module is generated in ranks $\leq d$ and presented in ranks $\leq d+s+1$ if and only if it is generated in ranks $\leq d$ and centrally stable in the ranks $>d+s+1$. 3. (c)

Let $R$ be a ring with unitary stable rank $s$. A $U\operatorname{Sp}(R)$–module is generated in ranks $\leq d$ and presented in ranks $\leq d+s+2$ if and only if it is generated in ranks $\leq d$ and centrally stable in the ranks $>d+s+2$. 4. (d)

A $U\operatorname{AutF}$–module is generated in ranks $\leq d$ and presented in ranks $\leq d+2$ if and only if it is generated in ranks $\leq d$ and centrally stable in the ranks $>d+2$. 5. (e)

A $U\beta$–module is generated in ranks $\leq d$ and presented in ranks $\leq d+1$ if and only if it is generated in ranks $\leq d$ and centrally stable in the ranks $>d+1$. 6. (f)

A $U\operatorname{Mod}\Sigma$–module is generated in ranks $\leq d$ and presented in ranks $\leq d+2$ if and only if it is generated in ranks $\leq d$ and centrally stable in the ranks $>d+2$.

7. Polynomial degree

As we defined $S$ as the precomposition of $-\oplus 1$, for the next definition we will use $T$ defined as the precomposition of $1\oplus-$. The following definition is an adaptation of van der Kallen’s [vdK80] degree of a coefficient system. Our version is almost identical with [RWW15, Def 4.10].

Definition \theDefauto.

Let $V$ be a $U\mathcal{G}$–module then we define the $U\mathcal{G}$–modules

[TABLE]

We say $V$ has polynomial degree $-\infty$ in the ranks $>d$ if $V_{n}=0$ for all $n>d$. For $r\geq 0$ we say $V$ has polynomial degree $\leq r$ in the ranks $>d$ if $(\ker V)_{n}=0$ for all $n>d$ and $\operatorname{coker}V$ has polynomial degree $\leq r-1$ in the ranks $>d$.

Remark \theremauto.

Assuming that the dimension of $V_{n}$ is finite for all $n>d$, $V$ having polynomial degree $\leq r$ in the ranks $>d$ implies that the dimension grows polynomially of degree $\leq r$ in the ranks $>d$.

Lemma \thelemauto.

1. (a)

Let

[TABLE]

be an exact sequence of $U\mathcal{G}$–modules in ranks $>d$, i.e.

[TABLE]

is a short exact sequence for all $n>d$. Then if two of the three $U\mathcal{G}$–modules $V^{\prime},V,V^{\prime\prime}$ have polynomial degrees $\leq r$ in ranks $>d$, then so does the third. 2. (b)

Assume the base ring $R$ is a field. If $V$ and $W$ are $U\mathcal{G}$–modules have polynomial degrees $\leq r$ and $\leq s$ in ranks $>d$, respectively, then $V\otimes W$ has polynomial degree $\leq r+s$ in ranks $>d$, where $V\otimes W$ is defined by $(V\otimes W)_{n}=V_{n}\otimes W_{n}$.

Proof.

1. (a)

From the snake lemma we get an exact sequence

[TABLE]

for all $n>d$. If $V^{\prime}$ has polynomial degree $r$ in ranks $>d$,

[TABLE]

for all $n>d$. Hence this sequence actually splits into two short exact sequences. If $V$ and $V^{\prime\prime}$ have finite polynomial degree, then the above sequence still splits into two short exact sequences. The assertion is shown by induction on the degree. 2. (b)

If $-\infty\in\{r,s\}$, also $V\otimes W$ has polynomial degree $-\infty$ in ranks $>d$. Thus $\ker(V\otimes W)_{n}=0$ for all large enough $n$. For an induction argument we consider the following commutative diagram.

[TABLE]

Clearly the three columns are exact. By chasing this diagram, we can prove that the top and the bottom row are also exact. Even more,

[TABLE]

is a short exact sequence for all $n>d$ because $V_{n}\otimes TW_{n}\to TV_{n}\otimes TW_{n}$ is injective for all $n>d$. By induction $\operatorname{coker}(V\otimes W)$ has polynomial degree $\leq r+s-1$ in ranks $>d$ using (a).∎

The following lemma is essentially [PS14, Lem 3.11].

Lemma \thelemauto.

Let $V$ be a $U\mathcal{G}$–module then the map $\mathrm{id}_{n}\oplus\iota_{1}\colon n\to n\oplus 1$ induces the zero map on $\widetilde{H}_{-1}V$. This implies that every morphism of $U\mathcal{G}$ that is not an isomorphism induces the zero map on $\widetilde{H}_{-1}V$.

Proof.

Let

[TABLE]

be given by the inclusion $v\mapsto 1\otimes v$. Then the following diagram commutes by the calculations from Section 4.

[TABLE]

The exactness of the lower row implies that $V_{n}\to\widetilde{H}_{-1}V_{n+1}$ must be the zero map. Thus surjectivity of $V_{n}\to\widetilde{H}_{-1}V_{n}$ implies the assertion. ∎

Proposition \thepropauto.

Let $V$ be a $U\mathcal{G}$–module and $d\in\mathbb{N}$. If $\operatorname{coker}V$ is generated in ranks $\leq d-1$, then $V$ is generated in ranks $\leq d$.

Proof.

We consider the following diagram.

[TABLE]

The map $\widetilde{H}_{-1}V\to T\widetilde{H}_{-1}V$ is the zero map because of Section 7. Thus there is an epimorphism

[TABLE]

This implies that $T\widetilde{H}_{-1}V$ is generated in ranks $\leq d-1$. Using Section 7 again, we infer that $\widetilde{H}_{-1}V_{n+1}=T\widetilde{H}_{-1}V_{n}=0$ for all $n>d$. Therefore $V$ is generated in ranks $\leq d$ by Section 5. ∎

Definition \theDefauto.

Let $\ell,b\geq 1$. We define the following condition.

**H4($N$): **

$\widetilde{H}_{i}(\operatorname{coker}R\operatorname{Hom}(m,-))_{n}=0$* for all $i<N$ and all $n>\ell\cdot(i+m)+b$.*

Remark \theremauto.

Knowing that $\operatorname{coker}R\operatorname{Hom}(m,-)$ is stably acyclic for all $m\in\mathbb{N}$, one can also follow that $T^{r}R\operatorname{Hom}(m,-)$ is stably acyclic for all $m,r\in\mathbb{N}$.

Proposition \thepropauto.

Let $N\in\mathbb{N}$. Assume H3($N$) and H4($N$) with $b\geq\max(k,a)$. Let $d_{0},\dots,d_{N}\leq-1$ with $d_{i}\geq\max(\ell\cdot d_{i-1}+b,d_{i-1}+b)$. If $\ker V_{n}=0$ for all $n>d_{1}$ and $\widetilde{H}_{i}(\operatorname{coker}V)_{n}=0$ for all $i\leq N-1$ and all $n>d_{i+1}$, then $\widetilde{H}_{i}(V)_{n}=0$ for all $i\leq N-1$ and $n>d_{i+1}+1$.

Proof.

The case $N=0$ is proved in Section 7.

Let $N\geq 1$. By induction

[TABLE]

for all $n>d_{i+1}+1$ for $i<N-1$. Section 5 implies the existence of a resolution

[TABLE]

by $U\mathcal{G}$–modules $P_{i}$ that are freely generated in ranks $\leq d_{i}+1$, because $d_{i}-d_{i-1}\geq b\geq\max(k,a)$. If we can prove that $K_{N-1}=\ker(P_{N-1}\to P_{N-2})$ is generated in ranks $\leq d_{N}+1$, we are done. Let

[TABLE]

with the convention $P_{-1}=K_{-1}=V$. Because $T$ is exact, the snake lemma gives us the exact sequence

[TABLE]

If $i\geq 0$, $\ker K_{i}$ is a submodule of $\ker P_{i}=0$. Therefore there is a long exact sequence

[TABLE]

By the assumption on $\widetilde{H}_{N-i-3}(\operatorname{coker}P_{i+1})_{n}$, we see that

[TABLE]

is surjective for all $i\leq N-2$ and all $n>\ell\cdot(d_{i+1}+1+{N-i-3})+b$.

If $i=-1$, ie $\ker K_{i}=\ker V$, this method only infers

[TABLE]

is surjective for all $n>\ell\cdot(d_{0}+N-1)+b$. Because $\ker V_{n}=0$ for all $n>d_{1}$,

[TABLE]

for all $n>d_{1}+N-1$. Therefore in the same range

[TABLE]

is injective.

Thus the assumption that $\widetilde{H}_{N-1}(\operatorname{coker}V)_{n}=0$ for all $n>d_{N}$ implies

[TABLE]

in the same range because

[TABLE]

and

[TABLE]

By Section 7, $K_{N-1}$ is generated in ranks $\leq d_{N}+1$. ∎

A direct consequence of this proposition is Theorem B:

Corollary \thecorauto.

Assume H3($\infty$) and H4($\infty$) with $b\geq\max(k,a)$. Let $r\geq 1$, $d\geq-1$ and let $V$ be of polynomial degree $\leq r$ in ranks $>d$, then $\widetilde{H}_{i}V_{n}=0$ for all $i\geq-1$ and $n>\ell^{i+1}(d+r)+(\ell^{i}+\dots+1)b+1$.

Proof.

For $r=0$, $\ker V_{n}=\operatorname{coker}V_{n}=0$ for all $n>d$. Then $\widetilde{H}_{i}(\operatorname{coker}V)_{n}=0$ for all $n>d+i+1$. Let $d_{0}^{0}=d$ and for $i>0$ let

[TABLE]

if $d\geq 0$ and

[TABLE]

if $d=-1$. Then $d_{i}^{0}\geq\max(\ell\cdot d_{i-1}^{0}+b,d_{i-1}^{0}+b)$, $\ker V_{n}=0$ for all

[TABLE]

and $\widetilde{H}_{i}(\operatorname{coker}V)_{n}=0$ for all

[TABLE]

Therefore by Section 7,

[TABLE]

for all $n>d^{0}_{i+1}+1$. This proves the case $r=0$.

Let $r>0$ and define

[TABLE]

By induction $\widetilde{H}_{i}\operatorname{coker}V_{n}=0$ for all

[TABLE]

We also have that

[TABLE]

and

[TABLE]

for all

[TABLE]

Thus we can apply Section 7 to get

[TABLE]

for all $n>d^{r}_{i+1}+1$. This proves the case $r>0$. ∎

From H3($N$) and a long exact sequence we get that

[TABLE]

is surjective for all $i\leq N-1$ and all $n>k\cdot(i-1)+a+m$. With the next proposition we will express $K_{\bullet}T\operatorname{Hom}(m,-)_{n}$ in form of the semisimplicial sets $K_{\bullet}\operatorname{Hom}(0,-)_{n}$ that are analyzed in [RWW15].

Proposition \thepropauto.

The augmented semisimplicial set $K_{\bullet}T\operatorname{Hom}(m,-)_{n}$ is isomorphic to the disjoint union of augmented semisimplicial sets

[TABLE]

with the embedding of $K_{\bullet}\operatorname{Hom}(0,-)_{n}\subset K_{\bullet}\operatorname{Hom}(0,-)_{n+1}$ by $\sigma\mapsto(\iota_{1}\oplus\mathrm{id}_{n})\circ\sigma$.

Proof.

From Section 4 we know that $K_{p}T\operatorname{Hom}({m},-)_{n}$ is isomorphic to

[TABLE]

with

[TABLE]

where $\sigma^{\prime}=f\circ(\mathrm{id}_{n-p-1}\oplus\iota_{p+1})$ is a complement of $\sigma=fG_{n-p-1}$. The equation

[TABLE]

and

[TABLE]

together with Section 5 gives that $\tau\in\operatorname{Lk}^{L}((\iota_{1}\oplus\mathrm{id}_{n})\circ\sigma)$, hence

[TABLE]

Example \theexauto.

Here we want to provide an example of a stability category that satisfies H3($\infty$) but does not even satisfy H4([math]). We will explain why $T\operatorname{Hom}_{U\beta}(1,-)$ is not generated in finite ranks. Let $\sigma=b\beta_{n}\in T\operatorname{Hom}_{U\beta}(1,-)_{n}=\operatorname{Hom}_{U\beta}(1,1\oplus n)$ which is the coset that is represented by a braid $b\in\beta_{n+1}$. Now

[TABLE]

which is depicted in the following diagram.

[TABLE]

Our claim, that $T\operatorname{Hom}_{U\beta}(1,-)$ is not generated in finite ranks, amounts to showing

[TABLE]

This follows because the braid

[TABLE]

is not contained in the LHS.

8. Short exact sequences

In this section will deal with short exact sequences of groups and stability groupoids. We restate some well-known properties of group homology regarding these short exact sequences in the language of modules over stability categories. Most of the arguments of this section have already appeared in [PS14] but we hope that they become more accessible in the language of this paper.

Let

[TABLE]

be a short exact sequence of groups. We want to describe the group homology of $N$ as a $Q$–module. For this let us first review how an automorphism $\phi\in\operatorname{Aut}(N)$ acts on $H_{i}(N;M)$ for some $RN$–module $M$ (given an $RN$–homomorphism $\psi\colon M\to\operatorname{Res}_{\phi}M$, ie $\psi(n\cdot m)=\phi(n)\cdot\psi(m)$.) Let $E_{*}N$ be a free (right) $RN$–resolution of the trivial representation $R$. Let $\xi\colon E_{*}N\to\operatorname{Res}_{\phi}E_{*}N$ be an $RN$–homomorphism. (This can always be found and any two are chain homotopic.) Then $\phi$ induces the map

[TABLE]

Note that the RHS is canonically isomorphic to $E_{*}N\otimes_{N}M$ because $\phi$ is a bijection.

Let $\phi$ be an inner automorphism of $N$, say conjugation by $n\in N$. Then $\xi(x)=xn^{-1}$ and $\psi(m)=nm$ fulfill the above requirements. But then $\xi\otimes\psi$ is in fact the identity on $E_{*}N\otimes_{N}M$.

Let $E_{*}G$ be a free (right) $RG$–resolution of the trivial representation $R$. Assume $M$ is the restriction of an $RG$–module and $\phi$ the conjugation by an element $g\in G$. ($N$ is normal in $G$.) Then $\xi(x)=xg^{-1}$ and $\psi(m)=gm$ fulfill the above requirements as before. Note that $\xi\otimes\psi$ is not the identity on $E_{*}G\otimes_{N}M$. Summarizing, we have seen that $E_{*}G\otimes_{N}M$ is an $RQ$–module by

[TABLE]

This induces an $RQ$–module structure on $H_{*}(N;M)$ for every $RG$–module $M$.

Now we generalize this concept.

In this section, we deal with multiple stability groupoids $\mathcal{N}=(N_{n})_{n\in\mathbb{N}},\mathcal{G}=(G_{n})_{n\in\mathbb{N}},\mathcal{Q}=(Q_{n})_{n\in\mathbb{N}}$.

For the most time we assume that $\mathcal{G}$ and $\mathcal{Q}$ are braided, in which their stability categories $U\mathcal{G}$ and $U\mathcal{Q}$ are monoidal and prebraided, whence it makes sense to take of their central stability complex and homology, which we denote by $\widetilde{C}_{*}^{\mathcal{G}},\widetilde{H}_{*}^{\mathcal{G}}$ and $\widetilde{C}_{*}^{\mathcal{Q}},\widetilde{H}_{*}^{\mathcal{Q}}$, respectively.

Definition \theDefauto.

Let $F\colon\mathcal{N}\to\mathcal{G}$ and $F^{\prime}\colon\mathcal{G}\to\mathcal{Q}$ be homomorphisms of stability groupoids. We call this data a stability SES if

[TABLE]

is a short exact sequence for all $n\in\mathbb{N}$.

Proposition \thepropauto.

Let $\mathcal{G},\mathcal{Q}$ be stability groupoids and $F^{\prime}\colon\mathcal{G}\to\mathcal{Q}$ a homomorphism of stability groupoids and $F^{\prime}_{n}\colon G_{n}\to Q_{n}$ is surjective for every $n\in\mathbb{N}$. Then there is a stability groupoid $\mathcal{N}$ and a homomorphism of stability groupoids $F\colon\mathcal{N}\to\mathcal{G}$ such that

[TABLE]

is a stability SES.

Proof.

Let $N_{n}$ be the kernel of $F^{\prime}_{n}$ and $F_{n}$ the embedding of $N_{n}$ in $G_{n}$. Let $\mathcal{N}$ be the groupoid formed by these groups and $F$ the functor given by all $F_{n}$’s. Because

[TABLE]

we get an injective group homomorphism

[TABLE]

This proves that $\mathcal{N}$ is a stability groupoid. All other assertions are immediate. ∎

Lemma \thelemauto.

Let

[TABLE]

be a stability SES and $V$ a $U{\mathcal{G}}$–module. Then for every $i\geq 0$ there is a $U{\mathcal{Q}}$–module $W$ with

[TABLE]

We denote this $U{\mathcal{Q}}$–module by $H_{i}(\mathcal{N};V)$ and abbreviate $H_{i}(\mathcal{N})$ if $V=R\operatorname{Hom}(0,-)$.

Proof.

We will use Section 5 for this proof.

We already know that for every $i\geq 0$ we have a sequence of $RQ_{n}$–representations. Let

[TABLE]

be induced by $\iota_{1}\oplus\mathrm{id}_{n}$. On the chain complex level this is

[TABLE]

induced by $\phi=V(\iota_{1}\oplus\mathrm{id}_{n})\colon V_{n}\to V_{n+1}$.

That means that

[TABLE]

is induced by the map

[TABLE]

Because $x\mapsto xg^{\prime}$ for some $g^{\prime}\in G_{n-m}$ is a $G_{m}$–automorphism of the free resolution $E_{*}G_{n}$, it is homotopic to the identity. By functoriality the homotopy descends to $E_{*}G_{n}\otimes_{N_{m}}V_{m}$. Therefore $x\otimes m\mapsto xg^{\prime}\otimes m$ induces the identity on $H_{i}(N_{m};V_{m})$. By the following commutative diagram, the action of $G_{n-m}$ on $\phi(H_{i}(N_{m};V_{m}))$ is trivial. This finishes the proof by application of Section 5.

[TABLE]

The central technical tool of this section is the following spectral sequence.

Proposition \thepropauto.

Let

[TABLE]

be a stability SES. Assume that $\mathcal{G}$, $\mathcal{Q}$, and $\mathcal{G}\to\mathcal{Q}$ are braided. Let $V$ be a $U\mathcal{G}$–module. Then the two spectral sequences constructed from the double complex

[TABLE]

are

[TABLE]

and

[TABLE]

Proof.

The first spectral sequence is given by

[TABLE]

Therefore

[TABLE]

The other spectral sequence is given by

[TABLE]

The differential $d^{1}$ is induced by the map

[TABLE]

with $g_{i}=\mathrm{id}_{n-p-1}\oplus b_{1,i}\oplus\mathrm{id}_{p-i}$ and $\phi^{\prime}=V(\mathrm{id}_{n-p}\oplus\iota_{1})$.

Then there is a $Q_{n}$–equivariant isomorphism

[TABLE]

given by

[TABLE]

Therefore

[TABLE]

Because the following diagram commutes, $d^{1}$ coincides with the differential of $\widetilde{C}^{\mathcal{Q}}_{*}$.

[TABLE]

is given by

[TABLE]

The idea for this spectral sequence can be found in [PS14]. The following immediate consequence illustrates how it can be used.

Corollary \thecorauto.

Let

[TABLE]

be a stability SES. Assume that $\mathcal{G}$, $\mathcal{Q}$, and $\mathcal{G}\to\mathcal{Q}$ are braided. For each $n$, there is a spectral sequence

[TABLE]

that converges to zero for $p+q\leq\frac{n-a-1}{k}$ if $U\mathcal{G}$ satisfies H3($\infty$).

Proof.

The two spectral sequences in Section 8 converge to the same limit. The first will converge to zero for $n\geq k\cdot q+a+1$ by H3($\infty$) if we set $V$ to be $R\operatorname{Hom}(0,-)$. In particular, the diagonals $p+q\leq\frac{n-a-1}{k}$ converge to zero. Thus the same is true for the second spectral sequence. ∎

The proof of Theorem C uses the same methods as Putman and Sam in [PS14, Theorem 5.13].

Proof of Theorem C.

We may use the spectral sequences from Section 8. The first spectral sequence implies that both converge stably to zero, because $V$ is stably acyclic. We will now prove the result by induction on $i$.

For $i=0$ note that because

[TABLE]

we get

[TABLE]

Because $H_{0}$ is right exact it follows that $H_{0}(\mathcal{N};V)$ is a finitely generated $U{\mathcal{Q}}$–module.

Let $i>0$ and assume $H_{q}(\mathcal{N};V)$ is finitely generated for all $q<i$. From the Noetherian condition we get that $H_{q}(\mathcal{N};V)$ is stably acyclic for all $q<i$. Now using the second spectral sequence from Section 8, we infer $E^{2}_{-1i}=\widetilde{H}_{-1}(H_{i}(\mathcal{N};V))$ is stably zero, because the only incoming and outgoing differentials are all stably zero. Section 5 and the assumption that $H_{i}(\mathcal{N};V)_{n}$ is a finitely generated $R$–module for all $n$ implies that $H_{i}(\mathcal{N};V)$ is finitely generated. ∎

9. Quillen’s argument revisited

In this section we will solely prove Theorem D from the introduction.

Proof of Theorem D.

We consider the stability SES

[TABLE]

where $\mathbb{N}$ is the category whose objects are the natural numbers and whose morphisms are only the identity maps. In Section 8, we constructed two spectral sequences for this situation. From the first

[TABLE]

which is zero for $n+1>k\cdot q+a$, we see that both spectral sequences converge to zero when $p+q\leq\frac{n-a}{k}$. The other spectral sequence is given by

[TABLE]

The differentials are easy to understand:

[TABLE]

is zero if $p$ is odd and it is $\phi_{*}$ if $p$ is even. In particular $E^{1}_{0,i}\to E^{1}_{-1,i}$ is the stabilization map

[TABLE]

Assume $n\geq ki+a-1$, we want to prove that $\phi_{*}$ is surjective. We know that $E^{\infty}_{-1,i}=0$ if $i-1\leq\frac{n-a}{k}$, in particular when $n\geq ki+a-1$. We want to use induction to show that $E^{2}_{pq}=0$ when $p+q=i$ and $q<i$. This would imply that $E^{2}_{-1,i}$ already vanishes and $\phi_{*}$ is surjective. If $p$ is even, to show that $E^{2}_{pq}=0$ it suffices to show that

[TABLE]

is surjective. By induction this is the case if $n-p-1\geq kq+a-1$. If $p$ is odd, it suffices to show that

[TABLE]

is injective. By induction this is true if $n-p\geq kq+a$, which is the same condition. We know that

[TABLE]

which is what we need.

Assume $n\geq ki+a$, we want to prove that $\phi_{*}$ is injective. We know that $E^{\infty}_{0,i}=0$ for $i\geq\frac{n-a}{k}$, in particular when $n\geq ki+a$. Again we prove $E^{2}_{pq}=0$ when $p+q=i+1$ and $q<i$ by induction, which implies that $E^{2}_{0,i}=0$ and $\phi_{*}$ is injective. We already computed that $E^{2}_{pq}$ vanishes when $n-p\geq kq+a$. And we calculate

[TABLE]