aa r X i v : . [ m a t h . G R ] O c t Relative cohomology theory for profinite groups
Gareth WilkesOctober 2, 2017
Abstract
In this paper we define and develop the theory of the cohomology of a profinite group relative toa collection of closed subgroups. Having made the relevant definitions we establish a robust theoryof cup products and use this theory to define profinite Poincar´e duality pairs. We use the theoryof groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give resultsconcerning decompositions of 3-manifold groups. Finally we discuss the relationship between discreteduality pairs and profinite duality pairs, culminating in the result that profinite completion of thefundamental group of a compact aspherical 3-manifold is a profinite Poincar´e duality group relativeto the profinite completions of the fundamental groups of its boundary components.
Introduction
The classical theory of the cohomology of groups has developed in several directions. Our principal sourcefor the theory of relative group cohomology and Poincar´e duality for group pairs will be the paper [BE78]by Bieri and Eckmann. For the cohomology theory of profinite groups we have relied on Serre [Ser13] andSymonds and Weigel [SW00]. The principle aim of this paper is to establish a relative cohomology theoryfor profinite groups which exhibits the salient features of both these theories. Many of the outcomesclosely parallel those of classical relative group cohomology, and naturally some of the arguments in thispaper are broadly similar to those in [BE78]—albeit with extra effort needed to cope with the topology.One major way in which the profinite theory differs from the classical theory is in the absence of asufficiently broad cap product. Thus most results are stated and proved in terms of cup products andmaps derived from them.We will now describe the general plan of the paper. In the first section we will discuss the variouspreliminary concepts which will be required for our theory. Particular attention will be paid to thedifferent categories involved. The homology theory of a profinite group naturally takes coefficients amongcompact modules, while the cohomology theory takes coefficients in discrete modules. The theory ofduality groups requires, under some conditions, extension of the cohomology theory to compact modulesand the homology theory of discrete modules. We will spend some time describing the way in which thesecategories fit together, as well as detailing the functors we wish to consider.Section 2 we define the (co)homology of a profinite group pair and derive some basic properties, suchas the appropriate version of an Eckmann-Shapiro lemma. We also define the cohomological dimensionof a profinite group pair with respect to a choice of prime.In Section 3 we will define and derive the properties of a cup product sufficiently broad for our needs.One might expect an article concerned with duality groups to rely heavily on cap products as [BE78].However as will be discussed at the time, the theory of cap products does not fit comfortably with theframework of profinite groups, and we will exclusively use cup products and mappings derived from them.Section 4 develops various excision and Mayer-Vietoris-type results for profinite amalgamated productsand HNN extensions, as well as other consequences for relative cohomology of an action on a profinite.1n Section 5 we define the notion of a profinite duality pair at a prime p and prove various foundationalresults concerning such pairs. We also use the Mayer-Vietoris sequences from the previous section todiscuss graphs of Poincar´e duality pairs.In Section 6 we explore the relationship between classical duality pairs and profinite duality pairs, inthe context of ‘cohomological goodness’ properties of discrete group pairs. We pay particular attention tothe case of 3-manifold groups and conclude with an application of our theory, proving results concerningthe Kneser-Milnor and JSJ decompositions of 3-manifolds. We intend to explore this direction further ina future paper.The article concludes with two appendices proving technical results about profinite groups and mod-ules. These results are not really necessary for an understanding of the rest of the paper, and theconsequences deriving from them are plausible enough that the reader will believe them without subject-ing themselves to the proofs. However the proofs should appear somewhere, so we leave them at the endof the paper. Conventions.
The following conventions will be in force through the paper. • Unless otherwise specified G will be a profinite group and π will be a non-empty set of primes. • For a set of primes π , the ring of π -adic integers (that is, the pro- π completion of Z ) will be denoted Z π . If π “ t p u then we will write Z p rather than Z t p u . The field with p elements will be denoted F p . • Maps of topological groups or modules should be assumed to be continuous homomorphisms in theappropriate sense. • If G is a profinite group, H is a subgroup of G and M is a G -module of some kind, then res GH p M q denotes the H -module obtained by restriction. • If G is a group and M is a left G -module then M K will denote the canonical right G -moduleassociated to M —that is, the module with underlying abelian group M and right G -action m ¨ g “ g ´ m for g P G and m P M . • For π a set of primes, I π will denote Q π { Z π . A model for this is the group of rational numberswhose denominators are π -numbers. • Conjugation in groups will be a right action, so that x y “ y ´ xy . There are several different categories in play in the theory to be developed. Let G be a profinite groupand let π be a set of primes. Recall that an abelian group is called π -primary if it is a torsion group andthe order of every element is divisible only by primes in π . Define the following categories. All modulesare left modules unless otherwise specified. Definition 1.1.
Let G be a profinite group and let π be a set of primes. • F π p G q is the category of finite π -primary G -modules with G -linear morphisms • C π p G q is the category of compact pro- π G -modules—that is, inverse limits of modules in F π p G q .The morphisms are continuous G -linear group homomorphisms. Similarly define C π p G q K to be thecategory of compact pro- π right G -modules. These are abelian categories with enough projectivesbut not enough injectives. 2 D π p G q is the category of discrete π -primary G -modules—that is, direct limits of modules in F π p G q .The morphisms are the G -linear group homomorphisms. This is an abelian category with enoughinjectives but not enough projectives. • P π p G q is the category of modules either in C π p G q or in D π p G q . The morphisms are those G -lineargroup homomorphisms which are continuous and strict , in the sense that the quotient topologyon the image of a map agrees with the subspace topology induced from the codomain. This isnot an abelian category as it lacks a direct sum, but satisfies all the other axioms of an abeliancategory. Such a category is called an exact category . The injectives and projectives are preciselythe injectives and projectives in D π p G q and C π p G q respectively.When π is the set of all primes we omit π from the notation. We will similarly omit G when G isthe trivial group. We remark that our usage of the symbol D p G q disagrees with the usage in [SW00],where all discrete modules are allowed. Our restriction to torsion modules does therefore restrict thedomain of definition of the cohomology theory. However allowing non-torsion modules does not seem tobe particularly useful and does cause certain issues, so we will feel no qualms about ignoring them.Since the category P π p G q is not quite an abelian category, and does not have enough projectivesor injectives, one must consider a restricted notion of functor. Once this is done, the machinery ofhomological algebra functions in a satisfactory way. Effectively one reduces all proofs to deal first withthe case when the argument lies in D π p G q (or C π p G q , depending on context) and take limits to deal with C π p G q (respectively D π p G q ). Definition 1.2.
A covariant additive functor T : P π p G q Ñ P π is called continuous if • T p M q P C π and T p A q P D π for M P C π p G q , N P D π p G q • the restriction of T to D π p G q commutes with direct limits • the restriction of T to C π p G q commutes with inverse limitsDually, one has Definition 1.3.
A contravariant additive functor T : P π p G q Ñ P π is called co-continuous if • T p M q P D π and T p A q P C π for M P C π p G q , N P D π p G q • the restriction of T to D π p G q commutes with direct limits in the sense that T | D π p G q ˝ lim ÝÑ “ lim
ÐÝ ˝ T | D π p G q • the restriction of T to C π p G q commutes with inverse limits in the sense that T | C π p G q ˝ lim ÐÝ “ lim
ÝÑ ˝ T | C π p G q One defines notions of connected sequences of continuous functors, and continuous cohomologicaland homological functors in the natural way. A continuous cohomological functor T ‚ is co-effaceable if T i p J q “ i ą J P D π p G q . Similarly for effaceability. Proposition 1.4 (Proposition 3.6.1 of [SW00]) . Let S , T : P π p G q Ñ P π be continuous functors and let f : S | F π p G q Ñ T | F π p G q be a natural transformation. Then there is a unique extension of f to a naturaltransformation f : S Ñ T . If f is a natural isomorphism, so is f . Proposition 1.5 (Corollary 3.6.3 of [SW00]) . Let T ‚ be a continuous co-effaceable cohomological P π p G q -functor, and U ‚ a non-negative connected sequence of continuous functors. Then every natural transfor-mation f : T Ñ U lifts uniquely to a mapping of sequences f ‚ : T ‚ Ñ U ‚ . If U ‚ is also a co-effaceablecohomological functor and f is a natural isomorphism, then all the f ‚ are natural isomorphisms. • For M P C π p G q K and N P C π p G q one may form the completed tensor product M p b G N P C π . For thetheory of completed tensor products see [RZ00b], Section 5.5. This functor commutes with inverselimits in both variables. • If M P C π p G q K is a finitely generated module (over Z π rr G ss ) and N P F π p G q the tensor product M p b G N is finite and agrees with the usual tensor product. Therefore defining M p bbb G A for A P D π p G q by endowing the usual tensor product with the discrete topology yields a continuous functor M p bbb G ´ from P π p G q to P π . • The discrete tensor product functors on D π p G q are defined using their classical definitions. • For M P C π p G q the complete tensor product with diagonal action gives a functor M p b ´ from C π p G q to itself. Similarly for D π p G q .We endow the various Hom-groups with the compact-open topology. • For M P C π p G q , A P D π p G q the group Hom p M, A q is a discrete π -primary torsion module and maybe equipped with the diagonal G -action. So Hom p´ , ´q is a functor C π p G q ˆ D π p G q Ñ D π p G q which is contravariant-covariant. It commutes with inverse limits in the first variable and directlimits in the second. • If M P C π p G q is finitely generated (over Z π rr G ss ) and N P C π p G q then Hom G p M, N q is a compactmodule in C π p G q , and so we have a functor Hom G p M, ´q from C π p G q to itself. This commuteswith inverse limits in the second variable. Taken together with the previous point Hom G p M, ´q isnow a continuous functor from P π p G q to P π . • If J P D π p G q is co-finitely-generated in the sense that it is a direct limit of finite modules with auniform bound on the size of a generating set then Hom G p M, J q is finite for any finite module M . Inthis case Hom G p A, J q lies in C π for A P D π p G q and Hom G p´ , J q is a co-continuous contravariantfunctor from P π p G q to P π . This notion is the Pontrjagin dual (see below) of the previous item. • These last two points have analogous functors
Hom p´ , ´q defined when one of the arguments is(co-)finitely generated as a module over the trivial group. Remark.
The most correct notation for the tensor product of a compact right G -module M and a compactleft G -module N would be M p b Z π rr G ss N . This is rather cumbersome so, as above, we will abbreviatevarious notations: M p b Z π rr G ss N “ M p b G N, Hom Z π rr G ss p M, A q “
Hom G p M, A q and so on where M, N P C π p G q and A P D π p G q . Furthermore when the group involved is the trivialgroup it will be omitted, so that for example Hom p A, B q without qualification would mean Hom Z π p A, B q .These contractions will be extended in the obvious way to the derived Tor and Ext functors.4 roposition 1.6 (‘Pontrjagin duality’, see Proposition 2.2.1 of [SW00] or Section 5.1 of [RZ00b]) . Let I π “ Q π { Z π P D π . Then the operation ˚ from P π p G q to itself given by M ˚ “ Hom p M, I π q with G -action given by p g ¨ f qp m q “ f p g ´ m q is a contravariant exact additive functor. The composition ˚ ˝ ˚ is naturally isomorphic to the identity functor. Furthermore there is an isomorphism of topological Z π -modules Hom p A, B q –
Hom p B ˚ , A ˚ q for A, B P P π p G q . Proposition 1.7 (Corollary 2.2.2 of [SW00]) . If M P C p is p -torsion-free then it is a free abelian pro- p group and hence free and projective in C p . With the above functors are associated derived functors, defined in the usual way.
Definition 1.8.
For M P C π p G q K one has the left-derived functors of M p b G ´ , viz.Tor G ‚ p M, ´q : C π p G q Ñ C π For M P C π p G q one has the right-derived functors of Hom G p M, ´q , viz.Ext ‚ G p M, ´q : D π p G q Ñ D π One result that we note for future reference is that Pontrjagin duality induces isomorphismsExt ‚ G p M, N ˚ q – Tor G ‚ p M K , N q ˚ for M P C π p G q , where M K is the canonical right G -module associated to M . This follows from theidentification Hom G p M, N ˚ q – p M K p b G N q ˚ , f ÞÑ p m b n ÞÑ f p m qp n qq (where f P Hom G p M, N ˚ q , m P M , n P N ) and the exactness of the functor p´q ˚ . The reader is left tocheck that this morphism is actually well defined with regard to the specified G -actions.Since the categorical properties of the module categories involved are very similar to the classicalcase, these functors have properties closely analogous to those for modules over discrete groups. See[Ser13, SW00] and [RZ00b], Chapter 6. An important property of these functors, which we shall useconstantly and without explicit citation, is the following. Proposition 1.9 (Corollaries 6.1.8 and 6.1.10 of [RZ00b]) . Tor G ‚ p M, ´q commutes with inverse limitsand Ext ‚ G p M, ´q commutes with direct limits. Definition 1.10.
A module M P C π p G q is of type FP n if it has a projective resolution P ‚ Ñ M with P k finitely generated for k ď n , where n may be an integer or infinity.Given a module M of type FP and a projective resolution as above, we have continuous P π p G q -functors Hom G p P n , ´q for each n and can therefore define continuous right derived functors of the functor Hom G p M, ´q to be Ext ‚ G p M, N q “ H n p Hom p P ‚ , N qq for N P P π p G q . Similarly one may define Tor G ‚ p M, ´q for a right module of type FP .These functors also satisfy Pontrjagin duality. See Sections 3.7 and 4.2 of [SW00] for more details.Finally one defines homology and cohomology groups of G in the usual way.5 efinition 1.11. For N P C π p G q and A P D π p G q define H ‚ p G, N q “
Tor G ‚ p Z π , C q , H n p G, A q “
Ext ‚ G p Z π , A q If G is of type π -FP —that is, the G -module Z π is of type FP —then we may define H ‚ p G, N q “
Tor G ‚ p Z π , N q , H n p G, A q “
Ext ‚ G p Z π , N q for N P P π p G q . The above definitions have been made with respect to a choice of some set of primes π . One could askwhether these definitions are invariant under enlarging the set of primes. For instance for A P D π p G q one could ask whether the cohomology groups H ‚ p G, A q depend on whether A is considered as an objectof D π p G q or of D p G q . As one may imagine from the fact that we have not troubled to include π in thenotation, all the relevant notions are canonically isomorphic for different choices of π in a way we shallnow describe.All finite G -modules, being a fortiori finite abelian groups, have a unique p -Sylow subgroup for each p . So the module M splits as a direct sum over p P π of modules in F p p G q in a canonical way. Thedirect sum of those components for p P π is called the π -primary component M p π q . By taking limits allmodules in C p G q have a similar canonical decomposition as a direct product of p -primary components,and all modules in D p G q have a direct sum decomposition. For instance we have I π “ Q π { Z π “ à p P π Q p { Z p Not only G -modules but also the maps between them split into p -primary components. This is aneasy consequence of the fact that no non-trivial maps exist between a finite p -group and a finite q -groupfor distinct primes p and q .One consequence of this that will be buried in the notation is that Pontrjagin duality is independentof π —that is, for M P P π p G q we have M ˚ “ Hom Z π p M, Q π { Z π q “ Hom p Z p M, Q { Z q If π Ď π are non-empty sets of primes then all of this ultimately boils down to a statement that ‘taking π -primary components is an exact functor from P π p G q to P π p G q taking projectives to projectives andinjectives to injectives’. One may easily check that this exact functor behaves well with respect to tensorproducts and Hom-functors. Therefore the different possible definitions of, for instance, H ‚ p G ; A q for A P D π p G q are canonically isomorphic. In the same way take the π -primary component any of the exactsequences we will describe will yield an analogous exact sequence. Hence every definition or theoremstatement we make will be true independent of which theory we will choose to develop.We also remark that the notion of ‘finitely generated projective’ is invariant under change of π in acertain sense. We state these as formal propositions as, unlike most of the remarks in this section, wewill refer to it later. Proposition 1.12.
Let π Ď π be non-empty sets of primes. • Let P be a projective in C π p G q . Let Q “ P p π q be the π -primary component of P . Then Q isprojective, both in C π p G q and C π p G q . • If M P C π p G q is finitely generated (over Z π rr G ss ) then M p π q is finitely generated (both over Z π rr G ss and over Z π rr G ss ). roof. By an inverse limit starting from finite modules one finds that for any M P C π p G q we have M “ M p π q ‘ M p π r π q From this both the first statement and finite generation over Z π rr G ss rapidly follow. For finite generationover Z π rr G ss note that if φ : Z π rr G ss ‘ n Ñ M p π q is a surjection then φ vanishes on the p π r π q -primarycomponent of Z π rr G ss ‘ n , so the induced map on π -primary components φ π : Z π rr G ss ‘ n Ñ M p π q is alsoa surjection. Proposition 1.13.
Let M P C π p G q . Suppose that for all p P π the p -primary component M p p q isprojective in C p p G q . Then M is projective in C π p G q .Proof. Let φ : S ։ T be a surjection of modules in C π p G q and let f : M Ñ T . Then each of φ and f splitas the direct product of their p -primary components, and each φ p p q : S p p q Ñ T p p q is a surjection. Thenby assumption each f p p q lifts to a map ˜ f p p q : M p p q Ñ S p p q . The direct product of all these lifts is a liftof f to a map M Ñ S as required. We be frequently manipulating modules and subgroups, and we list here several identities of use and giveindications of the proofs. We will use these often, and sometimes without explicit reference. The readeris warned that we will not always describe the G -actions on modules which appear later in the paper ifthe module is described in this section, as to do so would rather clutter the paper.In cases where modules are equipped with extra finiteness properties yielding continuous (‘bold’) P π p G q -functors as in Section 1.1, one may take limits of the identities in this section to establish similaridentities for the continuous P π p G q -functors. In all cases, the identites in this section can largely bederived from identities for finite modules via limiting process.Throughout this section let G be a profinite group and S a closed subgroup of G . Definition 1.14.
Let M P C π p S q and A P D π p S q . The induced and coinduced G -modules are the modulesind SG p M q “ Z π rr G ss p b S M P C π p G q , coind SG p A q “ Hom S p Z π rr G ss , A q P D π p G q where the G -actions are given by g ¨ p x b m q “ p gx q b m, p g ¨ f qp x q “ f p xg q for g, x P G , m P M , f P Hom S p Z π rr G ss , A q .In the case where U is open in G so that Z π rr G ss is finitely generated as a U -module, one may alsodefine ind UG p A q “ Z π rr G ss p bbb U A P D π p G q , coind UG p M q “ Hom U p Z π rr G ss , M q P C π p G q for M P C π p U q and A P D π p U q . In this case ind and coind are continuous functors P π p U q Ñ P π p G q .There is an alternative formulation of these modules, in the case when M and A are in fact G -modules,with the S -module structure obtained by restriction. Proposition 1.15.
Let M P C π p G q and A P D π p G q . Then there are isomorphisms of G -modules ind SG p M q – Z π rr G { S ss p b M, coind SG p A q – Hom p Z π rr G { S ss , A q where the modules on the right hand side have the diagonal G -actions, given by g ¨ p x b m q “ p gx q b p gm q , p g ¨ f qp x q “ gf p g ´ x q for g, x P G , m P M , f P Hom S p Z π rr G ss , A q . roof. The isomorphisms are given by the maps x b m Ñ p xS q b p xm q , f ÞÑ ` xS ÞÑ xf p x ´ q ˘ where x P G , m P M , f P Hom S p Z π rr G ss , A q . The reader is left to verify that these are well-definedisomorphisms. Proposition 1.16.
Let G be a profinite group and let U be an open subgroup of G . Then for any M P C π p U q there is a natural isomorphism of G -modules ind UG p M q “ coind UG p M q “ Hom U p Z π rr G ss , M q Proof.
Given a choice of section σ : G { U Ñ G we may define a natural isomorphism of G -modulesHom U p Z π rr G ss , M q ÝÑ Z π rr G ss p b U M, f ÞÑ ÿ x P G { U σ p x q b f p σ p x q ´ q where M P F π p U q . This is independent of the choice of σ . Now taking an inverse limit (noting that Z π rr G ss is finitely generated over Z π rr U ss ) gives the result for all M P C π p U q . Proposition 1.17.
Let
M, N P C π p G q and A P D π p G q . Then there is an isomorphism of G -modules Hom p M p b N, A q –
Hom p M, Hom p N, A qq where all Hom and tensor modules inherit the diagonal G -action from their constituent modules. Inparticular, taking the submodules of G -invariants, there are isomorphisms of p Z -modules Hom G p M p b N, A q –
Hom G p M, Hom p N, A qq In particular, using the formulations of induced and coinduced modules from Proposition 1.15 we havethe following.
Corollary 1.18.
Let M P C π p G q and A P D π p G q . There are natural isomorphisms Hom G p ind SG p M q , A q – Hom G p M, coind SG p A qq Proposition 1.19.
Let M P C π p S q and C P D π p G q . There is a natural identification of G -modules coind SG p Hom p M, C qq –
Hom p ind SG p M q , C q where Hom -modules have diagonal actions.Proof.
The isomorphism is given by the mapΦ : Hom S p Z π rr G ss , Hom p M, C qq Ñ
Hom p Z π rr G ss p b S M, C q , Φ p f qp x b m q “ xf p x ´ qp m q for f P Hom S p Z π rr G ss , Hom p M, C qq , x P G and m P M . One may readily verify that this is well-definedand G -linear. Proposition 1.20.
Let P P C π p G q and A P D π p S q . Then there is a natural isomorphism of p Z -modules Hom S p P, A q –
Hom G p P, coind SG p A qq Proof.
The isomorphism is given by the map h ÞÑ p p ÞÑ p x ÞÑ h p xp qqq where h P Hom S p P, A q , p P P and x P G . 8inally there are several statements about derived functors arising from the above identites, whichtogether are known by the collective name of ‘the Shapiro lemma’ or ‘Shapiro-Eckmann identities’. Proposition 1.21. [‘(Absolute) Shapiro Lemma’] Let M P C π p S q and A P D π p S q . There are naturalisomorphisms H ‚ p G, ind SG p M qq – H ‚ p S, M q , H ‚ p G, coind SG p A qq – H ‚ p S, A q Proof.
Take a projective resolution P ‚ of Z π by right G -modules. By Proposition 1.25 this is also aprojective resolution in C p S q by restriction. The identity P ‚ p b G Z π rr G ss p b S M “ P ‚ p b S M gives the first part of the proposition upon passing to homology.For the cohomology identity take a projective resolution P ‚ of Z π by left G -modules. ApplyingProposition 1.20 and passing to cohomology gives the result. Remark.
In the case U is open in G and when Z π has type FP as a G -module—and hence as a U -module by Proposition 1.26 below—then all the homology, cohomology functors and the induction andcoinduction functors are continuous functors and one may take limits to find natural isomorphisms H ‚ p G, ind UG p M qq – H ‚ p U, M q , H ‚ p G, coind UG p M qq – H ‚ p U, M q for all M P P π p U q .Let S be a normal subgroup of G and let M P C π p G q , A P D π p G q . Then the cohomology groups of S acquire G -actions as follows. Let P ‚ be a projective resolution of Z π in C π p G q . For g P G , p P P r and f P Hom S p P r , A q define p g ‹ f qp p q “ gf p g ´ p q One may verify that this is well defined and so gives a left action of G on H r p S, res GS p A qq . Fur-thermore if we identify coind SG p A q with Hom p Z π rr G { S ss , A q by Proposition 1.15 then for g P G and f P Hom G p P r , coind SG p A qq then we may define p g ‹ f qp p qp xS q “ gf p p qp xgS q for p P P r and x P G . This gives a G -action on H r p G, coind SG p A qq . We may define similar constructs inhomology. Let P ‚ be a projective resolution of Z π in C π p G q K . Define a right G -action on P r p b S res GS p M q by p p b m q ‹ g “ p pg q b p g ´ m q for p P P r , m P M and g P G . Identifying ind SG p M q with Z π rr G { S ss p b M there is a right G -action on P ‚ p b G ind SG p M q given by p p b xS b m q ‹ g “ p b xgS b m for p P P , g, x P G and m P M . These constructions give right actions on H r p S, res GS p M qq and H r p G, ind SG p M q . Proposition 1.22.
Suppose that S is a normal subgroup of G and let M P C π p G q , A P D π p G q . Then theShapiro isomorphisms H ‚ p G, ind SG p M qq – H ‚ p S, res GS p M qq , H ‚ p G, coind SG p A qq – H ‚ p S, res GS p A qq are isomorphisms of G -modules where all modules are endowed with the natural G -actions as definedabove. G -actions match up. These constructions are unsurprisingly Pontrjagin dual to one another. In particularwe note the special case that H p , res G p M qq – M K , H p , res G p A qq – A are isomorphisms of G -modules. Proposition 1.23.
Let M P C π p G q and A P D π p G q . There are natural isomorphisms Tor G ‚ p Z π rr S z G ss , M q – Tor S ‚ p Z π , M q , Ext ‚ G p Z π rr G { S ss , A q – Ext ‚ S p Z π , A q Proof.
Take an projective resolution P ‚ of M by left G -modules. The identity Z π rr S z G ss p b G P ‚ “ Z π p b S Z π rr G ss p b G P ‚ “ Z π p b S P ‚ gives the required isomorphism on passing to homology.For the second identity take an injective resolution J ‚ of A in D π p G q . By Proposition 1.25 andPontrjagin duality res GS p J ‚ q is still an injective resolution of A in D π p S q by restriction. Using Propositions1.15 and 1.20 and Corollary 1.18 we haveHom G p Z π rr G { S ss , J ‚ q “ Hom G p ind SG p Z π q , J ‚ q – Hom G p Z π , coind SG p J ‚ qq – Hom S p Z π , J ‚ q Now pass to cohomology to get the result. We mention some results about modules of type FP which we shall need. The first is a consequence ofthe generalised Schanuel’s Lemma. See Section VIII.4 of [Bro12]. Proposition 1.24.
Let M be a finitely generated module in C π p G q . Then the following are equivalent: • M has type FP n • there exists a partial projective resolution F n Ñ ¨ ¨ ¨ Ñ F Ñ M with each F i free of finite rank • for every partial projective resolution P k Ñ ¨ ¨ ¨ Ñ P Ñ M for k ă n with each P i finitely generated, ker p P k Ñ P k ´ q is finitely generated. Proposition 1.25 (Proposition 3.3.1 of [SW00]) . Let P P C π p G q be projective and S a closed subgroupof G . Then the restriction res GS p P q is projective in C π p S q . Proposition 1.26.
Let G be a profinite group and U an open subgroup of G . Let M P C π p G q . Then M has type FP n if and only if res GU p M q is of type FP n as a U -module.Proof. Since res GU takes finitely generated modules to finitely generated modules and projective modulesto projective modules, one direction is clear. For the other, assume M “ res GU p M q is of type FP n asa U -module. First note that since M is finitely generated as a U -module, M is a finitely generated G -module. Suppose we have a partial resolution P k Ñ ¨ ¨ ¨ Ñ P Ñ M (0 ď k ă n ) with each P k a finitelygenerated projective G -module. We must show that the kernel ker p P k Ñ P k ´ q is finitely generated over G . But since res GU p P ‚ q is a partial resolution of M by finitely generated U -modules, the kernel is finitelygenerated over U and we are done. Proposition 1.27.
Let G be a profinite group and S a closed subgroup of G . Then ind SG p´q is an exactfunctor taking finitely generated projectives to finitely generated projectives. roof. Since Z π rr G ss is free as an S -module, it is projective and hence Z π rr G ss p b S ´ is an exact functor. Afinitely generated projective is by definition a summand of some finitely generated free module Z π rr S ss ‘ m .Taking induced modules gives a direct summand of the finitely generated free module Z π rr G ss ‘ m , hencea finitely generated projective module in C π p G q . Proposition 1.28.
Let G be a profinite group and S a closed subgroup of G . Let M P C π p S q . If M isfinitely generated as an S -module then ind SG p M q is finitely generated as a G -module.Proof. If F Ñ M is an epimorphism from a finitely generated free S -module then applying ind SG gives anepimorphism from a finitely generated free G -module to ind SG p M q .In the world of discrete groups the converse of this proposition is true, and indeed not difficult. Anyelement of the induced module may be written as a finite sum of basic tensors, hence finite generatingset for the induced module gives a finite generating set consisting only of basic tensors. From this it isnot difficult to derive a finite generating set for M .It is much less obvious whether the converse to Proposition 1.28 is true—indeed it appears to beunknown. Therefore we will coin the following property, while hoping that it turns out to be vacuous. Definition 1.29.
Let G be a profinite group and let π be a set of primes. Then the pair p G, S q hasproperty FIM (with respect to π ) if the following property holds:For every M P C π p S q the induced module ind SG p M q is finitely generated as a G -module if andonly if M is finitely generated as an S -module.We say G has property FIM (with respect to π ) if p G, S q has property FIM with respect to π for all closedsubgroups S of G .We will usually omit the set of primes when clear from the context. Here ‘FIM’ stands for ‘finiteness(properties) of induced modules’. There is however no obstruction for open subgroups. Proposition 1.30.
Let G be a profinite group and U a open subgroup of G . Then p G, U q has propertyFIM with respect to any set of primes.Proof. Since U is open, if ind UG p M q is finitely generated as a G -module then it is also finitely generatedas a U -module. Since M is a direct summand of res GU p ind UG p M qq (see Section 6.11 of [RZ00b]) there is a U -linear retraction to U and we are done.When one only considers one prime p at a time and considers a (virtually) pro- p group one candispense with some of the difficulty of the finiteness conditions for modules. Proposition 1.31 (Proposition 4.2.3 of [SW00]) . Let G be a virtually pro- p group and M P C p p G q . Then M is of type FP n if and only if Ext kG p M, F q is finite for every (simple) F P F p p G q and all k ď n . Proposition 1.32.
Let G be a pro- p group. Then G has property FIM with respect to π “ t p u .Proof. This follows from Proposition 1.31 and the identityHom G p ind SG p M q , F p q “ Hom S p M, F p q noting that the only simple p -primary G (or S ) module is F p (Lemma 7.1.5 of [RZ00b]). Proposition 1.33.
Let G be a profinite group and S a closed subgroup of G . Let M P C π p S q . If M hastype FP as an S -module then ind SG p M q has type FP as a G -module. If p G, S q has property FIM withrespect to π then the converse also holds. roof. The forwards direction follows immediately from Proposition 1.27. For the other direction proceedby induction. Suppose ind SG p M q has type FP as a G -module. Suppose we have proved that M has typeFP n and take a projective resolution P n Ñ ¨ ¨ ¨ Ñ P Ñ M with each P i finitely generated. We must provethat K “ ker p P n Ñ P n ´ q is finitely generated. Applying ind SG gives a partial resolution of ind SG p M q finitely generated in each dimension. Then by Proposition 1.24 the kernel ker p ind SG p P n q Ñ ind SG p P n ´ qq is a finitely generated G -module. By exactness of ind SG p´q this kernel is precisely ind SG p K q so by propertyFIM we are done. Proposition 1.34.
Let M be a finite direct sum of G -modules M , . . . , M r P C π p G q . Then M has typeFP if and only if each M i does.Proof. If all M i are of type FP then taking a direct sum of projective resolutions of the M i , finitelygenerated in each degree, gives the required resolution of M .So suppose M has type FP . Without loss of generality we will prove that M has type FP ; andby combining the other summands we may assume that r “
2. We will prove by induction that both M and M have type FP . Suppose we know that they both have type FP n and let P n Ñ ¨ ¨ ¨ Ñ P Ñ M and Q n Ñ ¨ ¨ ¨ Ñ Q Ñ M be partial resolutions with all P k and Q k finitely generated projectives. Thedirect sum p P k ‘ Q k q ď k ď n is a partial resolution of M finitely generated in each dimension. Hence byProposition 1.24 the kernel P n ‘ Q n Ñ P n ´ ‘ Q n ´ is finitely generated. But this kernel is simplythe sum of the kernels P n Ñ P n ´ and Q n Ñ Q n ´ which are therefore finitely generated and we aredone. Proposition 1.35.
Let Ñ M Ñ M Ñ M Ñ be a short exact sequence in C π p G q . If M has typeFP then M has type FP if and only if M does.Proof. Suppose that M and M have type FP and take projective resolutions P and P of M and M which are finitely generated in each dimension. By the Horseshoe Lemma ([Wei95], Lemma 2.2.8) thereis a projective resolution of M whose objects are P n ‘ P n , hence is finitely generated in each dimension.So M has type FP as required.Suppose that M and M have type FP . Take projective resolutions P ‚ and Q ‚ of M and M whichare finitely generated in each dimension and lift the map M Ñ M to a chain map f ‚ : P ‚ Ñ Q ‚ . Themapping cone C ‚ of f ‚ is a chain complex with modules C n “ P n ´ ‘ Q n and is therefore projective andfinitely generated in each dimension. By the long exact sequence for the mapping cone C ‚ is exact ateach degree n ě M in degree zero. So the mapping cone is the required projectiveresolution. Refer to [Wei95], Section 1.5 for the mapping cone construction. We will now set up our conventions for tensor products of chain complexes. Take two chain complexes P ‚ and Q ‚ , with P r “ r ă r and Q s “ s ă s for some integers r , s , in an abelianmonoidal category with monoidal product b , for instance C π p G q with product p b Z π (and with diagonalactions on the product). Their tensor product P ‚ b Q ‚ will be the double complex R ‚‚ with objects R rs “ P r b Q s and with differentials d hor rs “ d Pr b id : P r b Q s Ñ P r ´ b Q s d ver rs “ p´ q r id b d Qs : P r b Q s Ñ P r b Q s ´ The total complex tot p R ‚‚ q is of course the complex C ‚ with C n “ à r ` s “ n R rs , d Cn “ à r ` s “ n p d hor rs ` d ver rs q p b denotes p b Z π in accordancewith our standard conventions. Proposition 1.36.
Let G be a profinite group and let π be a set of primes. Let P ‚ Ñ Z π and Q ‚ Ñ Z π beprojective resolutions of Z π in C π p G q . Then tot p P ‚ p b Q ‚ q with augmentation P p b Q Ñ Z π p b Z π “ Z π is a projective resolution of Z π .Proof. This follows immediately from (the proof of) Theorem 2.7.2 of [Wei95], noting that ´ p b Z π isexact (indeed it is the identity functor). Definition 1.37.
Let G be a profinite group, and S “ t S x u x P X a family of subgroups of G indexed bya profinite space X . We say that S is continuously indexed by X if whenever U is an open subset of G the set t x P X | S x Ď U u is open in X . An equivalent definition is that the set tp g, x q P G ˆ X | g P S x u is a closed subset of G ˆ X . Proposition 1.38.
Let G be a profinite group and S “ t S x u a family of subgroups continuously indexedby X . Consider the equivalence relation on X ˆ G given by p x, g q „ S p x , g q ðñ x “ x and g ´ g P S x Then the quotient space G { S “ X ˆ G { „ S , equipped with the quotient topology, is a profinite space.Proof. The quotient is of course compact, so we must show that it is Hausdorff and totally disconnected.Note also that the map X ˆ G Ñ X factors through G { S . Take two distinct points rp x , g qs and rp x , g qs in G { S , where we use square brackets to denote an equivalence class under „ S . If x ‰ x then takepreimages of X -clopen sets which separate x and x ; these give clopen sets in G { S separating our twopoints.So suppose x “ x “ x so that g S x X g S x “ H . Then since S x is a closed subgroup of G there isan open normal subgroup W of G such that W g ´ g X W S x “ H . Since S is continuously indexed by X there is a clopen neighbourhood Y of x in X such that S y Ď W S x for all y P Y . The map W ˆ W Ñ G, p w , w q ÞÑ p g w q ´ p g w q being continuous, and mapping p , q to a point outside W , there is an open normal subgroup U of G such that p g u q ´ p g u q is not in W —and hence not in any S y for y P Y —for all u , u P U . Thus theclopen sets Y ˆ g U and Y ˆ g U of X ˆ G are disjoint upon passing to the quotient by „ S , and providedisjoint clopen neighbourhoods of rp x, g qs and rp x, g qs as required. Proposition 1.39.
Let G be a profinite group and let S “ t S x u x P X be a family of subgroups of G continuously indexed over a profinite set X . Let γ : X Ñ G be a continuous function. Let S be thefamily of subgroups S “ ! S γ p x q x | x P X ) Then S is continuously indexed by X and there is homeomorphism G { S – G { S compatible with the left G -actions on these spaces. roof. Consider the homeomorphism X ˆ G Ñ X ˆ G, p x, g q ÞÑ p x, gγ p x qq This takes the closed subset tp g, x q P G ˆ X | g P S x u to the corresponding one for S , thus proving that S is continuously indexed by X . Furthermore thehomeomorphism takes the equivalence relation „ S to „ S , hence induces the claimed homeomorphism ofquotient spaces. It will unfortunately prove necessary for certain applications to consider infinite collections of subgroups,or more generally infinite collections of compact modules. However there is no sensible compact topologyon the abstract direct sum of infinitely many compact G -modules, so one must use a slightly modifiednotion of ‘profinite direct sum of a sheaf of modules’. For the most part this has similar properties tothe abstract direct sum, so the casual reader may choose to omit this section. In this section all moduleswill be compact . Definition 1.40.
Let R be a profinite ring. A sheaf of R -modules consists of a triple p M , µ, X q with thefollowing properties. • M and X are profinite spaces and µ : M Ñ X is a continuous surjection. • Each ‘fibre’ M x “ µ ´ p x q is endowed with the structure of a compact R -module such that themaps R ˆ M Ñ M , p r, m q Ñ r ¨ m M p q “ tp m, n q P M | µ p m q “ µ p n qu Ñ M , p m, n q Ñ m ` n are continuous.A morphism of sheaves p α, ¯ α q : p M , µ, X q Ñ p M , µ , X q consists of continuous maps α : M Ñ M and ¯ α : X Ñ X such that µ α “ ¯ αµ and such that the restriction of α to each fibre is a morphism of R -modules M x Ñ M ¯ α p x q .We often contract ‘the sheaf p M , µ, X q ’ to simply ‘the sheaf M ’. Regarding an R -module as a sheafover the one-point space one may talk of a sheaf morphism from a sheaf to an R -module. Definition 1.41. A profinite direct sum of a sheaf M consists of an R -module Ð X M and a sheafmorphism ω : M Ñ Ð X M (sometimes called the ‘canonical morphism’ such that for any R -module N and any sheaf morphism β : M Ñ N there is a unique morphism of R -modules ˜ β : Ð X M Ñ N suchthat ˜ βω “ β .We will sometimes call this an ‘external direct sum’. Note that as any compact module is an inverselimit of finite modules it is sufficient to verify this universal property for finite N . We may also denotethe profinite direct sum as Ð x P X M x .In [Rib17] this sum is simply denoted with À . We prefer to use a different notation to remind thereader that the notion is in a certain sense ‘more rigid’ than a traditional direct sum. For example anidentification of each fibre M x with some module N x , depending on some choices, need not give a sensibleidentification of the entire direct sum if the N x do not form a sheaf in any natural way such that theidentifications depend continuously on X . A more precise difference from the classical direct sum is thatthere may no longer be a projection to each fibre M x which vanishes on all the other fibres. Slightlyweaker statements, for instance Corollary A.5, must be used instead.14ith certain exceptions such as this, the properties of the profinite direct sum are fairly intuitive andmirror those of the abstract direct sum in those respects which will be useful to us. Therefore we willleave the statements of properties of the direct sum that we will use, and their proofs, to Appendix A. Definition 2.1.
Let G be a profinite group and let S “ t S x u x P X be a family of (closed) subgroups of G continuously indexed by a non-empty profinite space X . We will often abbreviate this to ‘let p G, S q bea profinite group pair’. Recall from Section 1.6 that there is a natural topology on the set G { S “ ğ x P X G { S x making it into a profinite space such that the natural map X ˆ G Ñ G { S is continuous. Here the symbol Ů denotes ‘disjoint union of sets’. Let Z π rr G { S ss be the free Z π -module on G { S , viewed as a Z π rr G ss -module via the natural G -action on G { S . Note that by Proposition A.14 another way of expressing Z π rr G { S ss is as the profinite internal direct sum Z π rr G { S ss “ ð x P X Z π rr G { S x ss Now consider the augmentation map Z π rr G { S ss Ñ Z π . The kernel, an object of C π p G q , will be denoted∆, or ∆ G, S when more precision is desired. This will be the crucial actor in the development of the theory.Note that it is topologically generated by the elements gS x ´ S y for g P G and x, y P X , as may be readilyseen by an inverse limit argument from the finite case. For modules A P D π p G q , M P C π p G q define H k p G, S ; M q “ H k ´ p G ; ∆ p b M q , H k p G, S ; A q “ H k ´ p G ; Hom p ∆ , A qq Note the dimension shift. Here ∆ p b M and Hom p ∆ , A q are equipped with diagonal actions, i.e. g ¨ p δ b m q “ p gδ q b p gm q , p g ¨ f qp δ q “ gf p g ´ δ q where g P G, m P M, δ P ∆ , f P Hom p ∆ , A q .There is another characterisation of these functors, for which we require the following lemma. Lemma 2.2. ∆ is projective as a Z π -module.Proof. Consider the p -primary component of ∆—this is the kernel of the augmentation map Z p rr G { S ss Ñ Z p . This is a submodule of the free module Z p rr G { S ss , hence is p -torsion-free and hence Z p -free byProposition 1.7. So each p -primary component of ∆ is a projective Z p -module; hence ∆ is a projective Z π -module by Proposition 1.13. Proposition 2.3.
There are natural isomorphisms of functors
Tor G ‚ p ∆ K , M q – H ‚` p G, S ; M q – Tor G ‚ p M K , ∆ q , Ext ‚ G p ∆ , M q – H ‚` p G, S ; M q where ∆ K denotes ∆ with the canonical right G -action δ ¨ g “ g ´ δ .Proof. We will prove the statement for cohomology. By the previous lemma, ∆ is projective over Z π .Thus the functor Hom p ∆ , ´q is an exact functor on Z π -modules, and hence also on Z π rr G ss -modules.15herefore the right derived functors of Hom G p Z π , Hom p ∆ , ´qq are precisely Ext ‚ G p Z π , Hom p ∆ , ´qq . Butsince there is an isomorphism Hom G p ∆ , ´q – Hom G p Z π , Hom p ∆ , ´qq these are also the right derived functors of Hom G p ∆ , ´q . So we have the required isomorphismsExt ‚ G p ∆ , ´q – Ext ‚ G p Z π , Hom p ∆ , ´qq “ : H ‚` p G, S ; ´q The homology statement is similar, or may be deduced using Pontrjagin duality.If ∆ happens to be a module of type FP then we may extend these definitions to P π p G q -functors H ‚` p G, S ; M q “ Tor G ‚ p ∆ K , M q , H ‚` p G, S ; M q “ Ext ‚ G p ∆ , M q Judicious use of Propositions 1.4 and Proposition 1.5 allows us to extend the various propositions we willprove to the case of these ‘bold’ functors when all required modules have type FP . Proposition 2.4.
Let p G, S q be a profinite group pair. There is a natural long exact sequence ¨ ¨ ¨ Ñ H k p G, A q Ñ H k p S , A q Ñ H k ` p G, S ; A q Ñ H k ` p G, A q Ñ ¨ ¨ ¨ (2.1) where A P D π p G q . Similarly in homology.Remark. For brevity we have defined H k p S ; M q “ H k p G, Z π rr G { S ss p b M q , H k p S ; A q “ H k p G, Hom p Z π rr G { S ss , A qq When S is a finite collection these are simply the direct sums of the homology and cohomology of the S i .Note that due to the dimension shift inherent in the definition of relative cohomology, the ‘connectinghomomorphisms’ in this sequence are actually the maps from H ‚ p G, S q to H ‚ p G q . Proof.
Let A P D π p G q . The short exact sequence0 ∆ Z π rr G { S ss Z π Z π -modules, hence remains exact when the functor Hom p´ , A q is applied.So we have a short exact sequence of G -modules0 Hom p ∆ , A q Hom p Z π rr G { S ss , A q A H ‚ p G, ´q gives the familiar long exact sequence for relative homology. Remark.
Via Proposition 2.3—and a similar one for the module Z π rr G { S ss —one may also see this as thelong exact sequence deriving from the application of the functor Ext ‚ G p´ , A q to the short exact sequence0 ∆ Z π rr G { S ss Z π Definition 2.5.
Let G be a profinite group and let S “ t S x u x P X be a family of (closed) subgroups of G continuously indexed by a non-empty profinite space X and let H be a profinite group with a family T “ t T y u y P Y of closed subgroups, continuously indexed over a profinite set Y . A map of profinite grouppairs p φ, f q : p H, T q Ñ p G, S q consists of a group homomorphism φ : H Ñ G and a continuous function f : Y Ñ X such that φ p T x q Ď S f p y q for each y P Y . 16 roposition 2.6. The relative (co)homology functors, and the long exact sequence (2.1) , are naturalwith respect to maps of group pairs p φ, f q : p H, T q Ñ p G, S q .Proof. One sees immediately from the definitions that there is a commuting diagram of H -modules0 ∆ H, T Z π rr H { T ss Z π
00 ∆ G, S Z π rr G { S ss Z π H ‚ p G, S ; A q Ñ H ‚ p H, T ; A q when A P D π p G q is regarded as an H -module via φ . Indeed, we obtain maps of the entire long exactsequence (2.1). Similarly for homology.This is of course not the only ‘relative cohomology sequence’. We mention in particular that there isa long exact sequence which allows one to carry out certain inductions when S is a finite family. Proposition 2.7.
Suppose that G is a profinite group and S i is a family of subgroups of G continuouslyindexed over a non-empty profinite set X i for i “ , . Let S “ S \ S be the natural family of subgroupsof G continuously indexed over the disjoint union of X and X . Then there is a natural long exactsequence ¨ ¨ ¨ Ñ H k p G, S ; A q Ñ H k p S ; A q Ñ H k ` p G, S ; A q Ñ H k ` p G, S ; A q Ñ ¨ ¨ ¨ A P D π p G q . Similarly in homology.Proof. Apply the functor Ext ‚ G p´ , A q to the top row of the commuting diagram with exact rows andcolumns ∆ G, S ∆ G, S Z π rr G { S ss Z π rr G { S ss Z π rr G { S ss Z π rr G { S ss Z π Z π (2.2)This exact sequence also of course has the expected naturality properties. Proposition 2.8.
Pontrjagin duality holds for relative cohomology. That is, for M P C p G q there arenatural isomorphisms H n p G, S ; M q ˚ – H n p G, S ; M ˚ q Proof.
For M P C p G q we have H n p G, S ; M q ˚ “ H n ´ p G, ∆ p b M q ˚ “ H n ´ p G, p ∆ p b M q ˚ q“ H n ´ p G, Hom p ∆ , M ˚ qq“ H n p G, S ; M ˚ q where the manipulation from the second to the third lines is an application of Proposition 1.17.17e also remark that the relative cohomology is ‘invariant up to conjugacy of S ’ in the followingmanner. Proposition 2.9.
Let G be a profinite group and let S “ t S x u x P X be a family of subgroups of G contin-uously indexed over a profinite set X . Let γ : X Ñ G be a continuous function. Let S be the family ofsubgroups S γ “ ! S γ p x q x | x P X ) Then S γ is continuously indexed by X and there is a commuting diagram of isomorphisms G, S Z π rr G { S ss Z π
00 ∆ G, S γ Z π rr G { S γ ss Z π – – id and hence isomorphisms of functors H ˚ p G, S ; ´q – H ˚ p G, S γ ; ´q , H ˚ p G, S ; ´q – H ˚ p G, S γ ; ´q Proof.
This follows immediately from Proposition 1.39.
Let G be a profinite group and let S be a family of subgroups continuously indexed by the profinitespace X . Let H be a closed subgroup of G . Fix a continuous section σ : H z G Ñ G of the quotient map G Ñ H z G . Such a section exists by Proposition 2.2.2 of [RZ00b]. There is a family of subgroups S Hσ “ H X σ p y q S x σ p y q ´ | x P X, y P H z G { S x ( indexed over the profinite set H z G { S “ ğ x P X H z G { S x This indexing set has a natural topology making it a profinite space such that the natural quotient G { S Ñ H z G { S is continuous. Here we have abused notation by writing σ for the section H z G { S Ñ G { S induced by σ : H z G Ñ G .Before we proceed further we must check that S Hσ is continuously indexed by H z G { S . This is trueprovided that the subset p HgS x , h q P H z G { S ˆ H | h P H X σ p y q S x σ p y q ´ ( is a closed subset of the constant sheaf H z G { S ˆ H . This subset is the preimage of the closed subset t S x | x P X u Ď G { S under the continuous map H z G { S ˆ H Ñ G { S , p HgS x , h q ÞÑ σ p HgS x q ´ hσ p HgS x q S x hence is closed.If σ : H z G Ñ G is another section then we have a continuous function γ p Hg q “ σ p Hg q σ p Hg q ´ from H z G to H . Furthermore for every y P H z G we have H X σ p y q S x σ p y q ´ “ ` H X σ p y q S x σ p y q ´ ˘ γ p y q Hence by Proposition 2.9 the module ∆ H, S Hσ does not depend on σ (up to canonical isomorphism). Wewill usually therefore drop σ from the notation.So we may consider the relative (co)homology of H relative to S H . To compare this to the relative(co)homology of G we note the following important result.18 roposition 2.10. Let G be a profinite group, S a family of subgroups continuously indexed by theprofinite space X . Let H be a closed subgroup of G . Then for any continuous section σ : H z G Ñ G wehave a canonical isomorphism of H -modules ∆ H, S Hσ – res GH p ∆ G, S q Proof.
Consider the continuous map H { S Hσ Ñ G { S , h p H X σ p y q S x σ p y q ´ q ÞÑ hσ p y q S x which is easily checked to be a bijection, hence a homeomorphism as both spaces are profinite. It is alsocompatible with the left H -action, so there is an isomorphism of H -modules Z π rr H { S Hσ ss – Z π rr G { S ss This isomorphism commutes with the augmentation maps to Z π so the kernels of these maps are isomor-phic H -modules. This is precisely the statement of the proposition. Corollary 2.11. If U is an open subgroup of G then ∆ G, S is of type FP as a G -module if and only if ∆ U, S U is of type FP as a U -module.Proof. This now follows immediately from Proposition 1.26.As a result of Proposition 2.10, we obtain a ‘relative Shapiro lemma’.
Proposition 2.12 (‘Relative Shapiro Lemma’) . Let p G, S q be a group pair and H a closed subgroup of G . Then for any M P C π p H q and any A P D π p H q there are natural isomorphisms H ‚ p H, S H ; M q – H ‚ p G, S ; ind HG p M qq , H ‚ p H, S H ; A q – H ‚ p G, S ; coind HG p A qq Proof.
Let ∆ “ ∆ G, S , so that ∆ H, S H “ res GH p ∆ q . There is an isomorphism of left G -modules Z π rr G ss p b H p res GH p ∆ q p b M q – ∆ p b p Z π rr G ss p b H M q g b δ b m ÞÑ p gδ q b g b m Here, as in Section 1.3, the tensor products over Z π are given the diagonal action and the tensor productsover H have the left G -action given by the left action on Z π rr G ss . From this isomorphism and theusual Shapiro isomorphisms (Proposition 1.21) we have the first isomorphism from the statement of theproposition. The second follows from the isomorphismHom H p Z π rr G ss , Hom p res GH p ∆ q , A qq – Hom p ∆ , Hom H p Z π rr G ss , A qq f ÞÑ ` δ ÞÑ p g ÞÑ f p g qp gδ qq ˘ and the absolute Shapiro lemma. Remark.
Suppose U is open in G and ∆ G, S is of type FP as a G -module. Then as for the absoluteShapiro lemma one may take limits to find natural isomorphisms H ‚ p G, S ; ind UG p M qq – H ‚ p U, S U ; M q , H ‚ p G, S ; coind UG p M qq – H ‚ p U, S U ; M q for all M P P π p U q .We may now define restriction and corestriction maps.19 efinition 2.13. Let p G, S q be a profinite group pair and H a closed subgroup of G . For M P C π p G q the restriction map on homology is given by the mapres HG : H ‚ p H, S H ; M q – H ‚ p G, S ; ind HG p M qq Ñ H ‚ p G, S ; M q corresponding to the map Z π rr G ss p b H M Ñ M, g b m ÞÑ gm For A P D π p G q the restriction map on cohomology is the mapres GH : H ‚ p G, S ; A q Ñ H ‚ p G, S ; coind HG p A qq – H ‚ p H, S H ; A q induced by the module map A Ñ Hom H p Z π rr G ss , A q , a ÞÑ p g ÞÑ ga q One may verify that these agree with the functorial maps arising from the obvious map of group pairs p H, S H q Ñ p G, S q . Definition 2.14.
Let p G, S q be a profinite group pair and U a open subgroup of G . For M P C π p G q the corestriction map on homology is given by the mapcor GU : H ‚ p G, S ; M q Ñ H ‚ p G, S ; ind UG p M qq – H ‚ p U, S U ; M q corresponding to the map M Ñ Z π rr G ss p b U M, m ÞÑ ÿ i g i b p g ´ i m q where t g i u is a complete set of coset representatives of G { U . The map above does not depend on thechoice of these representatives.For A P D π p G q the corestriction map on cohomology is the mapcor UG : H ‚ p U, S U ; A q – H ‚ p G, S ; coind UG p A qq Ñ H ‚ p G, S ; A q induced by the module map Hom U p Z π rr G ss , A q Ñ A, ´ f ÞÑ ÿ g i f p g ´ i q ¯ where again t g i u is an (irrelevant) choice of coset representatives of G { U .The corestriction map is also often known as a transfer map . The following proposition followsimmediately from the definitions. Proposition 2.15.
For a profinite group pair p G, S q and an open subgroup U of G , the composition cor UG ˝ res GU : H ‚ p G, S ; A q Ñ H ‚ p G, S ; A q is multiplication by r G : U s for any A P D π p G q . Similarly for any M P C π p G q the composition res UG ˝ cor GU : H ‚ p G, S ; M q Ñ H ‚ p G, S ; M q is multiplication by r G : U s . Proposition 2.16.
Let p G, S q be a profinite group pair and let K i be a nested descending sequence ofclosed subgroups of G . For each i let A i P D π p K i q and let A i Ñ A i ` be a sequence of maps compatiblewith the inclusions K i ` Ñ K i . For each i let M i P C π p K i q be a K i -module and let M i ` Ñ M i be asequence of maps compatible with the inclusions K i ` Ñ K i . Then if L “ Ş K i we have H ‚ p L, S L ; lim ÐÝ M i q “ lim ÐÝ H ‚ p K i , S K i ; M i q , H ‚ p L, S L ; lim ÝÑ A i q “ lim ÝÑ H ‚ p K i , S K i ; A i q roof. There is a natural isomorphism of G -moduleslim ÝÑ Hom K i p Z π rr G ss , A i q – ÝÑ Hom L p Z π rr G ss , lim ÝÑ A i q and the cohomology statement follows from the relative Shapiro Lemma. The homology statement followsby Pontrjagin duality. Let p be a prime. By analogy with the absolute case one may definehd p p G, S q “ max t n | D M P C p p G q such that H n p G, S ; M q ‰ u cd p p G, S q “ max t n | D A P D p p G q such that H n p G, S ; A q ‰ u These are in fact equal by the Pontrjagin duality of the last section, and we will use them interchangeably.By convention we set the dimension to be zero if the (co)homology vanishes in every dimension for allmodules, and if the defining sets are unbounded.Of course by the definition of relative cohomology we have cd p p G, S q ď cd p p G q `
1. Furthermore fromthe long exact sequence in relative homology we see that if cd p p S x q ă cd p p G q ´ x P X thencd p p G, S q ď cd p p G q . Proposition 2.17.
Let p G, S q be a profinite group pair and H a closed subgroup of G . Then cd p p H, S H q ď cd p p G, S q Proof.
This follows immediately from the relative Shapiro lemma of the last section.The next propositions are simply relative forms of Propositions 21 and 21 of [Ser13], and the proofsare much the same. We will give a sketch here for the convenience of the reader, but the details will beleft out. Proposition 2.18.
For a profinite group pair p G, S q , assume that G is a pro- p group. Then the followingare equivalent:(1) cd p p G, S q ď n (2) H n ` p G, S ; F p q “ Proof.
This follows from a standard d´evissage argument. All non-simple modules in F π p G q are successiveextensions by smaller finite modules, and the only simple module is F p (see Lemma 7.1.5 of [RZ00b]).Now apply the long exact sequence for Ext ‚ G p ∆ , ´q to find H n ` p G, S ; M q “ M P F π p G q . Takinga direct limit gives the result. Proposition 2.19.
The following are equivalent, for a profinite group pair p G, S q .(1) cd p p G, S q ď n (2) H n ` p H, S H ; F p q “ for all closed subgroups H of G (3) H n ` p U, S U ; F p q “ for all open subgroups U of G roof. That (1) implies (2) is an immediate consequence of Proposition 2.18. That (2) implies (3)is trivial. That (3) implies (2) follows from Proposition 2.16 applied to the trivial module F p and adescending sequence of open subgroups of G intersecting in H .Finally assume that (2) holds and let A P D p p G q . Taking H to be a p -Sylow subgroup of G , theprevious proposition implies that cd p p H, S H q ď n . So H n ` p H, S H ; A q “
0. Now write H as theintersection of a decreasing sequence of open subgroups U i . By Proposition 2.15 the compositioncor U i G ˝ res GU i : H n ` p G, S ; A q Ñ H n ` p G, S ; A q is multiplication by r G : U i s , hence is an injective map since r G : U i s is coprime to p . Hence for every i the restriction map H n ` p G, S ; A q Ñ H n ` p U i , S U i ; A q is injective. Hence the restriction map H n ` p G, S ; A q Ñ lim ÝÑ H n ` p U i , S U i ; A q “ H n ` p H, S H ; A q “ Lemma 2.20.
Suppose that the family S is such that at most one subgroup S is non-trivial. Then forevery k ą and every M P C π p G q , A P D π p G q , we have H k p G, S ; M q “ H k p G, t S u ; M q , H k p G, S ; A q “ H k p G, t S u ; A q In particular, cd p p G, S q “ cd p p G, t S uq for all p P π except possibly if one dimension is 1 and the other0.Proof. Note that for all kH k p S ; M q “ Tor Gk p Z π , Z π rr G { S ss p b M q “ ð S x P S Tor Gk p Z π , ind S x G p M qq “ Tor Gk p Z π , ind S G p M qq “ H k p S ; M q for k ě
1. The second equality follows from Proposition A.13. Using the absolute Shapiro Lemma all thegroups Tor Gk p p Z , ind S G p M qq “ Tor Gk p p Z , ind G p M qq “ H k p , M q vanish for S x ‰ S so the third equality follows from Lemma A.10.The result on homology now follows from the 5-Lemma and the map of long exact sequences in relativehomology corresponding to the map of pairs p G, t S uq Ñ p G, S q . The cohomology statement follows byPontrjagin duality. Remark.
The discrepancy in the cohomological dimension statement is just about possible—for examplecd p p , t uq “ p p , t u \ t uq “ We briefly mention that there is a spectral sequence associated to an ‘extension of group pairs’ in asuitable sense. Recall first that the Lyndon-Hochschild-Serre spectral sequence exists in the world ofprofinite groups.
Proposition 2.21 (see Theorem 7.2.4 of [RZ00b]) . Let G be a profinite group and N a closed normalsubgroup of G with G { N “ Q . For A P D π p G q there is a natural convergent first quadrant cohomologicalspectral sequence E rs “ H r p Q, H s p N ; A qq ñ H r ` s p G, A q roposition 2.22. Let p G, S q be a profinite group pair where S is continuously indexed by a profinitespace X and let N be a closed normal subgroup of G contained in S for all S P S . Let Q “ G { N and let T “ t S { N | S P S u be a family of subgroups of Q ; then T is continuously indexed by X . The group N acts trivially on ∆ G, S and ∆ G, S , regarded as a Q -module, is naturally isomorphic to ∆ Q, T .Proof. The statement that T is continuously indexed follows immediately from the definition. For x P G , n P N , S P S the formula n ¨ xS “ xx ´ nxS “ xS shows that N acts trivially on Z π rr G { S ss , hence on ∆ G, S . Finally the natural continuous bijections G { S – p G { N q{p S { N q show that Z π rr G { S ss is naturally isomorphic as a G -module to Z π rr Q { T ss in a waycompatible with the augmentation to Z π . Hence ∆ G, S is naturally isomorphic to ∆ Q, T . Proposition 2.23.
Let p G, S q be a profinite group pair and let N be a closed normal subgroup of G contained in S for all S P S . Let Q “ G { N and let T “ t S { N | S P S u . For A P D π p G q there is anatural convergent first quadrant cohomological spectral sequence E rs “ H r p Q, T ; H s p N ; A qq ñ H r ` s p G, S ; A q Proof.
This follows from the spectral sequence in Proposition 2.21 via the natural isomorphism H s p N ; Hom p ∆ , A qq “ Hom p ∆ , H s p N, A qq where ∆ “ ∆ G, S . This isomorphism holds because ∆, viewed as an N -module, is simply a free Z π -modulewith trivial N -action. Hence the functor Hom p ∆ , ´q is an exact functor withHom N p P, Hom p ∆ , A qq “ Hom p ∆ , Hom N p P, A qq (as G -modules) for every P P C π p N q and the cohomology isomorphisms follow. Remark.
The classical theory of Poincar´e duality pairs in [BE78] makes extensive use of cap products,particularly with respect to a ‘fundamental class’ in the top-dimensional homology. The cap product inits most general form consists of a family of homomorphisms H r ` s p G, C q b H s p G, A q Ñ H r p G, C b A q However here we run into an issue. Without additional conditions on G , group homology is defined onlywith compact coefficients, and cohomology with discrete torsion coefficients. So to make sense of theabove homomorphisms in general one needs to make sense of the tensor product of a compact module C with a discrete module A in such a way that C b A is a compact module. There is no way to do this in thegenerality required, so we must consider cap products to be largely unavailable to us. Therefore we willdevelop a duality theory only making use of cup products—and therefore only using discrete modules.One may note in passing that such a cap product is well defined and behaves as one would expect inthe case when A is finite. This is not really sufficient for our purposes, but does suffice to treat absoluteduality groups. The reader interested in the theory of absolute profinite duality groups developed in thisway is directed to [Ple80a, Ple80b]. 23 .1 Definition and basic properties Let
C, A, B P D π p G q and B P Hom G p C b A, B q . We define a cup product as follows. Let P ‚ Ñ Z π be aprojective resolution of Z π by left G -modules. Then the complex Q ‚ “ tot p P ‚ p b P ‚ q is also a projectiveresolution of Z π . For ζ P Hom G p P r , C q and ξ P Hom G p P s , A q we may define a cup product ζ ! ξ “ ζ ! B ξ : Q r ` s pr ÝÑ P r p b P s ζ b ξ ÝÑ C b A B ÝÑ B at the level of cochains. One may readily check that this descends to a map on cohomology using theformula d p ζ ! ξ q “ dζ ! ξ ` p´ q r ζ ! dξ Standard arguments from homological algebra show that this definition is independent of the chosenresolution P ‚ .We will mostly be utilising the cup product to explore duality notions, and hence require a species ofadjoint cup product. In the classical case one does this by taking cap products with respect to a chosenclass in H n . Here we use the dual notion. Again given C, A, B P D π p G q and B P Hom G p C b A, B q ,choose also a ‘coclass’ e : H n p G, B q Ñ I π . Now define the ‘adjoint cup product’Υ e “ Υ e, B : H k p G, C q Ñ H n ´ k p G, A q ˚ by means of the composition H k p G, C q b H n ´ k p G, A q ! ÝÑ H n p G, B q e ÝÑ I π One basic example of a cup product that we will encounter frequently arises from the pairing B : Hom p ∆ , C q b A Ñ Hom p ∆ , C b A q f b a ÞÑ p δ ÞÑ f p δ q b a q which yields, by definition of relative cohomology, a cup product H r p G, S ; C q b H s p G, A q Ñ H r ` s p G, S ; C b A q The cup product is natural with respect to group homomorphisms in the following sense.
Proposition 3.1.
Let
G, H be profinite groups. Let
C, A, B P D π p G q and B P Hom G p C b A, B q . Let C , A , B P D π p H q and B P Hom H p C b A , B q . Let f : H Ñ G be a group homomorphism, and let c : C Ñ C , a : A Ñ A , b : B Ñ B be continuous group homomorphisms compatible with f in the obviousway, and assume that b ˝ B “ B ˝ p c b a q . Then the diagram H r p G, C q b H s p G, A q H r p H, C q b H s p H, A q H r ` s p G, B q H r ` s p H, B q ! B ! B commutes, so that cup products are natural with respect to group homomorphisms.Proof. One may readily check this on the level of cochains.We will of course be requiring maps not just of single cohomology groups, but of entire exact sequences.Given short exact sequences of modules in D π p G q Ñ C γ ÝÑ C γ ÝÑ C Ñ Ñ A α ÝÑ A α ÝÑ A Ñ B i P Hom G p C i b A i , B q , we say that these pairings are compatible with the maps in theshort exact sequences if the diagrams C b A C b A C b A C b A C b A B C b A B id b α γ b id B id b α γ b id B B B commute. Theorem 3.2.
Consider short exact sequences of modules in D π p G q and compatible pairings as above.Assume also that these short exact sequences split as sequences of Z π -modules (i.e. as abelian groups).Choose a coclass e : H n p G, B q Ñ I π . Then the diagram of long exact sequences H k p G, C q H k p G, C q H k p G, C q H k ` p G, C q H n ´ k p G, A q ˚ H n ´ k p G, A q ˚ H n ´ k p G, A q ˚ H n ´ k ´ p G, A q ˚ Υ e, B Υ e, B Υ e, B Υ e, B p´ q k (3.1) commutes where the lower sequence is the dual of the coefficient sequence for the A i , with an occasionalsign change in the connecting map as shown.Remark. The restriction that the short exact sequences are Z π -split will not hinder the results of the restof the paper, but is necessary here. Proof.
Take a projective resolution P ‚ Ñ Z π and let Q ‚ “ tot p P ‚ p b P ‚ q . Equipped with the augmentationmap P p b P Ñ Z π p b Z π “ Z π we know that Q ‚ is also a projective resolution of Z π by Proposition 1.36.Commutativity of diagrams such asHom p P r , C q b Hom p P s , A q Hom p P r , C q b Hom p P s , A q Hom p Q r ` s , C b A q Hom p Q r ` s , C b A q Hom p P r , C q b Hom p P s , A q Hom p Q r ` s , C b A q Hom p Q r ` s , B q id b α γ b id ! ! id b α γ b id B ! B follows immediately from the definitions. Taking homology gives a commuting diagram H r p G, C q b H s p G, A q H r p G, C q b H s p G, A q H r p G, C q b H s p G, A q H r ` s p G, B q γ b idid b α ! B ! B (3.2)Passing to the adjoint cup product, this diagram guarantees that the first square in (3.1) does indeedcommute. Similarly for the second square.To deal with the square involving the connecting homomorphisms we must show that the diagram H r p G, C q b H s p G, A q H r ` p G, C q b H s p G, A q H r ` s p G, C b A q H r ` s ` p G, C b A q H r p G, C q b H s ` p G, A q H r ` s ` p G, C b A q H r ` s ` p G, B q p´ q r id b δ δ b id ! ! δ δ B ! B δ denote connecting homomorphisms. That the uppermost andleftmost quadrilaterals commute may be deduced very quickly from the definitions of cup products andconnecting maps. The sign in the leftmost quadrilateral arises since the relevant part of the connectinghomomorphism on the right vertical side of the quadrilateral derives via the snake lemma from thedifferentials P r p b P s ` Ñ P r p b P s in the double complex, which are given by p´ q r id b d Ps ` .The lower-right square requires considerably more effort. Viewing the given short exact sequences ofmodules as chain complexes with A and C in degree zero, consider the double complex below. All rowsand columns are exact by the splitness assumption. The signs on the arrows are to remind the reader ofthe sign convention from Section 1.5.0 0 00 C b A C b A C b A C b A C b A C b A C b A C b A C b A
00 0 0 ´ ´´ ´
One may verify by a direct diagram chase that there is a commuting diagram in which both rows areexact0 C b A ‘ C b A C b A ‘ C b A C b A ‘ C b A C b A ‘ C b A ker p d ´ q C b A diag (3.3)where ker p d ´ q is the kernel of the differential map of the total complex, and is mapped to C b A viathe map C b A ‘ C b A Ñ C b A Ñ C b A . There is also a commuting diagram0 C b A ‘ C b A ker p d ´ q C b A C b A ‘ C b A p C b A ‘ C b A ‘ C b A q { im p d q C b A B B – B ` B B ` B ` B where the bottom left square commutes because of the compatibility of the pairings with the maps in theshort exact sequences. Applying H ‚ p G, ´q to the lower two rows gives us a commuting diagram H r ` s p G, C b A q H r ` s ` p G, C b A ‘ C b A q H r ` s p G, q H r ` s ` p G, B q δ B ‘ B (3.4)26inally note that due to the long exact sequences associated to the diagram (3.3) and additivity ofconnecting homomorphisms, the top arrow in diagram (3.4) is simply the difference (because of the signsin the double complex) between the connecting homomorphisms H r ` s p G, C b A q Ñ H r ` s ` p G, C b A q and H r ` s p G, C b A q Ñ H r ` s ` p G, C b A q Since the composition of this difference with the map induced by B ` B vanishes, this gives thecommuting square H r ` s p G, C b A q H r ` s ` p G, C b A q H r ` s ` p G, C b A q H r ` s ` p G, B q δ δ B B that we required.One particular sequence that we will need later is the following. Proposition 3.3.
Let J P D π p G q be such that J ˚ is finitely generated and free as a Z π -module, and let j : H n p G, S ; J q Ñ I π be a coclass. Then for any short exact sequence Ñ C Ñ C Ñ C Ñ in D π p G q the cup product induces a map of long exact sequences H k p G, C q H k p G, C q H k p G, C q H k ` p G, C q H n ´ k p G, C _ q ˚ H n ´ k p G, C _ q ˚ H n ´ k p G, C _ q ˚ H n ´ k ´ p G, C _ q ˚ Υ j, ev1 Υ j, ev2 Υ j, ev3 Υ j, ev1 p´ q k where C _ i “ Hom p C i , J q and ev i : C i b C _ i Ñ J is the evaluation map.Proof. Firstly note that since J ˚ is finitely generated as a Z p -module, Hom p M, J q is finite for any finitemodule M , and hence Hom p´ , J q is a well-defined functor from D p p G q to itself. Since J ˚ is a free Z p -module, this functor is exact. Hence the lower of the long exact sequences is well-defined. The mapof long exact sequences commutes by Theorem 3.2 when the short exact sequence C ‚ is Z p -split. In thissituation we have not just a map of one pair of long exact sequences, but a natural transformation offunctors H ‚ p G, ´q Ñ H n ´‚ p G, p´q _ q ˚ which commutes with connecting homomorphisms for split short exact sequences. Proposition 3.5.2 of[SW00] then guarantees that the diagram of long exact sequences commutes even when the sequence doesnot split. The key example to which we apply the long exact sequence of (adjoint) cup products is the exactsequence for relative cohomology. We will primarily be interested in using cup products in connectionwith Poincar´e duality pairs. As we shall see in Section 5 in this case this will force S to be a finite familyof subgroups, and therefore we will make this simplifying assumption here.Let C, A P D π p G q and consider the short exact sequences0 Ñ C γ ÝÑ Hom p Z π rr G { S ss , C q γ ÝÑ Hom p ∆ , C q Ñ Ñ A α ÝÑ Hom p Z π rr G { S ss , A q α ÝÑ Hom p ∆ , A q Ñ Ñ ∆ ÝÑ Z π rr G { S ss ÝÑ Z π Ñ Z π -modules. Define pairings B : Hom p ∆ , C q b A Ñ Hom p ∆ , C b A q f b a ÞÑ p δ ÞÑ f p δ q b a q B : C b Hom p ∆ , A q Ñ Hom p ∆ , C b A q c b f ÞÑ p δ ÞÑ c b f p δ qq and define B : Hom p Z π rr G { S ss , C q b Hom p Z π rr G { S ss , A q Ñ Hom p ∆ , C b A q to be the mapcoev : Hom p Z π rr G { S ss , C q b Hom p Z π rr G { S ss , A q Ñ Hom p Z π rr G { S ss , C b A q f b g ÞÑ p xS i ÞÑ f p xS i q b g p xS i qq where x P G followed by the mapHom p Z π rr G { S ss , C b A q Ñ Hom p ∆ , C b A q induced by inclusion. Note that the first map is only given by this formula on the ( Z π -)basis vectors xS i of Z π rr G { S ss , where x P G and is extended to the whole module linearly. In particular a generator xS i ´ S y of ∆ is mapped to xS i ´ S y ÞÑ f p xS i q b g p xS i q ´ f p S y q b g p S y q One may now readily check that these pairings are compatible with the maps in the short exact sequence.Hence given a coclass e : H n ´ p G, Hom p ∆ , C b A qq Ñ I π we obtain a map of long exact sequences ¨ ¨ ¨ H k p G, C q H k p G, Hom p Z π rr G { S ss , C qq H k ` p G, S ; C q ¨ ¨ ¨¨ ¨ ¨ H n ´ k p G, S ; A q ˚ H n ´ k ´ p G, Hom p Z π rr G { S ss , A qq ˚ H n ´ k ´ p G, A q ˚ ¨ ¨ ¨ Υ e, B Υ e, B Υ e, B (3.5)Now the middle vertical map is identified with a map à i H k p S i , C q Ñ à i H n ´ k ´ p S i , A q ˚ which the reader is no doubt expecting to correspond with the cup product on each S i . This followsimmediately from the following proposition. When this proposition is used M will be either Z π or ∆ S, T for some family of subgroups T of S . Proposition 3.4.
Let G be a profinite group and S a closed subgroup of G . Let C, A P D π p G q and M P C π p S q . Let coev be the map coev : Hom p ind SG p M q , C q b Hom p Z π rr G { S ss , A q Ñ Hom p ind SG p M q , C b A q f b g ÞÑ p x b m ÞÑ f p x b m q b g p xS qq here x P G and m P M . Let τ r be the Shapiro isomorphism τ r : H r p S, ´q – ÝÑ H r p G, coind SG p´qq and recall from Proposition 1.19 that there is a natural identification of G -modules σ : coind SG p Hom p M, C qq –
Hom p ind SG p M q , C q Then there is a commuting diagram H r p S, Hom p M, C qq b H s p S, A q H r p G, Hom p ind SG p M q , C qq b H s p G, Hom p Z π rr G { S ss , A qq H r ` s p S, Hom p M, C b A qq H r ` s p G, Hom p ind SG p M q , C b A qq p στ r qbp στ s q– ! id C b A ! coev στ r ` s – In particular if B P D π p G q and B is any pairing Hom p ind SG p M q , C q b Hom p Z π rr G { S ss , A q Ñ B which factors as h ˝ coev where h is a module map h : Hom p ind SG p M q , C b A q Ñ B and e : H n p G, B q Ñ I π is a coclass then there is a commuting diagram of adjoint cup products H r p S, Hom p M, C qq H r p G, Hom p ind SG p M q , C q H n ´ r p S, A q ˚ H n ´ r p G, Hom p Z p rr G { S ss , A qq ˚ στ r – Υ B e, id C b A Υ e, B στ n ´ r – where B e is the coclass defined by H n p S, Hom p M, C b A qq στ n ÝÑ H n p G, Hom p ind SG p M q , C b A q e ˝ B ÝÝÝÑ I π Proof.
Let P ‚ Ñ Z π be a projective resolution of Z π by left G -modules. From the proofs in Section 1.3,the Shapiro isomorphism is given by the map induced on cohomology by the chain isomorphism τ r : Hom S i p P r , E q Ñ Hom G p P r , Hom S p Z π rr G ss , E qq given by τ r p f qp p qp x q “ f p xp q p f P Hom S i p P r , C q , p P P r , x P G q where E P D π p S q . Furthermore σ is given by σ p f qp x b m q “ xf p x ´ qp m q for f P Hom S p Z π rr G ss , Hom p M, C qq , x P G and m P M . One may now readily check that the first diagramin the statement commutes at the level of cochains. The second statement follows immediately.Thus we have the familiar chain of adjoint cup products for relative cohomology.29 roposition 3.5. Let
C, A P D π p G q and let e : H n ´ p G, S ; C b A q Ñ I π be a coclass. Then there is acommutative diagram of long exact sequences and adjoint cup product maps ¨ ¨ ¨ H k p G, C q H k p S , C q H k ` p G, S ; C q H k ` p G, C q ¨ ¨ ¨¨ ¨ ¨ H n ´ k p G, S ; A q ˚ H n ´ k ´ p S , A q ˚ H n ´ k ´ p G, A q ˚ H n ´ k ´ p G, S ; A q ˚ ¨ ¨ ¨ Υ e Υ B e Υ e Υ e p´ q k where B e “ B i e : À i H n ´ p S i ; C b A q Ñ I π .Let p φ, f q : p H, T q Ñ p G, S q be a map of group pairs, let c , A P D π p H q and let e H : H n ´ p H, T ; C b A q Ñ I π be a coclass. Assume that the coclass e on H n ´ p G, S q is the composition H n ´ p G, S ; C b A q ÝÑ H n ´ p H, T ; C b A q e H ÝÑ I π Then the commuting diagram above is natural with respect to p φ, f q in the obvious sense. The final naturality statement follows from fact that these sequences are induced from applyingfunctors to coefficient sequences, together with the naturality statements in Propositions 2.6 and 3.1.We also note that putting this result together together with the commuting square (3.2) one acquiresa commuting pentagon for each iH r p S i , C q b H s p G, A q H r p S i , C q b H s p S i , A q H r ` p G, S ; C q b H s p G, A q H r ` s p S i , C b A q H r ` s ` p G, S ; C b A q ! Si ! G (3.6) π trees Let C be an variety of finite groups closed under taking isomorphisms, subgroups, quotients and extensions.Let π p C q be the set of primes which divide the order of some finite groups in C . A pro- C group is a profinitegroup which is an inverse limit of groups in C .The full development of the theory of profinite graphs and trees is well beyond the scope of this paper.The material here is mostly to be found in [ZM89]. The full theory of profinite trees may be found in[Rib17], or distributed around various papers in the literature, mainly by Gildenhuys, Ribes, Zalesskiiand Mel’nikov. The theory for pro- p groups is given in [RZ00a]. We will adopt the following definition. Definition 4.1. An abstract graph T is a set with a distinguished subset V p T q and two retractions d , d : T Ñ V p T q . Elements of V p T q are called vertices, and elements of E p T q “ T r V p T q are callededges. Note that a graph comes with an orientation on each edge.If an abstract graph is in addition a profinite space (that is, an inverse limit of finite discrete topologicalspaces), V p T q and E p T q are closed and d , d are in addition continuous, then T is called a profinite graph .A morphism of profinite graphs T , T is a continuous function f : T Ñ T such that d i f “ f d i for each i . An action of a profinite group on a graph is a continuous action by graph morphisms. For a set ofprimes π , a profinite graph is a pro- π tree if the chain complex0 Ñ Z π rr E p T qss d ´ d ÝÑ Z π rr V p T qss ǫ ÝÑ Z π Ñ ǫ is the augmentation. 30 emark. The most general definition of a profinite graph does not require that E p T q is closed. Ourrestricted definition simplifies the exposition, but does not materially alter the results we will state.Moreover the cases that usually arise in applications have E p T q closed. Therefore we will make thissimplification. To develop the theory with E p T q not closed one must work with pointed profinite spacesand the free modules over them; specifically in various places E p T q must be replaced with the pointedprofinite space p T { V p T q , V p T q{ V p T qq .The theory of profinite graphs of groups can be defined over general profinite graphs; we shall onlyconsider finite graphs here as this considerably simplifies the theory and is sufficient for our needs. Definition 4.2. A finite graph of pro- C groups G “ p Y, G ‚ q consists of a connected finite graph Y , apro- C group G y for each y P Y , and (continuous) monomorphisms B i : G y Ñ G d i p y q for i “ , y P V p Y q . Definition 4.3.
Given a finite graph of pro- C groups p Y, G ‚ q , choose a maximal subtree Y of Y . A pro- C fundamental group of the graph of groups with respect to Y consists of a pro- π group ∆, and amap φ : ž y P Y G y > ž e P E p Y q x t e y Ñ H such that φ p t e q “ e P E p Y q and φ p t ´ e B p g q t e q “ φ pB p g qq for all e P E p Y q , g P G e and with p H, φ q universal with these properties, within the category of pro- C groups. The pro- C group H will be denoted Π p G q or Π p Y, G ‚ q . Here š denotes the free profinite product; see [RZ00b], Chapter 9.The group so defined exists and is independent of the maximal subtree Y (see Section 3 of [ZM89],Section 6.2 of [Rib17]). Note that in the category of discrete groups this universal property is preciselythe same as the classical definition of π G as group with a certain presentation. Also notice that freeproducts are a special case of a graph of groups in which all edge groups are trivial.We use the notation G “ G > L G for a pro- C amalgamated free product, i.e. the fundamental pro- C group of a graph of groups with two vertex groups G and G and one edge group L , which is a commonsubgroup of the two vertex groups. We by convention orient the edge from G to G .If we have a pro- C group G and two subgroups L and L which are isomorphic via an isomorphism τ then we denote by G > L,τ the pro- C HNN extension; that is, the fundamental pro- C group of the graphof groups with vertex group G , edge group L and monomorphisms B “ id : L Ñ L Ď G and B “ τ : L Ñ L Ď G Remark.
The above notion of fundamental group does of course depend on the variety C . For instancea pro- p amalgamated free product of pro- p groups is very different from the profinite amalgamated freeproduct of those same groups viewed as profinite groups. One could introduce π into the notation, butthis would clutter it rather. All the theorems in this paper will make clear which is meant, although itis likely that the context would be sufficient to tell which category is in use.In the classical Bass-Serre theory, a graph of discrete groups p Y, G ‚ q gives rise to a fundamental group π p Y, G ‚ q and an action on a certain tree T whose vertices are cosets of the images φ p G v q of the vertexgroups in π p Y, G ‚ q and whose edge groups are cosets of the edge groups. Putting a suitable topologyand graph structure on the corresponding objects in the profinite world and proving that the result is aprofinite tree, is rather more involved than the classical theory; however the conclusion is much the same.We collate the various results into the following theorem.31 heorem 4.4 (Proposition 3.8 of [ZM89], Section 6.3 of [Rib17]) . Let G “ p Y, G ‚ q be a finite graph ofpro- C groups. Let Π “ Π p G q and set Π p y q “ im p G y Ñ Π q . Then there exists an (essentially unique)pro- π p C q tree T p G q , called the standard tree of G , on which Π acts with the following properties. • The quotient graph Π z T p G q is isomorphic to Y . • The stabiliser of a point t P T p G q is a conjugate of Π p ζ p t qq in Π , where ζ : T p G q Ñ Y is the quotientmap. In fact T satisfies a stronger property, of being C p π q -simply connected. Conversely (see Section 6.6 of[Rib17]) an action of a profinite group on a C p π q -simply connected profinite tree with quotient a finitegraph gives rise to a decomposition as a finite graph of profinite groups. In particular open subgroupsof fundamental groups of graphs of groups, which act on the standard tree, are themselves fundamentalgroups of graphs of groups formed in a way closely analogous to the discrete theory. However no analogousresults hold when the quotient graph is infinite.In the classical theory one tacitly identifies each G y with its image in the fundamental group π p Y, G ‚ q of a graph of groups. In general in the world of profinite groups the maps φ y : G y Ñ Π p G q may not beinjective, even for simple cases such as amalgamated free products. We call a graph of groups proper ifall the maps φ y are in fact injections. Let G be a profinite group and let S “ t S x | x P X u be a family of subgroups of G continuously indexedby a profinite space X , and let π be a set of primes. Suppose G acts on a pro- π tree T on the right. Let η : T Ñ T { G be the quotient map. Note that by Proposition A.14 we have natural identifications Z π rr V p T qss “ ð ¯ v P V p T q{ G Z π rr η ´ p ¯ v qss , Z π rr E p T qss “ ð ¯ e P E p T q{ G Z π rr η ´ p ¯ e qss Hence applying the functor Tor G ‚ p´ , ∆ G, S p b M q to the short exact sequence (4.1), where M P C π p G q ,and applying Proposition A.13 gives a long exact sequence ¨ ¨ ¨ Ñ ð ¯ e P E p T q{ G Tor Gn p Z π rr η ´ p ¯ e qss , ∆ G, S p b M q Ñ ð ¯ v P V p T q{ G Tor Gn p Z π rr η ´ p ¯ v qss , ∆ G, S p b M qÑ Tor Gn p Z π , ∆ G, S p b M q Ñ ð ¯ e P E p T q{ G Tor Gn ´ p Z π rr η ´ p ¯ e qss , ∆ G, S p b M q Ñ ¨ ¨ ¨ Now for each ¯ v P V p T q{ G , given a choice of lift to v P V p T q the map g ÞÑ vg gives an identification Z π rr η ´ p ¯ v qsss – Z π rr G v z G ss where G v is the stabiliser of v ; hence using Proposition 1.23 and Proposition 2.10 we have identificationsTor Gn p Z π rr η ´ p ¯ v qss , ∆ G, S p b M q – Tor G v n p Z π , ∆ G v , S Gv p b M q “ H n ` p G v , S G v ; M q (4.2)Similarly for ¯ e P E p T q{ G . This immediately yields: Proposition 4.5.
Let G be a profinite group acting on a pro- π tree from the right, and let S “ t S x | x P X u be a family of subgroups of G continuously indexed by a profinite space X . Let G t denote thestabiliser of t P T . Then hd p p G, S q ď max v P V p T q ,e P E p T q hd p p G v , S G v q , hd p p G e , S G e q ` ( for every p P π . roof. If the right hand side is finite, denote it by n . If it is infinite we have nothing to prove. Then forany M P C p p G q and any k ą n we have, for every v P V p T q and e P E p T q : H k p G v , S G v ; M q “ , H k ´ p G e , S G e ; M q “ H k p G, S ; M q “ Ð H n p G v , S G v q . The answer is that the latter groups do notform a particularly well-defined sheaf in the greatest generality. More precisely the identifications in (4.2)required a choice of lift v P V p T q , so that the subgroups G v may not be continuously indexed by V p T q{ G .This was not an issue for the above proposition since there is no difficulty manipulating a sheaf all ofwhose fibres are the zero module.If one has a continuous section σ : T { G Ñ T of η then one can indeed make the required identificationsin a continuous manner and recover the expected long exact sequence. ¨ ¨ ¨ Ñ ð ¯ e P E p T q{ G H n ` p G σ p ¯ e q , S G σ p ¯ e q ; M q Ñ ð ¯ v P V p T q{ G H n ` p G σ p ¯ e q , S G σ p ¯ v q ; M qÑ H n ` p G, S ; M q Ñ ð ¯ e P E p T q{ G H n p G σ p ¯ e q , S G σ p ¯ e q ; M q Ñ ¨ ¨ ¨ In particular if T { G is finite such a section exists. Let C be an variety of finite groups closed under taking isomorphisms, subgroups, quotients and extensions.Let π “ π p C q be the set of primes which divide the order of some finite groups in C .In this section we will record several long exact sequences associated with injective graphs of profinitegroups. Given the set-up in the previous sections, these derivations are generally similar those in Section3 of [BE78] and we shall not reproduce them all here. An exception is Theorem 4.9 which is a theorem(though not proof) related to Proposition 2.3 of [Rib69] and does not appear in [BE78].In all cases the plan is much the same. In the case of a proper pro- C amalgamated free product G “ G > L G one starts from the short exact sequence0 Ñ Z π rr G { L ss p´ res , res q ÝÝÝÝÝÝÑ Z π rr G { G ss ‘ Z π rr G { G ss Ñ Z π Ñ Z π -free modules, applies an appropriate Ext orTor functor, and uses Shapiro isomorphisms to translate between the cohomology of the various groupsinvolved.Suppose we have a proper pro- C HNN extension G “ G > L,τ where L is a subgroup of G and τ : L Ñ L is an isomorphism to another subgroup L of G . In this case one takes as a starting point the sequence0 Ñ Z π rr G { L ss res τ ˚ ´ res ÝÝÝÝÝÝÑ Z π rr G { G ss Ñ Z π Ñ Z π rr G { L ss Ñ Z π rr G { G ss , gL ÞÑ gG and τ ˚ denotes the map Z π rr G { L ss Ñ Z π rr G { L ss , gL ÞÑ t ´ gtL t is some stable letter for the HNN extension. One can check that when this last map is translatedvia a Shapiro isomorphism into a map on the (co)homologies of L and L it does in fact agree with themap induced functorially from τ . Theorem 4.6 (Theorem 3.2 of [BE78]) . Let G “ G > L G be a proper pro- C amalgamated free product.Let S i be a family of subgroups of G i continuously indexed by X i for each i , where X i may possibly beempty. Let S be the family of subgroups S \ S be the family of subgroups of G continuously indexed by X \ X . Then there is a natural long exact sequence ¨ ¨ ¨ H k ´ p L q Ñ H k p G, S q p res , res q ÝÝÝÝÝÑ H k p G , S q ‘ H k p G , S q p´ res q‘p res q ÝÝÝÝÝÝÝÝÑ H k p L q ¨ ¨ ¨ with coefficients in an arbitrary A P D π p G q . Here relative cohomology with respect to an empty familyshould be interpreted as absolute cohomology. Similarly for homology.Proof. In the case when S and S are non-empty this derives from the commuting diagram of shortexact sequencesind G G p ∆ G , S q ‘ ind G G p ∆ G , S q ∆ G, S Z π rr G { L ss Z π rr G ss p b G ∆ G , S ‘ Z π rr G ss p b G ∆ G , S Z π rr G { S ss ‘ Z π rr G { S ss Z π rr G { G ss ‘ Z π rr G { G ss Z π Z π For the other cases see [BE78].
Theorem 4.7 (Theorem 3.3 of [BE78]) . Let p G , S q be a profinite group pair with G a pro- C group and S possibly empty. Let L, L be subgroups of G isomorphic via an isomorphism τ . Suppose G “ G > L,τ is a proper pro- C HNN extension. Then one has a natural long exact sequence ¨ ¨ ¨ H k ´ p L q Ñ H k p G, S q res ÝÑ H k p G , S q p´ res q‘p τ ˚ ˝ res q ÝÝÝÝÝÝÝÝÝÝÝÑ H k p L q ¨ ¨ ¨ with respect to any A P D π p G q . Here res is the restriction map H k p G , S q Ñ H k p L q . Similarly forhomology. Theorem 4.8 (“Excision”, Proposition 3.4 of [BE78]) . (a) Let G “ G > L G be a proper pro- C amalga-mated free product. Let S be a family of subgroups of G continuously indexed by X which may possiblybe empty. Then the map of pairs p G , S \ L q Ñ p G, S \ G q induces isomorphisms H ‚ p G , S \ L ; ´q – H ‚ p G, S \ G ; ´q , H ‚ p G, S \ G ; ´q – H ‚ p G , S \ L ; ´q (b) Let G be a pro- C group, and let S be a family of subgroups of G continuously indexed by X which may possibly be empty. Let L, L be subgroups of G isomorphic via an isomorphism τ . Suppose G “ G > L,τ is a proper pro- C HNN extension. Then there are natural isomorphisms H ‚ p G , S \ L \ L ; ´q – H ‚ p G, S \ L ; ´q , H ‚ p G, S \ L ; ´q – H ‚ p G , S \ L \ L ; ´q induced by the obvious maps of pairs. Theorem 4.9.
Let G “ G > L G be a proper pro- C amalgamated free product, and S i a family ofsubgroups of G i continuously indexed by a possibly empty profinite set X i for each i . Consider the familyof subgroups S “ S \ t L u \ S of G continuously indexed by X \ t˚u \ X . Then there are naturalisomorphisms H ˚ p G, S \ L \ S ; A q – H ˚ p G , S \ L ; A q ‘ H ˚ p G , S \ L ; A q induced by the maps of pairs p G i , S i \ L q Ñ p G, S \ L q . roof. Consider the commutative diagram below. Z π rr G { L ss Z π rr G { L ss À i “ , Z π rr G ss p b G i ∆ G i , S i \ L Z π rr G {p S \ t L u \ t L u \ S qss Z π rr G { G ss ‘ Z π rr G { G ss ∆ G, S \ L \ S Z π rr G {p S \ t L u \ S qss Z π p´ , q ‘ where the middle column arises from the short exact sequence0 Z π rr G { L ss Z π rr G { L ss ‘ Z π rr G { L ss Z π rr G { L ss p´ , q ‘ by taking a direct sum with Z π rr G { S \ S ss . The middle row is the result of applying the exact functors Z π rr G ss p b G i ´ to the definitions of the ∆ G i , S i \ L . The bottom row and final column are already knownto be exact. From this it follows that the leftmost vertical map is an isomorphism, whence the theorem.Note also that this isomorphism, via a Shapiro isomorphism, agrees with the inclusion map defined inSection 2.1. Remark.
Using the theorems above there are several maps one can define from H n ´ p L q to H n p G, S q .One has the map from Theorem 4.6; one has the map induced by∆ G, S \ S Ñ ∆ S \ G – ind G G p ∆ G , S \ L q Ñ Z π rr G { L ss where the second map is the inverse of the excision isomorphism (Theorem 4.8); and the map∆ G, S \ S Ñ ∆ G, S \ L \ S – à i “ , ind G i G ∆ G i , S i \ L Ñ ind G G p ∆ G , S \ L q Ñ Z π rr G { L ss where the second map is the inverse of the isomorphism in Theorem 4.9. One may check by a simplediagram chase that these three maps agree. Denote this map by B . Similarly for HNN extensions.The following result appears as Theorems 3.5 and 3.7 of [BE78]. We will give a different proof. Theorem 4.10.
Let G “ G > L G be a proper pro- C amalgamated free product. Let S i be a family ofsubgroups of G i continuously indexed by X i for each i , where X i may possibly be empty. Let S be thefamily of subgroups S \ S be the family of subgroups of G continuously indexed by X \ X . Then thereis a natural long exact sequence ¨ ¨ ¨ Ñ H k ´ p L q Ñ H k p G , S \ L q ‘ H k p G , S \ L q ÝÑ H k p G, S q ÝÑ H k p L q Ñ ¨ ¨ ¨ with coefficients in an arbitrary A P D π p G q . Here relative cohomology with respect to an empty familyshould be interpreted as absolute cohomology. Similarly for homology.Furthermore this sequence is natural with respect to cup products in the following sense. Let C, A, B P D π p G q and let B : C b A Ñ B be a pairing. Choose a coclass e : H n p G, S ; B q Ñ I π . Then the followingdiagram ¨ ¨ ¨ H k ´ p L, C q H k p G , S \ L ; C q ‘ H k p G , S \ L ; C q H k p G, S ; C q H k p L ; C q ¨ ¨ ¨¨ ¨ ¨ H n ´ k p L, A q ˚ H n ´ k p G , A q ˚ ‘ H n ´ k p G , A q ˚ H n ´ k p G, A q ˚ H n ´ k ´ p L ; A q ˚ ¨ ¨ ¨ ommutes up to sign—the sign being a p´ q k ´ in the third square if S ‰ H or in the second square if S “ H . Here the first vertical map is the cup product map with respect to B and the coclass B e : H n ´ p L, B q ´B ÝÝÑ H n p G, S ; B q Ñ I π The second vertical map is the sum of the cup products with respect to coclasses H n p G i , S i \ L ; C q Ñ H n p G, S ; B q Ñ I π induced from e . The third vertical map is the usual cup product with respect to e .Proof. In the case when S is not empty the long exact sequence is derived, via Theorem 4.9, from thetop row of the commuting diagram of exact sequences below.∆ G, S \ S ∆ G, S \ L \ S Z π rr G { L ss Z π rr G { S ss ‘ Z π rr G { S ss Z π rr G {p S \ t L u \ S qss Z π rr G { L ss Z π Z π When S “ H the long exact sequence is derived from the long exact sequence for the pair p G, t L uq viaTheorem 4.9.Next we derive naturality properties with respect to the cup product. Suppose first that S ‰ H . Let C, A, B P D π p G q and let B : C b A Ñ B be a pairing. Choose a coclass e : H n p G, S ; B q Ñ I π . We have( Z π -split) exact sequences0 ∆ G, S \ S ∆ G, S \ L \ S Z π rr G { L ss Ñ Z π rr G { L ss Ñ Z π rr G { G ss ‘ Z π rr G { G ss Ñ Z π Ñ p´ , C q and Hom p´ , A q and define the following pairings on the groups involved. B : Hom p Z π rr G { L ss , C q b Hom p Z π rr G { L ss , A q Ñ Hom p ∆ G, S , B q is the pairing coev : Hom p Z π rr G { L ss , C q b Hom p Z π rr G { L ss , A q Ñ Hom p Z π rr G { L ss , C b A q f b g ÞÑ p xL ÞÑ f p xL q b g p xL qq followed by the natural map to Hom p ∆ G, S , B q induced by ´B . By Proposition 3.4, via the Shapiroisomorphisms the cup product map induced on cohomology is simply the usual adjoint cup product on H ‚ p L q with co-class B e “ e ˝p´B q : H n ´ p L, B q Ñ I π induced from e . As noted in the theorem statementthis is the same as a map appearing in Theorem 4.6.Next we have the pairing B : Hom p ∆ G, S \ L \ S , C q b Hom p Z π rr G { G ss ‘ Z π rr G { G ss , A q – Hom p ind G G p ∆ G , S \ L q ‘ ind G G p ∆ G , S \ L q , C q b Hom p Z π rr G { G ss ‘ Z π rr G { G ss , A q coev ÝÝÝÑ
Hom p Z π rr G { G ss ‘ Z π rr G { G ss , C b A q Ñ Hom p ∆ G, S , B q p G i , S i q , with coclasses H n p G i , S i ; B q Ñ I π induced from the coclass e via a map of pairs and the excision isomorphisms.Finally, B : Hom p ∆ G, S , C q b A Ñ Hom p ∆ G, S , B q is the standard cup product pairing.If one has sufficient tenacity one may check that these pairings are compatible with the maps in theshort exact sequence. We can now apply Theorem 3.1 (noting that both short exact sequences terminatein a free module so split as Z π -modules) to obtain the long exact sequence of cup product maps as in thestatement of the theorem, which commutes except for a sign p´ q k ´ in the third square.In the case when S “ H the long exact sequence in the theorem is derived from the long exactsequence in cohomology for the group pair p G, L q via Theorem 4.9, and the cup product diagram is atranslation of Proposition 3.5. This time however the diagram commutes up to a sign p´ q k ´ in thesecond square, as the ‘connecting homomorphism’ in this case is the second map.Of course there is also a version for HNN extensions. Theorem 4.11 (Theorems 3.6 and 3.8 of [BE78]) . Let p G , S q be a profinite group pair with G a pro- C group and S possibly empty. Let L, L be subgroups of G isomorphic via an isomorphism τ . Suppose G “ G > L,τ is a proper pro- C HNN extension. Then there is a natural long exact sequence ¨ ¨ ¨ Ñ H k ´ p L q Ñ H k p G , S \ L \ L q ÝÑ H k p G, S q ÝÑ H k p L q Ñ ¨ ¨ ¨ with coefficients in an arbitrary A P D π p G q . Here relative cohomology with respect to an empty familyshould be interpreted as absolute cohomology. Similarly for homology.Furthermore this sequence is natural with respect to cup products in the following sense. Let C, A, B P D π p G q and let B : C b A Ñ B be a pairing. Choose a coclass e : H n p G, S ; B q Ñ I π . Then the followingdiagram ¨ ¨ ¨ H k ´ p L, C q H k p G , S \ L \ L ; C q H k p G, S ; C q H k p L ; C q ¨ ¨ ¨¨ ¨ ¨ H n ´ k p L, A q ˚ H n ´ k p G , A q ˚ H n ´ k p G, A q ˚ H n ´ k ´ p L ; A q ˚ ¨ ¨ ¨ commutes up to sign—the sign being a p´ q k ´ in the third square if S ‰ H or in the second square if S “ H . Here the first vertical map is the cup product map with respect to B and the coclass B e “ e ˝ p´B q : H n ´ p L, B q Ñ H n p G, S ; B q Ñ I π The second vertical map is the cup product with coclass H n p G , S \ L \ L ; C q Ñ H n p G, S ; B q Ñ I π induced from e . The third vertical map is the usual cup product with respect to e . Let p be a prime. 37 .1 Definitions and basic properties Recall that a G -module M P C p p G q is of type p - FP if there exists a projective resolution of M byfinitely generated Z p rr G ss -modules. In this case both Tor G ‚ p M K , ´q and Ext ‚ G p M, ´q are functors whichtake finite modules to finite modules. Definition 5.1.
Let G be a profinite group and S a family of subgroups of G continuously indexed overa set X . Then G is of type p - FP if Z p is a module of type FP , and the pair p G, S q is of type p - FP if the module ∆ G, S is of type FP .When the pair p G, S q is of type p -FP we may define P p p G q -functors H ‚ p G, S ; ´q “ Tor G ‚ p ∆ K , ´q , H ‚ p G, S ; ´q “ Ext ‚ G p ∆ , ´q These functors of course extend the functors we have been working with.
Proposition 5.2.
Let p G, S q be a profinite group pair and consider the following statements.(1) p G, S q is of type p - FP .(2) S is a finite collection of subgroups and for each S x P S the module Z π rr G { S x ss is of type FP .(3) G is of type p - FP .If (1) holds then (2) and (3) are equivalent. Note that if G has property FIM then the second conditionis equivalent to:(2’) S is a finite collection of subgroups each of type p - FP .Proof. We will consider the short exact sequence of modules0 Ñ ∆ G, S Ñ Z p rr G { S ss Ñ Z p Ñ . This is tautologous except for Z p rr G { S ss . If (2) holds then by Proposition 1.34 Z p rr G { S ss has type FP . On the other hand, if Z p rr G { S ss is finitely generated then S is a finite family byProposition A.15 and each summand is of type FP by Proposition 1.34.The result now follows from Proposition 1.35.Now when cd p p G q “ n ą G is of type p -FP , then H n p G, ´q ˚ is a representable functor. Moreprecisely there exists a (non-zero) module I p p G q P D p p G q and a map e : H n p G, I p p G qq Ñ I p such that forany M P F p p G q the assignment ´ f : M Ñ I p p G q ¯ ÞÝÑ ´ H n p G, M q f ÝÑ H n p G, I p p G qq e ÝÑ I p ¯ gives a natural isomorphism of F p p G q -functorsHom G p´ , I p p G qq – H n p G, ´q ˚ The pair p I p p G q , i q is unique up to unique isomorphism. See [Ser13], Section 3.5, Lemma 6 andProposition I.17.Precisely the same arguments show that when ∆ G, S is of type p -FP and cd p p G, S q “ n ą dualising module I p p G, S q and a map e : H n p G, S ; I p p G, S qq Ñ I p which again givesan isomorphism of F p p G q -functors Hom G p´ , I p p G, S qq – H n p G, S ; ´q ˚ (5.1)38ote that by uniqueness of the dualising module if I p p G, S q ‰ p p G, S q “ n . We will also havecause to consider the compact dualising module D p p G, S q : “ I p p G, S q ˚ .Since p G, S q is of type p -FP , the right hand side is a restriction of a continuous P p p G q -functor.The left hand side commutes with inverse limits in the first variable, so we in fact have isomorphisms of C p p G q -functors Hom G p M, I p p G, S qq – H n p G, S ; M q ˚ for M P C p p G q .In particular one has the following of isomorphisms. H n p G, S ; Z p rr G ssq ˚ – Hom G p Z p rr G ss , I p p G, S qq– Hom p Z p , I p p G, S qq “ I p p G, S q One may check that these are isomorphisms of G -modules when the first two modules are given theactions deriving from the right action of G on Z p rr G ss . Hence we have an identification H n p G, S ; Z p rr G ssq “ D p p G, S q (5.2)Finally if D p p G, S q is a finitely generated G -module both sides of (5.1) are co-continuous P π p G q -functorsagreeing on F π p G q so this is in fact an isomorphism of P π p G q -functors. Definition 5.3.
The pair p G, S q is a duality pair of dimension n at the prime p (or more briefly a D n pair at p ) if p G, S q is of type p -FP and cd p p G, S q ď n , and H k p G, S ; Z p rr G ssq “ D p p G, S q ‰ k “ n k ‰ n where the compact dualising module D p p G, S q is isomorphic as a Z p -module to a finitely generated free Z p -module. If in addition D p p G, S q – Z p as a Z p -module then we say p G, S q is a Poincar´e duality pair ofdimension n at the prime p , or more briefly a PD n pair at p . We refer to the map i from the definition ofthe dualising module as the fundamental coclass of the pair—this is the dual notion to the fundamentalclass in the top-dimensional homology.For a PD n pair at the prime p the orientation character of p G, S q is the homomorphism χ : G Ñ Aut p Z p q given by the action on D p p G, S q and (any) identification D p p G, S q – Z p . We say p G, S q is orientable if im p χ q is trivial, and virtually orientable if im p χ q is finite.The obvious analogous definitions for absolute cohomology give the definition of a profinite duality(or Poincar´e duality) group. Proposition 5.4. If p G, S q is a PD n pair at the prime p then G is of type p - FP . Hence S is a finitecollection of subgroups and for each S P S the module Z π rr G { S ss is of type FP . If in addition G hasproperty FIM then each S P S has type p - FP .Proof. Take a resolution P ‚ of ∆ by finitely generated projective G -modules. By [Bru66], Proposition3.1, the kernel of P n ´ Ñ P n ´ is actually projective (and is finitely generated, being the image of P n ) sowe may truncate P ‚ and assume that P k “ k ą n . Now apply the functor Hom G p´ , Z p rr G ssq . Onemay readily see that Hom G p Z p rr G ss , Z p rr G ssq “ Z p rr G ss so that Hom G p´ , Z p rr G ssq preserves the propertyof being finitely generated free, hence the property of being finitely generated projective. The complex Hom p P n ´ ´‚ , Z p rr G ssq is a complex of finitely generated projective modules. Furthermore the homologyof this complex is H k p Hom p P n ´ ´‚ , Z p rr G ssqq “ H n ´ k p G, S ; Z p rr G ssq “ D p p G, S q if k “
00 if k ‰ D p p G, S q by finitely generated free G -modules—that is, of Z p with some G -action. If χ : G Ñ Aut p Z p q “ Z ˆ p is the G -action, then let ¯ χ p g q “ χ p g q ´ be the inverse G -action—whichis well defined as Aut p Z p q is abelian. For a homomorphism ρ : G Ñ Aut Z p let Z p p χ q denote Z p with G -action given by ρ . Then Z p p χ q p b Z p p ¯ χ q (with diagonal action) is isomorphic to the trivial module Z p .Finally note that, via the isomorphism Z p rr G ss p b Z p p ¯ χ q Ñ Z p rr G ss , g b ÞÑ ¯ χ p g q ´ ¨ g we see that ´ p b Z p p ¯ χ q is an exact functor taking finitely generated projectives to finitely generatedprojectives. Hence Hom p P n ´ ´‚ , Z p rr G ssq p b Z p p ¯ χ q is the required resolution of Z p by finitely generatedprojective modules.The statement about S follows from Proposition 5.2.In the next two propositions there are maps of P π p G q -functors induced by the cup product. In Section3 we defined cup product maps on the category D π p G q . These are extended to the maps in the theoremusing Propositions 1.4 and 1.5. Theorem 5.5 (cf Theorem 4.4.3 of [SW00]) . Suppose that G has type p - FP . Let J P D p p G q be anon-zero module such that J ˚ is finitely generated as a Z p -module, and let j : H m p G, J q Ñ I p be a map. Then the following are equivalent:(1) G is a D n group at the prime p with dualising module J and fundamental coclass j .(2) For every k there is an isomorphism of P p p G q -functors (equivalently of F p p G q -functors) Υ : H k p G, C q Ñ H n ´ k p G, Hom p C, J qq ˚ induced by the cup product maps on D π p G q with coclass H n p G, C p bbb Hom p C, J qq ev ÝÑ H n p G, J q j ÝÑ I p We will not prove this proposition, as the proof is very similar to the proof of the next proposition.
Theorem 5.6.
Suppose that p G, S q has type p - FP and G has type p - FP . Let J P D p p G q be a non-zeromodule such that J ˚ is finitely generated as a Z p -module, and let j : H m p G, S ; J q Ñ I p be a map. Then the following are equivalent:(1) p G, S q is a D n pair at the prime p with dualising module J and fundamental coclass j .(2) For every k the cup product map Υ : H k p G, S ; C q Ñ H n ´ k p G, Hom p C, J qq ˚ induced by the coclass H n p G, S ; C p bbb Hom p C, J qq ev ÝÑ H n p G, S ; J q j ÝÑ I p is an isomorphism of connected sequences of P p p G q -functors, or equivalently of F p p G q -functors. emark. Recall from Section 1.1 that when J ˚ is finitely generated as a Z p -module, Hom p C, J q is finitefor C P F p p G q and it makes sense to speak of Hom p´ , J q as a co-continuous functor in P p p G q . Proof.
The fact that isomorphisms of F p p G q -functors are equivalent to isomorphisms of P p p G q -functorsfollows from Propositions 1.4 and 1.5.First suppose that (1) holds. By Propositions 1.4, 1.5 and 3.3 the map Υ is a morphism of connected P p p G q -functors. For C P F π p G q note thatHom G p C, J q “
Ext G p C, J q “
Ext G p Z p , Hom p C, J qq where the last identification holds since J ˚ is free over Z p by assumption. Furthermore by looking at thedefinition of cup product we see that under this identification the mapHom G p C, J q Ñ H m p G, S ; C q ˚ agrees with the cup product map H p G, Hom p C, J qq Ñ H m p G, S ; C q ˚ Then the map Υ of connected sequences of cohomological coeffaceable P p p G q -functors is an isomorphismon finite modules at degree zero. Therefore by Propositions 1.4 and 1.5 the functor Υ is an isomorphismfor all k .If (2) is true, then the k “ n case of the cup product isomorphism gives the defining property of thedualising module so J is indeed the dualising module for the pair p G, S q . Now notice that H n ´ p G, S ; F p p b Z p rr G ssq – H p G, Hom p F p p b Z p rr G ss , J qq– H p G, Hom p Z p rr G ss , Hom p F p , J qqq“ H p G, coind G p F p b J qq “ Ñ Z p rr G ss p ÝÑ Z p rr G ss Ñ F p p b Z p rr G ss Ñ J ˚ Ñ J ˚ given by multiplication by p is an injection.Proposition 1.7 implies that J ˚ is actually a free Z p -module. Furthermore we have H k p G, S ; Z p rr G ssq – H n ´ k p G, Hom p Z p rr G ss , J qq “ H n ´ k p G, coind G p J qq “ k ‰ n . Furthermore the cup product isomorphisms immediately show cd p p G, S q ď n . So p G, S q is aD n pair as required. Proposition 5.7.
Suppose that p G, S q has type p - FP and G has type p - FP . Let J P D p p G q be a G -module isomorphic to I p as an abelian group and let j : H m p G, S ; J q Ñ I p be a map. Then the following are equivalent:(1) p G, S q is a PD n pair at the prime p with dualising module J and fundamental coclass j .(2) For every k the cup product map Υ : H k p G, C q Ñ H m ´ k p G, S ; Hom p C, J qq ˚ induced by the coclass H n p G, S ; C p bbb Hom p C, J qq ev ÝÑ H n p G, S ; J q j ÝÑ I p is an isomorphism of connected sequences of P p p G q -functors. roof. By Proposition 5.4, G also has type p -FP . Since J is isomorphic to I p as an abelian group, thena form of ‘Pontrjagin duality with G -action’—that is, C – Hom p Hom p C, J q , J q as G -modules, where Hom groups have diagonal actions—implies that this condition is equivalent to (2)of Proposition 5.6. Corollary 5.8. If p G, S q is a PD n pair at the prime p then n “ cd p p G, S q “ cd p p G q ` Proof.
The only non-trivial part is that cd p p G q ď n ´
1, which follows easily from the duality in the lastproposition.
Proposition 5.9.
Suppose that p G, S q is a PD n pair at the prime p with dualising module J and funda-mental coclass j . Assume that either S is of type p - FP for all S x P S or that G has property FIM. Theneach subgroup S x P S is a PD n ´ group at the prime p with dualising module res GS x p J q and fundamentalcoclass given by B j : H n ´ p S x , J q Ñ H n ´ p S , J q Ñ H n p G, S ; J q j ÝÑ I p Proof.
This follows immediately from Theorems 5.5 and 5.6 and Proposition 5.7 combined with Propo-sition 3.5 and the Five Lemma. The assumptions in the second sentence of the statement are neededto guarantee that each S x has type p -FP —when G has property FIM this follows from Proposition5.4. Proposition 5.10.
Let p G, S q be a pair of type p - FP and cd p p G, S q “ n . Then for any open subgroup U of G , the pair p U, S U q is of type p - FP and I p p U, S U q “ res GU I p p G, S q Proof.
The first statement is Corollary 2.11. Now let M P F p p U q . Recall that by Proposition 1.16, since U is open in G the induced and coinduced modules on M agree. Therefore we have, using the relativeShapiro Lemma, H n p U, S U ; M q ˚ – H n p G, S ; coind UG p M qq ˚ – Hom G p coind UG p M q , I p p G, S qq– Hom G p ind UG p M q , I p p G, S qq– Hom U p M, I p p G, S qq where the last isomorphism follows from Corollary 1.18 and Proposition 1.20. Hence the result. Proposition 5.11.
Let p G, S q be a profinite group pair with G a p -torsion-free profinite group and let U be an open subgroup of G . Then p G, S q is a PD n pair at the prime p if and only if p U, S U q is a PD n pairat the prime p , and their dualising modules agree as U -modules.Proof. The fact that p G, S q has type p -FP if and only if p U, S U q has type p -FP is Proposition 2.11.Next we prove that cohomological dimension n if and only if p U, S U q does. Given Proposition 5.6, if p U, S U q is a PD n pair at the prime p thencd p p G, S q ě cd p p U, S U q “ cd p p U q ` “ cd p p G q ` ě cd p p G, S q where the equality cd p p U q “ cd p p G q derives from Serre’s Theorem (Proposition 14 of [Ser13]). If p G, S q is a PD n pair at the prime p then cd p p U, S U q “ n by Lemma 5.12.42inally we have isomorphisms H k p G, S ; Z p rr G ssq “ H k p G, S ; ind UG p Z p rr U ssqq “ H k p G, S ; coind UG p Z p rr U ssqq “ H k p U, S U ; Z p rr U ssq using Proposition 1.16 and the relative Shapiro Lemma. This concludes the proof. Lemma 5.12. If p G, S q is a profinite group pair of cohomological dimension n and U is an open subgroupof G then for any A P D p p G q the corestriction map cor UG : H n p U, S U ; A q Ñ H n p G, S ; A q is a surjection. In particular cd p p U, S U q “ cd p p G, S q .Proof. Consider the map Σ : Hom U p Z p rr G ss , A q Ñ A which defines the corestriction, which is a surjection. The short exact sequence0 Ñ ker p Σ q Ñ Hom U p Z p rr G ss , A q Ñ A Ñ H n p U, S U ; A q cor ÝÑ H n p G, S ; A q Ñ H n ` p G, S ; ker p Σ qq “ p p G, S q ď cd p p U, S U q ; the other inequalityfollows from the relative Shapiro Lemma. Proposition 5.13.
Suppose that p G, S q is a PD n pair at p and that H is a closed subgroup of G with p | r G : H s (see Section I.1.3 of [Ser13] for the definition of this index). Then cd p p H, S H q ă cd p p G, S q Proof.
Let n “ cd p p G, S q . By Proposition 2.19 it suffices to show that for any H with p | r G : H s wehave H n p H, S H ; F p q “
0. Since p | r G : H s we may find a descending chain t U i u i ě of open subgroupsof G with p | r U i , U i ` s for all i . Since by Proposition 2.16 we have H n p H, S H ; F p q – lim ÝÑ H n p U i , S U i ; F p q it suffices to show that each restriction map H n p U i , S U i ; F p q Ñ H n p U i ` , S U i ` ; F p q vanishes for each i . Since for all open subgroups U of G we have that p U, S U q are PD n pairs at the prime p by Proposition 5.10, both the domain and codomain of the corestriction are isomorphic to subgroups ofHom p F p , Q p { Z p q – F p , hence either one of them is zero (so that the restriction map vanishes) or both areisomorphic to F p . In this last case consider the corestriction map cor U i ` U i in the other direction, which isa surjection by Lemma 5.12. Hence the corestriction is an isomorphism.Finally recall that the mapcor U i ` U i ˝ res U i U i ` : H n p U i , S U i q Ñ H n p U i , S U i q is multiplication by r U i : U i ` s . Here since p | r U i : U i ` s this map is zero. Since the corestriction is anisomorphism, the restriction map is zero as required.43 orollary 5.14. Let p G, S q be a pro- π group pair which is a PD n pair at every prime p P π . Supposethat G acts on a pro- π tree T . Suppose that for every edge e of T we have cd p p G e , S G e q ă n ´ for all p P π , where G e denotes the stabiliser of e . Then G fixes a vertex of T .Proof. By Proposition 2.4.12 of [Rib17] one may pass to a minimal invariant subtree of T so that G actsirreducibly. We aim to prove that there exists p P π such that p | r G : G x s for all x P T , for then theresult now follows from Propositions 4.5 and 5.13.By factoring out the kernel of the action one may assume that G acts faithfully on T . Then byTheorem 4.2.10 of [Rib17] if G does not fix a vertex then either G admits a non-abelian free pro- p subgroup acting freely on T —which forces p | r G : G x s for all x P T —or for some p P π , there is anabelian normal subgroup of G isomorphic to Z p . By Lemma 4.2.6(c) of [Rib17] this subgroup acts freelyso again p | r G : G x s for all x P T as required.Finally we conclude with the result that for pro- p groups the property of being a PD n pair is de-termined by the behaviour of the cohomology with coefficients in F p . Recall that for pro- p group pairshaving type p -FP is equivalent to having finite cohomology by Proposition 1.31. Theorem 5.15.
Let p G, S q be a pro- p group pair with cd p p G, S q “ n . Then the following are equivalent.(1) p G, S q is a PD n pair at the prime p .(2) H k p G, S ; F p q is finite for all k , dim F p p H n p G, S ; F p qq “ and the pairing H k p G, S ; F p q b H n ´ k p G, F p q Ñ H n p G, S ; F p q induced by the cup product (with respect to the multiplication pairing F p b F p Ñ F p ) is non-degeneratefor all ď k ď n .Proof. Assume first that p G, S q is a PD n pair. Then H n p G, S ; F p q – Hom G p F p , I p p G, S qq ˚ “ F p noting that since G is a pro- p group it must act trivially on the submodule x p ´ y of I p under any G -action,so the natural map F p Ñ I p p G, S q is G -linear. The non-degeneracy follows from Theorem 5.6, notingthat the multiplication pairing coincides with the evaluation pairing F p b Hom p F p , I p p G, S qq Ñ I p p G, S q under any identification of F p with the submodule of I p p G, S q killed by p .Now assume that G satisfies (2). Let F p C p p G q be the subcategory of modules annihilated by p .Consider, for C P F p C p p G q , the mapΥ k : H k p G, S ; C q Ñ H n ´ k p G, C ˚ q ˚ induced by the cup product with respect to the evaluation pairing C b C ˚ Ñ F p and a choice of iden-tification coclass H n p G, S ; F p q Ñ F p . Note that since C is in F p C p p G q the evaluation map on C b C ˚ does indeed have image in F p . By Proposition 3.3 this map gives a map of connected sequences of con-tinuous F p F p p G q -functors. The unique simple G -module in F p F p p G q is C “ F p ([RZ00b], Lemma 7.1.5).Using the long exact sequence and induction starting from F p we find that Υ k is an isomorphism for all C P F p F p p G q . Both sides commute with inverse limits, so Υ k is an isomorphism of F p C p p G q -functors.In particular we have H n ´ k p G, S ; F p rr G ssq – H k p G, F p rr G ss ˚ q ˚ “ H k p G, F p rr G ssq “ H k p G, ind G p F p qq “ F p if k “
00 if k ‰ Ñ Z p rr G ss p Ñ Z p rr G ss Ñ F p rr G ss Ñ H n p G, S ; Z p rr G ssq “ D p p G, S q , we find that the sequence0 Ñ D p p G, S q p Ñ D p p G, S q Ñ F p Ñ D p p G, S q is p -torsion-free and hence a free abelian pro- p group by Proposition 1.7, andis also of rank 1. So I p p G, S q is isomorphic to I p as an abelian group. Furthermore the rest of this longexact sequence shows that for every k ‰ n the map H k p G, S ; Z p rr G ssq p Ñ H k p G, S ; Z p rr G ssq is an isomorphism. For a compact p -primary module multiplication by p is only an isomorphism whenthe module is trivial. This concludes the proof. n pairs Let C be an variety of finite groups closed under taking isomorphisms, subgroups, quotients and extensions.Let π p C q be the set of primes which divide the order of some finite groups in C . Theorem 5.16.
Let G “ G > L G be a proper pro- C amalgamated free product. Assume that L does notequal either G or G . Let S i be a (possibly empty) finite family of subgroups of G i . Let S be the familyof subgroups S \ S of G , which is continuously indexed by X \ X . Then the following hold.(1) If each pair p G i , S i \ L q is a PD n pair at the prime p , so is p G, S q .(2) Suppose p G, S q is a PD n pair at the prime p and L is a PD n ´ group at the prime p . Assume eitherthat G i and p G i , S i \ L q have type p - FP for each i or that G has property FIM. Then each pair p G i , S i \ L q is a PD n pair at the prime p .Moreover the dualising modules are all isomorphic to appropriate restrictions of I p p G, S q . In particular p G, S q is orientable if and only if both p G i , S i \ L q are orientable.Proof. We first note that in each case all necessary groups and group pairs have type p -FP . In case(1) the conditions we know that G and G and are of type p -FP by Proposition 5.4, and furthermorethat Z p rr G i { L ss has type p -FP as a G i -module. The short exact sequence0 Ñ Z p rr G { L ss Ñ Z p rr G { G ss ‘ Z p rr G { G ss Ñ Z p Ñ G to have type p -FP —note that Z p rr G { L ss “ ind G G p Z p rr G { L ssq . Furthermore applying Propositions 5.4 and 1.33 to the short exact sequence0 ind G G p ∆ G , S q ‘ ind G G p ∆ G , S q ∆ G, S Z π rr G { L ss p G, S q also has type p -FP .In the case (2) we must show that each pair p G i , S i \ L q and group G i have type p -FP , assumingthat G has property FIM. We know that G and L and all the finitely many groups in S have type p -FP .The short exact sequence0 ∆ G, S ∆ G, S \ L Z π rr G { L ss p G, S \ L q to be of type p -FP .The isomorphism à i “ , Z π rr G ss p b G i ∆ G i , S i \ L – ∆ G, S \ L \ S p G i , S i q havetype p -FP . It now follows that G and G also have type p -FP by Proposition 5.2.We now move on to the main part of the theorem. First suppose the conditions of (1) hold. By Proposi-tion 5.9 the group L is a PD n ´ group whose dualising module I p p L q is isomorphic to res G i L p I p p G i , S i \ L qq for both i “ ,
2, hence the actions G i Ñ Aut p Q p { Z p q agree on L and by the universal property of amal-gamated free products there is a natural G -module J with underlying abelian group Q p { Z p so thatres GG i p J q “ I p p G i , S i \ L q for each i .Since any fundamental coclass e i of p G i , S i \ L q restricts to a fundamental coclass of L , choosecoclasses e i which restrict to the same fundamental coclass of L . There is also an induced coclass e : H n p G, S ; J q Ñ I p given by e ´ e : H n p G , S ; I p p G , S \ L qq ‘ H n p G , S ; I p p G , S \ L q Ñ I p Note that e vanishes on H n ´ p L q since the e i both restrict to a fundamental coclass on H n ´ p L, I p p L qq .Therefore by the long exact sequence in Theorem 4.10 we do indeed have a well-defined coclass on H n p G, S ; J q .We may now use Theorems 4.10 and 5.6 and the Five Lemma applied to the sign-commutative diagramof long exact sequences (4.10) with A “ Hom p C, J q and coclass e to prove that p G, S q is a PD n pair atthe prime p with dualising module J as required.Next assume the conditions of part (2) and assume also that the dualising module for L is in factres GL p I p p G, S qq . We will prove this later. Let I p p G, S q “ J . It follows from the Five Lemma and (4.10)that for each i and k and for any C P P p p G q the cup product induces an isomorphismΥ : H k p G i , S i ; res GG i p C qq Ñ H m ´ k p G i , Hom p res GG i p C q , J qq ˚ Let G i z G Ñ G be a continuous section of the quotient map. This yields an identification of G i -modules Z p rr G ss “ Z p rr G i ssrr G i z G ss and a G i linear epimorphism: q : Z p rr G ss “ Z p rr G i ssrr G i z G ss Ñ Z p rr G i ss sending each element G i g of G i z G to 1. Furthermore we have the natural G i -linear map i : Z p rr G i ss Ñ Z p rr G ss such that qi “ id M . Now the commuting diagram H k p G i , S i ; Z p rr G i ssq H k p G i , S i ; res GG i p Z p rr G ssqq H n ´ k p G i , Hom p Z p rr G i ss , J qq ˚ H n ´ k p G i , Hom p res GG i p Z p rr G ssq , J qq ˚ i Υ q – iq combined with the fact that applying various functors to q and i preserves the property that qi is anidentity map shows that the cup product map on the left is in fact an isomorphism. Then the naturalmap H k p G i , S i ; Z p rr G i ssq – H n ´ k p G i , coind G i p J qq ˚ “ k ‰ n res GG i p D p p G, S qq if k “ n is an isomorphism as required. This is an isomorphism of G i -modules by Proposition 1.22.We now prove that I p p L q “ res GL p I p p G, S qq . Consider the map B : H n ´ p L, I p p G, S qq Ñ H n p G, S ; I p p G, S qq from the long exact sequence in Theorem 4.6. By Poincar´e duality for L and p G, S q both sides are iso-morphic to (subgroups of) Q p { Z p as abelian groups. Therefore the map B is either an isomorphism46r factors through the multiplication-by- p map on H n ´ p L, I p p G, S qq . If B is an isomorphism thensince the left hand side is isomorphic to Hom L p I p p L q , I p p G, S qq and the right hand side is isomorphicto Hom G p I p p G, S q , I p p G, S qq – I p there exists a non zero L -linear map between I p p L q and I p p G, S q .Since these are both isomorphic to Q p { Z p , one may show that there must be an L -module isomorphismbetween the two modules and indeed I p p L q “ res GL p I p p G, S qq .We now turn our attention to the other case and show that it is impossible. The fundamental coclass e of p G, S q yields a coclass B e : H n ´ p L, I p p G, S qq Ñ I p which factors through multiplication by p . If C is a finite G -module killed by p then the cup product mapΥ B e, ev : H k p L, C q b H n ´ ´ k p L, Hom p C, I p p G, S qqq Ñ H n ´ p L, I p p G, S qq Ñ I p (given by the evaluation pairing and the coclass B e ) is a map of modules killed by p which factors throughmultiplication by p . That is, it is zero. This cup product is exactly the right-hand map appearing in thefollowing piece of the map of long exact sequences (4.10): H n ´ p G, S ; C q H n ´ p L, C q H p G, Hom p C, I p p G, S qq ˚ H p L, Hom p C, I p p G, S qqq ˚– Therefore the bottom map vanishes. Since I p p G, S q is isomorphic to I p as an abelian group, by Pontrjagin-duality-with-action the coefficient group Hom p C, I p p G, S qq of the bottom row ranges over all finite G -modules killed by p . Therefore, considering the rest of the long exact sequence on the bottom row andusing Shapiro isomorphisms, for all finite modules A killed by p the sequenceHom G p Z p , A q Ñ Hom G p Z p rr G { G ss ‘ Z p rr G { G ss , A q Ñ Hom G p Z p rr G { L ss , A q Ñ Z p rr G { G i ss is finitely generated we obtain an exact sequence Hom G p Z p , M q Ñ Hom G p Z p rr G { G ss ‘ Z p rr G { G ss , M q Ñ Hom G p Z p rr G { L ss , M q Ñ G -modules M which are killed by p . In particular the natural quotient map from Z p rr G { L ss Ñ F p rr G { L ss extends to a G -linear map Z p rr G { G ss ‘ Z p rr G { G ss Ñ F p rr G { L ss . That is, the sequence0 Ñ F p rr G { L ss Ñ F p rr G { G ss ‘ F p rr G { G ss Ñ F p Ñ F p is a trivial module there must be a G -invariant point of F p rr G { G i ss for each i ,which is non-zero for at least one i . This is impossible by Propositions B.1 and B.2.In the classical case one would simply assert that unless L “ G ´ i , the group G i has infinite index in G “ G > L G and therefore the existence of a G -invariant point of F p rr G { G i ss is absurd. In our case wemust work a little harder—for instance, absurd as it is, one may show that F rr Z ss has a Z -invariantpoint. This analysis does not really belong here and we leave it to Appendix B.Similarly to the above one has the analogous result for HNN extensions, relying upon Theorem 4.11. Theorem 5.17.
Let p G , S q be a profinite group pair with G a pro- C group and S a possibly emptyfinite family of subgroups of G . Let L, L be subgroups of G isomorphic via an isomorphism τ . Suppose G “ G > L,τ is a proper pro- C HNN extension. Then the following hold.(1) If p G , S \ L \ L q is a PD n pair at the prime p , so is p G, S q .
2) Suppose p G, S q is a PD n pair at the prime p and L is a PD n ´ group at the prime p . Assume eitherthat G and p G , S \ L \ L q have type p - FP or that G has property FIM. Then p G , S \ L \ L q is a PD n pair at the prime p .Moreover the dualising modules are all isomorphic to appropriate restrictions of I p p G, S q . In particular p G, S q is orientable if and only if p G, S \ L \ L q is orientable. By a standard induction one deduces a theorem valid for finite graphs of pro- C groups with more thanone edge. By a ‘reduced’ graph of groups below we mean that any edge whose edge group coincides withthe adjacent vertex groups is a loop. By collapsing any non-loops with this property one may alwaysmake a graph of groups reduced. Theorem 5.18.
Let G “ p Y, G ‚ q be a reduced proper finite graph of pro- C groups. For each y P V p Y q let S y be a possibly empty finite family of subgroups of G y . Let E y denote the set of subgroups of G y which are images of the edge groups B i p G e q for adjacent edges with d i p e q “ y . Finally let G denote thefundamental pro- C group of G and let S “ Ů y P V p Y q S y . Then the following hold.(1) If each pair p G y , S y \ E y q is a PD n pair at the prime p , so is p G, S q .(2) Suppose p G, S q is a PD n pair at the prime p and G e is a PD n ´ group at the prime p for each e P E p Y q . Assume either that G y and p G y , S y \ E y q have type p - FP for all y P V p Y q or that G hasproperty FIM. Then each pair p G y , S y \ E y q is a PD n pair at the prime p .Moreover the dualising modules are all isomorphic to appropriate restrictions of I p p G, S q . In particular p G, S q is orientable if and only if p G y , S y \ E y q is orientable for all y P V p Y q . Let Γ be a discrete group and G a profinite group. Let φ : Γ Ñ G be a group homomorphism. Let M be a Γ-module and N P C π p G q , and let f : M Ñ N be a group homomorphism compatible with φ inthe natural way, i.e. f p γm q “ φ p γ q f p m q for γ P Γ , m P M . Let A be a Γ-module and B P D π p G q andlet g : B Ñ A be a group homomorphism compatible with φ in the sense that γg p n q “ g p φ p γ q n q for γ P Γ , n P N . Take a projective resolution P ‚ of M by Γ-modules and a projective resolution Q ‚ of N in C π p G q . Viewing Q ‚ simply as a complex of Γ-modules via φ , the map f : M Ñ N lifts to a chain map f ‚ : P ‚ Ñ Q ‚ , unique up to chain homotopy. Any continuous G -module map Q n Ñ B gives a Γ-modulemap P n Ñ A via f n and g , yielding a chain mapHom Z π rr G ss p Q ‚ , B q Ñ Hom Z Γ p P ‚ , A q and thus, passing to cohomology, a mapExt ‚ Z π rr G ss p N, B q Ñ
Ext ‚ Z Γ p M, A q which as usual is independent of the choices of P ‚ and Q ‚ and is natural with respect to maps of modulesand maps of short exact sequences.Similarly let M be a right Γ-module and M is a left Γ-module, and N P C π p G q K , N P D π p G q . Thengiven homomorphisms M Ñ N , M Ñ N which are compatible with φ there are canonical morphismsTor Γ ‚ p M , M q Ñ Tor G ‚ p N , N q
48n our particular case of study, let Γ be a discrete group and Σ a finite family of subgroups of Γ. Let p G, S q be a profinite group pair and assume φ is a map of group pairs (in the same sense as in Section2.1). There is then a commuting diagram of short exact sequences0 ∆ Γ , Σ Z r Γ { Σ s Z
00 ∆ G, S Z π rr G { S ss Z π A P D π p G q and take (continuous) module morphisms from the above diagram to A —regarding A as a Γ-module via φ . Combining this with the canonical map Z Ñ Z π and applying theabove construction gives a commuting diagram of long exact sequences ¨ ¨ ¨ H k p G, A q H k p S , A q H k p G, S ; A q ¨ ¨ ¨¨ ¨ ¨ H k p Γ , A q H k p Σ , A q H k p Γ , Σ; A q ¨ ¨ ¨ where the bottom row is the relative cohomology sequence for discrete groups. Definition 6.1.
Let Γ be a discrete group and Σ be a finite family of subgroups of Γ. Let G “ p Γ p π q be the pro- π completion relative to some set π of primes, and let i : Γ Ñ G be the canonical map. Let S be the finite family of subgroups i p S q where S P Σ, where the bar denotes closure in G . We call thepair p G, S q the pro- π completion of the pair p Γ , Σ q . Note that the groups i p S q may not be the pro- π completions of the discrete groups S P Σ.We say that p Γ , Σ q is (cohomologically) π -good if for every A P F π p G q the map H ‚ p G, S ; A q Ñ H ‚ p Γ , Σ; A q is an isomorphism. We also recall that Γ is (cohomologically) π -good if for every A P F π p G q the map H ‚ p G, A q Ñ H ‚ p Γ , A q is an isomorphism.When π is the set of all primes, any finite Γ-module is a G -module in a natural way and the definitionscould be (and usually are) phrased in terms of ‘all finite Γ-modules’. When π is not the set of all primes,not every finite π -primary Γ module M need be a G -module: one also requires that the image of the mapΓ Ñ Aut p M q is a π -group.The definition is stated in terms of cohomology. Suppose p Γ , Σ q is of type FP , in the sense that∆ Γ , Σ has a resolution by finitely generated projectives P ‚ . Then for any M P F π p G q , we have H n p Γ , Σ; M q ˚ “ H n p P K‚ b Γ M q ˚ “ H n pp P K‚ b Γ M q ˚ q “ H n p Hom Γ p P ‚ , M ˚ qq “ H n p Γ , Σ; M ˚ q noting that p´q ˚ “ Hom p´ , I π q is exact when applied to the sequences of finite π -modules P ‚ b Γ M .Thus Pontrjagin duality holds for the discrete group pair. Since it holds for the profinite group pair aswell we have the following proposition. Proposition 6.2. If p Γ , Σ q is a π -good pair of type FP and p G, S q is its pro- π completion then forevery M P F π p G q the map of pairs i : p Γ , Σ q Ñ p G, S q induces isomorphisms H ‚ p Γ , Σ; M q – H ‚ p G, S ; M q
49 basic proposition that derives immediately from the above diagram of exact sequences and the5-Lemma is the following.
Proposition 6.3.
Let Γ be a discrete group and Σ be a finite family of subgroups of Γ . Let p G, S q bethe pro- π completion of p Γ , Σ q and assume that for every S P Σ the natural map p S p π q Ñ i p S q is anisomorphism, where p S p π q is the pro- π completion. Assume further that each S P Σ is π -good. Then Γ is π -good if and only if the pair p Γ , Σ q is π -good. We now proceed towards the expected result that the pro- π completion of a PD n pair of discretegroups is a PD n pair at the prime p where p P π , under certain conditions on the pro- π topology on Γ. Definition 6.4.
Let Γ be a discrete group and G its profinite completion. For a Γ-module P , let p P bethe G -module defined by p P p π q “ lim ÐÝ K,m Z { m b K P where K runs over the finite index normal subgroups of Γ with Γ { K a π -group and m runs over thoseintegers divisible only by primes from π . Here each module in the inverse limit acquires the Γ { K -action(and hence G -action) given by the left action on P . This is well-defined since K is normal in G . Bypassing to a subsequence one may assume that this limit is indexed over a totally ordered set of pairs p K, m q . Lemma 6.5.
The map P Ñ p P p π q is an additive functor from the category of finitely generated G -modulesto C π p G q taking finitely generated projective modules to finitely generated projective modules.Proof. The additivity statement follows immediately from the fact that an inverse limit of additive func-tors to F π p G q is an additive functor to C π p G q . By definition we have Z π rr G ss “ lim ÐÝ K,m Z { m b K Z r Γ s (see Section 5.3 of [RZ00b] for information on complete group rings). So our functor takes finitelygenerated free modules to finitely generated free modules. Since in both categories the finitely generatedprojectives are precisely the direct summands of the finitely generated free modules we are done. Proposition 6.6.
Suppose that the pair p Γ , Σ q is π -good, and let p G, S q be its pro- π completion. Supposethat P ‚p π q is a resolution of ∆ Γ , Σ by finitely generated projective Γ -modules. Then the complex p P ‚p π q is aresolution of ∆ G, S by finitely generated projective Z π rr G ss -modules, where p G, S q is the pro- π completionof p Γ , Σ q .Proof. We compute the homology of the chain complex p P ‚ to verify that it is a resolution of ∆ G, S . Wehave H k p p P ‚p π q q “ H k p lim ÐÝ Z { m b K P ‚ q“ lim ÐÝ H k p Z { m b K P ‚ q by the Mittag-Leffler condition “ lim ÐÝ Tor Kk p Z { m, ∆ K, Σ K q“ lim ÐÝ H k p K, Σ K ; Z { m q by Proposition 2.3 “ lim ÐÝ H k p K, S K ; Z { m q by Proposition 6.2 “ H k p , S ; Z π q by Propostion 2.16, since č K “ “ Tor k p Z π , res G p ∆ G, S qq“ k ‰ G, S if k “ π completion, theintersection of all the subgroups K Ď G (that is, of all the open subgroups) is trivial. For the Mittag-Leffler condition see Section 3.5 of [Wei95]. This is in fact a chain of isomorphisms of (right) G -moduleswhen ∆ and p P ‚p π q are given the canonical right action dual to their left G -actions and the Tor groupshave the right G -action induced as in Proposition 1.22. Corollary 6.7.
Suppose that the pair p Γ , Σ q is π -good, and let p G, S q be its pro- π completion. If p Γ , Σ q has type FP then p G, S q has type p - FP for all p P π .Proof. The previous proposition shows that Z π is of type FP in C π p G q . If we take a projective resolutionof Z π which is finitely generated in each dimension then taking p -primary components (which is an exactfunctor) gives a resolution of Z p in C p p G q by finitely generated projectives by Proposition 1.12. Corollary 6.8.
Suppose p Γ , Σ q is a p -good pair of discrete groups with H k p Γ , Σ; F q finite for all p -primary Γ -modules F and all k . If p G, S q is its pro- p completion then p G, S q has type p - FP .Proof. This follows immediately from Proposition 1.31.
Theorem 6.9.
Suppose that the pair p Γ , Σ q is of type FP and let p G, S q be its pro- π completion. Supposethat Γ and p Γ , Σ q are π -good. Assume that p Γ , Σ q is orientable if R π . If p Γ , Σ q is a PD n pair then p G, S q is a PD n pair at the prime p for every p P π .Proof. First note that by the previous proposition p G, S q is of type p -FP and by goodness cd p p G, S q ď cd p Γ , Σ q “ n . Let r Z be the dualising module of p Γ , Σ q (in the sense of Definition 4.1 of [BE78]). Eitherthe action on r Z is trivial or it factors through a map from Γ to Z { Z . Hence by the assumption onorientability the action of Γ on r Z factors through G , so we may take the pro- p completion to acquire a G -module Ă Z p . The Pontrjagin dual of Ă Z p is a module r I p isomorphic to I p as an abelian group.For a finite module M let Ă M “ r Z b M with the diagonal action—that is, the abelian group M withan appropriately twisted action. Note that r Z b r Z “ Z .We have the following chain of isomorphisms induced by π -goodness and Poincar´e duality for thepair p Γ , Σ q and its finite index normal subgroups K with Γ { K a π -group (goodness is inherited by suchsubgroups by Proposition 6.10 below). We also note that by the analogue of Proposition 5.4 for discretegroups, Γ is of type FP and we may also use homological goodness as in Proposition 6.2. The inverselimits are indexed over pairs p U, m q where U is an open normal subgroup of G with G { U a π -group and m is a π -number. H ˚ p G, S ; Z p rr G ssq “ lim ÐÝ H ˚ p G, S ; Z { p m r G { U sq– lim ÐÝ H ˚ p Γ , Σ; Z { p m r G { U sq– lim ÐÝ H ˚ p Γ , Σ; Z { p m r Γ { Γ X U sq– lim ÐÝ H ˚ p Γ , Σ; ind Γ X U Γ p Z { p m qq– lim ÐÝ H ˚ p Γ , Σ; coind Γ X U Γ p Z { p m qq– lim ÐÝ H ˚ p Γ X U, Σ Γ X U ; Z { p m q– lim ÐÝ H n ´˚ p Γ X U, Č Z { p m q– lim ÐÝ H n ´˚ p U, Č Z { p m q– H n ´˚ p , r Z p q“ ˚ ‰ n r Z p if ˚ “ n p G, S q is indeed a PD n pair. These are all isomorphisms of G -modules where the actions in the firstfour lines are derived from the right action of G on Z p rr G ss or on Z { p m r G { U s and the actions in theremaining lines are given by the conjugation action of G . See Proposition 1.22. The second to last linefollows by the absolute version of Proposition 2.16.The absolute version of Theorem 6.9 was first proved by Pletch [Ple80b]. Here we state and give brief proofs of some properties of goodness which mirror those of absolute goodness.
Proposition 6.10.
Let p Γ , Σ q be a π -good pair, and let K be a finite index normal subgroup of Γ suchthat Γ { K is a π -group. Then p K, Σ K q is also π -good.Proof. Let p G, S q be the pro- π completion of p Γ , Σ q and let U “ K be the closure of K in G . Then p U, S U q is the pro- π completion of p K, Σ K q and for any M P F π p U q there is a commuting diagram H r p U, S U ; M q H r p G, S ; Z π rr G ss p b U M q H r p K, Σ K ; M q H r p Γ , Σ; Z r Γ s b K M q Since G { K is a finite π -group the coefficient modules in the right-hand column are isomorphic and lie in F π p G q , hence the right vertical map is an isomorphism by hypothesis and we are done. Definition 6.11.
Let Γ be a discrete group and let Λ ď Γ. Let π be a set of primes. Then we say that Λis π -separable in Γ if for every g P Γ r Λ there is a map φ from Γ to a finite π -group such that φ p g q R φ p Λ q .We say that Γ induces the full pro- π topology on Λ if for every finite index normal subgroup U ofΛ with Λ { U a π -group there is a finite index normal subgroup V of Γ with Γ { V a π -group and with V X Λ ď U .We say that Λ is fully π -separable in Γ if it is π -separable in Γ and Γ induces the full pro- π topologyon Λ.As usual we omit the symbol π when π is the set of all primes. An immediate consequence of full π -separability of a subgroup Λ ď Γ is that the natural map from the pro- π completion of Λ to the pro- π completion of Γ is an isomorphism to its image. Proposition 6.12.
Let Ñ N Ñ E Ñ Γ Ñ be an extension of groups, such that N is finitelygenerated, E induces the full pro- π topology on N , and H n p N, M q is finite for any n and any finite π -primary N -module M . Let Π be a finite family of subgroups of E each containing N and let Σ be theimage of this family in Γ . Assume p Γ , Σ q is a π -good pair and that N is π -good. Then p E, Π q is π -good.Proof. The conditions of the theorem imply the existence of a short exact sequence of pro- π completions1 Ñ p N π Ñ p E π Ñ p Γ π Ñ M P C π p p E π q the natural maps of the discrete groups to their pro- π completions induce a naturalmap from the relative Lyndon-Hochschild-Serre spectral sequence (Proposition 2.23) to the analogousspectral sequence for the extension of discrete groups. By the goodness assumptions this map is anisomorphism H r p p Γ π , p Σ π ; H s p p N π , M qq Ñ H r p Γ , Σ; H s p N, M qq on the second page, hence gives an isomorphism H r ` s p p E π , p Π π ; M q Ñ H r ` s p E, Π; M q in the limit as required. 52e will need some terminology before moving on. Definition 6.13.
Let G “ p X, Γ ‚ q be a finite graph of finitely generated groups with fundamental groupΓ. We say that G is π -efficient if Γ is residually π -finite and each group Γ x is fully π -separable in Γ.One immediate consequence of π -efficiency is that the corresponding graph of pro- π completions isproper and has pro- π fundamental group equal to the pro- π completion of Γ. Proposition 6.14.
Let p X, Γ ‚ q be a π -efficient finite graph of finitely generated groups with fundamentalgroup Γ . Let Σ x be a finite family of subgroups of Γ x for x P V p X q and let E x be the set of edge groupsincident to Γ x . Let Σ “ Ů x P V p X q Σ x . Suppose that p Γ x , Σ x \ E x q is a π -good pair for each x P V p X q andthat G e is π -good for each e P E p X q . Then p Γ , Σ q is π -good.Proof. This may be proved by induction from the cases of amalgamated free products and HNN ex-tensions. The conditions of the theorem give a map from the Mayer-Vietoris sequences from Theorems4.10 and 4.11 to the analagous sequences for discrete groups. The result then follows from the FiveLemma. n pairs In this section we give the expected examples of good group pairs and profinite PD n pairs. First we givea sufficient condition for a group pair to have type FP which the author could not immediately find inthe literature. Proposition 6.15.
Suppose that X is a connected aspherical cell complex with finitely many cells in eachdimension with fundamental group Γ , and suppose that t Y i u ď i ď n p n ě q is a collection of connecteddisjoint subcomplexes. Suppose further that each Y i is an aspherical complex with fundamental group S i and that the inclusions of complexes induce injective maps S i Ñ Γ . Then if Σ “ t S i u ď i ď n , the grouppair p Γ , Σ q is of type FP .Remark. We have been somewhat sloppy where basepoints are concerned; however the basepoints areentirely irrelevant to the outcome as the module ∆ Γ , Σ is independent of the choices of the S i up toconjugacy by Proposition 2.9. Proof.
We may assume that every vertex in X lies in one of the Y i in the following manner. The 1-skeletonof X is some connected finite graph and therefore admits a ‘maximal subforest relative to the Y i ’. Thatis, a collection of disjoint trees which together include all vertices of X not in any Y i and each includingexactly one vertex of some Y i . If we enlarge each Y i by attaching those trees incident to it we do notchange any of the homotopy-theoretic properties of the statement of the proposition. So we may proceedwith this assumption.For a cell complex Z let C ‚ p Z q be its cellular chain complex. Consider the universal cover ˜ X withprojection map p : ˜ X Ñ X . The cellular chain complex E ‚ “ C ‚ p ˜ X q is a free resolution of Z by finitelygenerated free Z Γ-modules. This chain complex has a subcomplex D ‚ “ À ni “ C ‚ p p ´ p Y i qq deriving fromthe pre-images of the Y i . This subcomplex is isomorphic as a complex of Γ-modules to n à i “ Z Γ b S i C ‚ p ˜ Y i q where ˜ Y i is the universal cover of Y i . Now D ‚ is a free resolution of À ni “ Z Γ b S i Z and the inclusion map D ‚ Ñ E ‚ induces the augmentation map À ni “ Z Γ b Z Ñ Z . Finally note that since D ‚ Ñ E ‚ is inducedby inclusions of subcomplexes, E ‚ { D ‚ is a complex of free finitely generated modules. From the long exactsequence in homology deriving from the short exact sequence of chain complexes D ‚ Ñ E ‚ Ñ E ‚ { D ‚
53e find that E ‚ { D ‚ is exact in degree at least 2 and that H p E ‚ { D ‚ q – ∆ Γ , Σ . By the first part ofthe construction E { D “
0. Thus E ‚ { D ‚ provides a free resolution of ∆ Γ , Σ by finitely generated freemodules as required. Proposition 6.16.
Let X be a compact surface with boundary components l , . . . , l n which is not a disc.Let Γ “ π X and Σ “ t i ˚ p π l i qu ni “ where i ˚ denotes an inclusion map. Let π be a set of primes andassume that X is orientable if R π . Then the pair p Γ , Σ q is π -good and the pro- π completion of p Γ , Σ q is a PD pair at every prime p P π .Proof. Note that Γ is a free group and therefore π -good for every π (see for example Exercise 2.6.2 of[Ser13]). Furthermore boundary subgroups are fully π -separable for every π (see for instance Propositions3.1 and 3.2 of [Wil16] where this is done for π “ t p u ; much the same arguments work for general π ).Thus we may apply Proposition 6.3 and Theorem 6.9 to find the result.One cannot quite give such a sweeping statement as this in dimension 3, since the pro- π topology ona 3-manifold group may be very poorly behaved. We will therefore restrict our attention to the set of allprimes—that is, we will consider the profinite completion. We must first establish separability conditionsfor boundary components, or more generally fundamental groups of embedded surfaces. These are notentirely new results; see the historical note at the end of the section. Theorem 6.17.
Let G “ p X, Γ ‚ q and L “ p Y, Λ ‚ q be finite graphs of finitely generated groups. Let f : Y Ñ X be a map of graphs and for every y P Y let φ y : Λ y Ñ Γ f p y q be an injective group homomorphismcompatible with the boundary maps in L and G . Suppose further that the induced map on the fundamentalgroups φ : Λ “ π p L q Ñ π p G q “ Γ is injective.If G is an efficient graph of groups and for each y P Y the subgroup Γ f p y q induces the full profinitetopology on φ y p Λ y q then Γ induces the full profinite topology on φ p Λ q .Proof. Consider a finite index normal subgroup U of Λ. Since each Λ y is fully separable in Γ f p y q thereare finite index subgroups V x of the Γ x such that V x X Λ y ď U X Λ y for every y with f p y q “ x . Since thegraph of groups G is efficient we may assume, by passing to deeper subgroups V x if necessary, that thereis a finite index normal subgroup r Γ of Γ such that V x “ r Γ X Γ x for all x P X .Now r Γ is the fundamental group of a graph of groups r G “ p r X, r Γ ‚ q whose vertex and edge groups arerepresentatives of the conjugacy classes of the V x in Γ and where r X is some finite cover of the graph X .Similarly r Λ “ r Γ X Λ is the fundamental group of a graph of groups r L “ p r Y , r Λ ‚ q whose vertex and edgegroups are representatives of the conjugacy classes of the V x X Λ y “ r Γ X Λ y in Λ and r Y is some finitecover of the graph Y . Note that if we find a finite index subgroup W of r Γ with W X r Λ ď U X r Λ then weare done; for then W is finite index in Γ and W X Λ “ p W X r Γ q X Λ “ W X r Λ ď U X r Λ ď U Now by construction U X r Λ contains all the vertex groups of r L which are (conjugates in Λ of) V x X Λ y ď U X Λ y . Therefore the quotient map f : r Λ Ñ r Λ {p U X r Λ q factors through the map to the graph fundamentalgroup π p r Y q . r Λ π p L q π p ˜ Y q r Λ { U X r Λ r Γ π p G q π p r X q π p r X q{ W Now π r Y is a finitely generated subgroup of the free group π r X , which is subgroup separable by a theoremof Marshall Hall [Hal49]. Therefore there is a finite index normal subgroup W of π r X with W X π p r Y q contained in the kernel of the map π p r Y q Ñ r Λ {p U X r Λ q . The preimage of W in r Γ is the required subgroup W . 54 heorem 6.18. Let M be a closed 3-manifold and let L be an embedded π -injective surface in M . Then π L is fully separable in π M .Proof. Separability is a theorem of Przytycki and Wise [PW14]. We consider the profinite topology.First note that we are free to pass to a finite index cover of M so we may assume that both M and L are orientable. Consider the spheres in the Kneser-Milnor decomposition of M and perturb them tomake them transverse to L . Since L is incompressible by the Loop Theorem, any intersection curve of L with a sphere bounds a disc in L ; so by performing surgeries on L we may find a surface disjoint from allthe spheres carrying the same fundamental group as L . Since the profinite topology on a free product ofresidually finite groups is efficient, hence induces the full profinite topology on any free factor, we havenow reduced to the case when M is an irreducible 3-manifold.The JSJ tori of M induce an efficient graph of groups decomposition of π M ([WZ10], Theorem A).After possibly performing a small isotopy of the tori, the intersections of the tori with L are a finitecollection of simple closed curves which split L as a finite graph of groups with finitely generated vertexgroups. The inclusion of F into M induces exactly such a map of graphs of groups as in Theorem 6.17.Each piece of the JSJ decomposition is either Seifert fibred or cusped hyperbolic, hence the fundamentalgroups are subgroup separable. In the Seifert fibred case this is a theorem of Scott [Sco78]. The cuspedhyperbolic case is part of the recent seminal advances in 3-manifold theory pioneered by Wise, Agol,Przytycki and many others. The reader is directed to [AFW15], Section 5.2 for a complete account andthe appropriate citations. Thus we may apply Theorem 6.17 and this completes the theorem. Theorem 6.19.
Let M be a compact orientable 3-manifold with incompressible boundary and let L be aproperly embedded, incompressible and boundary incompressible surface in M . Then π L is fully separablein π M .Proof. Let DM be the double of M along its boundary and let DL be the double of L along its boundary,canonically embedded in DM . If L is a closed surface let DL “ L . Since the boundary of M isincompressible π M injects into π DM . Then DL satisfies all the conditions of Theorem 6.18 and so π DL is fully separable in π DM . If g P π M r π L then g P π DM r π DL and a homomorphism from π DM to a finite group separating g from π DL separates g from π L in π M . So π L is separable.Now let U be a finite index subgroup of π L . The preimage DU of U under the ‘folding map’ π DL Ñ π L is a finite index subgroup of π DL meeting π L precisely in U . Since π DL is fullyseparable there exists a finite index subgroup V of π DM such that V X π DL ď DU . Then V X π M is a finite index subgroup of π M such that p V X π M q X π L “ V X π DL X π L ď DU X π L “ U as required. Corollary 6.20.
Let M be a compact 3-manifold with π -injective boundary and let L be a boundarycomponent of M . Then π L is fully separable in π M .Proof. We are free to pass to a double cover so that M is orientable. Then the boundary component L is orientable, incompressible (by the Loop Theorem) and hence also boundary incompressible. Theorem6.19 now applies and we are done. Theorem 6.21.
Let M be a compact aspherical 3-manifold with incompressible boundary components B M , . . . B M r . Let Γ “ π M , let Σ “ t π B M i u ď i ď r . Then p Γ , Σ q is a good pair (with respect to the setof all primes) and its profinite completion is a PD pair at every prime p .Proof. Goodness follows immediately from Corollary 6.20 and Propositions 6.3 once we know that 3-manifold groups are good, which is well-known and may be found in work of various authors. See[AFW15], Section 5.2 for a full account. That the profinite completion is a PD pair at every prime nowfollows from Proposition 6.15 and Theorem 6.9. 55 emark (Historical Note) . The full separability result above (Theorem 6.19) has various precursors inthe literature, although the result in its greatest strength seems to be new. The fact that boundarycomponents are separable was first proved by Long and Niblo [LN91]. In the case when the boundary istoroidal, full separability was established by Hamilton [Ham01]. As cited in the proof, separability forembedded surface subgroups is a theorem of Przytycki and Wise [PW14].
We conclude with a couple of simple applications of the theory of relative profinite duality groups, whichparallel and extend the results of [WZ17b].
Theorem 6.22.
Let M and N be compact orientable 3-manifolds with incompressible boundary. Let theKneser-Milnor decompositions of M and N be M “ M ¨ ¨ ¨ M r F and N “ N ¨ ¨ ¨ N s F whereeach M i and N j is irreducible and F k and F l are connect sums of k and l copies of ß21 respectively.Assume that there is an isomorphism Φ : z π M – z π N . Then r “ s , k “ l and, up to reordering, each z π M i is isomorphic to (a conjugate of ) z π N i via Φ .Remark. For the case of closed 3-manifolds this is Theorem 2.2 of [WZ17b].
Proof.
Consider the profinite tree T dual to the splitting z π N “ z π N i > ¨ ¨ ¨ > z π N i > z π F of z π N as an efficient graph of groups, and consider the action of each z π M i on T via Φ. If Σ i denotesthe family of fundmental groups boundary components of M lying in M i (the boundary components areincompressible, so the Kneser-Milnor decomposition only involves spheres which do not meet boundarycomponents) then by Theorem 6.21 the profinite completion p z π M i , S i q of the pair p π M i , Σ i q is a PD pair at every prime p . Therefore, since the edge stabilisers of the action are trivial, Corollary 5.14 impliesthat z π M i fixes some vertex of T , and hence is conjugate into some z π N j (it is not conjugate into x F s by reason of cohomological dimension). By symmetry every z π N j is conjugate into some z π M i . Assubgroups of profinite groups cannot be conjugate into proper subgroups of themselves, every z π M i isisomorphic via Φ to a conjugate of some z π N j . By considering the action on T one may readily see thatthe different z π M i are not conjugate to each other. Therefore r “ s and after reordering we have thateach z π M i is isomorphic to a conjugate of z π N i . Finally taking the quotient by the normal subgroupgenerated by the z π M i and the z π N i gives an isomorphism y π F – z π F whence k “ l as different freegroups are distinguished by their profinite completions. Proposition 6.23.
Let M be a compact hyperbolic 3-manifold with empty or toroidal boundary. If z π M acts on a profinite tree T with abelian edge stabilisers then z π M fixes a (unique) vertex.Remark. The statement of this proposition is identical with Lemma 4.4 of [WZ17b], where it is provedusing Dehn filling techniques. In that paper it is suggested that a theory of profinite duality pairs wouldprovide an alternative proof, and we include it here to illustrate that relative cohomology can indeed byused for this purpose.
Proof.
Let P , . . . , P r be representatives of the peripheral subgroups of π M , let G “ z π M and let S “ t p P , . . . , p P r u . Then by Theorem 6.21 the pair p G, S q is a PD pair at every prime. Suppose that G acts on a profinite tree T with abelian edge stabilisers. If G fixes a vertex then it fixes a uniquevertex since G is non-abelian and edge stabilisers are abelian (Corollary 2.9 of [ZM89]). So assume for acontradiction that G does not fix any vertex. Then by Corollary 5.14 there is some edge e P E p T q suchthat cd p p G e , S G e q ě
2. 56e will prove that G e is contained in some conjugate p P γi of a peripheral subgroup. First we recallthat for every g P G and p P i X p P gj ‰ i “ j and g P p P i . This critical fact—the malnormality of theperipheral subgroups—is Lemma 4.5 of [WZ17a].Now consider the group pair p G e , S G e q . If cd p p G e q ě G e is abelian) then by Theorem9.3 of [WZ17a] G e is conjugate into some peripheral subgroup. If on the other hand cd p p G e q “ p p G e , S G e q ě S G e . By definition of S G e thismeans that there exists some γ P G and some i such that G e X p P γi contains some non-trivial element h .Since G e is abelian, h is therefore also an element of p P γgi for every g P G e . By malnormality this forces g P p P γi . So G e is indeed contained in the conjugate p P γi of a peripheral subgroup.Consider the family of subgroups S G e “ ! G e X σ p y q p P i σ p y q ´ | x P X, y P G e z G { p P i , ď i ď r ) where σ : G e z G Ñ G is some continuous section. By malnormality, if the left hand side of G e X σ p y q p P j σ p y q ´ ď p P γi X σ p y q p P j σ p y q ´ is non-trivial for any y and j then i “ j and γσ p y q P p P i , and hence G e y p P j “ G e γ p P i . So precisely one ofthe groups in the family S G e is non-trivial, and equals G e . Therefore by Lemma 2.20 we have H k p G e , S G e ; M q “ H k p G e , t G e u ; M q “ k ě M P C p G e q and therefore cd p p G e , S G e q ă
2. This contradiction completes theproof.
Appendix A Profinite Direct Sums
We have required several facts about profinite direct sums of modules. To the author’s knowledge thesehave not yet appeared in these precise forms in published literature. It has seemed expedient to includethem here so that the curious reader has access to the proofs. Analogous facts for free profinite productsof groups appear in [Rib17]. One may adapt these to the case of modules, and this is what we dohere. The author claims no particular credit for most of these results or proofs, which are, apart fromProposition A.14 onwards, taken and adapted from material in Chapter 5 and Section 9.1 of [Rib17]. Inthis section all modules are compact.First we recall the definition.
Definition A.1.
Let R be a profinite ring. A sheaf of R -modules consists of a triple p M , µ, X q with thefollowing properties. • M and X are profinite spaces and µ : M Ñ X is a continuous surjection. • Each ‘fibre’ M x “ µ ´ p x q is endowed with the structure of a compact R -module such that themaps R ˆ M Ñ M , p r, m q Ñ r ¨ m M p q “ tp m, n q P M | µ p m q “ µ p n qu Ñ M , p m, n q Ñ m ` n are continuous.A morphism of sheaves p α, ¯ α q : p M , µ, X q Ñ p M , µ , X q consists of continuous maps α : M Ñ M and ¯ α : X Ñ X such that µ α “ ¯ αµ and such that the restriction of α to each fibre is a morphism of R -modules M x Ñ M ¯ α p x q . 57e often contract ‘the sheaf p M , µ, X q ’ to simply ‘the sheaf M ’. Regarding an R -module as a sheafover the one-point space one may talk of a sheaf morphism from a sheaf to an R -module. Definition A.2. A profinite direct sum of a sheaf M consists of an R -module Ð X M and a sheafmorphism ω : M Ñ Ð X M (sometimes called the ‘canonical morphism’) such that for any R -module N and any sheaf morphism β : M Ñ N there is a unique morphism of R -modules ˜ β : Ð X M Ñ N suchthat ˜ βω “ β .Note that as any compact module is an inverse limit of finite modules it is sufficient to verify thisuniversal property for finite N . Proposition A.3.
The profinite direct sum exists and is unique up to canonical isomorphism.Proof.
Uniqueness follows immediately from the universal property. Now define A “ à x P X M x to be the abstract direct sum. Let f : M Ñ A be the natural function and let U be the set of all finiteindex submodules U of A such that M Ñ A Ñ A { U is continuous. Denote by M the completion of A with respect to the topology U . That is, M is the inverse limit of the A { U for U P U . Then M is acompact R -module and f extends to a continuous sheaf morphism ω : M Ñ M . Note that since f p M q generates A the image of ω topologically generates M .Suppose that N is a finite R -module and β : M Ñ N is a morphism. Then there is a natural map ofabstract R -modules g : A Ñ N such that g ˝ f “ β , which is continuous. Hence ker p g q P U so g extendsto a continuous map of R -modules ˜ β : M Ñ N such that ˜ βω “ β . This map is unique since the image of ω generates M . So M satisfies the relevant universal property and is a profinite direct sum of M . Lemma A.4.
Let M be a sheaf of R -modules and let ν : M Ñ N be a continuous map to a finite R -module such that for some x P X the restriction ν x P M x Ñ N is a ring homomorphism. Then thereexists a neighbourhood Y of x in X such that ν y is a ring homomorphism for all y P Y .Proof. Consider the continuous maps η : R ˆ M p q Ñ N ˆ N , ρ : R ˆ M p q Ñ X given by η p r, m , m q “ p rν p m q ` ν p m q , ν p rm ` m qq , ρ p r, m , m q “ µ p m q “ µ p m q Since these maps are continuous and N is finite, the preimage of the diagonal DN “ tp n , n q P N ˆ N | n “ n u under η is open. Now consider the subset Y of X consisting of those y P X such that ρ ´ p y q Ď η ´ p DN q —that is, those y P X for which ν is an R -module morphism when restricted to M y .We must show that Y is open; but the complement of Y is simply ρ ` R ˆ M p q r η ´ p DN q ˘ which iscompact, hence closed as required. Proposition A.5.
Let M be a sheaf of R -modules, let x P X and let W be a clopen neighbourhood of x .Then every continuous morphism σ x : M x Ñ N to a finite module N can be extended to a morphism ofsheaves σ : M Ñ N such that σ y “ for all y R W .Proof. For every n P N the set σ ´ x p n q is compact. Since M is a profinite space there exists a clopensubset U n of X for each n such that the U n are disjoint for different n and such that σ ´ x p n q Ď U n . Definea continuous map ν from M to N by setting ν p m q “ n if m P U n m R Ť n P N U n This agrees with the R -module morphism σ x on M x . Applying Lemma A.4 there is a clopen neigh-bourhood Z of x in X , which without loss of generality is contained in W , such that ν restricts to an R -module morphism on M z for all z P Z . Finally define σ : M Ñ N to agree with ν on µ ´ p Z q and bythe zero map elsewhere. This is the required sheaf morphism.58 roposition A.6. Let p M , µ, X q be a sheaf of R -modules, let M be the profinite direct sum of M andlet ω : M Ñ M be the canonical morphism. Then • M is generated by the subgroups M x “ ω p M x q • If x ‰ y then M x X M y “ • ω maps M x isomorphically onto M x for all x P X Proof.
The first point holds by the explicit construction in Proposition A.3. For the second, take m P M x r t u and let m be a preimage of m in M x . There is then a finite module N and a module morphism σ x : M x Ñ N such that the image of m is non-trivial. By Proposition A.5 (taking W to be any clopenneighbourhood of x that does not include y ) we may extend this to a sheaf morphism σ : M Ñ N vanishing on M y . By the universal property this induces a morphism M Ñ N sending M y to 0 and m to σ x p m q ‰
0. This shows that M x X M y “
0. A similar argument shows that for any m P M x r t u there is a morphism M Ñ N to a finite module sending ω p m q to a non zero element, which proves thefinal item.We now discuss the connections between profinite direct sums and inverse limits. Lemma A.7.
Let p M i , µ i , X i q i P I be an inverse system of sheaves indexed over a directed poset p I, ě q and let p M , µ, X q be the inverse limit sheaf. Then every sheaf morphism β : M Ñ N to a finite module N factors through one of the M k .Proof. By assumption β is a continuous map from the profinite space M to N . Since M is the inverselimit of the M i as a topological space β factors through a continuous function β i : M i Ñ N for some i P I . Of course at this stage β i need not be a sheaf morphism. Set I “ t i P I | i ě i u . For every i P I define β i : M i Ñ N to be β i “ β i φ ii where φ ii is the transition map M i Ñ M i . We claim thatfor some k the map β k is a sheaf morphism, which will give the required factorisation of β . Note that β “ lim ÐÝ i ě i β i .Consider the map η : R ˆ M p q Ñ N ˆ N, η p r, m , m q “ p rβ p m q ` β p m qq and similar maps η i : R ˆ M p q i Ñ N ˆ N for each i P I with β i in place of β . It is readily seen that M p q “ lim ÐÝ i ě i M p q i , that η “ lim ÐÝ i ě i η i and that η p M p q q “ Ş i ě i η i p M p q i q . Now N ˆ N is a finiteset and I is a directed poset so at some point this intersection must stabilise and there exists k P I such that η p M p q q “ η k p M p q k q . Now β is a sheaf morphism so η p M p q q is contained in the diagonal DN “ tp n , n q P N ˆ N | n “ n u . Hence the image of η k is also contained in this diagonal. This isprecisely the statement that β k is a sheaf morphism and we are done. Proposition A.8.
For an inverse system of sheaves p M i , µ i , X i q indexed over a directed poset p I, ě q wehave ð X lim ÐÝ i P I M i “ lim ÐÝ i P I ð X i M i Proof.
Let M i “ Ð X i M i with canonical morphism ω i : M i Ñ M i . If i, j P I with i ě j let φ ij : M i Ñ M j be the transition map in the inverse system and let ψ ij : M i Ñ M j be the canonical morphisminduced by ω j φ ij : M i Ñ M j . Then t M i , ψ ij , I u is an inverse system of compact R -modules. Let M beits inverse limit and let ω “ lim ÐÝ ω i : M Ñ M , ψ i : M Ñ M i be the natural maps. We shall prove that M “ Ð X M with universal morphism ω .Let N be a finite R -module and let β : M Ñ N be a morphism. By the previous lemma, β factorsthrough some M k , and thereby induces a map ˜ β k : M k Ñ N . Define ˜ β “ ˜ β k ψ k . This is a map M Ñ N such that ˜ βω “ β . It is the unique such map since ω i p M i q generates M i for each i by Lemma A.6, so ω p M q generates M . Thus M has the appropriate universal property and we are done.59et M be an R -module and suppose F “ t M x u is a family of submodules continuously indexed by aprofinite space X (in the same sense as in Definition 1.37). We say M is an internal direct sum of F if: • M x X M y “ x ‰ y • If β : Ť x P X M x Ñ N is a continuous function to an R -module N which restricts to a morphism oneach M x then there is a unique extension of β to a morphism M Ñ N .Recall that since F is continuously indexed by X if and only if the triple p M , µ, X q is a sheaf of R -moduleswhere M “ tp m, x q P M ˆ X | g P M x u and µ is the restriction of the projection map—that is, if and only if M is a closed subset of M ˆ X . Proposition A.9.
The definitions of internal and external direct sums agree. More precisely, if M isthe internal direct product of the continuously indexed family F “ t M x u x P X then M is the external directsum of the sheaf M “ tp x, m q P X ˆ M | m P M x u Conversely if M is the external direct sum of a sheaf M with canonical morphism ω : M Ñ M then M is the internal direct sum of F “ t ω p M x qu x P X .Proof. First suppose that M is the internal direct sum of F . Let N be an R -module and let β : M Ñ N be a morphism. Noting that by definition Ť x P X M x is the quotient space of M under collapsing X ˆ t u to a point and X ˆ t u is mapped to 1 P N by β , the map β factors through a continuous map Ť M x Ñ N which agrees with β on each M x . Therefore β extends uniquely to a map ˜ βM Ñ N with ˜ βω “ β . So M is the external direct sum of the sheaf M .Conversely suppose that M is the external direct sum of M (with canonical morphism ω ). That thefamily F is continuously indexed by X follows easily from the definition of a sheaf. Let β : Ť x P X M x Ñ N be a continuous function to an R -module N which restricts to a morphism on each M x . Then βω is asheaf morphism M Ñ N and so there is a unique ring morphism M Ñ N extending βω —and hence aunique morphism extending β itself, since any such morphism would extend βω . It only remains to checkthat M x X M y “ x ‰ y . This follows from Proposition A.6.We record the following simple lemma for later use. Lemma A.10.
Let p M , µ, X q be a sheaf of R -modules, and suppose that there is at most one X forwhich M x is not the zero module. Then Ð X M “ M x .Proof. One may readily see that M x is the internal direct sum of the M y for y P Y . So the result followsfrom Lemma A.9. Proposition A.11.
Let M be an R -module and F “ t M x u be a family of submodules indexed by aprofinite space X . Suppose there exist inverse systems p A i , φ i,j , I q and p X i , f i,j , I q where the A i arecompact R -modules and the X i are profinite spaces such that(1) X “ lim ÐÝ X i (2) For every i P I we have A i “ Ð x i P X i A i,x i for some collection of submodules A i,x i continuouslyindexed by X i .(3) φ i,j p A i,t q Ď A j,f i,j p x i q for every i ě j and every x i P X i .(4) For every x “ p x i q P X “ lim ÐÝ X i we have M x “ lim ÐÝ A i,x i (5) M “ lim ÐÝ A i hen F is continuously indexed by X and M is the profinite direct sum of F .Proof. This is a straightforward translation of Proposition A.8 into the language of internal direct sumsusing Proposition A.9. The various conditions of the proposition guarantee that the natural sheaf corre-sponding to the family F is an inverse limit of sheaves corresponding to the families t A i,x i u .The converse statement is that there always exists an expression of a profinite direct sum as an inverselimit of finite direct sums. Furthermore the summands may always be taken to be finite. Proposition A.12.
Let M be an R -module and F “ t M x u be a family of submodules indexed by aprofinite space X . Suppose F is continuously indexed by X and M is the profinite direct sum of F . Thenthere exist inverse systems p A i , φ i,j , I q and p X i , f i,j , I q where the A i are finite R -modules and the X i arefinite discrete spaces such that(1) X “ lim ÐÝ X i (2) For every i P I we have A i “ À x i P X i A i,x i for some collection of finite R -modules A i,x i indexed by X i .(3) φ i,j p A i,x i q Ď A j,f i,j p x i q for every i ě j and every x i P X i .(4) For every x “ p x i q P X “ lim ÐÝ X i we have M x “ lim ÐÝ A i,x i (5) M “ lim ÐÝ A i Proof.
Let R be the set of all equivalence relations on X whose equivalence classes are clopen. Let I “ t i “ p R, U q | R P R , U an open submodule of M u and make I into a directed poset by declaring p R, U q ě p R , U q if and only if U Ď U and R Ď R (that is, xRy implies xR y ). For i “ p R, U q P I define X i “ X { R , let x i “ r x s R be the equivalence class of x P X under R , let A i,x i “ M p x i q U { U and A i “ À x i P X i A i,x i . Here M p x i q is the submodule of M generatedby all M y for y P x i “ r x s R .For i “ p R, U q ě p R , U q “ j define f ij : X i Ñ X j to be the natural map of quotient spaces.Note that the inclusion r x s R Ď r x s R induces an inclusion M p x i q Ď M p x j q and therefore induces maps A i,x i Ñ A j,f ij p x i q for all x i P X i . Thus we also have a natural continuous morphism ψ ij : A i Ñ A j whichis an epimorphism since both sides are generated by the images of the M y as y ranges over all y P X .Certainly t A i , ψ ij , I u and t X i , f ij , I u are inverse systems and X “ lim ÐÝ X i . So we have conditions (1)–(3).Let x P X and let x i be the image of x in X i . We claim that M x “ lim ÐÝ A i,x i “ lim ÐÝ M p x i q U { U . Firstnote that I “ i “ p R , U q | M pr x s R q U { U “ M x U { U ( is a directed sub-poset of I . For let i “ p R , U q and i “ p R , U q be in I . Set U “ U X U . Now sincethe family F is continuously indexed by X the set of y such that M y Ď M x U is open in X . Then we maychoose R P R such that R Ď R X R and M y Ď M x U for all y P r x s R . Then certainly i “ p R, U q ě i , i .Furthermore we certainly have M pr x s R q U “ M x U so i P I . The sub-poset I is also cofinal in I . Forif i “ p R, U q P I then again t y P X | M y Ď M x U u is open so we may find R P R with R Ď R and r x s R Ď t y P X | M y Ď M x U u . Then p R , U q ě p R, U q and p R , U q P I . It follows thatlim ÐÝ i P I A i,x i “ lim ÐÝ i P I A i,x i “ lim ÐÝ p R,U qP I M pr x s R q U { U “ lim ÐÝ U M x U { U “ M x as claimed.Finally we have a natural epimorphism ρ : M Ñ lim ÐÝ A i . By point (4) each M x embeds in the inverselimit under this map, and by Proposition A.11 the inverse limit of the A i is the profinite direct sum ofthe image of the family F . Since M is also this direct sum, the map ρ is an isomorphism and M “ lim ÐÝ A i as required. 61 roposition A.13. Let p M , µ, X q be a sheaf of compact R -modules where R “ Z π rr G ss and let M “ Ð X M . As usual identify each fibre M x with its image M x in the profinite direct sum M . Let F be afunctor from C π p G q to itself which is additive and commutes with inverse limits. Then: • for each x P X , the natural map M x Ñ M gives a canonical embedding of F p M x q in F p M q • F p M q “ Ð x P X F p M x q In particular this holds for
Tor functors.Proof.
Write M as an inverse limit of finite direct sums A i,x i as in Proposition A.12. Since both finitedirect sums and inverse limits commute with F we have F p M q “ F ˜ lim ÐÝ i à t P X i A i,x i ¸ “ lim ÐÝ i à x i P X i F p A i,x i q “ ð x P X F p M x q as required. The fact that the final result is a well-defined (internal) direct sum follows since the finalinverse limit satisfies the conditions of Proposition A.11. In particular the canonical map F p M x q Ñ F p M q is an embedding by Lemma A.6.Next we remark upon relations between profinite direct sums and free modules. For the theory of freeprofinite modules over profinite spaces see Section 5.2 of [RZ00b]. Proposition A.14.
Let µ : Y Ñ X be a surjection of profinite spaces and let R be a profinite ring. Then R rr Y ss is the internal profinite direct sum R rr Y ss “ ð x P X R rr µ ´ p x qss Proof.
Write µ as an inverse limit of maps of finite spaces µ i : Y i Ñ X i . Then we have R rr Y ss “ lim ÐÝ R rr Y i ss “ lim ÐÝ à x P X i R rr µ ´ i p x qss “ ð x P X R rr µ ´ p x qss noting that when Y i is a disjoint union of finitely many clopen sets µ ´ i p x q the second equality followsimmediately from the universal properties of finite direct sums and free modules. The fact that thefinal result is a well-defined (internal) direct sum follows since the inverse limit satisfies the conditions ofProposition A.11.There is one final proposition we will need. Proposition A.15.
Let R “ Z π rr G ss where π is a set of primes and G is a profinite group. Let p M , µ, X q be a sheaf of compact R -modules and suppose that there exists p P π such that the fibres M x admits asurjection to the trivial R -module F p for infinitely many x P X . Then M “ Ð X M is not finitelygenerated over R .Proof. Let n be a natural number. It is sufficient to prove that for any n there is a surjection M Ñ F np ,since such a sum cannot be generated by fewer than n elements and hence neither can M .Take points x , . . . , x n in X with M x i admitting a surjection φ i : M x i Ñ F p and take disjoint clopenneighbourhoods W i of each x i . By Proposition A.5 there is then a sheaf morphism β i : M Ñ F p extending φ i and vanishing outside W i . This induces an epimorphism ˜ β i : M Ñ F p which is a surjection whenrestricted to M x i and vanishes on all M x j for j ‰ i . The product map to F np is therefore also a surjectionand we are done. 62 emark. The requirement in the proposition is stronger than ‘infinitely many M X are non-zero’, andis necessary. For instance suppose there exist at least countably many distinct finite simple R -modules S n p n P N q . This is the case unless G is virtually pro- p by Corollary 5.3.5 of [Ben98]. Then one may formthe inverse limit M “ lim ÐÝ n ` S ‘ ¨ ¨ ¨ ‘ S n ˘ where the maps in the inverse limit are the obvious projections. One may see by Proposition A.11 that M is the direct sum of a sheaf whose base space is the one-point compactification of N and with infinitelymany non-zero fibres. However each p S ‘ ¨ ¨ ¨ ‘ S n q in the limit may be generated by a single element,hence so can M .For arbitrary G and a collection S “ t S x u x P X of subgroups continuously indexed by a space X wenote that every fibre of Z π rr G { S ss “ ð x P X Z π rr G { S x ss surjects to the simple module F p for any p P π and so Z π rr G { S ss is finitely generated if and only if X isfinite. Appendix B Supplement on Graphs of Profinite Groups
Here we will prove some propositions which are necessary for the results in Section 5.2. We did notinclude it in that section as it would have rather disrupted the flow of the paper.
Proposition B.1.
Let p be a prime, G be a profinite group and S a closed subgroup of G such that p divides r G : S s . Then F p rr G { S ss has no non-zero G -invariant elements. In particular the naturalaugmentation map F p rr G { S ss Ñ F p does not split.Proof. Write S as the intersection of a sequence of open subgroups U i of G such that p | r U i : U i ` s forall i . Then F p rr G { S ss “ lim ÐÝ F p rr G { U i ss “ lim ÐÝ F p r G { U i s Now assume that F p rr G { S ss has a non zero G -invariant element m , and let m i be its image in F p r G { U i s for each i . Then the m i map to each other under the maps in the inverse system and for some i , theelement m i is non-zero. Now the G -invariant elements of F p r G { U i s are precisely the elements λN i : “ ÿ gU i P G { U i λgU i for λ P F p . However the image of λN i ` in F p r G { U i s is exactly λ r U i : U i ` s N i for all i and all λ , whichvanishes since p | r U i : U i ` s . We have reached a contradiction. Remark.
The assumption that p | r G : S s is of course necessary. One may readily see that the element ` q ´ k N k ˘ k ě P lim ÐÝ F p p Z { q k q “ F p p Z q q is Z q -invariant where p and q are any two distinct primes.Before the next proposition we must recall a fact about the structure of fundamental groups of graphsof pro- π groups. Specifically take a finite graph of profinite groups p X, G ‚ q . Let G abs be the fundamentalgroup of p X, G ‚ q as a graph of abstract groups . Then the pro- π fundamental group of p X, G ‚ q is exactlythe completion of G abs with respect to the topology U “ N Ÿ f G abs | G i X N Ÿ o G i and G abs { N is a π -group ( Ÿ f and Ÿ o mean normal subgroups of finite index and open normal subgroups respectively. SeeProposition 6.5.1 of [Rib17] and the discussion leading to it.Let C be an variety of finite groups closed under taking isomorphisms, subgroups, quotients andextensions. Let π p C q be the set of primes which divide the order of some finite groups in C . Proposition B.2.
Let G be either a proper pro- C HNN extension G “ G > L or a proper pro- Ci amal-gamated free product G “ G > L G where L ‰ G i for each i . Then p | r G : G s for all p P π p C q .Proof. In the HNN extension case there is a map G Ñ Z π whose kernel is the normal subgroup generatedby G and the result follows immediately. For the amalgamated free product case we must work a littleharder. By the assumption of the theorem there is a map from φ : G Ñ P to a finite π -group such that φ p L q is a proper subgroup of φ p G i q for each G . Then G “ ker φ is the fundamental group of a graph ofgroups whose base graph is not a tree. More precisely the Euler characteristic of the graph is χ “ r P : φ p G qs ` r P : φ p G qs ´ r P : φ p L qs which is non-positive. One may see this either by considering the action of G on the standard tree of thegraph of pro- C groups decomposition of G , or by translating to the language of abstract graphs of groupsand back again using the discussion prior to the theorem. Since this graph now has a loop, the quotientof G by all its vertex groups is a non-trivial free pro- C group of rank 1 ´ χ . This follows for exampleby the argument in Theorem 6.2.4 of [Rib17], or using the above translation to the theory of graphs ofabstract groups. Hence all vertex groups G v of G have p | r G : G v s . Since for each i , G and G i arefinite index overgroups of G and G v for some v they also have this property as required. References [AFW15] Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton. . EuropeanMathematical Society, Zurich, 2015.[BE78] Robert Bieri and Beno Eckmann. Relative homology and Poincar´e duality for group pairs.
Journal of Pure and Applied Algebra , 13(3):277–319, 1978.[Ben98] David J. Benson.
Representations and cohomology I , volume 30 of
Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Cambridge, 1998.[Bro12] Kenneth S. Brown.
Cohomology of groups , volume 87. Springer Science & Business Media,2012.[Bru66] Armand Brumer. Pseudocompact algebras, profinite groups and class formations.
Journal ofAlgebra , 4(3):442–470, 1966.[Hal49] Marshall Hall. Coset representations in free groups.
Transactions of the American MathematicalSociety , 67(2):421–432, 1949.[Ham01] Emily Hamilton. Abelian Subgroup Separability of Haken 3-Manifolds and Closed Hyperbolic n -Orbifolds. Proceedings of the London Mathematical Society , 83(3):626–646, 2001.[LN91] Darren D. Long and Graham A. Niblo. Subgroup separability and 3-manifold groups.
Mathe-matische Zeitschrift , 207(1):209–215, 1991.[Ple80a] Andrew Pletch. Profinite duality groups I.
Journal of Pure and Applied Algebra , 16(1):55–74,1980. 64Ple80b] Andrew Pletch. Profinite duality groups II.
Journal of Pure and Applied Algebra , 16(3):285–297, 1980.[PW14] Piotr Przytycki and Daniel T. Wise. Separability of embedded surfaces in 3-manifolds.
Com-positio Mathematica , 150(9):1623–1630, 2014.[Rib69] Luis Ribes. On a cohomology theory for pairs of groups.
Proceedings of the American Mathe-matical Society , 21(1):230–234, 1969.[Rib17] Luis Ribes.
Profinite Graphs and Groups , volume 66 of
A Series of Modern Surveys in Math-ematics . Springer, 2017.[RZ00a] Luis Ribes and Pavel Zalesskii. Pro- p trees and applications. In New horizons in pro- p groups ,pages 75–119. Springer, 2000.[RZ00b] Luis Ribes and Pavel Zalesskii. Profinite groups . Springer, 2000.[Sco78] Peter Scott. Subgroups of surface groups are almost geometric.
Journal of the London Math-ematical Society , 2(3):555–565, 1978.[Ser13] Jean-Pierre Serre.
Galois cohomology . Springer Science & Business Media, 2013.[SW00] Peter Symonds and Thomas Weigel. Cohomology of p -adic analytic groups. In New horizonsin pro- p groups , volume 184, pages 347–408. Birkh¨auser Boston, 2000.[Wei95] Charles A. Weibel. An introduction to homological algebra . Number 38 in ‘Cambridge studiesin advanced mathematics’. Cambridge university press, 1995.[Wil16] Gareth Wilkes. Virtual pro- p properties of 3-manifold groups. Journal of Group Theory , 2016.[WZ10] Henry Wilton and Pavel Zalesskii. Profinite properties of graph manifolds.
Geometriae Dedi-cata , 147(1):29–45, 2010.[WZ17a] Henry Wilton and Pavel Zalesskii. Distinguishing geometries using finite quotients.
Geometry& Topology , 21(1):345–384, 2017.[WZ17b] Henry Wilton and Pavel Zalesskii. Profinite detection of 3-manifold decompositions. arXivpreprint arXiv:1703.03701 , 2017.[ZM89] PA Zalesskii and Oleg Vladimirovich Mel’nikov. Subgroups of profinite groups acting on trees.