aa r X i v : . [ m a t h . K T ] A p r CENTRAL STABILITY HOMOLOGY
PETER PATZT
Abstract.
We give a new categorical way to construct the central stabilityhomology of Putman and Sam and explain how it can be used in the contextof representation stability and homological stability. In contrast to them, wecover categories with infinite automorphism groups. We also connect centralstability homology to Randal-Williams and Wahl’s work on homological stabil-ity. We also develop a criterion that implies that functors that are polynomialin the sense of Randal-Williams and Wahl are centrally stable in the sense ofPutman.
Contents
1. Introduction 12. Central stability homology 113. Stability categories 144. Central stability and stability categories 185. Central stability and finiteness properties 226. Notions of central stability 287. Polynomial degree 318. Short exact sequences 389. Quillen’s argument revisited 42References 441.
Introduction
In 1960 Nakaoka [Nak60] proved the pioneering result that for the symmetricgroups H i p S n ´ q ÝÑ H i p S n q is an isomorphism for all n large enough in comparison to i . Quillen [Qui70] wasable to prove this result, using a new method that has since been generalized inmany ways. He was considering a highly connected simplicial complex X n on which Date : April 2017.2010
Mathematics Subject Classification. S n acts and the groups that stabilizes simplices pointwise are isomorphic to S m forlarge enough m ď n . For the symmetric groups X n “ ∆ n ´ the p n ´ q –dimensionalstandard simplex works. Randal-Williams and Wahl [RWW15] turned this gamearound: They construct a semisimplicial set W n whose p –simplices are given by S n { S n ´ p ´ such that it is a transitive S n –set whose stabilizers are S n ´ p ´ . Then they provethat W n is highly connected. More precisely, let ∆ be the category of finite orderedsets (including the empty set) and strictly monotone maps and FI the category offinite sets and injection. Being a semisimplicial set, W n is a functor from p ∆ q op to the category of sets with t , , . . . , p u ÞÝÑ S n { S n ´ p ´ . Because S n { S n ´ p ´ is as an S n –set the set of injective maps t , , . . . , p u ã ÝÑ t , , . . . , n u , this functor factors through FI op the opposite category of FI : p ∆ q op ÝÑ FI op Hom FI p´ , t ,...,n uq ÝÑ Set
The category FI has taken up a prominent role in the theory of representationstability. Established by Church and Farb in [CF13], representation stability char-acterizes sequences p V n q n P N of representations of 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 Q n is the directsum of the trivial representation and the standard representation for n ě
2. Thepermutation representation can also be realized as an FI –module, by which we meana functor from FI to Q –vector spaces (and later more generally a functor from FI to R –modules). This can be done by sending a finite set S to the finite dimen-sional vector space with basis S . For an FI –module V , we will denote the imagesof t , . . . , n u 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, Ellenbergand Farb [CEF15] proved that a finitely generated FI –module is representationstable. Putman [Put15] noticed that for an FI –module V one cannot expect that V n – Ind S n S n ´ V n ´ , because the transposition of p n ´ n q acts trivially on theimage of V n ´ in V n . But he calls an FI –module V centrally stable if V n is iso-morphic to the largest quotient of Ind S n S n ´ V n ´ such that p n ´ n q acts trivially ENTRAL STABILITY HOMOLOGY 3 on the image of V n ´ for all n large enough. This quotient can be expressed as acoequalizer coeq ` Ind S n S n ´ V n ´ Ñ Ind S n S n ´ V n ´ ˘ . Let FI ď N be the full subcategory of FI whose objects all have cardinality at most N and denote the inclusion by inc N : FI ď N ã ÝÑ FI . Church, Ellenberg, Farb and Nagpal [CEFN14] then proved that an FI –module iscentrally stable if and only if it is presented in FI ď N (having a presentation bysums of representable functors from FI ď N , also see Proposition 6.1(a)) for somesufficiently large N . Later Putman and Sam [PS14] also gave a categorical conceptthe name central stability. They call an FI –module V is centrally stable if there isan N large enough such that V is the left Kan extension of its restriction to FI ď N : V – Lan inc N p V ˝ inc N q In other words, for any FI –module W and any morphism (ie natural transformation) f : V ˝ inc N ÝÑ W ˝ inc N between the restrictions, there is a unique morphism V Ñ W which induces f .In a more general setup, Djament [Dja16] proves that an FI –module fulfills thiscondition if and only if it is presented in FI ď N and is thereby equivalent to theoriginal definition from [Put15]. Unfortunately the two notions of central stabilitydiffer in natural generalizations of the category FI , we consider later. We comparemany 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 p V n q n P N of S n –representations.He defined it by(1.1) ¨ ¨ ¨ Ñ Ind S n S n ´ p ´ ˆ S p ` V n ´ p ´ b A p ` Ñ Ind S n S n ´ p ˆ S p V n ´ p b A p Ñ . . . where A p is the sign representation of S p . It turns out, that for an FI –module V this chain complex is given by ¨ ¨ ¨ Ñ à S Ăt ,...,n u| S |“ p ` V pt , . . . , n uz S q Ñ à S Ăt ,...,n u| S |“ p V pt , . . . , n uz S q Ñ . . . where the differential is the alternated sum of face maps d i : V ` t , . . . , n uzt j ă ¨ ¨ ¨ ă j p u ˘ ÝÑ V ` t , . . . , n uzt j ă ¨ ¨ ¨ ă ˆ j i ă ¨ ¨ ¨ ă j p u ˘ that is induced by the inclusion of sets. This complex also computes FI –homology – the derived functor analyzed by Church and Ellenberg [CE16]. PETER PATZT
There is a related chain complex(1.2) ¨ ¨ ¨ Ñ à f : t , ,...,p u ã ÝÑt ,...,n u V pt , . . . , n uz im f q Ñ à f : t , ,...,p ´ u ã ÝÑt ,...,n u V pt , . . . , n uz im f q Ñ . . . for 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(1.3) à f : t , ,...,p u ã ÝÑt ,...,n u V pt , . . . , n uz im f q – Ind S n S n ´ p V pt , . . . , n ´ p uq . To recover Putman’s chain complex (1.1), one factors out the S p ` –action. Thesecond complex (1.2) computes what we will call central stability homology of an FI –module in this paper.Both complexes are related to the notions of representation stability. For exam-ple, the augmentation map Ind S n S n ´ V n ´ ÝÑ V n is surjective for all n ą N if and only if V is generated in FI ď N .A similar connection can be found for central stability. By definition V n – coeq p Ind S n S n ´ V n ´ Ñ Ind S n S n ´ V n ´ q if and only if the partInd S n S n ´ V n ´ Ñ Ind S n S n ´ V n ´ Ñ V n Ñ S n S n ´ ˆ S V n ´ b A Ñ Ind S n S n ´ ˆ S V n ´ b A Ñ V n ÝÑ FI is an exampleof a complemented category.Let us plug in the constant FI –module V p S q “ Q that sends all maps to the iden-tity. 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 ∆ n ´ . Randal-Williams and Wahl [RWW15] gave a general construction of the semisimplicial set W n for a homogeneous category (see Definition 3.6). FI is also an example of ahomogeneous category. Moreover, they have a general construction of a simplicialcomplex S n whose simplicial chain complex is a generalization of (1.1).Let C be a small monoidal category with initial unit and let D be a cocompletecategory. In Section 2, we give a construction of a functorFun p C , D q ÝÑ Fun p C , Fun pp ∆ q op , D qq ENTRAL STABILITY HOMOLOGY 5 that turns a functor from C to D into a functor from C to augmented semisimplicialobjects in D . This generalizes Randal-Williams and Wahl’s construction of theaugmented semisimplicial set W n and the augmented semisimplicial module in (1.2).For example plugging in the constant functorHom FI pH , ´q : FI ÝÑ Set , we get a functor from FI to augmented semisimplicial sets with t , . . . , n u ÞÝÑ W n . In our construction functorial dependencies become more visible than in previouslyexisting literature. If D is abelian, we denote the associated chain complex for afunctor F : C Ñ D by r C ˚ p F q and its homology by r H ˚ p F q . This is what we callthe central stability chain complex and the central stability homology in this paper.When V is an FI –modules it computes to be r C p p V qp S q – à f : t , ,...,p u ã ÝÑ S V p S z im f q as in (1.2). Stability categories. [PS14] and [RWW15] generalize FI by weakly complementedcategories (see Definition 3.10) and homogeneous categories (see Definition 3.6),respectively. We will work with a different generalization of FI that we call stabilitycategories which are both weakly complemented and homogeneous. In Section 3,we define stability groupoids , that are sequences of groups G “ p G n q n P N togetherwith a monoidal structure ‘ : G m ˆ G n ã ÝÑ G m ` n and some mild condition (see Definition 3.1). We then use a construction of Quillen,which is also used in [RWW15], to get a category U G whose objects are the naturalnumbers and whose homomorphisms areHom U G p m, n q – $&% G n { G n ´ m m ď n H m ą n. We call U G a stability category if G is braided. All examples considered by [PS14]and [RWW15] are covered in this setup: Examples 1.1.
Here is a list of some braided stability groupoids G and their sta-bility categories U G . (a) The symmetric groups S “ p S n q n P N is a symmetric stability groupoid andits stability category U S is a skeleton of FI . PETER PATZT (b)
Fix a commutative ring R . The general linear groups GL p R q “ p GL n p R qq n P N is a symmetric stability groupoid and its stability category U GL p R q is askeleton of VIC R whose objects are finitely generated free R –modules andits morphisms R m Ñ R n are given by pairs p f, C q of a monomorphism f : R m Ñ R n and a free complement C ‘ im f “ R n . (c) Fix a commutative ring R . The symplectic groups Sp p R q “ p Sp n p R qq n P N isa symmetric stability groupoid and its stability category U Sp p R q is a skele-ton of SI R whose objects are finitely generated free symplectic R –modulesand its morphisms are isometries. (d) The automorphism groups of free groups
AutF “ p
Aut F n q n P N is a sym-metric stability groupoid and its stability category U AutF is given by pairs p f, C q of a monomorphism f : F m Ñ F n and a complement C ˚ im f “ F n . (e) There is a braided stability groupoid for the braid groups β “ p β n q n P N . Itsmonoidal structure is described in Example 3.9. (f) There is a braided stability groupoid for the mapping class groups
Mod Σ “p Mod Σ g, q g P N of connected, oriented surfaces of genus g P N and one bound-ary component. Its monoidal structure is described in [RWW15, Sec 5.6] .A description of the corresponding stability categories of β and Mod Σ canalso be found in [RWW15, Sec 5.6] . From now on, we call a functor from U G to the category of R –modules for somefixed commutative ring R a U G –module . In Section 4, we compute the centralstability complex of a U G –module V very concretely to be isomorphic to r C p p V q n – à f P Hom U G p p ` ,n q V n ´ p ´ – Ind G n G n ´ p ´ V n ´ p ´ . This complex clearly generalizes (1.2) and (1.3).The stability category U G of a braided stability groupoid G is monoidal (see[RWW15, Prop 2.6] or Proposition 3.8) and has an initial unit 0 asHom U G p , n q – G n { G n ´ – t˚u . That means R Hom U G p , ´q : U G ÝÑ R ´ mod is the functor sending all objects to R and all morphisms to the identity on R . Theconnectivity condition on U G used in [RWW15] can be described by the followingcondition. H3:
The central stability homology r H i p R Hom U G p , ´qq n “ i ě ´ n large enough in comparison to i .All stability categories in Examples 1.1 satisfy H3 . Also, Putman and Sam’s[PS14, Thm 3.7] implies H3 for all stability categories U G that fulfill the noetherian ENTRAL STABILITY HOMOLOGY 7 condition that every submodule of a finitely generated U G –module is again finitelygenerated. Using H3 in Section 5, we relate resolutions by U G –modules freelygenerated in finite ranks (see Definition 5.1) with the central stability homology.We prove a quantitive version of the following theorem as Theorem 5.7, where wealso consider partial resolutions. Theorem A.
Let U G be a stability category. Assume H3 . Let V be a U G –module,then the following statements are equivalent. (a) There is a resolution ¨ ¨ ¨ Ñ P Ñ P Ñ V Ñ of V by U G –modules P i that are freely generated in finite ranks. (b) The homology r H i p V q n “ for all i ě ´ and all n large enough in comparison to i . Let U G ď N denote the full subcategory of U G whose objects (which are nat-ural numbers) are exactly t , . . . , N u . Previously it has only been known that r H ´ p V q n “ n ą d if and only if V is generated in ranks ď d . For FI ,our r H ´ is what is denoted by H in [CEF15]. [CEFN14] and [PS14] have thenused a noetherian property of the categories, they worked with, to show that allcentral stability homology vanishes for large enough n . Theorem A explains thisphenomenon very clearly. It also makes the existence of free resolutions tangible,when such a noetherian condition is not true. For example for VIC Z and SI Z .We call a U G –module V centrally stable if r H ´ p V q n “ r H p V q n “ n . This notion and terminology was introduced by Putman [Put15]for sequences of representations of the symmetric groups. Theorem A proves thatunder the assumption of H3 , a U G –module V is centrally stable if and only if V ispresented in U G ď N for some sufficiently large N . A U G –module V is presented in afull subcategory with finitely many objects if and only if it is the left Kan extensionof its restriction along the inclusion. Unfortunately this notion was also dubbedcentral stability in [PS14] and [CEFN14]. See Section 6 for more clarification ondifferent notion used for central stability. Polynomial functors.
Many interesting examples of U G –modules satisfy polyno-miality conditions. For example the dimensions of the permutation representationsgrow polynomially (even linearly). In fact [CEF15] proved that, assuming R “ Q ,dimensional growth is eventually polynomial for all finitely generated FI –modules.In Section 7, we recall the definition of the polynomial degree of a U G –module V PETER PATZT from [RWW15]. We say V has polynomial degree ´8 if it is zero. For r ě V has polynomial degree ď r if the map V p ‘ n q ÝÑ V p ‘ n q is always injective and its cokernel is a U G –module of polynomial degree ď r ´
1. Forexample the constant U G –module has polynomial degree ď ď R Hom U G p m, ´q “ coker ´ R Hom U G p m, ´q Ñ R Hom U G p m, ‘ ´q ¯ of the representable functors R Hom U G p m, ´q . H4:
The central stability homology r H i p coker R Hom U G p m, ´qq n “ m P N , all i ě ´ n large enough in comparison to m and i .It is easy to check that coker R Hom U S p m, ´q is free (eg see [CEFN14, Prop 2.2]) and thus H4 follows from H3 for U S (whichis equivalent to FI ). We prove together with Jeremy Miller and Jennifer Wilson[MPW17] that U GL p R q and U Sp p R q satisfy H4 if R is a PID. In Example 7.11we show that the stability category U β of the braid groups does not satisfy H4 .It is unknown whether U AutF and U Mod Σ satisfy H4 . Assuming H4 , we canmake a statement about the central stability homology of U G –module with finitepolynomial degree. Theorem B.
Let U G be a stability category. Assume H3 and H4 . Let V be a U G –module with finite polynomial degree, then r H i p V q n “ for all i ě ´ and all n large enough in comparison to i . We prove a quantified version of this theorem with Corollary 7.9. This theoremis the main new ingredient of [MPW17], where we together with Jeremy Miller andJennifer Wilson prove central stability for the second homology groups of Torellisubgroups of automorphism groups of free groups and mapping class groups as wellas certain congruence subgroups of general linear groups.
Stability SES.
In Section 8, we consider families of short exact sequences1 ÝÑ N n ÝÑ G n ÝÑ Q n ÝÑ N “ p N n q n P N ÝÑ G “ p G n q n P N ÝÑ Q “ p Q n q n P N . ENTRAL STABILITY HOMOLOGY 9
We call these families stability SES . For every U G –module V , there is a U Q –module H i p N ; V q such that H i p N ; V q n – H i p N n ; V n q . Examples 1.2.
Here is a (very incomplete) list of interesting stability SES’s. (a)
The pure braid groups
P β “ p
P β n q n P N are the kernels in the sequence ÝÑ P β ÝÑ β ÝÑ S ÝÑ . (b) Fix a commutative ring R and an ideal I Ă R . The congruence subgroups GL p R, I q “ p GL n p R, I qq n P N are the kernels in the sequence ÝÑ GL p R, I q ÝÑ GL p R q ÝÑ GL p R { I q ÝÑ , where GL p R { I q “ p GL n p R { I qq n P N are the images of GL n p R q in GL n p R { I q . (c) Fix a commutative ring R and an ideal I Ă R . The congruence subgroups Sp p R, I q “ p Sp n p R, I qq n P N are the kernels in the sequence ÝÑ Sp p R, I q ÝÑ Sp p R q ÝÑ Sp p R { I q ÝÑ , where Sp p R { I q “ p Sp n p R { I qq n P N are the images of Sp n p R q in Sp n p R { I q . (d) The Torelli subgroups IA “ p IA n q n P N of the automorphism groups Aut F n are the kernels in the sequence ÝÑ IA ÝÑ AutF ÝÑ GL p Z q ÝÑ . (e) The Torelli subgroups I “ p I g, q g P N of the mapping class groups Mod Σ g, are the kernels in the sequence ÝÑ I ÝÑ Mod Σ ÝÑ Sp p Z q ÝÑ . (f) If there is an inclusion of S Ă Q , we can twist N to receive a braidedstability groupoid r N “ p r N n q n P N , where r N n Ă G n is the preimage of S n .Then ÝÑ N ÝÑ r N ÝÑ S ÝÑ is a stability SES. This has been done by Putman [Put15] for (b) and bythe author together with Wu [PW16] for the Houghton groups. It could alsobe done for (c) , (d) , and (e) . Note that in all these examples G and Q are braided, but N is not. Assume forthe following that G and Q both are braided. The central technical tools of thispaper are the following spectral sequences. Let V be a U G –module and n P N , thenthere are two spectral sequences E pq “ E p G n b RN n r H q p V q n and E pq “ r H p p H q p N ; V qq n that converge to the same limit. In particular when r H q p V q n “ n largeenough in comparison to q , then both spectral sequences converge to zero in eachdiagonal p ` q for large enough n .As an application of this spectral sequence, we prove a generalization of a theo-rem of Putman and Sam [PS14, Thm 5.13]. Theorem C.
Let ÝÑ N ÝÑ G ÝÑ Q ÝÑ be a stability SES. Assume that G , Q , and G Ñ Q are braided. Let V be a U G –module. Assume furthermore: (a) All submodules of finitely generated U Q –modules are finitely generated. (Noe-therian condition) (b) H i p N ; V q n is a finitely generated R –module for all i, n P N . (c) V is a finitely generated U G –module. (d) r H i p V q n “ for all n large enough in comparison to i .Then H i p N ; V q is a finitely generated U Q –module for every i P N . This spectral sequence is also an important tool in [MPW17], where we apply itto the stability SES’s in Examples 1.2 (b), (d), and (e).Finally in Section 9, we revisit Quillen’s argument for homological stability. Weprove a theorem on homological stability with twisted coefficients. Randal-Williamsand Wahl [RWW15, Thm A] prove a similar result but require their coefficientsto have finite polynomial degree. If we additionally assume H4 , our Theorem Bimplies that our condition is weaker, but in general our ranges are worse than theirs. Theorem D.
Let U G be a stability category. Assume H3 . Let V be a U G –modulewith r H i p V q n “ for all n ě k ¨ i ` a for some a P Z and k ě , then the stabilization map φ ˚ : H i p G n ; V n q ÝÑ H i p G n ` ; V n ` q is an epimorphism for all n ě k ¨ i ` a ´ and an isomorphism for all n ě k ¨ i ` a . Remark 1.3.
Notice that Theorem D and Remark 7.7 give criterions to checkwhether the double cosets G n ´ a z G n { G n ´ b stabilize with growing n and fixed a, b . ENTRAL STABILITY HOMOLOGY 11
Acknowledgement.
The author was supported by the Berlin Mathematical Schooland the Dahlem Research School. The author also wants to thank Aur´elien Dja-ment, Daniela Egas Santander, Reiner Hermann, Henning Krause, Daniel L¨utgehet-mann, Jeremy Miller, Holger Reich, Steven Sam, Elmar Vogt, Nathalie Wahl, JennyWilson for helpful conversations. Special thanks to Reiner Hermann and his invi-tation to NTNU where the idea for this project was born.2.
Central stability homology
Let p C , ‘ , q be a small monoidal category with unit object 0. In particular, ‘ : C ˆ C Ñ C is a bifunctor. Let D be cocomplete category and Fun p C , D q the category of functorsfrom C to D . Then the monoidal structure gives rise to a functorFun p C , D q Ñ Fun p C ˆ C , D q by precomposition of ‘ . This is the same as giving a functor S : C Ñ End p Fun p C , D qq from C to the category of endofunctors of Fun p C , D q . Explicitly it sends an object X P C to the suspension endofunctor S X : F ÞÑ F p´ ‘ X q . Because S X is just the precomposition of ´ ‘ X , the functor S X has a leftadjoint, namely the left Kan extension along ´ ‘ X (see [ML98, Cor X.3.2]). Wewill call its left adjoint the desuspension endofunctor and denote it by Σ X . Thereis an antiequivalence of categories between left and right adjoint endofunctors. (Cf[Kan58, Thm 3.2]) Consequently, there is a functorΣ : C op Ñ End p Fun p C , D qq that sends X to Σ X .Clearly S X ‘ Y “ S Y ˝ S X . Because Σ X is the right adjoint of S X Σ X ‘ Y – Σ X ˝ Σ Y . Thus Σ is in fact a monoidal functor, where End p Fun p C , D qq is equipped with themonoidal structure ˝ . Example 2.1.
Consider the monoidal category C “ FI (which is not small but hasa small skeleton) and D “ Z ´ mod the cocomplete category of abelian groups. Then Σ X is given by p Σ X F qp A q – à f : X ã ÝÑ A F p A z im f q , for finite sets X and A and an FI –module F : FI Ñ Z ´ mod . This Z –module isisomorphic to Ind S n S n ´ m F pt , . . . , n ´ m uq when X “ t , . . . , m u and A “ t , . . . , n u with m ď n . Definition 2.2.
Let the augmented semisimplicial category ∆ be the categorywith objects the sets t , . . . , n u for n ě ´ and morphisms strictly increasing setmaps. t , . . . , m u ‘ t , . . . , n u “ t , . . . , m, m ` , . . . , m ` ` n u defines a monoidal structure on ∆ with the unit H , which is additionally initial. The following theorem describes how ∆ behaves universally among all monoidalcategories with initial unit object. Let for the remainder of this section p C , ‘ , q bea small monoidal category with initial unit object 0 and denote the initial maps by ι X : 0 Ñ X . Theorem 2.3.
Let X be an object in C . Then there exists a unique monoidalfunctor ∆ Ñ C that sends t u to X .Proof. Similar to [Wei94, Ex 8.1.6], a functor ∆ Ñ C is given by a sequence ofobjects K ´ , K , . . . and maps d i : K p ´ Ñ K p p i “ , . . . , p q such that if i ă j then d j d i “ d i d j ´ . To be monoidal K p must be X ‘ p ` for all n ě ´
1. Because 0 P C is initial, there is exactly one candidate for d : K ´ “ Ñ K “ X , which is theinitial map ι X : 0 Ñ X . Further the other maps must have the following form. d i “ id X ‘ i ‘ ι X ‘ id X ‘ p ´ i : K p ´ “ X ‘ i ‘ ‘ X ‘ p ´ i Ñ K p “ X ‘ i ‘ X ‘ X ‘ p ´ i One easily checks that this satisfies the condition given. (cid:3)
Definition 2.4.
Let X an object of C . Then we get an augmented semisimplicialobject ∆ op Ñ C op Ñ End p Fun p C , D qq . This is the same as a functor K X : Fun p C , D q ÝÑ Fun p ∆ op , Fun p C , D qq . Thus for every functor F : C Ñ D we get an augmented semisimplicial object K X ‚ F in Fun p C , D q with K Xp F “ Σ X ‘ p ` F – Σ p ` X F. Definition 2.5.
Assume D is an abelian category, we denote the associated chaincomplex of K X ‚ F by r C X ˚ F and the homology of this chain complex by r H X ˚ F . Wecall this the central stability homology of F . ENTRAL STABILITY HOMOLOGY 13
A well known consequence of the Yoneda Lemma is thatΣ X Hom C p A, ´q – Hom C p A ‘ X, ´q . Example 2.6.
Let us consider C “ FI and D “ Set , and let us fix X “ t˚u asingleton. Given F “ Hom FI pH , ´q the representable functor, K X ‚ F is given by K Xp F – Hom FI pt , . . . , p u , ´q with face maps given by precomposition of the morphisms t , . . . , ˆ i, . . . , p u ã ÝÑ t , . . . , p u where the hat indicates that i is omitted.Evaluating at t , . . . , n u , we get the (augmented) semisimplicial set W n – p K X ‚ F qpt , . . . , n uq from [RWW15, Def 2.1] associated to the homogeneous category FI . More generally we can describe Σ X F for a functor F : C Ñ D as a colimit. Let p´‘ X Ó B q be the comma category whose objects are the pairs p A, ψ : A ‘ X Ñ B q ,then there is a forgetful functor p A, ψ : A ‘ X Ñ B q ÞÑ A. Precomposing F with this functor we get p Σ X F qp B q – colim p´‘ X Ó B q F. For a morphism X Ñ Y in C , the natural transformation Σ Y F Ñ Σ X F is givenby the functor p´ ‘ Y Ó B q ÝÑ p´ ‘ X Ó B q mapping to the composition p A ‘ Y Ñ B q ÞÝÑ p A ‘ X Ñ A ‘ Y Ñ B q . Example 2.7.
For C “ FI , D “ Z ´ mod and X “ t˚u , the central stability homol-ogy is given by the semisimplicial FI –module K X ‚ F with K Xp F – à f P Hom FI pt ,...,p u , ´q F p C f q where C f is the complement of the image of f and the face maps given by theprecomposition of the morphisms above and the inclusion of the complements. Stability categories
In this section we introduce stability categories that are a specific kind of homo-geneous categories (see [RWW15, Def 1.3] or Definition 3.6) and weakly comple-mented categories (see [PS14] or Definition 3.10).
Definition 3.1.
Let p G , ‘ , q be a monoidal groupoid whose monoid of objects isthe natural numbers N . The automorphism group of the object n P N is denoted G n “ Aut G p n q . Then G is called a stability groupoid if it satisfies the followingproperties. (a) The monoidal structure ‘ : G m ˆ G n ã ÝÑ G m ` n is injective for all m, n P N . (b) The group G is trivial. (c) p G l ` m ˆ q X p ˆ G m ` n q “ ˆ G m ˆ Ă G l ` m ` n for all l, m, n P N . Definition 3.2. A homomorphism G Ñ H of stability groupoids is a monoidalfunctor sending to . Remark 3.3.
A stability groupoid G is given by a sequence of groups p G n q n P N andgroup homomorphisms ‘ : G m ˆ G n ÝÑ G m ` n satisfying additional properties that would require spelling out what a monoidalstructure is. In notation we will usually suppress the monoidal structure and justwrite G “ p G n q n P N for a stability groupoid.Similarly a homomorphism of stability groupoids f : G “ p G n q n P N ÝÑ H “ p H n q n P N is given by a sequence of group homomorphisms f n : G n ÝÑ H n for which the diagrams G m ˆ G n / / f m ˆ f n (cid:15) (cid:15) G m ` nf m ` n (cid:15) (cid:15) H m ˆ H n / / H m ` n commute. ENTRAL STABILITY HOMOLOGY 15
Recall that a braiding b of a monoidal category p C , ‘ , q is a natural isomorphismbetween the functors ‘ : C ˆ C Ñ C and ‘ precomposed with a swap that has someadditional conditions. In particular, there are isomorphisms b A,B : A ‘ B Ñ B ‘ A for all objects A, B P C , such that for all morphisms f : A Ñ C and g : B Ñ D thediagram A ‘ B b A,B / / f ‘ g (cid:15) (cid:15) B ‘ A g ‘ f (cid:15) (cid:15) C ‘ D b C,D / / D ‘ C commutes. For such natural isomorphism to be a braiding one also needs twohexagon identities. (Cf [ML98, Sec XI.1] for details.) If b ´ A,B “ b B,A the braiding is called a symmetry .For a stability groupoid G “ p G n q n P N , a braiding b is given by isomorphisms b m,n P G m ` n such that p g ‘ f q ˝ b m,n “ b m,n ˝ p f ‘ g q for all f P G m and g P G n and b l,m ` n “ p id m ‘ b l,n q ˝ p b l,m ‘ id n q and b l ` m,n “ p b l,n ‘ id m q ˝ p id l ‘ b m,n q which amounts to the hexagon identities.For any monoidal groupoid G , Randal-Williams and Wahl [RWW15, Sec 1.1]give a construction originally due to Quillen that yields the category U G . Definition 3.4.
Let p G , ‘ , q be a monoidal groupoid. The category U G has thesame objects as G and its morphisms from A to B are equivalence classes of pairs p f, C q where C is an object in G and f is an (iso)morphism in G from C ‘ A Ñ B .Two of these pairs p f, C q and p f , C q are equivalent if there is an isomorphism g : C Ñ C (in G ) such that the diagram C ‘ A f / / g ‘ id A (cid:15) (cid:15) BC ‘ A f ; ; ①①①①①①①①① commutes. Definition 3.5. If G is a braided stability groupoid, we call p U G , ‘ , q the stabilitycategory of G . The following definition is due to Randal-Williams and Wahl [RWW15, Def 1.3].
Definition 3.6.
Let p C , ‘ , q be a monoidal category with initial . C is called a homogeneous category if it satisfies the following conditions for all objects A, B P C : H1:
Aut p B q acts transitively on Hom p A, B q by postcomposition. H2:
The map
Aut p A q Ñ Aut p A ‘ B q taking f to f ‘ id B is injective and itsimage is Fix p B, A ‘ B q : “ t f P Aut p A ‘ B q | f ˝ p ι A ‘ id B q “ ι A ‘ id B u , where ι A : 0 Ñ A denotes the initial morphism. C is called prebraided if its underlying groupoid is braided with a braiding b , thatsatisfies the equation b A,B ˝ p id A ‘ ι B q “ ι B ‘ id A for all objects A, B P C . Remark 3.7.
The condition H2 is not only symmetric for symmetric homogeneouscategory p C , ‘ , q ; prebraided suffices for this conclusion. Here is the reason. Clearly Aut p A q Ñ Aut p B ‘ A q is also injective and the image lies in Fix p B, B ‘ A q “ t f P Aut p B ‘ A q | f ˝ p id B ‘ ι A q “ id B ‘ ι A u . But assume f P Fix p B, B ‘ A q then b B,A ˝ f ˝ b ´ B,A “ g ‘ B for some g P Aut p A q by H2 and thus f “ b ´ B,A ˝ p g ‘ id B q ˝ b B,A “ id B ‘ g. The stability category of a braided stability groupoid is always a homogeneouscategory.
Proposition 3.8.
Let G be a braided (symmetric) stability groupoid, then U G is aprebraided (symmetric) homogeneous category. And the underlying groupoid of U G is G .Proof. In [RWW15, Prop 2.6(i)+(ii)] it is shown that 0 is initial in U G and that U 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. (cid:3) In [RWW15, Rem 1.4] it is already stated that in every homogeneous categoryHom p B, A ‘ B q – Aut p A ‘ B q{ Aut p A q . For U G this means Hom p n, m ‘ n q – G m ` n { G m . ENTRAL STABILITY HOMOLOGY 17
Because there is cancellation in N , the homomorphisms are given by equivalenceclasses of pairs p f, m q for some f P G m ` n and the above isomorphism is given by r f, m s ÞÝÑ f G m . The composition is then f G l ˝ gG m “ f p id l ‘ g q G l ` m for f G l : m ‘ n Ñ l ‘ m ‘ n and gG m : n Ñ m ‘ n .The monoidal structure on U G is then given by f G m ‘ f G m “ p f ‘ f qp id m ‘ b ´ n ,m ‘ id n q G m ` m for f G m P Hom p n , m ‘ n q and f G m P Hom p n , m ‘ n q . Example 3.9. S “ p S n q n P N is a braided stability groupoid with the braiding b m,n P S m ` n given by the permutation b m,n p i q “ $&% i ` n i ď mi ´ m i ą m. In fact, the braiding of S is a symmetry, wherefore it is also a symmetry on U S .An example of a braided stability groupoid that is not symmetric is given bythe braid groups β “ p β n q n P N . Its braiding b m,n P β m ` n is given by the followingdiagram. b m,n “ ¨ ¨ ¨ ¨ ¨ ¨ m nn m ¨ ¨ ¨ ¨ ¨ ¨ ‰ ¨ ¨ ¨ ¨ ¨ ¨ m nn m ¨ ¨ ¨ ¨ ¨ ¨ “ b ´ n,m A stability category of a braided stability groupoid is also a weakly complementedcategory as defined by Putman and Sam [PS14].
Definition 3.10.
Let p C , ‘ , q be a monoidal category with initial . C is called a weakly complemented category if it satisfies the following conditions for all objects A, B, C P C : (a) All morphisms are monomorphisms. (b)
The map
Hom p A ‘ B, C q Ñ
Hom p A, C q ˆ
Hom p B, C q given by ψ ÞÑ p ψ ˝ p id A ‘ ι B q , ψ ˝ p ι A ‘ id B qq is injective. (c) For every morphism ψ P Hom p A, B q there is an object D and an isomor-phism f : D ‘ A Ñ B such that ψ “ f ˝p ι D ‘ id A q . The pair p D, f q is uniquelydetermined up to isomorphisms in the comma category p´ ‘ A Ó B q . Proposition 3.11.
Let G “ p G n q n P N be a braided stability groupoid, then U G is aweakly complemented category.Proof. Let l, m, n P N , ie objects of G and U G . To prove (a), by H1 , we only haveto show that φ “ id l ‘ m ‘ n G l is a monomorphism. Let ψ “ f G m , ψ “ f G m : n Ñ m ‘ n such that p id l ‘ f q G l ` m “ φ ˝ ψ “ φ ˝ ψ “ p id l ‘ f q G l ` m . Then there is an automorphism g P G l ` m such thatid l ‘ f “ p id l ‘ f q ˝ p g ‘ id n q . By definition g ‘ id n “ id l ‘ p f ˝ f q P p G l ` m ˆ q X p ˆ G m ` n q “ ˆ G m ˆ . Therefore ψ and ψ must coincide.Using H1 to prove (b), it is enough to show that for every g P G l ` m ` n and φ “ id l ‘ m ‘ n G l the two equations φ ˝ p id m ‘ ι n q “ gφ ˝ p id m ‘ ι n q and φ ˝ p ι m ‘ id n q “ gφ ˝ p ι m ‘ id n q imply φ “ gφ . Using the second equation we get that ι l ‘ m ‘ id n “ g p ι l ‘ m ‘ id n q thus by H2 there is a g P G l ` m with g “ g ‘ id n . Applying the same trick to thefirst equation we get ι l ‘ id m “ g p ι l ‘ id m q and therefore the existence of a g P G l with g “ g ‘ id m . For g “ g ‘ id m ‘ id n clearly φ “ gφ .(c) is true by definition of U G . (cid:3) For the remainder of this paper we assume that G “ p G n q n P N is a braided stabilitygroupoid and thus U G is its prebraided stability category. Given a functor F from U G to some category D , we denote the images F p n q by F n .Further for the object P U G , we abbreviate S , Σ , K ‚ , r C ˚ , r H ˚ by S, Σ , K ‚ , r C ˚ , r H ˚ ,respectively. Central stability and stability categories
In this section, we explain some basic properties of the functor K ‚ from Definition 2.5in the context of stability categories. ENTRAL STABILITY HOMOLOGY 19
Proposition 4.1.
Let l, m, n be objects in U G . Then the map Hom p m, l ‘ m q Ñ Hom p m ‘ n, l ‘ m ‘ n q given by ψ ÞÑ ψ ‘ id n is injective and its image is t χ P Hom p m ‘ n, l ‘ m ‘ n q | χ ˝ p ι m ‘ id n q “ ι l ‘ m ‘ id n u . (Because of Remark 3.7, the symmetric version of this statement is also true.)Proof. Given two morphisms from f G l , f G l : m Ñ l ‘ m , and assume that p f ‘ id n q G l “ f G l ‘ id n “ f G l ‘ id n “ p f ‘ id n q G l . Then there is an automorphism g P G l such that f ‘ id n “ p f ‘ id n q ˝ p g ‘ id m ‘ n q which implies that g ‘ id m ‘ n “ p f ˝ f q ‘ id n . Because G l ` m Ñ G l ` m ` n is injective, g ‘ id m “ f ˝ f. Thus f G l “ f G l and the map ψ ÞÝÑ ψ ‘ id n is injective.If ψ P Hom p m, m ‘ n q , p ψ ‘ id n q ˝ p ι m ‘ id n q “ p ψ ˝ ι m q ‘ id n “ ι l ‘ m ‘ id n . Let on the other hand χ “ hG l : m ‘ n Ñ l ‘ m ‘ n such that χ ˝ p ι m ‘ id n q “ ι l ‘ m ‘ id n . Then hG l ` m “ id l ‘ m ‘ n G l ` m , whence there is an isomorphism g P G l ` m with h “ g ‘ id n . Thus gG l ‘ id n “ hG l “ χ. (cid:3) Proposition 4.2.
Let F : U G Ñ Set . Then there is an isomorphism p K p ´ F q n “ p Σ p F q n – G n ˆ G n ´ p F n ´ p . For a map η “ hG p ´ q P Hom p q, p q the corresponding morphism p Σ p F q n – G n ˆ G n ´ p F n ´ p ÝÑ G n ˆ G n ´ q F n ´ q – p Σ q F q n is given by r g, x s ÞÝÑ r g p id n ´ p ‘ h q , φ p x qs with φ “ F p id n ´ p ‘ ι p ´ q q . This morphism is independent of the choice of h P η .Proof. We know from Section 2 that p Σ p F q n – colim p´‘ p Ó n q F. Note that by construction, every morphism m ‘ p Ñ n factors through an auto-morphism p n ´ p q ‘ p Ñ n . Therefore there is a surjection G n ˆ F n ´ p Ý ։ colim p´‘ p Ó n q F. There are still some relations, which are given by the precomposition by elementsin G n ´ p . Therefore p Σ p F q n – G n ˆ G n ´ p F n ´ p together with the (choice-free) maps F p m, ψ q “ F m ÝÑ G n ˆ G n ´ p F n ´ p given by x ÞÑ r g, φ p x qs where φ “ F p ι n ´ p ´ m ‘ id m q and ψ “ gG n ´ p ´ m .To understand the functoriality, we see that p´ ‘ p Ó n q ÝÑ p´ ‘ q Ó n q is given by p m, ψ q ÞÝÑ p m, ψ ˝ p id m ‘ η qq . Therefore r g, x s P G n ˆ G n ´ p F n ´ p maps to colim p´‘ q Ó n q F via F n ´ p corresponding to p n ´ p, g ˝ p id n ´ p ‘ η qq P p´ ‘ q Ó n q . Thus r g, x s ÞÝÑ r g p id n ´ p ‘ h q ˝ p b ´ n ´ p,p ´ q ‘ id q q , φ p x qs “ r g p id n ´ p ‘ h q , φ p x qs where h P G p with η “ hG p ´ q . (cid:3) A similar proposition can be made for functors U G Ñ R ´ mod : Proposition 4.3.
Let V : U G Ñ R ´ mod . Then there is an isomorphism p Σ p V q n – RG n b RG n ´ p V n ´ p . For a map η “ hG p ´ q P Hom p q, p q the corresponding morphism p Σ p V q n – RG n b RG n ´ p V n ´ p ÝÑ RG n b RG n ´ q V n ´ q – p Σ q V q n is given by g b v ÞÝÑ g p id n ´ p ‘ h q b φ p x q ENTRAL STABILITY HOMOLOGY 21 with φ “ V p id n ´ p ‘ ι p ´ q q . This morphism is independent of the choice of h P η . This allows us to describe the face maps of the semisimpicial set p K ‚ F q n for a U G –set F . Recall that d i “ id i ‘ ι ‘ id p ´ i “ p b ,i ‘ id p ´ i q G : p Ñ p ` . Then p K p F q n “ G n ˆ G n ´ p ´ F n ´ p ´ ÝÑ G n ˆ G n ´ p F n ´ p “ p K p ´ F q n is given by r g, x s ÞÝÑ r g p id n ´ p ´ ‘ b ,i ‘ id p ´ i q , φ p x qs . Example 4.4.
The functor F “ Hom p , ´q : C Ñ Set sends every object to asingleton. Then p K p F q n – G n ˆ G n ´ p ´ F n ´ p ´ “ G n { G n ´ p ´ “ Hom p p ` , n q . Which is the semisimplicial set W n p , q n from [RWW15, Def 2.1] . Example 4.5.
We can now pick up Example 2.7 again. Let V : FI Ñ Z ´ mod . As U S from Example 3.9 is a skeleton of FI , we have p K p V q n – Z S n b Z S n ´ p ´ V n ´ p ´ – à f P Hom FI pt n ´ p,...,n u , t ,...,n uq V pt , . . . , n uz im f q by sending g b v to g | t ,...,n ´ p ´ u p v q P V pt , . . . , n uz g pt n ´ p, . . . , n uqq in the summand corresponding to f “ g | t n ´ p,...,n u . Functors F : U G Ñ Set that preserve monomorphisms, such as representablefunctors, send all maps in U G to injective set maps. For such functors we can split K ‚ F into a disjoint union. Proposition 4.6.
Let F : U G Ñ Set preserve monomorphisms. Then the semisim-plicial set p K ‚ F q n splits disjointly p K ‚ F q n – ž x P F n L x ‚ into augmented semisimplicial sets L x ‚ Ă p K ‚ Hom p , ´qq n with L xp “ t σ P p K p Hom p , ´qq n | x P im F p σ qu , where σ “ f ˝ p id n ´ p ´ ‘ ι p ` q is a complement of σ “ f G n ´ p ´ . Proof.
Because p K ´ F q n “ F n , the augmented semisimplicial set splits into p K ‚ F q n – ž x P F n L x ‚ with L xp Ă p K p F q n the preimage of x P F n which is L xp “ tr g, y s P G n ˆ G n ´ p ´ F n ´ p ´ | gφ p y q “ x u using Proposition 4.2. Because F preserves monomorphims, the set map φ “ F p id n ´ p ´ ‘ ι p ` q is injective. Thus for every g P G n there is at most one y P F n ´ p ´ with gφ p y q “ x . Therefore L x ‚ can be included into p K ‚ Hom p , ´qq n by sending r g, y s to gG n ´ p ´ .Finally, note that F p σ q “ gφ . (cid:3) Central stability and finiteness properties
Definition 5.1.
We call a functor from a category C to the category of sets a C –set and write C ´ Set : “ Fun p C , Set q . A U G –set V is generated in ranks ď m if there isan epimorphism P Ý ։ V of U G –sets where P is a disjoint union P “ ž i P I Hom p m i , ´q of representable functors such that m i ď m for all i P I .Fix a ring R . We call a functor from a category C to the category of R –modulesa C –module and write C ´ mod : “ Fun p C , R ´ mod q . A U G –module V is generatedin ranks ď m if there is an epimorphism P Ý ։ V of U G –modules where P is adirect sum P “ à i P I R Hom p m i , ´q such that m i ď m for all i P I .In both situations we call such a P freely generated in ranks ď m . Remark 5.2.
For U G –modules, a similar notion has been called “freely generated”by for example [CEF15] . The U G –module R Hom p m, ´q has a linear right actionby G m . Let W m be a not necessarily free RG m –module, then R Hom p m, ´q b RG m W m is a U G –module. These modules are projective over U G if and only if W m is a pro-jective RG m –module. Therefore we will not call these freely generated as [CEF15] do in the case of FI –modules. ENTRAL STABILITY HOMOLOGY 23
Let V be a U G –set or a U G –module, then V n “ V p n q is a sequence of G n –sets or G n –representations, respectively, and φ n “ V p ι ‘ id n q is G n –equivariant.The following lemma is a criterion when such a sequence is actually a U G –set or a U G –module. This criterion has been observed by Church–Ellenberg–Farb [CEF15,Rmk 3.3.1] for the special case of FI –modules but the argument easily generalizesas shown in [RWW15, Prop 4.2]. Lemma 5.3.
Let p V n , φ n q n P N be a sequence of RG n –modules or G n –sets (resp.) V n and G n –equivariant maps φ n : V n Ñ V n ` .There is a unique U G –module or U G –set (resp.) V with V p n q “ V n and V p ι ‘ id n q “ φ n if and only if for all m ď n and g P G n ´ m p g ‘ id m q ˝ φ m,n “ φ m,n , where φ m,n “ φ n ´ ˝ ¨ ¨ ¨ ˝ φ m . We want to connect the central stability homology to generation properties.
Proposition 5.4.
Let V : U G Ñ D be a functor to D “ Set or to D “ R ´ mod .Then V is generated in the ranks ď d if and only if the map p Σ V q n ÝÑ V n induced by ι P Hom p , q is surjective for all n ą d .Proof. We give the proof for D “ Set . For D “ R ´ mod the proof is analogous afterlinearlizing.Let p ď d ă n . Then for a representable functorHom p p ‘ , n q ÝÑ Hom p p, n q is given by f ÞÑ f ˝ p id p ‘ ι q is surjective because G n acts transitively on Hom p p, n q by postcomposition. Let there be an epimorphism P “ ž i P I Hom p m i , ´q Ý ։ V such that all m i ď d . The diagramΣ P n / / / / (cid:15) (cid:15) P n (cid:15) (cid:15) (cid:15) (cid:15) Σ V n / / V n commutes because Σ Ñ id is a natural transformation between endofunctors. Thisimplies the first implication.Let V be a functor for which p Σ V q n ÝÑ V n is surjective for all n ą d . Let P be a disjoint union P “ ž i P I Hom p m i , ´q of representable functors with m ď d together with a morphism of U G –sets P Ñ V such that P n Ý ։ V n is surjective for all n ď d . (The Yoneda Lemma lets us find such a P .)Let n ą d and assume that P n ´ Ñ V n ´ is surjective by induction. Then thefollowing commutative diagram show that P n Ñ V n is surjective. p Σ P q n – G n ˆ G n ´ P n ´ / / / / (cid:15) (cid:15) (cid:15) (cid:15) p Σ V q n – G n ˆ G n ´ V n ´ (cid:15) (cid:15) (cid:15) (cid:15) P n / / V n (cid:3) The previous proposition says that a U G –module V is generated in finite ranksif and only if r H ´ V is stably zero, ie r H ´ V n “ n large enough. We nextwant to generalize this concept. For this we need an additional condition. Definition 5.5.
Let a, k P N . We define the following condition. H3( N ): r H i p R Hom p , ´qq n “ for all i ă N and all n ą k ¨ i ` a . Remark 5.6.
Note that the condition
LH3 in [RWW15, Def 2.2] implies H3( ) with a “ rk A ` . For all of Examples 1.1, they prove or gather the following valuesfrom existing literature.stability category k a(a) U S U GL p R q for a ring R with stable rank s U Sp p R q for a ring R with unitary stable rank s U AutF
U β U Mod Σ
Theorem 5.7.
Assume
H3( N ) . Let V be a U G –module and d , . . . , d N P Z with d i ` ´ d i ě max p k, a q , then the following statements are equivalent. (a) There is a partial resolution P N Ñ P N ´ Ñ ¨ ¨ ¨ Ñ P Ñ V Ñ with P i that are freely generated in ranks ď d i . (b) The homology r H i p V q n “ for all ´ ď i ă N and all n ą d i ` . ENTRAL STABILITY HOMOLOGY 25
To prove this theorem we first compute the central stability homology of free U G –modules. This part is quite technical and can easily be skipped upon the firstread. Assuming Proposition 5.10, the proof of Theorem 5.7 is straight forward.To compute the central stability homology of free U G –modules, we investigate K ‚ Hom p m, ´q . We first introduce the concept of joins and links for augmentedsemisimplicial sets. We say ρ is a join τ ˚ σ of the p –simplex τ and the q –simplex σ in a semisimplicial object if B ¨ ¨ ¨ B p ρ “ σ and B p ` ¨ ¨ ¨ B p ` q ` ρ “ τ and let the left- and right-sided link beLk L σ “ t τ | D a join τ ˚ σ u and Lk R τ “ t σ | D a join τ ˚ σ u . In K ‚ Hom p , ´q n the p –simplices are morphisms from p ` Ñ n . And for ρ P K p ` q ` Hom p , ´q n B ¨ ¨ ¨ B p ρ “ ρ ˝ p ι p ` ‘ id q ` q and B p ` ¨ ¨ ¨ B p ` q ` ρ “ ρ ˝ p id p ` ‘ ι q ` q . Condition (b) of Definition 3.10 implies that τ ˚ σ is unique if it exists. Moregenerally we write τ ˚ σ “ ρ : l ‘ m Ñ n for τ : l Ñ n and σ : m Ñ n if ρ ˝ p ι l ‘ id m q “ σ and ρ ˝ p id l ‘ ι m q “ τ. Lemma 5.8.
Let f G n ´ m “ σ : m Ñ n , then Lk L σ “ t τ : l Ñ n | D τ : l Ñ n ´ m such that τ “ f ˝p id n ´ m ‘ ι m q˝ τ u – K ‚ Hom p , ´q n ´ m and Lk R σ “ t τ : l Ñ n | D τ : l Ñ n ´ m such that τ “ f ˝ b m,n ´ m ˝p ι m ‘ id n ´ m q˝ τ u – K ‚ Hom p , ´q n ´ m . Proof.
Assume τ “ f ˝ p τ ‘ ι m q then ρ “ f ˝ p τ ‘ id m q “ τ ˚ σ. Similarly if τ “ f ˝ b m,n ´ m ˝ p ι m ‘ τ q then ρ “ f ˝ b m,n ´ m ˝ p ι m ‘ τ q “ τ ˚ σ. For the opposite direction apply Proposition 4.1 to f ´ ˝ ρ : l ‘ m ÝÑ p n ´ m q ‘ m and f ´ ˝ b ´ m,n ´ m ˝ ρ : m ‘ l ÝÑ m ‘ p n ´ m q . (cid:3) Corollary 5.9.
Let σ P K p Hom p , ´q n , then both Lk L σ – Lk R σ – K ‚ Hom p , ´q n ´ p ´ . Proposition 5.10.
Assume
H3( N ) . Then r H i p R Hom p m, ´qq n “ for all i ă N and all n ą k ¨ i ` a ` m .Proof. From Proposition 4.6 we know that K p Hom p m, ´q n is isomorphic to ž τ P Hom p m,n q L τp with L τp “ t σ P K p Hom p , ´q n | τ P im Hom p m, ´qp σ qu where σ “ f ˝ p id n ´ p ´ ‘ ι p ` q is a complement of σ “ f G n ´ p ´ . Note thatim Hom p m, ´qp σ q Ă Hom p m, n q , that is the set of those morphisms that factor through σ , is independent of thechoice of f . Because τ P im Hom p m, ´qp σ q ðñ D τ : m Ñ n ´ p ´ τ “ σ ˝ τ , Lemma 5.8 gives that τ P Lk L σ , hence L τ ‚ “ Lk R τ – K ‚ Hom p , ´q n ´ m . This means that for every U G –module P freely generated in ranks ď m , r H i p P q n “ ´ ď i ă N and n ą k ¨ i ` a ` m . (cid:3) This proposition suffices to prove Theorem 5.7. First we will derive a corollarythat the modules described in Remark 5.2 have vanishing central stability homologyin the same range as the freely generated U G –module R Hom p m, ´q . Note thatin Theorem 5.7 (a) implies (b) for any partial resolution with vanishing centralstability homology in the same ranges as the P i that are freely generated in ranks ď d i . Corollary 5.11.
Assume
H3( N ) . Let W m be an RG m –module. Then r H i p R Hom p m, ´q b RG m W m q n “ for all i ă N and all n ą k ¨ i ` a ` m .Proof. Let Q N Ñ Q N ´ Ñ ¨ ¨ ¨ Ñ Q Ñ W m be a projective resolution of W m by RG m –modules. We consider the complex r C ˚ p R Hom p m, ´q b RG m Q ˚ q n – r C ˚ p R Hom p m, ´qq n b RG m Q ˚ ENTRAL STABILITY HOMOLOGY 27 and its two spectral sequences. The first spectral sequence E pq “ r C p p R Hom p m, ´qq n b RG m Q q is given by E pq “ E pq – $’&’% r H p ˆ R Hom p m, ´q b RG m W m ˙ n q “ q ą r C p p R Hom p m, ´qq n – Ind G n G n ´ m ´ p R is a free RG m –module.The second spectral sequence E pq “ r C q p R Hom p m, ´qq n b RG m Q p , that converges to the same limit, computes to E pq – r H q p R Hom p m, ´qq n b RG m Q p . Therefore the central stability homology of R Hom p m, ´q b RG m W m vanishes in thesame range as R Hom p m, ´q . (cid:3) Proof of Theorem 5.7.
Assume (a). Let K ´ “ P ´ : “ V and let K i : “ ker p P i Ñ P i ´ q for i ě
0. Then the long exact sequence of r H ˚ for the short exact sequence0 ÝÑ K i ÝÑ P i ÝÑ K i ´ ÝÑ r H j p K i q n ã ÝÑ r H j ´ p K i ` q n is injective for all i, j ď N ´ n ą kj ` a ` d i ` . Because P i ` Ý ։ K i forall i ď N ´
1, we know from Proposition 5.4 that r H ´ p K i q n “ n ą d i ` . Thus r H i p V q n ã ÝÑ H i ´ p K q n ã ÝÑ ¨ ¨ ¨ ã ÝÑ r H ´ p K i q n “ i ď N ´ n ą d i ` because d i ` ´ d j ě p i ` ´ j q max p k, a q ě k p i ´ j q ` a for j “ , . . . , i ` P N ´ Ñ ¨ ¨ ¨ Ñ P Ñ V Ñ be a partial resolution such that P i is freely generated in ranks ď d i for all i ď N ´ K ´ “ P ´ : “ V and let K i : “ ker p P i Ñ P i ´ q for i ě
0. We need to prove that K N ´ is generated in ranks d N . Similar as before r H j p K i q n Ý ։ r H j ´ p K i ` q n is surjective for all i, j ď N ´ n ą k p j ´ q ` a ` d i ` . Thus0 “ r H N ´ p V q n Ý ։ r H N ´ p K q n Ý ։ ¨ ¨ ¨ Ý ։ r H ´ p K N ´ q n for all n ą d N ě k p N ´ ´ j q ` a ` d j . (cid:3) Definition 5.12.
We call a U G –modules V stably acyclic if r H i p V q n “ for all i ě ´ and all n large enough. We want to conclude this section by giving an example that is by definition stablyacyclic, but the existence of the free resolution as in Theorem 5.7 is not a priori clear. More generally, in Section 7 we find a condition H4 on U G such that all U G –modules with finite polynomial degree (see Definition 7.1) are stably acyclic. Corollary 5.13.
Assume
H3( N ) . Let V be a U G –module such that V n “ forall n ą d . Then there is a resolution P N Ñ P N ´ Ñ ¨ ¨ ¨ Ñ P Ñ V Ñ with P i that are freely generated in ranks ď max p k, a q ¨ i ` d . Notions of central stability
Different notions of central stability have been used in the past. We want toclear this up for stability categories. In the following proposition we prove theequivalence of four conditions. The condition (a) is called “presented in degree ď d ” in [CEF15, CE16]. In [PS14], condition (b) is called “central stability”. Andconditions (c) and (d) are used in [CEFN14], although they phrase the colimit overan equivalent category. We will refer to this notion as in (a) by presented in theranks ď d . Proposition 6.1.
Let V be a U G –module, then the following are equivalent. (a) V is generated and presented in the ranks ď d , ie there are U G –modules P , P freely generated in the ranks ď d such that P ÝÑ P ÝÑ V ÝÑ is exact. ENTRAL STABILITY HOMOLOGY 29 (b)
Let inc d : U G ď d ã ÝÑ U G be the full subcategory with the objects , . . . , d .The left Kan extension Lan inc d p V ˝ inc d q – V is naturally isomorphic to V . (c) For all n ą d there is a natural isomorphism colim m Ñ nm ď d V m – V n . (d) For all n ą d there is a natural isomorphism colim m Ñ nm ă n V m – V n . Proof.
Djament [Dja16, Prop 2.14] proves that (a) and (b) are equivalent in aneven more general setting. Gan–Li [GL17, Thm 3.2] gave a different proof with theviewpoint of graded modules over graded nonunital algebra.The left Kan extension evaluated at n P N can be expressed as the colimit in (c)(see [ML98, Cor X.3.4]): p Lan inc d V ˝ inc d q n – colim p inc d Ó n q V “ colim m Ñ nm ď d V m This proves that (c) is equivalent to (b).Finally we want to prove that (c) and (d) are equivalent. First note thatcolim m Ñ nm ă n V m “ colim p inc n ´ Ó n q V is just a different way of writing the same colimit. Further there is a natural mapcolim p inc d Ó n q V ÝÑ V n . And because p inc d Ó n q is a full subcategory of p inc n ´ Ó n q there is a mapcolim p inc d Ó n q V ÝÑ colim p inc n ´ Ó n q V and similarly fixing a morphism m Ñ n there is a mapcolim p inc d Ó m q V ÝÑ colim p inc d Ó n q V. Let us prove the following claim.
Claim.
Let n ě d and assume that colim p inc d Ó m q V ÝÑ V m is an isomorphism for all m ă n , then colim p inc d Ó n q V ÝÑ colim p inc n ´ Ó n q V is an isomorphism. Proof.
For every morphism f : m Ñ n which is an object in p inc n ´ Ó n q , we getthe following diagram.colim p inc d Ó m q V – / / α (cid:15) (cid:15) V mβ (cid:15) (cid:15) colim p inc d Ó n q V γ / / colim p inc n ´ Ó n q V δ o o Here δ is the unique map such that α “ δβ . One checks also checks that β “ γα .This implies that δγα “ α and γδβ “ β. The universal property of the colimits implies that γ and δ are inverses. (cid:4) We now want to finish our proof of the equivalence of (c) and (d) by inductionover n . For n “ d ` p inc d Ó n q V ÝÑ colim p inc n ´ Ó n q V for all d ă m ă n . Then we can certainly assumecolim p inc d Ó m q V ÝÑ V m is an isomorphism for all m ă n . The claim then finishes the induction step. (cid:3) Putman’s [Put15] original notion of central stability for FI –modules is reflectedin condition (c) of the following proposition. In the case of symmetric stabilitycategories, this is equivalent to (a), which we call centrally stable in the ranks ą d . Proposition 6.2.
Let V be a U G –module, then the following are equivalent. (a) r H p V q n “ r H ´ p V q n “ for all n ą d . (b) V n – coeq ` Ind G n G n ´ V n ´ Ñ Ind G n G n ´ V n ´ ˘ for all n ą d .If G is symmetric, then the following condition is equivalent to the first two. (c) V n – coeq ` Ind G n G n ´ ˆ S V n ´ b A Ñ Ind G n G n ´ V n ´ ˘ for all n ą d , where S Ă G and A is its sign representation.Proof. Conditions (a) and (b) are equivalent because (b) is equivalent to the se-quence Ind G n G n ´ V n ´ ÝÑ Ind G n G n ´ V n ´ ÝÑ V n ÝÑ G n G n ´ V n ´ coincide. Let t P S be the transposition, then both mapsare given by v ÞÑ p ´ t q b φ p v q for v P V n ´ . This completes the proof. (cid:3) ENTRAL STABILITY HOMOLOGY 31
With Theorem 5.7 we can connect these two notions. Note that the assumptionof the theorem can be expressed as follows.
H3( ): b , ‘ id n ´ and 1 ˆ G n ´ generate G n for all n ą a . Corollary 6.3.
Assume
H3( ) . Let V be a U G –module. Then the following areequivalent. (a) V is generated in ranks ď d and presented in ranks ď d ` a . (b) V is generated in ranks ď d and centrally stable in the ranks ą d ` a . Corollary 6.4. (a) A U S –module is generated in ranks ď d and presented inranks ď d ` if and only if it is generated in ranks ď d and centrally stablein the ranks ą d ` . (b) Let R be a ring with stable rank s . A U GL p R q –module is generated in ranks ď d and presented in ranks ď d ` s ` if and only if it is generated in ranks ď d and centrally stable in the ranks ą d ` s ` . (c) Let R be a ring with unitary stable rank s . A U Sp p R q –module is generatedin ranks ď d and presented in ranks ď d ` s ` if and only if it is generatedin ranks ď d and centrally stable in the ranks ą d ` s ` . (d) A U AutF –module is generated in ranks ď d and presented in ranks ď d ` if and only if it is generated in ranks ď d and centrally stable in the ranks ą d ` . (e) A U β –module is generated in ranks ď d and presented in ranks ď d ` ifand only if it is generated in ranks ď d and centrally stable in the ranks ą d ` . (f) A U Mod Σ –module is generated in ranks ď d and presented in ranks ď d ` if and only if it is generated in ranks ď d and centrally stable in the ranks ą d ` . Polynomial degree
As we defined S as the precomposition of ´‘
1, for the next definition we will use T defined as the precomposition of 1 ‘ ´ . The following definition is an adaptationof van der Kallen’s [vdK80] degree of a coefficient system. Our version is almostidentical with [RWW15, Def 4.10]. Definition 7.1.
Let V be a U G –module then we define the U G –modules ker V “ ker p V Ñ T V q and coker V “ coker p V Ñ T V q . We say V has polynomial degree ´8 in the ranks ą d if V n “ for all n ą d . For r ě we say V has polynomial degree ď r in the ranks ą d if p ker V q n “ for all n ą d and coker V has polynomial degree ď r ´ in the ranks ą d . Remark 7.2.
Assuming that the dimension of V n is finite for all n ą d , V havingpolynomial degree ď r in the ranks ą d implies that the dimension grows polynomi-ally of degree ď r in the ranks ą d . Lemma 7.3. (a)
Let ÝÑ V ÝÑ V ÝÑ V ÝÑ be an exact sequence of U G –modules in ranks ą d , i.e. ÝÑ V n ÝÑ V n ÝÑ V n ÝÑ is a short exact sequence for all n ą d . Then if two of the three U G –modules V , V, V have polynomial degrees ď r in ranks ą d , then so does the third. (b) Assume the base ring R is a field. If V and W are U G –modules havepolynomial degrees ď r and ď s in ranks ą d , respectively, then V b W has polynomial degree ď r ` s in ranks ą d , where V b W is defined by p V b W q n “ V n b W n .Proof. (a) From the snake lemma we get an exact sequence0 ÝÑ ker V n ÝÑ ker V n ÝÑ ker V n ÝÑ coker V n ÝÑ coker V n ÝÑ coker V n ÝÑ n ą d . If V has polynomial degree r in ranks ą d ,ker V n / / coker V n ker V n / / ker coker V n “ O O for all n ą d . Hence this sequence actually splits into two short exactsequences. If V and V have finite polynomial degree, then the abovesequence still splits into two short exact sequences. The assertion is shownby induction on the degree.(b) If ´8 P t r, s u , also V b W has polynomial degree ´8 in ranks ą d . Thusker p V b W q n “ n . For an induction argument we ENTRAL STABILITY HOMOLOGY 33 consider the following commutative diagram.0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) V b ker W (cid:15) (cid:15) / / ker p V b W q (cid:15) (cid:15) / / ker V b T W (cid:15) (cid:15) V b W (cid:15) (cid:15) V b W (cid:15) (cid:15) / / V b T W (cid:15) (cid:15) V b T W (cid:15) (cid:15) / / T p V b W q (cid:15) (cid:15) T V b T W (cid:15) (cid:15) V b coker W (cid:15) (cid:15) / / coker p V b W q (cid:15) (cid:15) / / coker V b T W (cid:15) (cid:15) / /
00 0 0Clearly the three columns are exact. By chasing this diagram, we can provethat the top and the bottom row are also exact. Even more,0 ÝÑ V n b coker W n ÝÑ coker p V b W q n ÝÑ coker V n b T W n ÝÑ n ą d because V n b T W n Ñ T V n b T W n isinjective for all n ą d . By induction coker p V b W q has polynomial degree ď r ` s ´ ą d using (a). (cid:3) The following lemma is essentially [PS14, Lem 3.11].
Lemma 7.4.
Let V be a U G –module then the map id n ‘ ι : n Ñ n ‘ inducesthe zero map on r H ´ V . This implies that every morphism of U G that is not anisomorphism induces the zero map on r H ´ V .Proof. Let V n Ñ Σ V n ` – RG n ` b RG n V n be given by the inclusion v ÞÑ b v . Then the following diagram commutes by thecalculations from Proposition 4.3. V n { { ①①①①①①①①①① / / (cid:15) (cid:15) r H ´ V n / / (cid:15) (cid:15) V n ` / / V n ` / / r H ´ p V q n ` / / V n Ñ r H ´ V n ` must be the zero map.Thus surjectivity of V n Ñ r H ´ V n implies the assertion. (cid:3) Proposition 7.5.
Let V be a U G –module and d P N . If coker V is generated inranks ď d ´ , then V is generated in ranks ď d .Proof. We consider the following diagram. V / / (cid:15) (cid:15) (cid:15) (cid:15) T V / / (cid:15) (cid:15) (cid:15) (cid:15) coker V / / (cid:15) (cid:15) (cid:15) (cid:15) r H ´ V / / T r H ´ V / / coker r H ´ V / / r H ´ V Ñ T r H ´ V is the zero map because of Lemma 7.4. Thus there isan epimorphism coker V / / / / T r H ´ V .
This implies that T r H ´ V is generated in ranks ď d ´
1. Using Lemma 7.4 again,we infer that r H ´ V n ` “ T r H ´ V n “ n ą d . Therefore V is generated inranks ď d by Proposition 5.4. (cid:3) Definition 7.6.
Let ℓ, b ě . We define the following condition. H4( N ): r H i p coker R Hom p m, ´qq n “ for all i ă N and all n ą ℓ ¨ p i ` m q ` b . Remark 7.7.
Knowing that coker R Hom p m, ´q is stably acyclic for all m P N ,one can also follow that T r R Hom p m, ´q is stably acyclic for all m, r P N . Proposition 7.8.
Let N P N . Assume H3( N ) and H4( N ) with b ě max p k, a q .Let d , . . . , d N ď ´ with d i ě max p ℓ ¨ d i ´ ` b, d i ´ ` b q . If ker V n “ for all n ą d and r H i p coker V q n “ for all i ď N ´ and all n ą d i ` , then r H i p V q n “ for all i ď N ´ and n ą d i ` ` .Proof. The case N “ N ě
1. By induction r H i p V q n “ n ą d i ` ` i ă N ´
1. Theorem 5.7 implies the existence of a resolution P N ´ ÝÑ ¨ ¨ ¨ ÝÑ P ÝÑ V ÝÑ U G –modules P i that are freely generated in ranks ď d i `
1, because d i ´ d i ´ ě b ě max p k, a q . If we can prove that K N ´ “ ker p P N ´ Ñ P N ´ q is generated inranks ď d N `
1, we are done. Let K j “ ker p P j Ñ P j ´ q with the convention P ´ “ K ´ “ V . Because T is exact, the snake lemma givesus the exact sequence0 “ ker P i ` ÝÑ ker K i ÝÑ coker K i ` ÝÑ coker P i ` ÝÑ coker K i ÝÑ . ENTRAL STABILITY HOMOLOGY 35 If i ě
0, ker K i is a submodule of ker P i “
0. Therefore there is a long exactsequence ¨ ¨ ¨ Ñ r H N ´ i ´ p coker K i q n Ñ r H N ´ i ´ p coker K i ` q n Ñ r H N ´ i ´ p coker P i ` q n Ñ ¨ ¨ ¨ . By the assumption on r H N ´ i ´ p coker P i ` q n , we see that r H N ´ i ´ p coker K i q n Ý ։ r H N ´ i ´ p coker K i ` q n is surjective for all i ď N ´ n ą ℓ ¨ p d i ` ` ` N ´ i ´ q ` b .If i “ ´
1, ie ker K i “ ker V , this method only infers r H N ´ p coker V q n Ý ։ r H N ´ p coker K { ker V q n is surjective for all n ą ℓ ¨ p d ` N ´ q ` b . Because ker V n “ n ą d , p r C N ´ ker V q n – Ind G n G n ´ N ` ker V n ´ N ` “ n ą d ` N ´
1. Therefore in the same range r H N ´ p coker K q n ã ÝÑ r H N ´ p coker K { ker V q n is injective.Thus the assumption that r H N ´ p coker V q n “ n ą d N implies r H ´ p coker K N ´ q n “ d N ě ℓ N ´ i ´ ¨ d i ` ` p ` ¨ ¨ ¨ ` ℓ N ´ i ´ q b ě ℓ ¨ p d i ` ` N ´ i ´ q ` b and d N ě ℓ N ´ d ` p ` ¨ ¨ ¨ ` ℓ N ´ q b ě d ` N ´ ě ℓ ¨ d ` b ` N ´ . By Proposition 7.5, K N ´ is generated in ranks ď d N ` (cid:3) A direct consequence of this proposition is Theorem B:
Corollary 7.9.
Assume
H3( ) and H4( ) with b ě max p k, a q . Let r ě , d ě ´ and let V be of polynomial degree ď r in ranks ą d , then r H i V n “ for all i ě ´ and n ą ℓ i ` p d ` r q ` p ℓ i ` ¨ ¨ ¨ ` q b ` .Proof. For r “
0, ker V n “ coker V n “ n ą d . Then r H i p coker V q n “ n ą d ` i `
1. Let d “ d and for i ą d i “ ℓ i ¨ d ` p ℓ i ´ ` ¨ ¨ ¨ ` q b if d ě d i “ ´ ℓ i ´ ` p ℓ i ´ ` ¨ ¨ ¨ ` q b if d “ ´
1. Then d i ě max p ℓ ¨ d i ´ ` b, d i ´ ` b q , ker V n “ n ą d ě d, and r H i p coker V q n “ n ą d i ` ě d ` i ` . Therefore by Proposition 7.8, r H i V n “ n ą d i ` `
1. This proves the case r “ r ą d ri “ ℓ i p d ` r q ` p ℓ i ´ ` ¨ ¨ ¨ ` q b. By induction r H i coker V n “ n ą d ri ` ě d r ´ i ` ` . We also have that d ri ě ℓd ri ´ ` b and ker V n “ n ą d r ě d. Thus we can apply Proposition 7.8 to get r H i V n “ n ą d ri ` `
1. This proves the case r ą (cid:3) From
H3( N ) and a long exact sequence we get that r H i p T Hom p m, ´qq n Ý ։ r H i p coker Hom p m, ´qq n is surjective for all i ď N ´ n ą k ¨p i ´ q` a ` m . With the next propositionwe will express K ‚ T Hom p m, ´q n in form of the semisimplicial sets K ‚ Hom p , ´q n that are analyzed in [RWW15]. Proposition 7.10.
The augmented semisimplicial set K ‚ T Hom p m, ´q n is isomor-phic to the disjoint union of augmented semisimplicial sets ž τ P Hom p m,n ` q Lk R τ X K ‚ Hom p , ´q n with the embedding of K ‚ Hom p , ´q n Ă K ‚ Hom p , ´q n ` by σ ÞÑ p ι ‘ id n q ˝ σ . ENTRAL STABILITY HOMOLOGY 37
Proof.
From Proposition 4.6 we know that K p T Hom p m, ´q n is isomorphic to ž τ P Hom p m,n ` q L τp with L τp “ t σ P K p Hom p , ´q n | τ P im T Hom p m, ´qp σ qu where σ “ f ˝ p id n ´ p ´ ‘ ι p ` q is a complement of σ “ f G n ´ p ´ . The equationid ‘ σ “ ´ p ι ‘ id n q ˝ σ ¯ and τ P im T Hom p m, ´qp σ q ðñ D τ : m Ñ ‘ p n ´ p ´ q such that τ “ p id ‘ σ q˝ τ , together with Lemma 5.8 gives that τ P Lk L pp ι ‘ id n q ˝ σ q , hence L τ ‚ “ Lk R τ X K ‚ Hom p , ´q n . (cid:3) Example 7.11.
Here we want to provide an example of a stability category that sat-isfies
H3( ) but does not even satisfy H4( ) . We will explain why T Hom Uβ p , ´q is not generated in finite ranks. Let σ “ bβ n P T Hom Uβ p , ´q n “ Hom Uβ p , ‘ n q which is the coset that is represented by a braid b P β n ` . Now φ p σ q “ p id ‘ ι ‘ id n q ˝ σ “ p b ´ , ‘ id n q ˝ p id ‘ b q β n ` , which is depicted in the following diagram. p b ´ , ‘ id n q ˝ p id ‘ b q “ ¨ ¨ ¨ b ¨ ¨ ¨ β n ` ü ˝¨ ¨ ¨¨ ¨ ¨ ý β n ` Our claim, that T Hom Uβ p , ´q is not generated in finite ranks, amounts to showing p ˆ β n ` qp b ´ , ‘ id n qp ˆ β n ` qp β n ` ˆ q ‰ β n ` . This follows because the braid ¨ ¨ ¨ ¨ ¨ ¨ is not contained in the LHS. Short exact sequences
In this section will deal with short exact sequences of groups and stabilitygroupoids. We restate some well-known properties of group homology regardingthese 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 hopethat they become more accessible in the language of this paper.Let 1 ÝÑ N ÝÑ G ÝÑ Q ÝÑ N as a Q –module. For this let us first review how an automorphism φ P Aut p N q acts on H i p N ; M q for some RN –module M (given an RN –homomorphism ψ : M Ñ Res φ M , ie ψ p n ¨ m q “ φ p n q ¨ ψ p m q .) Let E ˚ N be a free (right) RN –resolution ofthe trivial representation R . Let ξ : E ˚ N Ñ Res φ E ˚ N be an RN –homomorphism.(This can always be found and any two are chain homotopic.) Then φ induces themap E ˚ N b N M ξ b ψ ÝÑ Res φ E ˚ N b N Res φ M. Note that the RHS is canonically isomorphic to E ˚ N b N M because φ is a bijection.Let φ be an inner automorphism of N , say conjugation by n P N . Then ξ p x q “ xn ´ and ψ p m q “ nm fulfill the above requirements. But then ξ b ψ is in fact theidentity on E ˚ N b 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 φ the conjugation by an element g P G .( N is normal in G .) Then ξ p x q “ xg ´ and ψ p m q “ gm fulfill the above require-ments as before. Note that ξ b ψ is not the identity on E ˚ G b N M . Summarizing,we have seen that E ˚ G b N M is an RQ –module by N g ¨ p x b m q “ xg ´ b gm. This induces an RQ –module structure on H ˚ p N ; M q for every RG –module M . ENTRAL STABILITY HOMOLOGY 39
Now we generalize this concept.
In this section, we deal with multiple stability groupoids N “ p N n q n P N , G “p G n q n P N , Q “ p Q n q n P N .For the most time we assume that G and Q are braided, in which their stabilitycategories U G and U Q are monoidal and prebraided, whence it makes sense to takeof their central stability complex and homology, which we denote by r C G ˚ , r H G ˚ and r C Q ˚ , r H Q ˚ , respectively. Definition 8.1.
Let F : N Ñ G and F : G Ñ Q be homomorphisms of stabilitygroupoids. We call this data a stability SES if ÝÑ N n ÝÑ G n ÝÑ Q n ÝÑ is a short exact sequence for all n P N . Proposition 8.2.
Let G , Q be stability groupoids and F : G Ñ Q a homomorphismof stability groupoids and F n : G n Ñ Q n is surjective for every n P N . Then thereis a stability groupoid N and a homomorphism of stability groupoids F : N Ñ G such that N F ÝÑ G F ÝÑ Q is a stability SES.Proof. Let N n be the kernel of F n and F n the embedding of N n in G n . Let N bethe groupoid formed by these groups and F the functor given by all F n ’s. Becauseker p F m ˆ F n q “ N m ˆ N n we get an injective group homomorphism N m ˆ N n ã ÝÑ N m ` n . This proves that N is a stability groupoid. All other assertions are immediate. (cid:3) Lemma 8.3.
Let ÝÑ N ÝÑ G ÝÑ Q ÝÑ be a stability SES and V a U G –module. Then for every i ě there is a U Q –module W with W n – H i p N n ; Res G n N n V n q . We denote this U Q –module by H i p N ; V q and abbreviate H i p N q if V “ R Hom p , ´q .Proof. We will use Lemma 5.3 for this proof.We already know that for every i ě RQ n –representations.Let φ n : H i p N n ; V n q ÝÑ H i p N n ` ; V n ` q be induced by ι ‘ id n . On the chain complex level this is E ˚ G n ` b N n V n ÝÑ E ˚ G n ` b N n ` V n ` induced by φ “ V p ι ‘ id n q : V n Ñ V n ` .That means that φ : H i p N m ; V m q ÝÑ H i p N n , V n q is induced by the map E ˚ G n b N m V m id b φ ÝÑ E ˚ G n b N n V n . Because x ÞÑ xg for some g P 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 b N m V m . Therefore x b m ÞÑ xg b m induces the identity on H i p N m ; V m q .By the following commutative diagram, the action of G n ´ m on φ p H i p N m ; V m qq istrivial. This finishes the proof by application of Lemma 5.3. E ˚ G n b N m V m id b φ / / ¨p g q ´ b id (cid:15) (cid:15) E ˚ G n b N n V n ¨p g q ´ b g ¨ (cid:15) (cid:15) E ˚ G n b N m V m id b φ / / E ˚ G n b N n V n (cid:3) The central technical tool of this section is the following spectral sequence.
Proposition 8.4.
Let ÝÑ N ÝÑ G ÝÑ Q ÝÑ be a stability SES. Assume that G , Q , and G Ñ Q are braided. Let V be a U G –module. Then the two spectral sequences constructed from the double complex E ˚ G n b RN n r C G ˚ p V q n are E pq – E p G n b RN n r H G q p V q n and E pq – r H Q p p H q p N ; V qq n . Proof.
The first spectral sequence is given by E pq “ E p G n b RN n r C G q p V q n . Therefore E pq – E p G n b RN n r H G q p V q n . ENTRAL STABILITY HOMOLOGY 41
The other spectral sequence is given by E pq – H q p N n ; r C G p p V q n q “ H q p N n ; RG n b RG n ´p p ` q V n ´p p ` q q“ H q p E ˚ G n b RN n RG n b RG n ´p p ` q V n ´p p ` q q . The differential d is induced by the map E ˚ G n b RN n RG n b RG n ´p p ` q V n ´p p ` q ÝÑ E ˚ G n b RN n RG n b RG n ´ p V n ´ p x b g b v ÞÝÑ ÿ p´ q i x b gg i b φ p v q with g i “ id n ´ p ´ ‘ b ,i ‘ id p ´ i and φ “ V p id n ´ p ‘ ι q .Then there is a Q n –equivariant isomorphism E ˚ G n b RN n RG n b RG n ´p p ` q V n ´p p ` q – RQ n b RQ n ´p p ` q ` E ˚ G n b RN n ´p p ` q V n ´p p ` q ˘ given by x b g b v ÞÝÑ N n g b p xg b v q xg ´ b g b v ÐÝß N n g b p x b v q . Therefore E pq – r C Q p p H i p N ; V qq n . Because the following diagram commutes, d coincides with the differential of r C Q ˚ . E ˚ G n b RN n RG n b RG n ´p p ` q V n ´p p ` q / / (cid:15) (cid:15) RQ n b RQ n ´p p ` q ` E ˚ G n b N n ´p p ` q V n ´p p ` q ˘ (cid:15) (cid:15) RQ n b RQ n ´p p ` q ` E ˚ G n b N n ´p p ` q V n ´p p ` q ˘ (cid:15) (cid:15) E ˚ G n b RN n RG n b RG n ´ p V n ´ p / / RQ n b RQ n ´p p ` q ` E ˚ G n b N n ´ p V n ´ p ˘ is given by x b g b v ✤ / / ❴ (cid:15) (cid:15) N n g b p xg b v q ❴ (cid:15) (cid:15) N n g b p xgg i b v q ❴ (cid:15) (cid:15) x b gg i b φ p v q ✤ / / N n gg i b p xgg i b φ p v qq . (cid:3) The idea for this spectral sequence can be found in [PS14]. The following imme-diate consequence illustrates how it can be used.
Corollary 8.5.
Let ÝÑ N ÝÑ G ÝÑ Q ÝÑ be a stability SES. Assume that G , Q , and G Ñ Q are braided. For each n , there isa spectral sequence E p,q – r H Q p p H q p N qq n that converges to zero for p ` q ď n ´ a ´ k if U G satisfies H3( ) .Proof. The two spectral sequences in Proposition 8.4 converge to the same limit.The first will converge to zero for n ě k ¨ q ` a ` H3( ) if we set V to be R Hom p , ´q . In particular, the diagonals p ` q ď n ´ a ´ k converge to zero. Thusthe same is true for the second spectral sequence. (cid:3) 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 Proposition 8.4. Thefirst spectral sequence implies that both converge stably to zero, because V is stablyacyclic. We will now prove the result by induction on i .For i “ N n H G n M G n ´ m – Q n M Q n ´ m we get H p N ; R Hom U G p m, ´qq – R Hom U Q p m, ´q . Because H is right exact it follows that H p N ; V q is a finitely generated U Q –module.Let i ą H q p N ; V q is finitely generated for all q ă i . From theNoetherian condition we get that H q p N ; V q is stably acyclic for all q ă i . Now usingthe second spectral sequence from Proposition 8.4, we infer E ´ i “ r H ´ p H i p N ; V qq is stably zero, because the only incoming and outgoing differentials are all stablyzero. Proposition 5.4 and the assumption that H i p N ; V q n is a finitely generated R –module for all n implies that H i p N ; V q is finitely generated. (cid:3) Quillen’s argument revisited
In this section we will solely prove Theorem D from the introduction.
Proof of Theorem D.
We consider the stability SES1 ÝÑ G id ÝÑ G ÝÑ N ÝÑ ENTRAL STABILITY HOMOLOGY 43 where N is the category whose objects are the natural numbers and whose mor-phisms are only the identity maps. In Proposition 8.4, we constructed two spectralsequences for this situation. From the first E pq “ E p G n ` b G n ` r H G q V n ` which is zero for n ` ą k ¨ q ` a , we see that both spectral sequences converge tozero when p ` q ď n ´ ak . The other spectral sequence is given by E pq – r C N p p H q p G ; V qq n ` – H q p G n ` ´p p ` q ; V n ` ´p p ` q q . The differentials are easy to understand: H q p G n ` ´p p ` q ; V n ` ´p p ` q q Ñ H q p G n ` ´ p ; V n ` ´ p q is zero if p is odd and it is φ ˚ if p is even. In particular E ,i Ñ E ´ ,i is thestabilization map φ ˚ : H i p G n ; V n q ÝÑ H i p G n ` ; V n ` q . Assume n ě ki ` a ´
1, we want to prove that φ ˚ is surjective. We know that E ,i “ i ´ ď n ´ ak , in particular when n ě ki ` a ´
1. We want to useinduction to show that E pq “ p ` q “ i and q ă i . This would imply that E ´ ,i already vanishes and φ ˚ is surjective. If p is even, to show that E pq “ E p ` ,q – H q p G n ´ p ´ ; V n ´ p ´ q ÝÑ E pq – H q p G n ´ p ; V n ´ p q is surjective. By induction this is the case if n ´ p ´ ě kq ` a ´
1. If p is odd, itsuffices to show that E pq – H q p G n ´ p ; V n ´ p q ÝÑ E p ´ ,q – H q p G n ´ p ` ; V n ´ p ` q is injective. By induction this is true if n ´ p ě kq ` a , which is the same condition.We know that p ` kq ď i ` p k ´ qp i ´ q “ ki ´ k ` ď n ´ a which is what we need.Assume n ě ki ` a , we want to prove that φ ˚ is injective. We know that E ,i “ i ě n ´ ak , in particular when n ě ki ` a . Again we prove E pq “ p ` q “ i ` q ă i by induction, which implies that E ,i “ φ ˚ is injective.We already computed that E pq vanishes when n ´ p ě kq ` a . And we calculate p ` kq ď i ` ` p k ´ qp i ´ q “ ki ´ k ` ď n ´ a. (cid:3) References [CE16] Thomas Church and Jordan S. Ellenberg. Homology of FI-modules. Preprint, 2016,arXiv:1506.01022v2, to appear in Geom. Topol.[CEF15] Thomas Church, Jordan S. Ellenberg, and Benson Farb. FI-modules and stability forrepresentations of symmetric groups.
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Institut f¨ur Mathematik, Freie Universit¨at Berlin, Germany
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