Representation stability and outer automorphism groups
aa r X i v : . [ m a t h . R T ] F e b REPRESENTATION STABILITY AND OUTER AUTOMORPHISM GROUPS
LUCA POL AND NEIL P. STRICKLAND
Abstract.
In this paper we study families of representations of the outer automorphism groups indexed ona collection of finite groups U . We encode this large amount of data into a convenient abelian category AU which generalizes the category of VI-modules appearing in the representation theory of the finite generallinear groups. Inspired by work of Church–Ellenberg–Farb, we investigate for which choices of U the abeliancategory is locally noetherian and deduce analogues of central stability and representation stability resultsin this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibitrepresentation stability. Contents
1. Introduction 12. Preliminaries 53. Subcategories and their properties 74. Closed monoidal structure 95. Functors for subcategories 156. Simple objects 197. Finite groupoids 208. Projectives 229. Colimit-exactness 2410. Complete subcategories 2711. Finiteness conditions 2912. Torsion and torsion-free objects 3113. Noetherian abelian categories 3614. Representation stability 4315. Injectives 45References 461.
Introduction
In this paper we develop a framework for studying families of representations of the outer automorphismgroups. A common theme in representation theory is that there is a conceptual advantage in encodingthis large amount of (possibly complicated) data into a single object, which lives in a convenient abeliancategory. Using purely algebraic techniques we will deduce strong constraints on naturally occurring familiesof representations of the outer automorphism groups. We will then provide a range of examples for ourtheory coming from rational global homotopy theory.
The main character.
Fix k a field of characteristic zero and let G denote the category of finite groups andconjugacy classes of surjective group homomorphisms. We are interested in the category A = [ G op , Vect k ] f contravariant functors from G to the category of k -vector spaces. More generally, we will restrict ourattention to a replete full subcategory U ≤ G and then consider the smaller category AU = [ U op , Vect k ].Note that the endomorphism group of an object G ∈ U is the outer automorphism group U ( G, G ) = Out( G ).Therefore any object X ∈ AU gives rise to a collection of Out( G )-representations X ( G ) for G ∈ U . Thefunctoriality of X imposes further compatibility conditions on these representations. There are two mainexamples where all this data can be made very explicit. Example.
Consider the category C [2 ∞ ] of cyclic 2-groups. An object X ∈ AC [2 ∞ ] gives rise to a consistentsequence of representations of cyclic 2-groups: X (1) X ( C ) X ( C ) X ( C ) X ( C ) X ( C ) · · · C C C C where the horizontal maps are induced by the canonical projections. Example.
Fix a prime number p and consider category E [ p ] of elementary abelian p -groups. An object X ∈ AE [ p ] gives rise to a consistent sequence of representations of the finite general linear groups: X (1) X ( C p ) X ( C p ) X ( C p ) X ( C p ) X ( C p ) · · · GL ( F p ) GL ( F p ) GL ( F p ) GL ( F p ) GL ( F p ) where the horizontal maps are induced by the projection into the first coordinates.As we have already seen in the previous examples, it will often be convenient to restrict attention to specialsubcategories U (always full and replete) for which certain phenomena stand out more clearly. For example: • We might fix a prime p and restrict attention to p -groups. • We might restrict attention to solvable, nilpotent or abelian groups. • We might impose upper or lower bounds on the exponent, nilpotence class, order, or on the size ofa minimal generating set. • As special cases of the above, we might consider only cyclic groups, or only elementary abelian p -groups for some fixed prime p .To ensure good homological properties, we will impose additional conditions on U such as: • Closure under products: If
G, H ∈ U , then G × H ∈ U . If this holds, we say that U is multiplicative . • Closure under passage to subgroups: If G ∈ U and H ≤ G , then H ∈ U . • Downwards closure (i.e. closure under passage to quotients): If G ∈ U and G ( G, H ) = ∅ , then H ∈ U . • Upwards closure: If H ∈ U and G ( G, H ) = ∅ , then G ∈ U .We will see throughout this introduction that AU has its best homological behaviour when U is submulti-plicative (multiplicative and closed under passage to subgroups), or a global family (closed downwards andclosed under passage to subgroups). We refer the reader to Section 3 for a detailed list of all the closureproperties considered in this paper together with some examples.Before presenting our results we put the abelian category AU in the relevant context. Representations of combinatorial categories.
The abelian category AU is part of a larger family ofcategories appearing in representation theory and algebraic topology. Given a category I whose objects arefinite sets (with possibly extra structure) and whose morphisms are functions (possibly respecting the extrastructure), we can consider the associated diagram category A I = [ I , Vect k ]. Some examples of interestinclude: • Let FI be the category of finite sets and injections. The associated diagram category is the categoryof FI-modules which appears in [20] in the context of stable homotopy groups of symmetric spectra,and in [2, 3] in relation to the representation theory of the symmetric groups. Let VI be the category of finite dimensional F p -vector spaces and injective linear maps. The asso-ciated diagram category is the category of VI-modules which appears in [6, 13] in relation to therepresentation theory of the finite general linear groups. This category is equivalent by Pontryaginduality to the category AE [ p ] mentioned earlier. • Let VA be the category of finite dimensional F p -vector spaces and all linear maps. The associateddiagram category have been studied in relation to algebraic K -theory, rational cohomology, and theSteenrod algebra [11].Despite the similarities with other abelian categories appearing in representation theory, there is a majordifference between AU and all these categories. We are no longer considering a one-parameter family ofrepresentations but rather collections of representations which are indexed by a family of groups. Thisbrings into play group-theoretic properties of the family U and so introduces a new level of complexity intothe story which has so far not been explored. Noetherian condition.
The category AU is a Grothendieck abelian category with generators given by therepresentable functors e G = k [ U ( − , G )] G ∈ U . Many of the familiar notions from the theory of modules carry over to this setting. For example, there arenotions of finitely generated and finitely presented objects, see Definition 11.1 for the details. We then saythat abelian category AU is locally noetherian if all subobjects of e G are finitely generated for all G ∈ U . Itis then a formal consequence of the definition that subobjects of finitely generated objects are again finitelygenerated, and that any finitely generated object is also finitely presented.Work of Church–Ellenberg–Farb in the category of FI-modules showed that the noetherian condition plays afundamental role when working with sequences of representations [2]. This key technical innovation allowedthem to prove an asymptotic structure theorem for finitely generated FI-modules which gave an elegant expla-nation for the representation-theoretic patterns observed in earlier work [4]. Motivated by this, we investigatefor which choices of U the category AU is locally noetherian. The next result combines Proposition 13.3 andTheorem 13.15 in the body of the paper. Theorem A.
Let U be a subcategory of G and let p be a prime number. (a) If U is a multiplicative global family of finite abelian p -groups, then AU is locally noetherian. (b) If U is the global family of cyclic p -groups, then AU is locally noetherian.If U contains the trivial group and infinitely many cyclic groups of prime order, then AU is not locallynoetherian. In particular, A is not locally noetherian. There are several combinatorial criteria available in the literature to show that the category AU is locallynoetherian. We prove part (a) using the theory of Gröbner bases developed by Sam–Snowden [18], and part(b) using the criterion developed in [6]. Our result does not aim to give a complete classification of locallynoetherian categories, as this would be costly and highly non-trivial, but rather aims to give a good rangeof examples and counterexamples to which our theory applies.We then turn to study homological properties of our category of interest. Homological properties.
The levelwise tensor product of k -vector spaces gives AU a symmetric monoidalstructure in which the unit object is the constant functor with value k . For all X, Y ∈ AU , there exists aninternal hom object that we denote by Hom(
X, Y ) ∈ AU .We list a few interesting homological properties that our category enjoys.(i) As is typical for diagram categories, the finitely generated projective objects are not strongly dual-izable. In particular this means that the canonical map e G ⊗ Hom( e G , ) → Hom( e G , e G )is in general not an isomorphism, see Remark 4.3. However, the finitely generated projective ob-jects of AU still form a subcategory that is closed under tensor products and internal hom, seeProposition 4.13 and Theorem 4.20. ii) As is typical for diagram categories, any projective object is a retract of a direct sum of generators,see Lemma 8.2. However, under mild conditions on U (satisfied by G ) the projective objects of AU coincide with the torsion-free injective objects, see Proposition 15.1. In particular, the generators e G are injectives.(iii) Under mild conditions on U (which are satisfied by G itself), the only objects with a finite projectiveresolution are the projective ones, see Proposition 11.5.(iv) The abelian category AU is semisimple if and only if U is a groupoid, see Proposition 6.3. Representation stability.
A common goal in the representation theory of categories is to give an uniformdescription of the representations encoded into an object X ∈ AU . For example, an important result in thisfield is to show that finitely presented object can be recovered by finite amount of data via a “stabilizationrecipe”. This phenomenon is called central stability and it was first introduced by Putman [16] for describingcertain stability phenomena of the general linear groups. Since then, central stability has been shown to holdfor various diagram categories such as FI-modules [3] and complemented categories [17]. We will show thatcentral stability holds in our situation as well. The following summarizes some of the results in Section 14. Theorem B (Central stability) . Let U be a subcategory of G , and consider a finitely presented object X ∈ AU .Then there exists a natural number n ∈ N such that for all G ∈ U , we have X ( G ) = lim −→ H ∈ N ( G,n ) X ( G/H ) where N ( G, n ) = { H ⊳ G | |
G/H | ≤ m } . This result illustrates the fact that the representations encoded in a finitely presented object need to satisfystrong compatibility conditions. It tells us that we can recover the value X ( G ) from a finite amount of data,namely the poset N ( G, n ) and the representations X ( G/H ). We note that the poset N ( G, n ) is always finiteand often can be determined by purely combinatorial means. For instance, in the abelian p -group case itscardinality can be explicitly calculated using the Hall polynomials [1, 2.1.1].Given an epimorphism α : B → A , we also investigate the behaviour of the structure maps α ∗ : X ( A ) → X ( B )for sufficiently large groups A and B . In this case however, we need to restrict to the locally noetherian case.Consider the following families of finite abelian p -groups: F [ p n ] = { free Z /p n -modules } and C [ p ∞ ] = { cyclic p -groups } . The following is an adaptation in our setting of the injectivity and surjectivity conditions in the definitionof representation stability due to Church–Farb [4, 1.1].
Definition A.
Let U be either C [ p ∞ ] or F [ p n ] for some n ≥
1. Consider an object X ∈ AU . • We say that X is eventually torsion-free if there exists r ∈ N such that for every morphism α : B → A with | A | ≥ r , the induced map α ∗ : X ( A ) → X ( B ) is injective. • We say that X is stably surjective if there exists r ∈ N such that the canonical map X ( A ) ⊗ k [ U ( B, A )] → X ( B ) , ( x, α ) α ∗ ( x )is surjective, for all | B | ≥ | A | ≥ r .We are finally ready to state our second result which illustrates the fact that the structure maps of a finitelygenerated object need to satisfy strong compatibility conditions, see Theorem 14.6 in the body of the paper. Theorem C.
Fix a prime number p . Let Z [ p ∞ ] be the family of finite abelian p -groups and consider afinitely generated object X ∈ AZ [ p ∞ ] . Then the restriction of X to C [ p ∞ ] and F [ p n ] for n ≥ , is eventuallytorsion-free and stably surjective. lobal homotopy theory. A good source of examples of finitely generated objects satisfying representationstability comes from global stable homotopy theory: the study of spectra with a uniform and compatiblegroup action for all groups in a specific class. These are particular kind of spectra that give rise to cohomologytheories on G -spaces for all groups in the chosen class. The fact that all these individual cohomology theoriescome from a single object imposes extra compatibility conditions as the group varies. In this paper we willuse the framework of global homotopy theory developed by Schwede [21]. His approach has the advantageof being very concrete as the category of global spectra is the usual category of orthogonal spectra but witha finer notion of equivalence, called global equivalence. As any orthogonal spectrum is a global spectrum,this approach comes with a good range of examples. For instance, there are global analogues of the spherespectrum, cobordism spectra, K -theory spectra, Borel cohomology spectra and many others. It is a specialfeature of such a global spectrum X that the assignment G π (Φ G X ) ⊗ Q defines an object Φ ( X ) ∈ A ,where we put k = Q . The connection with global homotopy theory is even stronger as there is a triangulatedequivalence(1.0.1) Φ G : Sp Q G ≃ △ D ( A )between the homotopy category of rational G -global spectra and the derived category of A [21, 4.5.29]. Thisequivalence is compatible with geometric fixed points in the sense that π ∗ (Φ G X ) = H ∗ (Φ G ( X ))( G ).We obtain the following application to global homotopy theory which highlights the good behaviour of thegeometric fixed points functor on the full subcategory of compact global spectra. Recall that an object X in a triangulated category T is said to be compact if the representable functor T ( X, − ) preserves arbitrarysums. The proof of the following result can be found in Section 14. Theorem D.
Let Z [ p ∞ ] be the family of finite abelian p -groups and let X be a rational Z [ p ∞ ] -global spectrum.If X is compact, then for all k ∈ Z the geometric fixed points homotopy groups Φ k ( X ) ∈ AZ [ p ∞ ] satisfy theconditions of Theorems B and C. An interesting source of examples is given by the rational n -th symmetric product spectra. Example.
For n ≥
1, we let Sp n denote the orthogonal spectrum whose value at inner product space V isgiven by Sp n ( V ) = ( S V ) × n / Σ n . Its rationalization is a compact rational Z [ p ∞ ]-global spectrum by [8, 2.10, 5.1]. Therefore its geometricfixed points are eventually torsion-free, stably surjective and they satisfy central stability. Related work.
Our study of the representation theory and homological algebra of AU is inspired by earlierwork in the categories of FI-modules [2,3] and VI-modules [7,13]. Our Theorem A recovers the result that thecategory of VI-modules is locally noetherian, which was proved independently by Sam–Snowden [18, 8.3.3]and Gan–Li [6]. Versions of our representation stability theorems were already known to hold for the categoryof FI-modules [3], VI-modules [7] and complemented categories [17]. Finally our study of indecomposableinjective objects recovers part of the classification of injective VI-modules due to Nagpal [13].Nonetheless, to the best of our knowledge the results of this introduction are new and they generalize severalknown results to a wider class of examples of interest. Acknowledgements.
The first author thanks the SFB 1085 Higher Invariants in Regensburg for support,John Greenlees, Birgit Richter, Sarah Whitehouse and Jordan Williamson for reading preliminary versionsof this paper. 2.
Preliminaries
We start by introducing the main object of study of this paper, the abelian category A . Definition 2.1. • Let H be a group, and h an element of H . We write c h : H → H for the inner automorphism x hxh − . Let G be another group, and let ϕ, ψ : G → H be homomorphisms. We say that ϕ and ψ are conjugate if ψ = c h ◦ ϕ for some h ∈ H . This is easily seen to be an equivalence relation that iscompatible with composition. We write [ ϕ ] for the conjugacy class of ϕ . • We write G for the category whose objects are finite groups, and whose morphisms are conjugacyclasses of surjective homomorphisms. We also write Out( G ) = G ( G, G ). Lemma 2.2.
Let α : H → G be a surjective group homomorphism between finite groups. Then [ α ] is anepimorphism in G .Proof. Consider two surjective group homomorphisms β, γ : G → K , and suppose that [ βα ] = [ γα ]. Thismeans that c k βα = γα for some k ∈ K . Since α is surjective we have c k β = γ which shows that [ β ] = [ γ ]. (cid:3) Definition 2.3.
Fix a field k of characteristic zero, and set A = [ G op , Vect k ]. Given G ∈ G , we write ψ G : A →
Vect k for the evaluation functor X X ( G ). Definition 2.4.
Let U be a subcategory of G . Unless we explicitly say otherwise, such subcategories areassumed to be full and replete. (Replete means that any object of G isomorphic to an object of U is itself in U .) We then put AU = [ U op , Vect k ]. Remark 2.5.
The category AU is abelian and admits limits and colimits for all small diagrams. These(co)limits are computed pointwise, so they are preserved by the evaluation functors ψ G : AU →
Vect k . Definition 2.6.
Consider an object X ∈ AU . • The base of X is defined by base( X ) = min {| G | | X ( G ) = 0 } ∈ N . If X is zero, we set base( X ) = ∞ . • The support of X is defined by supp( X ) = { [ G ] | X ( G ) = 0 } where [ G ] denotes the isomorphismclass of the group G . We equipped the support with the partial order [ G ] ≫ [ H ] if and only if U ( G, H ) = ∅ . Definition 2.7.
Consider a subcategory
U ≤ G . We define objects of AU as follows. Most of them dependon an object G ∈ U , and possibly also a module V over k [Out( G )]. • We define e G by e G ( T ) = k [ G ( T, G )]. Yoneda’s Lemma tells us that AU ( e G , X ) = X ( G ) = ψ G ( X ). • We define objects e G,V and t G,V by e G,V ( T ) = V ⊗ k [Out( G )] k [ G ( T, G )] t G,V ( T ) = Hom k [Out( G )] ( k [ G ( G, T )] , V ) . • We put c G ( T ) = e G ( T ) Out( G ) = k [ G ( T, G ) / Out( G )] . Note that the basis set G ( T, G ) / Out( G ) here can be identified with the set of normal subgroups N ≤ T such that T /N ≃ G . Alternatively, we can regard k as a k -linear representation of Out( G )with trivial action, and then c G = e G,k . • The groups e G,V ( G ) and t G,V ( G ) are both canonically identified with V , and one can check thatthere is a unique morphism α : e G,V → t G,V with α G = 1. We write s G,V for the image of this.If T = G then s G,V ( T ) = V . If T ≃ G then s G,V ( T ) is canonically isomorphic to e G,V ( T ) or t G,V ( T ), but a choice of isomorphism T → G is needed to identify s G,V ( T ) with V . If T G then s G,V ( T ) = 0. • Now let C be a subcategory of U . Suppose that C is convex , which means that whenever G → H → K are surjective homomorphisms with G, K ∈ C and H ∈ U we also have H ∈ C . We then define the“characteristic function” χ C ∈ AU by χ C ( T ) = ( k if T ∈ C T
6∈ C . (Convexity ensures that this can be made into a functor in an obvious way: the map χ C ( T ) → χ C ( T ′ )is the identity if both groups are nonzero, and zero otherwise.)If we need to specify the ambient category U , we may write e U G rather than e G , and so on. emark 2.8. The abelian category AU is Grothendieck with generators given by e G for all G ∈ U . Thismeans that filtered colimits are exact and that any X ∈ A admits an epimorphism P → X where P is adirect sum of generators. Lemma 2.9.
For G ∈ U , we let M G denote the category of k [Out( G )] -modules. Then the evaluation functor ev G : AU → M G , X X ( G ) has a left and right adjoint which are respectively given by e G, • and t G, • . In particular, e G,V is projectiveand t G,V is injective.Proof.
The unit of the adjunction η V : V → e G,V ( G ) = V is the identity, and the counit is given by ǫ X ( T ) : e G,X ( G ) ( T ) → X ( T ) , x ⊗ [ α ] α ∗ ( x )for all T ∈ G . Similarly, the counit map t G,V ( G ) → V is the identity, and the unit is given by η X ( T ) : X ( T ) → t G,X ( G ) ( T ) , x ([ β ] β ∗ ( x ))for all T ∈ G . We leave to the reader to check that these maps are natural and that they satisfy the triangularidentities. The second part of the claim follows immediately from the fact that the evaluation functor isexact as colimits are computed pointwise. (cid:3) Remark 2.10. If C is a groupoid with finite hom sets, it is standard and easy that all objects in [ C op , Vect k ]are both projective and injective. (We will review these arguments in Section 7.) In some other cases where C is finite and an associated algebra is Frobenius, we find that the projectives and injectives are the same, butthat general objects do not have either property. For a typical small category, the projectives and injectivesare unrelated. For many of the categories U ≤ G arising in this paper, we will show that the projectivesin AU are a strict subset of the injectives, which are a strict subset of the full subset of objects. We arenot aware of any examples where this pattern has previously been observed; it has a number of interestingconsequences. 3. Subcategories and their properties
Throughout this paper we will consider a wide range of subcategories
U ≤ G , and we will impose differentconditions on U in different places. It is convenient to collect together the main examples and conditionshere. Definition 3.1.
Let U be a subcategory of G (assumed implicitly to be full and replete, as usual). • We say that U is subgroup-closed if whenever H ≤ G ∈ U we also have H ∈ U . • We say that U is closed downwards if whenever G → H is a surjective homomorphism with G ∈ U ,we also have H ∈ U . • We say that U is closed upwards if whenever H → K is a surjective homomorphism with K ∈ U , wealso have H ∈ U . • We say that U is convex if whenever G → H → K are surjective homomorphisms with G, K ∈ U ,we also have H ∈ U . • We say that U is multiplicative if 1 ∈ U , and G × H ∈ U whenever G, H ∈ U . Equivalently, U shouldcontain the product of any finite family of its objects, including the empty family. • We say that U is widely closed if whenever G ←− H −→ K are surjective homomorphisms with G, H, K ∈ U , the image of the combined morphism H → G × K is also in U . (We will show thatalmost all of our examples have this property.) • We say that U is finite if it has only finitely many isomorphism classes. • We say that U is groupoid if all morphisms in U are isomorphisms. • We say that U is colimit-exact if the functor X lim −→ G ∈U op X ( G ) is an exact functor AU →
Vect k .(We will show that almost all of our examples have this property.) • We say that U is submultiplicative if it is multiplicative and subgroup-closed. • We say that U is a global family if it is subgroup-closed and also closed downwards. Remark 3.2. If U is closed upwards or downwards or is a groupoid, then it is convex. • If U is submultiplicative then it is clearly widely closed. • If U is convex, then it is also widely closed. Indeed, if G ←− H −→ K are surjective homomorphismswith G, H, K ∈ U and L is the image of the resulting map H → G × K then we have evidentsurjective homomorphisms H → L → G , showing that L ∈ U . • In particular, if U is closed upwards or downwards or is a groupoid, then it is widely closed. Definition 3.3.
We define subcategories of G as follows. Some of them depend on a prime number p and/oran integer n ≥ • Z is the multiplicative global family of finite abelian groups. • C is the global family of finite cyclic groups. • G [ p ∞ ] is the subcategory of finite p -groups. • Z [ p ∞ ] = Z ∩ G [ p ∞ ] is the multiplicative global family of finite abelian p -groups. • C [ p ∞ ] = C ∩ G [ p ∞ ] is the global family of finite cyclic p -groups. • G [ p n ] is the multiplicative global family of finite groups of exponent dividing p n . • Z [ p n ] = Z ∩ G [ p n ] is the multiplicative global family of finite abelian groups of exponent dividing p n , which is equivalent to the category of finitely generated modules over Z /p n . • C [ p n ] = C ∩ G [ p n ] is the global family of finite cyclic groups of exponent dividing p n , which isequivalent to the category of cyclic modules over Z /p n . • F [ p n ] is the subcategory of groups isomorphic to ( Z /p n ) r for some r ≥
0, which is equivalent to thecategory of finitely generated free modules over Z /p n . • E [ p ] is the multiplicative global family of elementary abelian p -groups, which is the same as Z [ p ] or F [ p ].We also consider the following subcategories, primarily as a source of counterexamples: • W is the subcategory of finite simple groups, which is a groupoid. • W is the subcategory of (necessarily cyclic) groups of prime order, which is also a groupoid. • W is the subcategory of finite 2-groups in which every square is a commutator. This is easily seento be multiplicative and closed downwards. However, it contains the quaternion group Q but notthe cyclic group C < Q , so it is not subgroup-closed. • W is the subcategory of finite p -groups in which all elements of order p commute. This is clearlysubmultiplicative, but it is not closed downwards. Indeed, one can check that W contains the uppertriangular group U T ( Z /p ) (provided that p > U T ( Z /p ). (We thankYves de Cornulier, aka MathOverflow user YCor, for this example [10].)Given a subcategory U , we also define further subcategories as below, depending on an integer n > N ∈ U : • U ≤ n = { G ∈ U | | G | ≤ n } . This is always finite. If U is subgroup-closed, closed downwards, convex,widely-closed or a groupoid then U ≤ n inherits the same property. • U ≥ n = { G ∈ U | | G | ≥ n } . If U is closed upwards, convex, widely closed, finite or a groupoid then U ≥ n inherits the same property. • U = n = { G ∈ U | | G | = n } = U ≤ n ∩ U ≥ n . This is always a finite groupoid, and so is convex and widelyclosed. • U ≤ N = { G ∈ U | G ( N, G ) = ∅} . This is always finite. If U is closed downwards, convex, widely-closedor a groupoid then U ≤ N inherits the same property. • U ≥ N = { G ∈ U | U ( G, N ) = ∅} . If U is closed upwards, convex, widely closed, finite or a groupoidthen U ≥ N inherits the same property. • U ≃ N = { G ∈ U | G ≃ N } = U ≤ N ∩ U ≥ N . This is always a finite groupoid, and so is convex andwidely closed. Example 3.4.
Using Remark 3.2 we see that almost all of the specific subcategories listed above are widelyclosed. One exception is the subcategory F [ p n ] for n >
1. We will identify this with the category of finitelygenerated free modules over Z /p n and so use additive notation. We take G = K = Z /p n and H = ( Z /p n ) , nd we define maps G α ←− H β −→ K by α ( i, j ) = i and β ( i, j ) = i + pj . We find that the image of the combinedmap H → G × K is isomorphic to Z /p n × Z /p n − and so does not lie in F [ p n ].4. Closed monoidal structure
It is convenient to add a bit of structure on A . Definition 4.1.
We give AU the symmetric monoidal structure given by ( X ⊗ Y )( T ) = X ( T ) ⊗ Y ( T ). Theunit object is the constant functor with value k (so = e provided that 1 ∈ U ). We also putHom( X, Y )( T ) = A ( e T ⊗ X, Y ) . Standard arguments show that this defines an object of AU with AU ( W, Hom(
X, Y )) ≃ A ( W ⊗ X, Y ) , so AU is a closed symmetric monoidal category. We write DX for Hom( X, ), and call this the dual of X . Remark 4.2.
Note that the tensor product is both left and right exact, so all objects are flat.
Remark 4.3.
We warn the reader that DX is not obtained from X by taking levelwise duals, so thecanonical map X ⊗ DX → Hom(
X, X ) is usually not an isomorphism. To demonstrate this consider thecase X = e G for any non-trivial group G . If we evaluate at the trivial group, we find e G (1) ⊗ De G (1) = 0and Hom( e G , e G )(1) = k [Out( G )]. Therefore the map is far from being an isomorphism.For the rest of this section we study the effect of the tensor product and internal hom functor on thegenerators. The main results are Proposition 4.13 and Theorem 4.20 and they both rely on the followingnotion. Definition 4.4.
Let U be a subcategory of G . A permuted family of groups consists of a finite group Γ, afinite Γ-set A , a family of groups G a ∈ U for each a ∈ A , and a system of isomorphisms γ ∗ : G a → G γ ( a ) (for γ ∈ Γ and a ∈ A ) satisfying the functoriality conditions 1 ∗ = 1 and ( δγ ) ∗ = δ ∗ γ ∗ . The system ofisomorphisms gives maps stab Γ ( a ) → Aut( G a ) for each a ∈ A . We say that the family is outer if the imageof this map contains the inner automorphism group Inn( G a ) for all a . Given a permuted family G which isouter, we define the set e B ( G )( T ) = { ( a, α ) | a ∈ A, α ∈ Epi(
T, G a ) } . The group Γ acts on e B ( G )( T ) via the formula γ · ( a, α ) = ( γ ( a ) , γ ∗ ◦ α ). We define B ( G )( T ) = e B ( G )( T ) / Γand F ( G )( T ) = k [ B ( G )( T )]. This is contravariantly functorial in T , so F ( G ) ∈ AU . Proposition 4.5.
For all X ∈ AU there is a natural isomorphism AU ( F ( G ) , X ) = Y a ∈ A X ( G a ) ! Γ . If we choose a subset A ⊂ A containing one element of each Γ -orbit, we get an isomorphism F ( G ) = M a ∈ A e stab Γ ( a ) G a . Thus, F ( G ) is finitely projective (see Definition 11.1).Proof. We can reduce to the case where A is a single orbit, say A = Γ a ≃ Γ / ∆, where ∆ = stab Γ ( a ). We candefine φ : Epi( T, G a ) / ∆ → B ( G )( T ) by φ [ α ] = [ a, α ]. If [ b, β ] ∈ B ( G )( T ) then b = γ ( a ) for some a . We canthen put α = γ − ∗ ◦ β : T → G a and we find that [ b, β ] = φ [ α ]. On the other hand, if φ ( α ) = φ ( α ′ ) then thereexists γ ∈ Γ with ( γ ( a ) , γ ∗ ◦ α ) = ( a, α ′ ) which means that γ ∈ ∆ and [ α ] = [ α ′ ] in Epi( T, G a ) / ∆. It followsthat φ is a natural bijection. Thus, if we let Φ denote the image of ∆ in Out( G a ), we have F ( G ) ≃ e Φ G a . Notethat the inclusion e Φ G a ≤ e G a is split by the map x → | Φ | − P φ ∈ Φ φ · x . It follows that e Φ G a is projective. (cid:3) Definition 4.6.
Let ( G i ) i ∈ I be a finite family of groups in U with product P = Q i G i . • We say that a subgroup W ≤ P is wide if all the projections π i : W → G i are surjective. We say that a homomorphism f : T → P is wide if all the morphisms π i ◦ f are surjective, orequivalently f ( T ) is a wide subgroup of P .For G, H ∈ U , we let Wide(
G, H ) denote the set of wide subgroups of G × H which belong to U . Thisset is covariantly functorial in G and H with respect to morphisms in U . Given ϕ : G ′ → G in U and W ′ ∈ Wide( G ′ , H ), we put ϕ ∗ W ′ = ( ϕ × id H )( W ′ ) which is wide in G × H . This comes with a map j ϕ : W ′ → ϕ ∗ W ′ which makes the following diagram G ′ × H G × HW ′ ϕ ∗ W ′ ϕ × id j ϕ commute. The assignment W ′ ϕ ∗ W ′ defines a map ϕ ∗ : Wide( G ′ , H ) → Wide(
G, H ) between the set ofwide subgroups. Similar functoriality holds for H as well. Example 4.7.
Let G and G be finite groups.(a) The full group G × G is always wide. If G and G are nonisomorphic simple groups, then one cancheck (perhaps using Lemma 4.9 below) that this is the only example. Similarly, if | G | and | G | arecoprime, then G × G is the only wide subgroup.(b) If α : G → G is a surjective homomorphism, then the graphGr( α ) = { ( g, α ( g )) | g ∈ G } is always wide. If G and G are isomorphic simple groups, then one can check that every widesubgroup is of the form (a) or (b). Moreover, in (b) we see that α must be an isomorphism.(c) Now let U ≤ G be a groupoid, and suppose that G , G ∈ U . If W ≤ G × G is wide and lies in U ,we see easily that W is the graph of an isomorphism α : G → G .(d) Now consider the case U = C [ p ∞ ] = { cyclic p -groups } . If | G | ≥ | G | then it is not hard to see thatany cyclic wide subgroup of G × G is the graph of a surjective homomorphism α : G → G asin (b). Similarly, if | G | ≤ | G | then any cyclic wide subgroup of G × G is the graph of a surjectivehomomorphism β : G → G . Of course, if | G | = | G | then any surjective homomorphism α : G → G is an isomorphism, and the graph of α : G → G is the same as the graph of α − : G → G . Definition 4.8.
Suppose we have finite groups G and G , and normal subgroups N i ⊳ G i , and an isomor-phism α : G /N → G /N . We can then put H ( N , α, N ) = { ( x , x ) ∈ G × G | α ( x N ) = x N } ≤ G × G . This is easily seen to be a wide subgroup.
Lemma 4.9.
Every wide subgroup K ≤ G × G has the form H ( N , α, N ) for a unique triple ( N , α, N ) as above.Proof. Put N = { n ∈ G | ( n , ∈ K } , and similarly for N . If n ∈ N and g ∈ G then wideness gives g ∈ G such that ( g , g ) ∈ K . Itfollows that the element ( g n g − ,
1) = ( g , g )( n , g , g ) − lies in K and that N is normal. The sameargument shows that N is normal in G too. This means that K is the preimage in G × G of thesubgroup ¯ K = K/ ( N × N ) ≤ ( G/N ) × ( G/N ). We now find that the projections π i : ¯ K → G i /N i are bothisomorphisms, so we can define α : π π − : G/N → G/N . It is now easy to see that K = H ( N , α, N ), asrequired. (cid:3) Definition 4.10.
Any wide subgroup W ≤ G × H is of the form W = H ( N , α, N ) for some isomorphism α : G/N → H/N . We call G/N the left spread and H/N the right spread of W . We will write ls( W ) andrs( W ) for the left and right spread of W respectively. Remark 4.11.
The left and right spread of a wide subgroup W depend on the ambient group G × H . efinition 4.12. Given
G, H ∈ U , we let W ( G, H ) denote the tautological family indexed by Wide(
G, H ),so the group indexed by U ∈ Wide(
G, H ) is U itself. Then G × H acts on Wide( G, H ) by conjugation. We usethis to regard W ( G, H ) as a permuted family, and thus define a finitely projective object F ( W ( G, H )) ∈ AU .We now consider tensor products of generators. Proposition 4.13.
Let U be a widely closed subcategory of G , and suppose that G, H ∈ U . Then e G ⊗ e H is naturally isomorphic to F ( W ( G, H )) (and so is a finitely generated projective object of AU ).Proof. Consider another object T ∈ U and a pair ( α, β ) ∈ Epi(
T, G ) × Epi(
T, H ). This gives a wide subgroup U = h α, β i ( T ) ≤ G × H , which lies in U because U is assumed to be widely closed. We can regard h α, β i as a surjective homomorphism from T to U , so we have an element φ ( α, β ) = ( U, h α, β i ) ∈ e B ( W ( G, H ))( T ).This is easily seen to give a ( G × H )-equivariant natural bijection φ : Epi( T, G ) × Epi(
T, H ) → e B ( W ( G, H ))( T ) . It follows easily that we get an induced bijection U ( T, G ) × U ( T, H ) → B ( W ( G, H ))( T ) and an isomorphism e G ⊗ e H → F ( W ( G, H )) as required. (cid:3)
Remark 4.14. If G and H are abelian, then G × H acts trivially on W and so e G ⊗ e H = L U ∈ Wide(
G,H ) e U . Remark 4.15.
It is not true that e G ⊗ e H is always a direct sum of objects of the form e K . In particular,this fails when G = H = D . To see this, let N be the subgroup of G isomorphic to C , and put W = { ( g, h ) ∈ G × H | gN = hN } . This is wide, and has index 2 in G × H , so it is normal in G × H . Thegroup G × H acts by conjugation of the set Wide( G, H ) and the stabilizer of the conjugacy class of W isthe quotient Q = ( G × H ) /W . Then the summand in the tensor product e G ⊗ e H corresponding to theconjugacy class of W is given by e QW which is not of the form e K . Definition 4.16. A virtual homomorphism from G to H is a pair α = ( A, A ′ ) where A ′ ⊳ A ≤ G × H and A iswide and A ′ ∩ (1 × H ) = 1 and A/A ′ ∈ U . We write VHom( G, H ) for the set of virtual homomorphisms. Wethen let Q ( G, H ) be the parametrised family of groups with Q α = A/A ′ for all α = ( A, A ′ ) ∈ VHom(
G, H ).We call Q α the spread of α . Note that G × H acts compatibly on VHom( G, H ) and Q α by conjugation. We usethis to regard Q ( G, H ) as a permuted family, and thus to define a finitely projective object F ( Q ( G, H )) ∈ AU . Example 4.17.
Suppose that U contains the trivial group. For any surjective homomorphism u : G → H ,we can define A = A ′ = graph( u ) = { ( g, u ( g )) | g ∈ G } . This gives a virtual homomorphism with trivial spread. We claim that every virtual homomorphism withtrivial spread arises in this way from a unique homomorphism. Indeed, let α = ( A, A ) be any such virtualhomomorphism and consider the projection map A ≤ G × H → G . The condition A ∩ (1 × H ) = 1 ensuresthat every element g ∈ G has a unique preimage ( g, u ( g )) ∈ A under the projection. It is easy to check thatthe assignment u : G → H defines a surjective group homomorphism, and by construction A = graph( u ). Example 4.18.
Consider a virtual homomorphism α = ( A, A ′ ) ∈ VHom(1 , G ). The group A must be widein 1 × G , which just means that A = 1 × G . The group A ′ ≤ × G must satisfy A ′ ∩ (1 × G ) = 1, whichmeans that A ′ = 1. Thus, there is a unique virtual homomorphism α = (1 × G, G . Example 4.19.
Consider a virtual homomorphism α = ( A, A ′ ) ∈ VHom( G, A must beequal to G × G ) and A ′ can be any normal subgroup of G such that G/A ′ ∈ U . Theorem 4.20.
Let U be a multiplicative global family of finite groups. Fix groups G, H ∈ U and let Q ( G, H ) be the parametrised family of virtual homomorphisms from G to H . Then Hom( e G , e H ) is isomorphic to F ( Q ( G, H )) (and so is a finitely generated projective object of AU ) . The general structure of the proof is as follows. We will fix G and H , and define finite sets L ( T ), M ( T ) and N ( T ) depending on a third object T ∈ U . All of these will have actions of G × H by conjugation, and wewill construct equivariant bijections between them. We will also construct isomorphisms Hom( e G , e H )( T ) ≃ k [ N ( T )] G × H and F ( Q ( G, H ))( T ) ≃ k [ M ( T )] G × H . All of this is natural with respect to isomorphisms ′ → T , but unfortunately not with respect to arbitrary morphisms T ′ → T in U . However, we will introducefiltrations of all the relevant objects and show that the failure of naturality involves terms that shift filtration.It will follow that the associated graded object for Hom( e G , e H ) is isomorphic to F ( Q ( G, H )). As this objectis projective, we see that the filtration splits, so Hom( e G , e H ) itself is isomorphic to F ( Q ( G, H )), as claimed.
Definition 4.21.
Fix groups
G, H ∈ U . Let T be another group in U .(a) We define L ( T ) to be the set of wide subgroups V ≤ T × G × H such that V ∩ H = 1. (Here weidentify H with the subgroup 1 × × H ≤ T × G × H , and we will make similar identifications invarious places below.)(b) We define M ( T ) to be the set of triples ( A, A ′ , θ ) where ( A, A ′ ) ∈ VHom(
G, H ) and θ ∈ Epi(
T, A/A ′ ).(c) We define N ( T ) to be the set of pairs ( W, λ ) where W is a wide subgroup of T × G , and λ ∈ Epi(
W, H ).All of these sets have evident actions of G × H by conjugation. Definition 4.22.
Given a surjective homomorphism ϕ : T ′ → T , we define maps ϕ ∗ : L ( T ) → L ( T ′ ), andsimilarly for M and N , as follows:(a) ϕ ∗ ( V ) = ( ϕ × × − ( V ) = { ( t ′ , g, h ) ∈ T ′ × G × H | ( ϕ ( t ′ ) , g, h ) ∈ V } (b) ϕ ∗ ( A, A ′ , θ ) = ( A, A ′ , θϕ )(c) ϕ ∗ ( W, λ ) = (( ϕ × − ( W ) , λ ◦ ( ϕ × Construction 4.23.
We define a bijection µ : L ( T ) → M ( T ) as follows. Given V ∈ L ( T ) we put A = π G × H ( V ) ≤ G × H and A ′ = { ( g, h ) ∈ G × H | (1 , g, h ) ∈ V } . As V is wide in T × G × H , we see that A iswide in G × H . As V ∩ H = 1, we see that A ′ ∩ H = 1. This means that the pair ( A, A ′ ) is an element ofVHom( G, H ). Next, for t ∈ T we put θ ( t ) = { ( g, h ) ∈ G × H | ( t, g, h ) ∈ V } . This is a coset of A ′ in A , or in other word an element of A/A ′ . It is not hard to check that this gives ahomomorphism θ : T → A/A ′ . From the definition of A we see that θ is surjective. We have thus defined anelement µ ( V ) = ( A, A ′ , θ ) ∈ M ( T ).In the opposite direction, suppose we start with an element ( A, A ′ , θ ) ∈ M ( T ). We can then define V = { ( t, g, h ) ∈ T × A | θ ( t ) = ( g, h ) .A ′ } . It is clear that π T ( V ) = T and π G × H ( V ) = A . As A is wide in G × H , it follows that V is wide in T × G × H .Now suppose that (1 , , h ) ∈ V , so (1 , h ) ∈ A and the coset (1 , h ) .A ′ is the same as θ (1), or in other words(1 , h ) ∈ A ′ . It then follows from the definition of a virtual homomorphism that h = 1. This proves that V ∈ L ( T ). It is easy to check that this construction gives a map M ( T ) → L ( T ) that is inverse to µ . It is alsostraightforward to check that these bijections are natural with respect to the functoriality in Definition 4.22. Construction 4.24.
We define a bijection ν : L ( T ) → N ( T ) as follows. Given V ∈ L ( T ) we define W = π T × G ( V ) ≤ T × G . As V ∈ L ( T ) we have V ∩ H = 1, which means that the projection π T × G : V → W is an isomorphism. We define λ to be the composite W π − T × G −−−−→ V π H −−→ H. As V is wide in T × G × H , we see that λ is surjective, so we have an element ν ( V ) = ( W, λ ) ∈ N ( T ).In the opposite direction, suppose we start with an element ( W, λ ) ∈ N ( T ). We then put V = { ( t, g, h ) ∈ W × H | λ ( t, g ) = h } . As W is wide in T × G and λ : W → T is surjective, we see that V is wide in T × G × H . If (1 , , h ) ∈ V then we must have h = λ (1 ,
1) = 1. This proves that V ∈ L ( T ). It is easy to check that this constructiongives a map N ( T ) → L ( T ) that is inverse to ν . It is also easy to check that these bijections are natural withrespect to the functoriality in Definition 4.22. Remark 4.25.
It is straightforward to identify e B ( Q ( G, H ))( T ) with M ( T ), and so to identify F ( Q ( G, H ))( T )with k [ M ( T )] G × H . efinition 4.26. For each element x in L ( T ), M ( T ) or N ( T ) we define a positive integer σ ( x ) as follows.(a) For V ∈ L ( T ) we put V = { ( t, g, h ) ∈ V | ( t, , , (1 , g, h ) ∈ V } , and σ ( V ) = | V | / | V | .(b) For ( A, A ′ , θ ) ∈ M ( T ) we put σ ( A, A ′ , θ ) = | A/A ′ | .(c) For ( W, λ ) ∈ N ( T ) we put K ( W, λ ) = { t ∈ T | ( t, ∈ W and λ ( t,
1) = 1 } and σ ( W, λ ) = | T | / | K ( W, λ ) | .We then put F n L ( T ) = { x ∈ L ( T ) | σ ( x ) ≥ n } ⊆ L ( T ) F n k [ L ( T )] = k [ F n L ( T )] ≤ k [ L ( T )] Q n k [ L ( T )] = F n k [ L ( T )] /F n +1 k [ L ( T )] . Remark 4.27.
For (
A, A ′ , θ ) ∈ M ( T ) it is clear that σ ( A, A ′ , θ ) ≤ | G || H | . It follows that σ ( x ) ≤ | G || H | for x ∈ L ( T ) or x ∈ N ( T ) as well. Lemma 4.28.
Suppose that the elements V ∈ L ( T ) and ( A, A ′ , θ ) ∈ M ( T ) and ( W, λ ) ∈ N ( T ) are relatedas in Constructions 4.23 and 4.24. Then σ ( V ) = σ ( A, A ′ , θ ) = σ ( W, λ ) . Thus, those constructions givebijections F n L ( T ) ≃ F n M ( T ) ≃ F n N ( T ) .Proof. As in Construction 4.23, we have a surjective projection π : V → A , and it follows that | A/A ′ | = | V | / | π − ( A ′ ) | . Moreover, we have A ′ = { ( g, h ) | (1 , g, h ) ∈ V } , and it follows easily that π − ( A ′ ) = V ; thismakes it clear that σ ( V ) = σ ( A, A ′ , θ ). On the other hand, we also have a surjective projection π ′ : V → T ,and it follows that | T | / | K ( W, λ ) | = | V | / | ( π ′ ) − ( K ( W, λ )) | . Suppose we have ( t, g, h ) ∈ V with t ∈ K ( W, λ ).It then follows that ( t, , ∈ V , and thus that the product ( t, g, h ) . ( t, , − = (1 , g, h ) also lies in V , so( t, g, h ) ∈ V . This argument is reversible so we find that ( π ′ ) − ( K ( W, λ )) = V and σ ( V ) = σ ( W, λ ). (cid:3) We now want to define an isomorphism ζ : k [ N ( T )] G × H → Hom( e G , e H )( T ) = AU ( e T ⊗ e G , e H ) . One approach would be to split e T ⊗ e G as a sum over conjugacy classes of wide subgroups, but that involveschoices which are awkward to control. We will therefore define ζ in a different way, and then use the splittingof e T ⊗ e G to verify that it is an isomorphism. Construction 4.29.
Fix an element (
W, λ ) ∈ N ( T ). Now consider an object P ∈ U and a pair of surjectivehomomorphisms α : P → T and β : P → G , giving an element [ α ] ⊗ [ β ] ∈ ( e T ⊗ e G )( P ) and a wide subgroup h α, β i ( P ) ≤ T × G . If there exists an element ( t, g ) ∈ T × G such that c ( t,g ) ( h α, β i ( P )) = W , then we canform the composite P h c t α,c g β i −−−−−−→ W λ −→ H. This is a surjective homomorphism. Its conjugacy class depends only on the conjugacy classes of α and β ,and not on the choice of ( t, g ). Moreover, everything that we have done is natural for morphisms P ′ → P in U . We can thus define an element ζ ( W, λ ) ∈ AU ( e T ⊗ e G , e H ) by ζ ( W, λ )([ α ] ⊗ [ β ]) = λ ◦ h c t α, c g β i in the case discussed above, and ζ ( W, λ )([ α ] ⊗ [ β ]) = 0 in the case where h α, β i ( P ) is not conjugate to W .It is easy to see that if ( W , λ ) and ( W , λ ) lie in the same ( G × H )-orbit of N ( T ), then ζ ( W , λ ) = ζ ( W , λ ). We now extend linearly to get a map k [ N ( T )] → Hom( e G , e H )( T ), and restrict to get a map ζ : k [ N ( T )] G × H → Hom( e G , e H )( T ).We can now choose a list of wide subgroups W , . . . , W r ≤ T × G containing precisely one representativeof each conjugacy class, and let ∆ i be the normaliser of W i in T × G . We have seen that this gives adecomposition e T ⊗ e G = L i e ∆ i W i , and thus an isomorphism( e T ⊗ e G )( H ) = M k k [Epi( W i , H ) / ∆ i ] . Note here that ∆ i ≥ W i so the conjugation action of ∆ i on Epi( W i , H ) encompasses the action of innerautomorphisms.) From this it is not hard to see that ζ is an isomorphism. Definition 4.30.
We put F n Hom( e G , e H )( T ) = ζ ( F n k [ N ( T )]), and Q n Hom( e G , e H )( T ) = F n Hom( e G , e H )( T ) F n +1 Hom( e G , e H )( T ) . Now consider a surjective homomorphism ϕ : T ′ → T . This gives a map ϕ ∗ : M ( T ) → M ( T ′ ) given by ϕ ∗ ( A, A ′ , θ ) = ( A, A ′ , θϕ ), and this is straightforwardly compatible with our identification F ( Q ( G, H ))( T ) ≃ k [ M ( T )] G × H . However, the situation with N ( T ) and Hom( e G , e H )( T ) is more complicated. Definition 4.31.
Consider an element (
W, λ ) ∈ N ( T ), and a surjective homomorphism ϕ : T ′ → T . Let E ( ϕ, ( W, λ )) be the set of pairs ( W ′ , λ ′ ) ∈ N ( T ′ ) such that ( ϕ × W ′ ) = W and λ ′ is the same as thecomposite W ′ ϕ × −−−→ W −→ H. It is easy to see that the element ϕ ∗ ( W, λ ) = (( ϕ × − ( W ) , λ ◦ ( ϕ × E ( ϕ, ( W, λ )).
Lemma 4.32.
Suppose that ( W ′ , λ ′ ) ∈ E ( ϕ, ( W, λ )) . Then ϕ restricts to give a surjective homomorphism K ( W ′ , λ ′ ) → K ( W, λ ) . It follows that σ ( W ′ , λ ′ ) ≥ σ ( W, λ ) , with equality iff ( W ′ , λ ′ ) = ϕ ∗ ( W, λ ) .Proof. Suppose that t ′ ∈ K ( W ′ , λ ′ ), so that ( t ′ , ∈ W ′ and λ ′ ( t ′ ,
1) = 1. As ( ϕ × W ′ ) = W , we see that( ϕ ( t ′ ) , ∈ W . As λ ◦ ( ϕ ×
1) = λ ′ , we see that λ ( t,
1) = 1. This shows that t ∈ K ( W, λ ).Conversely, suppose that t ∈ K ( W, λ ). This means that ( t, ∈ W = ( ϕ × W ′ ), so there exists ( t ′ , g ) ∈ W ′ with ( ϕ ( t ′ ) , g ) = ( t, t ′ ∈ T ′ such that ( t ′ , ∈ W ′ and ϕ ( t ′ ) = t . Using therelation λ ◦ ( ϕ ×
1) = λ ′ again, we also see that λ ′ ( t ′ ,
1) = 1, so t ′ ∈ K ( W ′ , λ ′ ).We now see that | K ( W ′ , λ ′ ) | = | K ( W, λ ) || ker( ϕ ) ∩ K ( W ′ , λ ′ ) | ≤ | K ( W, λ ) | . | T ′ || T | . Rearranging this gives σ ( W ′ , λ ′ ) = | T ′ || K ( W ′ , λ ′ ) | ≥ | T || K ( W, λ ) | = σ ( W, λ ) . We have equality iff ker( ϕ ) ≤ K ( W ′ , λ ′ ). Because λ ′ factors through ϕ ×
1, we see that the second conditionin the definition of K ( W ′ , λ ′ ) is automatic, so we have equality iff ker( ϕ ) × ≤ W ′ . This clearly holds if W ′ = ( ϕ × − ( W ).Conversely, suppose that ker( ϕ ) × ≤ W ′ . We are given that ( ϕ × W ′ ) = W , so W ′ ≤ ( ϕ × − ( W ). Inthe other direction, suppose that ( t ′ , g ) ∈ ( ϕ × − ( W ), so ( ϕ ( t ′ ) , g ) ∈ W . As W = ( ϕ × W ′ ), we canchoose ( t ′ , g ) ∈ W ′ with ( ϕ × t ′ , g ) = ( t ′ , g ). In other words, we can find t ′ ∈ T ′ such that ϕ ( t ′ ) = ϕ ( t ′ )and ( t ′ , g ) ∈ W ′ . We now have t ′ = t ′ t ′ for some t ′ ∈ ker( ϕ ), so ( t ′ , ∈ W ′ by assumption. It follows thatthe product ( t ′ , g ) = ( t ′ , g )( t ′ ,
1) also lies in W ′ as required. (cid:3) Proposition 4.33.
The subspaces F n Hom( e G , e H )( T ) form a subobject of Hom( e G , e H ) in AU , so thequotient Q n Hom( e G , e H ) can also be regarded as an object of AU . Moreover, the sum Q ∗ Hom( e G , e H ) = L n Q n Hom( e G , e H ) is naturally isomorphic to F ( Q ( G, H )) .Proof. Consider an element m ∈ F n Hom( e G , e H )( T ) and a surjective homomorphism ϕ : T ′ → T . We canregard m as a morphism e T ⊗ e G → e H . Now suppose we have a surjective homomorphism ϕ : T ′ → T . Now ϕ ∗ m corresponds to the composite m ◦ ( e ϕ ⊗
1) : e T ′ ⊗ e G → e H . Consider a wide subgroup W ′ ≤ T ′ × G ,and the resulting map j ′ : e W ′ → e T ′ ⊗ e G . Put W = ( ϕ × W ′ ), which is wide in T × G , and let j be theresulting map e W → e T ⊗ e G . The composite mj : e W → e H can be expressed as a k -linear combination ofmorphisms λ ∈ Epi(
W, H ). The condition m ∈ F n Hom( e G , e H )( T ) means that for all λ appearing here, wehave σ ( W, λ ) ≥ n . It follows that ϕ ∗ ( m ) j ′ can be expressed as a k -linear combination of the correspondingmorphisms λ ′ = λ ◦ ( ϕ ×
1) : W ′ → H . Lemma 4.32 tells us that the resulting pairs satisfy σ ( W ′ , λ ′ ) ≥ n . Itfollows that F n Hom( e G , e H ) is a subobject, as claimed. Moreover, the edge case in Lemma 4.32 tells us that n the associated graded, we see only terms of the form ϕ ∗ ( W, λ ). This means that the associated graded isisomorphic in AU to k [ N ] or k [ L ] or k [ M ] or F ( Q ( G, H )), as claimed. (cid:3)
Proof of Theorem 4.20.
The subobjects F n Hom( e G , e H ) form a finite-length filtration of Hom( e G , e H ) withfinitely projective quotients, so the filtration must split. The claim follows easily from this. (cid:3) Remark 4.34.
We do not know whether there is a splitting of the filtration that is natural in G and H aswell as T . There may be some interesting group theory and combinatorics involved here.5. Functors for subcategories
In this section we study the formalism that relates the abelian category A to its smaller subcategories AU . Definition 5.1.
Let U and V be full and replete subcategories of G , with U ⊆ V . The inclusion i = i UV : U → V gives a pullback functor i ∗ : AV → AU . We write i ! and i ∗ for the left and right adjoints of i ∗ (so i ! , i ∗ : AU → AV ). These are given by the usual Kan formulae (in their contravariant versions):(a) ( i ! X )( G ) is the colimit over the comma category ( G ↓ U ) of the functor sending each object ( G u −→ iH )to X ( H ).(b) ( i ∗ X )( G ) is the limit over the comma category ( U ↓ G ) of the functor sending each object ( iK v −→ G )to X ( K ). Remark 5.2.
The above definition covers most of the inclusion functors that we need to consider, with oneclass of exceptions, as follows. Let U be a replete full subcategory of G . We then let U × be the categorywith the same objects, but with only group isomorphisms as the morphisms, and we let l : U × → U be theinclusion. Then U × is not a full subcategory of U , so definition 5.1 does not officially apply. Nonetheless,we still have functors l ∗ , l ! and l ∗ , whose behaviour is slightly different from what we see in the main case.Details will be given later. Lemma 5.3.
Let i : U → V be an inclusion of replete full subcategories of G . (a) The (co)unit maps i ∗ i ∗ ( X ) → X → i ∗ i ! ( X ) are isomorphisms, for all X ∈ AU . Thus, the functors i ! and i ∗ are full and faithful embeddings. (b) The essential image of i ! is { Y ∈ AV | ǫ Y : i ! i ∗ ( Y ) → Y is iso } .The essential image of i ∗ is { Y ∈ AV | η Y : Y → i ∗ i ∗ ( Y ) is iso } . (c) There are natural isomorphisms i ∗ ( ) = and i ∗ ( X ⊗ Y ) = i ∗ ( X ) ⊗ i ∗ ( Y ) giving a strong monoidalstructure on i ∗ . However, the corresponding map i ∗ Hom(
X, Y ) → Hom( i ∗ X, i ∗ Y ) is typically not anisomorphism. (d) There are natural maps i ! → → i ∗ and i ! ( X ⊗ Y ) → i ! ( X ) ⊗ i ! ( Y ) and i ∗ ( X ) ⊗ i ∗ ( Y ) → i ∗ ( X ⊗ Y ) giving (op)lax monoidal structures. (e) In all cases i ! preserves all colimits and i ∗ preserves all limits and i ∗ preserves both limits andcolimits. Also i ! preserves projective objects and i ∗ preserves injective objects. Both i ∗ and i ! preserveindecomposable objects. (f) If U is closed upwards in V , then i ! is extension by zero and so preserves all limits, colimits andtensors (but not the unit). (g) If U is closed downwards in V then i ∗ is extension by zero and so preserves all limits, colimits andtensors (but not the unit). (h) If U is submultiplicative, then i ! preserves the unit and all tensors; in other words, is stronglymonoidal. (i) If i has a left adjoint q : V → U then i ! = q ∗ (and so i ! preserves all (co)limits). (j) Suppose that G ∈ U and C ≤ U is convex. Then, for the objects defined in Definition 2.7 we have i ∗ ( e G,V ) = e G,V i ∗ ( t G,V ) = t G,V i ∗ ( s G,V ) = s G,V i ∗ ( χ C ) = χ C i ! ( e G,V ) = e G,V i ∗ ( t G,V ) = t G,V . If U is closed upwards, we also have i ! ( χ C ) = χ C i ! ( s G,V ) = s G,V i ! ( t G,V ) = χ U ⊗ t G,V . n the other hand, if U is closed downwards, we also have i ∗ ( e G,S ) = χ U ⊗ e G,S i ! ( s G,V ) = s G,V i ∗ ( χ C ) = χ C . Proof.
Almost all of this is standard, but we recall proofs for ease of reference.If G ∈ U then ( G −→ G ) is terminal in the comma category U ↓ G , so the Kan formula reduces to ( i ! X )( G ) = X ( G ). Using this, we see that the unit map X → i ∗ i ! ( X ) is an isomorphism for all X . It follows that themap i ! : AU ( X, Y ) → AV ( i ! X, i ! Y ) is an isomorphism, with inverse essentially given by i ∗ , so i ! is a full andfaithful embedding. A dual argument shows that the counit map i ∗ i ∗ ( X ) → X is also an isomorphism, andthat i ∗ is also a full and faithful embedding. This proves claim (a).Now put B = { Y ∈ AV | ǫ Y : i ! i ∗ ( Y ) → Y is iso }C = { Y ∈ AV | η Y : Y → i ∗ i ∗ ( Y ) is iso } . If Y ∈ B then Y ≃ i ! i ∗ ( Y ) so Y is in the essential image of i ! . Conversely, for X ∈ AU we have seen that theunit X → i ∗ i ! ( X ) is an isomorphism, so the same is true of the map i ! ( X ) → i ! i ∗ i ! ( X ). By the triangularidentities for the ( i ! , i ∗ )-adjunction, it follows that the counit i ! i ∗ i ! ( X ) → i ! ( X ) is also an isomorphism, so i ! ( X ) ∈ B , so any object isomorphic to i ! ( X ) also lies in B . This proves that B is the essential image of i ! ,and a dual argument shows that C is the essential image of i ∗ . This proves claim (b).We now consider claim (c). For all G ∈ U , we have ( i ∗ )( G ) = k = ( G ) and i ∗ ( X ⊗ Y )( G ) = X ( G ) ⊗ Y ( G ) =( i ∗ X ⊗ i ∗ Y )( G ) which proves that i ∗ is strongly monoidal as claimed. For the negative part of (c), considerthe case where U = { } . For any G we have ( i ∗ Hom( e G , e G ))(1) = AV ( e G , e G ) = k [Out( G )] = 0, butif G is nontrivial, then i ∗ ( e G ) = 0 and so Hom( i ∗ ( e G ) , i ∗ ( eG ))(1) = 0. This shows that the natural map i ∗ Hom(
X, Y ) → Hom( i ∗ ( X ) , i ∗ ( Y )) (adjoint to the evaluation map) is not always an isomorphism.From claim (c) we get a natural isomorphism i ∗ ( i ∗ ( X ) ⊗ i ∗ ( Y )) ≃ i ∗ i ∗ ( X ) ⊗ i ∗ i ∗ ( Y ) ǫ X ⊗ ǫ Y −−−−→ X ⊗ Y, and using the ( i ∗ , i ∗ )-adjunction we get a natural map i ∗ ( X ) ⊗ i ∗ ( Y ) → i ∗ ( X ⊗ Y ). A standard argumentshows that this makes i ∗ into a lax monoidal functor. By a dual construction, we get a natural map i ! ( X ⊗ Y ) → i ! ( X ) ⊗ i ! ( Y ) making i ! into an oplax monoidal functor. This proves (d).Most of claim (e) is formal and follows from the properties of adjunctions. If P is indecomposable, we seethat the only idempotent elements in End( P ) are 0 and 1, and that 0 = 1. As i ! is full and faithful, we seethat End( i ! P ) is isomorphic to End( P ), and so has the same idempotent structure. A similar proof worksfor i ∗ too.Now consider claim (f). Suppose that U is closed upwards in V , that X ∈ AU ,and that G ∈ V . If G ∈ U then i ! ( X )( G ) = X ( G ) by claim (a). If G
6∈ U then the upward closure assuption implies that G ↓ U = ∅ , sothe Kan formula reduces to i ! ( X )( G ) = 0. In other words, i ! is extension by zero, and the rest of claim (f)follows immediately. A dual argument proves (g).Now suppose instead that U is submultiplicative, as in (h), so in particular 1 ∈ U . We claim that the naturalmap i ! ( X ⊗ Y ) → i ! ( X ) ⊗ i ! ( Y ) is an isomorphism. As i ! and the tensor product preserve colimits, we canreduce to the case where X = e G and Y = e H for some G, H ∈ U . Recall that i ! ( e U G ) = e V G (or more briefly i ! ( e G ) = e G ), and similarly for H . Using Proposition 4.13 we see that i ! ( e G ) ⊗ i ! ( e H ) can be expressed interms of the wide subgroups W ≤ G × H such that W ∈ V , whereas i ! ( e G ⊗ e H ) is similar but involvesonly groups W that lie in U . However, the submultiplicativity condition ensures that any wide subgroup W ≤ G × H lies in U , so we see that i ! ( e G ) ⊗ i ! ( e H ) = i ! ( e G ⊗ e H ). We also have i ! ( ) = i ! ( e ) = e = .This shows that i ! is strongly monoidal.Now suppose that i has a left adjoint q as in (i). Then the comma category T ↓ U is equivalent to qT ↓ U which has a terminal object ( qT −→ qT ) giving Y ( T ) = X ( qT ). It follows that q ∗ and i ! are naturallyisomorphic as claimed. n claim (j), all the statements about i ∗ are straightforward. For any X ∈ AV we have AV ( i ! ( e G,V ) , X ) = AU ( e G,V , i ∗ ( X )) = M G ( V, X ( G )) = AV ( e G,V , X )where we used Lemma 2.9. It follows by the Yoneda Lemma that i ! ( e G,V ) = e G,V , and a similar proofgives that i ∗ ( t G,V ) = t G,V . The remaining claims in (j) follows from (f) and (g) as the functor i ! and i ∗ areextension by zero. (cid:3) Remark 5.4.
Part (f) of the lemma gives conditions under which i ! preserves tensor products. However,this does not always hold if we remove those conditions, as shown by the following counterexample. Take U = C [2 ∞ ] (the family of cyclic 2-groups). Note that the only wide subgroups of C × C are the whole group C × C and the graph subgroup Gr( π ) ≃ C of the canonical projection π : C → C . Using Proposition 4.13,we see that e C ⊗ e C ≃ e C × C ⊕ e Gr( π ) in A but e U C ⊗ e U C ≃ e U Gr( π ) in AU . Thus, the canonical map e Gr( π ) = i ! ( e U Gr( π ) ) ≃ i ! ( e U C ⊗ e U C ) → i ! ( e U C ) ⊗ i ! ( e U C ) = e C ⊗ e C ≃ e C × C ⊕ e Gr( π ) is not an isomorphism in A . Lemma 5.5.
Let V be a replete full subcategory of G . Let U and W be two replete full subcategories of V that are complements of each other, with inclusions i : U → V and j : W → V . Suppose that U is closedupwards, or equivalently, that V is closed downwards. Then: (a) The functor i ! : AU → AV admits a left adjoint i ! : AV → AU given by i ! ( Y ) = i ∗ (cok( j ! j ∗ ( Y ) → Y )) . (b) The functor j ∗ : AW → AV admits a right adjoint j ♯ : AV → AW given by j ♯ ( X ) = i ∗ (ker( X → j ∗ j ∗ X )) .Proof. We will only prove (a) as the argument for (b) is similar. Consider a morphism u : Y → i ! ( X ). Thisfits in a naturality square as follows: j ! j ∗ ( Y ) Yj ! j ∗ i ! ( X ) i ! ( X ) . ǫ Y j ! j ∗ ( u ) uǫ i !( X ) Lemma 5.3(f) tells us that i ! is extension by zero, so j ∗ i ! = 0, so the bottom left corner of the square iszero, so there is a unique morphism u : cok( ǫ Y ) → i ! ( X ) induced by u . We can now compose i ∗ ( u ) with theinverse of the unit map X → i ∗ i ! ( X ) to get a morphism u : i ! ( Y ) → X . We leave it to the reader to checkthat this construction gives the required bijection AV ( Y, i ! ( X )) ≃ AU ( i ! ( Y ) , X ). (cid:3) We can use the formalism of change of subcategory to construct functorial projective and injective resolutions.
Construction 5.6.
As in Remark 5.2, we let U × denote the subcategory with the same objects as U but onlyisomorphisms as morphisms. Let l : U × → U be the inclusion, and consider the functors l ! , l ∗ : AU × → AU .If we choose a skeleton U ′ ⊂ U , it is not hard to check that l ! ( W ) = M G ∈U ′ e G,W ( G ) l ∗ ( W ) = Y G ∈U ′ t G,W ( G ) for all W ∈ AU × . It follows that l ! ( W ) is always projective and l ∗ ( W ) is always injective. Moreover, we seethat the counit l ! l ∗ ( X ) → X is always an epimorphism for all X ∈ AU , and the unit X → l ∗ l ∗ ( X ) is alwaysa monomorphism.We now set P = l ! l ∗ ( X ) P = l ! l ∗ (ker( P → X )) P i +2 = l ! l ∗ (ker( P i +1 → P i )) ∀ i ≥ I = l ∗ l ∗ ( X ) I = l ∗ l ∗ (cok( X → I )) I i +2 = l ∗ l ∗ (cok( I i → I i +1 )) ∀ i ≥ P • → X and X → I • define functorial projective and injective resolutions of X , respectively. ecall the base of an object Remark 5.7.
Recall from Definition 2.6 thatbase( X ) = min {| G | | X ( G ) = 0 } . If | G | < base( X ) then we find that ( l ! l ∗ ( X ))( G ) = 0, and if | G | = base( X ) we find that the counit map( l ! l ∗ ( X ))( G ) → X ( G ) is an isomorphism. Using this, we see that base( P k ) ≥ base( X ) + k for all k ≥
0. Thus,our canonical projective resolution is convergent in a convenient sense.We now give other useful constructions and examples that we will use later on.
Construction 5.8.
Let
V ≤ G be a subcategory with only finitely many isomorphism classes. Let V ⋆ bethe submultiplicative closure of V , so G ∈ V ∗ iff G is isomorphic to a subgroup of Q ni =1 H i for some familyof groups H i ∈ V . For a finitely generated group F we put K ( F ; V ) = { N ⊳ F | F/N ∈ V} . We can choose a finite list of groups containing one representative of each isomorphism class in V , then eachgroup in K ( F ; V ) will occur as the kernel of one of the finitely many surjective homomorphisms from F toone of these groups. It follows that K ( F ; V ) is a finite collection of normal subgroups of finite index. Wedefine N ( F ; V ) to be the intersection of all the subgroups in K ( F ; V ), then we put q V ( F ) = F/ N ( F ; V ).This is isomorphic to the image of the natural map F → Y N ∈K ( F ; V ) F/N, so the submultiplicativity condition ensures that q V ( F ) ∈ V ⋆ . It is straightforward to check that anysurjective homomorphism φ : F → F has φ ( N ( F ; V )) ≤ N ( F ; V ) and so induces a homomorphism q V ( F ) → q V ( F ). This makes q V into a functor on the category of finitely generated groups and surjec-tive homomorphisms. If we restrict to finite groups, then the functor q V is the left adjoint to the inclusion i : V ⋆ → G . We therefore have i ! = q ∗V by Lemma 5.3(i). Example 5.9.
Let U be a submultiplicative subcategory of G and fix an integer n ≥
1. If we take V = U ≤ n then V ⋆ ⊆ U . In this case we will use the abbreviated notation q ≤ n ( F ) , K ≤ n ( F ) and N ≤ n ( F ). Example 5.10.
Let
U ≤ G be a submultiplicative subcategory and fix an integer n ≥
1. For any finite set X of cardinality n , let F X be the free group on X . Then we put T X = q ≤ n ( F X ) ∈ U ∗≤ n ⊆ U . This is finiteand functorial for bijections of X . If G is any group in U ≤ n , then we can choose a surjective map X → G ,and extend it to a surjective homomorphism F X → G . The kernel of this homomorphism is in K ≤ n ( F X )and so contains N ≤ n ( F X ), so we get an induced surjective homomorphism
T X → G . In particular, we cantake X = G and use the identity map to get a canonical epimorphism ǫ : T G → G .We now consider a natural filtration on objects of AU which will be useful later on. Construction 5.11.
Consider an object X ∈ AU . For n ≥
0, we let L ≤ n X denote the image of the counitmap i ≤ n ! i ∗≤ n X → X . By construction, L ≤ n X is the smallest subobject of X containing X ( H ) for all H ∈ U ≤ n This gives a filtration 0 = L ≤ X ≤ L ≤ X ≤ · · · ≤ L ≤ n X ≤ L ≤ n +1 X ≤ · · · ≤ X with subquotients denoted by L n X . Consider a map f : X → Y and an element x ∈ ( L ≤ n X )( G ). We canwrite x = P si =1 α ∗ i ( x i ) where x i ∈ X ( H i ) with | H i | ≤ n and α i ∈ U ( G, H i ). Note that f ( x ) = X i f α ∗ i ( x ) = X i α ∗ i f ( x ) ∈ ( L ≤ n Y )( G ) , so the filtration is natural in X . Therefore we also have induced maps L ≤ n f : L ≤ n X → L ≤ n Y and L n f : L n X → L n Y for all n . xample 5.12. For all G ∈ U , we have L ≤ n ( e G ) = ( n < | G | e G if n ≥ | G | . From this we see that L n ( e G ) = e G if | G | = n , and L n ( e G ) = 0 otherwise. Construction 5.13.
Consider an object X ∈ AU . We define( QX )( G ) = X ( G ) / X = N⊳G π ∗ X ( G/N )where π : G → G/N denotes the projection. Equivalently, if | G | = n then this is( QX )( G ) = cok( i We can now use Q to build minimal projective resolutions. Consider an object X ∈ AU .Then QX is a quotient of l ∗ X in the semisimple category AU × , so we can choose a section QX → l ∗ X .By passing to the adjoint, we get an morphism e : P ′ → X , where P ′ = l ! QX . We find that Qe is anisomorphism, so e is an epimorphism. We can iterate this in the same way as in Construction 5.6 to get aprojective resolution P ′• which is minimal in the sense that the differential d k : P ′ k → P ′ k − has Q ( d k ) = 0 forall k . As is familiar for minimal resolutions in other contexts, it follows that P ′• is a summand in any otherprojective resolution of X . 6. Simple objects In this section we classify the simple objects and show that AU is semisimple if and only if U is a groupoid. Definition 6.1. Let U be a replete full subcategory of G . • An object X ∈ AU is simple if the only subobjects are 0 and X . • An object X ∈ AU is semisimple if it is a sum of simple objects. • The abelian category AU is semisimple if every object is semisimple. • The abelian category AU is split if every short exact sequence in AU splits. Equivalently, everyobject of AU is both injective and projectiveWe immediately get the following result. Lemma 6.2. An object X ∈ AU is simple if and only if it is isomorphic to s G,V for some G and someirreducible k [Out( G )] -module V . roof. Consider a simple object X ∈ AU . Choose G of minimal order so that X ( G ) = 0. It is standardthat the category of k [Out( G )]-modules is semisimple, so we can choose a simple quotient V of X ( G ) inthis category. The projection X ( G ) → V is adjoint to a morphism X → t G,V in AU . As X ( H ) = 0 when | H | < | G | , we see that this factors through the subobject s G,V ≤ t G,V . The morphism X → s G,V is thenan epimorphism whose kernel is a proper subobject, and so must be zero by simplicity. Thus X ≃ s G,V asrequired. (cid:3) We are now ready to study when our abelian category is semisimple. Proposition 6.3. The following are equivalent: (a) U is a groupoid; (b) the abelian category AU is split; (c) the abelian category AU is semisimple.Proof. The fact that (b) and (c) are equivalent is well-known and proved for instance in [22, V.6.7]. It is alsostandard that (a) implies (b); the argument will be recalled as Proposition 7.3(a) below. Thus, we need onlyprove that (b) implies (a), or the contrapositive of that. Suppose that U is not a groupoid so there exists anepimorphism ϕ : G → H which is not an isomorphism. Consider the canonical epimorphism π : e H,k → s H,k .The map ϕ ∗ : e H,k ( H ) → e H,k ( G ) is easily seen to be injective. The map ϕ ∗ : s H,k ( H ) → s H,k ( G ) is of theform k → s H,k cannot be a retract of e H,k , so π cannot split. Thus, AU is not a split abelian category. (cid:3) Finite groupoids In this section we study the abelian category AU in the special case that U ≤ G is a finite groupoid. Forexample we could take U = { G ∈ G | | G | = n } . Lemma 7.1. Suppose we choose a list of groups G , . . . , G r containing precisely one representative of eachisomorphism class of groups in U , so G ( G i , G j ) = ∅ for i = j . Let M i be the category of modules for thegroup ring k [Out( G i )] and put M = Q ri =1 M i . Then the functor X ( X ( G i )) ri =1 gives an equivalence ofcategories AU → M .Proof. The inverse functor is given by ( V i ) ri =1 L ri =1 e G i ,V i . (cid:3) Remark 7.2. Let i : U → G denote the inclusion functor. After choosing a list of groups G , . . . , G r ∈ U asin Lemma 7.1, we have identifications i ! = r M i =1 e G i , • and i ∗ = r M i =1 t G i , • . Proposition 7.3. Suppose that U ⊆ V ⊆ G , and let i : U → V denote the inclusion. (a) All monomorphisms and epimorphisms in AU are split. (b) All objects in AU are both injective and projective. (c) All objects in the image of i ! are projective, and all objects in the image of i ∗ are injective. (d) The functor i ! preserves all limits and colimits, as does the functor i ∗ .Proof. We identify AU with M as in Lemma above. Maschke’s Theorem shows that ( a ) and ( b ) hold in M i ,and it follows that they also hold in M and AU . If X ∈ AU then the functor A ( i ! ( X ) , − ) is isomorphic to AU ( X, i ∗ ( − )). Here i ∗ and AU ( X, − ) preserve epimorphisms, so i ! ( X ) is projective. Similarly, we see that i ∗ ( X ) is injective, which proves ( c ).We next claim that i ∗ preserves all limits and colimits. As it is a right adjoint it is enough to show that itpreserves all colimits. By Remark 7.2, it is enough to show that the functor t G k , • preserves colimits for all ≤ k ≤ r . Choose f , . . . , f s ∈ G ( G k , G ), containing precisely one element from each Out( G k )-orbit. Let∆ s ≤ Out( G k ) be the stabiliser of f s . We find that t G k ,V ( G ) = Hom k [Out( G k )] ( k [ G ( G k , G )] , V ) = Y s V ∆ s , and this is easily seen to preserve all colimits as required. A similar argument shows that i ! preserves alllimits and colimits. As before, it is enough to show that the functor e G k , • preserves all limits. We find that e G k ,V ( G ) = k [ G ( G, G k )] ⊗ k [Out( G k )] V = M s V ∆ s and this is easily seen to preserve all limits. (cid:3) The following results are standard. Proposition 7.4. (a) The simple objects of AU are the same as the indecomposable objects, and these are precisely theobjects e G,V for some G ∈ U and irreducible k [Out( G )] -module V . (b) Every nonzero morphism to a simple object is a split epimorphism, and every nonzero morphismfrom a simple object is a split monomorphism. (c) If S and S ′ are non-isomorphic simple objects in AU , then AU ( S, S ′ ) = 0 . (d) If S is a simple object in AU , then End( S ) is a division algebra of finite dimension over k . (e) The category AU has finitely many isomorphism classes of simple objects. (f) Suppose that the list S , . . . , S s contains precisely one simple object from each isomorphism class,and put D j = End( S j ) . Let N j be the category of right modules over D j , and put N = Q j N j .Define functors AU φ −→ N ψ −→ AU by φ ( X ) j = AU ( S j , X ) and ψ ( N ) = L j N j ⊗ D j S j . Then φ and ψ are inverse to each other, and soare equivalences.Proof. The first part of (a) is clear from the fact that all monomorphisms are split. As any morphism in U is an isomorphism we see that e G,V = s G,V and this is simple when V is irreducible, see Lemma 6.2. For(b), suppose that α : X → S is nonzero, where S is simple. Then image( α ) is a nonzero subobject of S , so itmust be all of S , so α is an epimorphism, and all epimorphisms are split. This gives half of ( b ), and the otherhalf is similar. Now suppose that α : S → S ′ , where both S and S ′ are simple. If α = 0 then ( b ) tell us that α is both a split monomorphism and a split epimorphism, so it is an isomorphism. The contrapositive givesclaim (c), and the special case S ′ = S , gives most of (d), apart from the finite-dimensionality statement.For that, we choose a list of groups G i as in Lemma 7.1, and put U = L i e G i which is a generator for AU .We can decompose U as a finite direct sum of indecomposables, say U = L sj =1 S d j j with 0 < d j < ∞ and S j S k for j = k . If S is simple, there is an nonzero map U → S and so a nonzero map S j → S for some j , that has to be an isomorphism from (b). This proves (e). We also note that S is a summand in U , soEnd( S ) is a summand in End( U ) and hence it has finite dimension over k , completing the proof of (d).Now define φ and ψ as in (f). Put T m = ψ ( S m ) ∈ N , so ( T m ) m = D m and ( T m ) j = 0 for j = m . Define η N : N → φψ ( N ) = AU ( S j , M k N k ⊗ D k S k ) ǫ X : ψφ ( X ) = M j AU ( S j , X ) ⊗ D j S j → X as follows. First, any n ∈ N j gives a map D j → N j and thus a map S j = S j ⊗ D j D j → S j ⊗ D j N j ≤ M k N k ⊗ D k S k we take this to be the j -th component of η N . Similarly, there is an evaluation morphism AU ( S j , X ) ⊗ S j → X ,which is easily seen to factor through AU ( S j , X ) ⊗ D j S j . We combine these maps to give ǫ X . e claim that ǫ X is an isomorphism. Indeed, we know that the object U is a generator for AU , so the objects S j form a generating family. As all epimorphisms in AU split, we see that every object is a retract of a directsum of objects of the form S m . We also see that both φ and ψ preserve all direct sums. It will thereforesuffice to check that ǫ S m is an isomorphism, and this follows easily from our description of T m = ψ ( S m ).Because every module over a division algebra is free, we also see that every object of N is a direct sum ofobjects of the form T m . It is easy to see that η T m is an isomorphism, and it follows that η N is an isomorphismfor all N . (cid:3) Projectives In this section we study and classify the projective objects of AU for a replete full subcategory U of G . Lemma 8.1. Consider an object P in AU . Then the following are equivalent: (a) P is projective in AU . (b) P is isomorphic to a retract of a direct sum of objects of the form e G with G ∈ U .Proof. First suppose that ( a ) holds. Let U be a countable collection of objects of U that contains at leastone representative of every isomorphism class. Put F P = M G ∈U M x ∈ P ( G ) e G ∈ AU . Each pair ( G, x ) defines a morphism ǫ ( G,x ) : e G → P by the Yoneda Lemma. By combining these for allpairs ( G, x ), we get a morphism ǫ : F P → P which is an epimorphism by construction. As P is assumed tobe projective this epimorphism must split, so P is a retract of F P , so ( b ) holds.Next, i ! preserves colimits by Lemma 5.3(e), so it preserves all direct sums and retracts, and it sends e U G to e G by Lemma 5.3(i). It follows that ( b ) implies ( c ).Now suppose that ( c ) holds. We claim that ( a ) also holds, or equivalently that any epimorphism f : X → P in AU splits. As i ! preserves all colimits, it also preserves epimorphisms. Thus i ! ( f ) : i ! ( X ) → i ! ( P ) is anepimorphism with projective target, so there exists h : i ! ( P ) → i ! ( X ) with i ! ( f ) ◦ h = 1 . We now apply i ∗ ,recalling that i ∗ i ! ≃ 1. We find that the map h ′ = i ∗ ( h ) : P → X satisfies f ◦ h ′ = 1, as required. (cid:3) Lemma 8.2. Let i : U → V be an inclusion of replete full subcategories of G , and let P be an object of AU .Then P is projective in AU iff i ! ( P ) is projective in AV .Proof. First, if P is projective then the functor AV ( i ! ( P ) , − ) is isomorphic to the composite of the exactfunctors i ∗ : AV → AU and AU ( P, − ), so it is exact, so i ! ( P ) is projective.Conversely, suppose that i ! ( P ) is projective. We can certainly choose a projective object Q ∈ AU and anepimorphism u : Q → P . As i ! is a left adjoint, it preserves colimits and epimorphisms, so i ! ( u ) : i ! ( Q ) → i ! ( P )is an epimorphism. As i ! ( P ) is assumed projective, we can choose v : i ! ( P ) → i ! ( Q ) with i ! ( u ) ◦ v = 1. We nowapply i ∗ to this, recalling that i ∗ i ! ≃ 1; we find that i ∗ ( v ) gives a section for u , so u is a split epimorphism,so P is projective. (cid:3) Proposition 8.3. Let i : U → V be an inclusion of replete full subcategories of G , and let Q be an object of AV . Then the following are equivalent: (a) Q ≃ i ! ( P ) for some projective object P ∈ AU . (b) Q is a retract of i ! ( P ) for some projective object P ∈ AU . (c) Q is a retract of some direct sum of objects e G , with G ∈ U . (d) Q is projective, and the counit map i ! i ∗ Q → Q is an isomorphism.Moreover, if these conditions hold then i ∗ ( Q ) is projective in AU .Proof. From what we have seen already it is clear that ( a ) ⇒ ( b ) ⇔ ( c ) and that ( a ) ⇒ ( d ). Now supposethat ( b ) holds, so there is a projective object P ∈ AU and an idempotent e : i ! P → i ! P with Q = e. ( i ! P ) =cok(1 − e ). As i ! is full and faithful, there is an idempotent f : P → P with i ! ( f ) = e . As i ! preserves okernels, it follows that Q = i ! ( f.P ), and of course f.P is projective, so ( a ) holds. Also, if Q ≃ i ! P as in( a ) holds then i ∗ Q is isomorphic to P and so is projective.Now all that is left is to prove that ( d ) ⇒ ( b ). Suppose that Q is projective, and that the counit map i ! i ∗ Q → Q is an isomorphism. Choose a projective P ∈ AU and an epimorphism f : P → i ∗ Q . As i ! preserves epimorphisms, we see that i ! ( f ) : i ! P → i ! i ∗ Q ≃ Q is an epimorphism, but Q is projective, so Q isa retract of i ! P as required. (cid:3) Recall the functors L ≤ n and L n from Construction 5.11. Recall also that we put U n = { G ∈ U | | G | = n } (which is a groupoid), and we write i n for the inclusion U n → U . Proposition 8.4. If P is projective in AU , then the filtration L ≤∗ P can be split, so there is an unnaturalisomorphism P ≃ L n L n P , and the filtration quotients L n P are themselves projective. Furthermore, i ∗ n ( L n P ) is projective in AU n and ( i n ) ! ( i ∗ n L n P ) = L n P .Proof. We have seen that P can be written as a retract of a direct sum of generators. In more detail, wecan choose an object Q = L α e G α and an idempotent u : Q → Q such that P ≃ u.Q , so without loss ofgenerality P = u.Q . Let Q n be the sum of the terms e G α for which | G α | = n , so that Q = L n Q n . We canthen decompose u as a sum of morphisms u nm : Q m → Q n . When m < n we have AU ( Q m , Q n ) = 0 andso u nm = 0. Given this, the relation u = u implies that u nn = u nn . The object P ′ n = u nn .Q n is thereforeprojective. Put P ′ = L n P ′ n and let f : P ′ → P be the composite P ′ inc −−→ Q u −→ u.Q = P . We claim that thisis an isomorphism. By passing to the colimit, it will suffice to show that L ≤ n ( f ) is an isomorphism for all n .By an evident reduction, it will suffice to show that L n ( f ) is an isomorphism for all n . As L n is an additivefunctor we have L n ( P ) = L n ( u ) .L n ( Q ) = L n ( u ) .Q n = u nn .Q n = P ′ n , as required. All remaining claims arenow easy. (cid:3) Corollary 8.5. Suppose we choose a complete system of simple objects in AU n for all n , giving a sequence ( e G i ,S i | G i ∈ U n ) n of indecomposable projectives in AU . Then every projective object is a direct sum ofobjects of the form e G i ,S i . In particular, every indecomposable projective is isomorphic to some e G i ,S i .Proof. Because AU n is semisimple, we see that i ∗ n ( L n P ) splits in the indicated way. As L n P ≃ ( i n ) ! ( i ∗ n ( L n P )),we see that L n P also splits, as does L n L n P , which is isomorphic to P . (cid:3) Proposition 8.6. Any projective object P can be written as P ≃ Q n L n P . Furthermore, products ofprojective objects are projective.Proof. By Proposition 8.4 we can write P = L n L n P . Now note that for a fixed G ∈ U , there are onlyfinitely many indices n such that P n ( G ) is nonzero, so the inclusion L n L n P → Q n L n P is an isomorphism.For the second claim, let ( P α ) be a family of projectives, and put P = Q α P α . We can write P α = Q k L k P α as above, so P = Q k Q k where Q k = Q α L k P α . We know from Proposition 7.3 that ( i k ) ! preserves products,so Q k is in the image of ( i k ) ! . It follows that Q k is projective and also that P = Q k Q k is the same as L k Q k , so P is projective. (cid:3) Proposition 8.7. Let U be a widely closed subcategory of G . Then the full subcategory of projective objectsis closed under tensor products. If U is a multiplicative global family, then the full subcategory of projectiveobjects is also closed under the internal homs.Proof. Consider projective objects P, Q ∈ AU . We can write P as a retract of a direct sum of terms e G .The functor ( − ) ⊗ Q sends sums to sums, and the functor Hom( − , Q ) sends sums to products, and bothsums and products of projectives are projective. We can therefore reduce to the case P = e G . Next, we cansplit Q as a direct sum or product of terms L n Q . The functor e G ⊗ ( − ) preserves sums, and the functorHom( e G , − ) preserves products, so we can reduce to the case where Q = L n Q , or equivalently Q = ( i n ) ! ( M )for some M ∈ AU n . We can now write M as a retract of a sum of terms e H α with | H α | = n . We know fromProposition 4.13 that e G ⊗ e H α is projective, and it follows easily that e G ⊗ Q is projective as claimed.It also follows from Proposition 4.13, together with the formula Hom( e G , Z )( H ) = AU ( e G ⊗ e H , Z ), thatthe functor Hom( e G , − ) preserves sums. If U is a multiplicative global family, then Theorem 4.20 tells us hat Hom( e G , e H α ) is projective. From these two facts it follows that Hom( e G , Q ) is also projective, whichfinishes the proof. (cid:3) Colimit-exactness Let U be a subcategory of G . In this section we will write L for the colimit functor X lim −→ G ∈U op X ( G )from AU to Vect k . Recall that U is said to be colimit-exact if L is an exact functor. We will show that mostof our examples have this property. First, however, we give an equivalent condition. Proposition 9.1. There is a natural isomorphism AU ( X, ) ≃ Vect k ( LX, k ) . Thus, the object ∈ AU is injective if and only if U is colimit-exact. If so, then all objects of the form DX = Hom( X, ) are alsoinjective.Proof. The natural isomorphism AU ( X, ) ≃ Vect k ( LX, k ) is clear. The functor V V ∗ = Vect k ( − , k )is certainly exact, so if L is exact then AU ( − , ) is exact, so is injective. Conversely, suppose that is injective. For any short exact sequence X → Y → Z in AU , we deduce that the resulting sequence( LZ ) ∗ → ( LY ) ∗ → ( LZ ) ∗ is also short exact, and then linear algebra shows that LX → LY → LZ is shortexact as well. This proves that L is exact. Also, there is a natural isomorphism AU ( X, DW ) = AU ( W ⊗ X, ).The functors W ⊗ ( − ) and AU ( − , ) are exact, and it follows that DW is injective as claimed. (cid:3) Lemma 9.2. (a) For any G ∈ U we have Le G = k . In particular, if ∈ U then L = Le = k . (b) For any G ∈ U and any k [Out( G )] -module V we have L ( e G,V ) = V Out( G ) (the module of coinvari-ants). (c) Unless G is maximal in U we also have L ( t G,V ) = L ( s G,V ) = 0 .Proof. For T ∈ Vect k we haveVect k ( L ( e G,V ) , T ) = AU ( e G,V , T ⊗ ) = Mod k [Out( G )] ( V, T ) = Vect k ( V Out( G ) , T ) . By the Yoneda Lemma, we therefore have L ( e G,V ) = V Out( G ) . Taking V = k [Out( G )] gives L ( e G ) = k .Now consider the object L ( t G,V ). This is the colimit over H ∈ U of the groups t G,V ( H ) = Map Out( G ) ( U ( G, H ) , V ).If there are no morphisms G → H , then t G,V ( H ) = 0. If there is a morphism α : G → H , then by definitionthe limit map t G,V ( H ) → L ( t G,V ) factors through α ∗ . This makes it clear that the map t G,V ( G ) → L ( t G,V )is surjective. Now suppose that G is not maximal in U , so we can choose β : K → G in U that is not anisomorphism. The map t G,V ( G ) → L ( t G,V ) will then factor through β ∗ , but the codomain of β ∗ is zero, so L ( t G,V ) = 0. A simpler version of the same argument also gives L ( s G,V ) = 0. (cid:3) Remark 9.3. For X, Y ∈ AU there are natural unit maps X → ( LX ) ⊗ and Y → ( LY ) ⊗ . We cantensor these together and take adjoints to get a map L ( X ⊗ Y ) → ( LX ) ⊗ ( LY ). This gives an oplaxmonoidal structure on L . However, the map L ( X ⊗ Y ) → ( LX ) ⊗ ( LY ) is rarely an isomorphism. Forexample, we have Le G ⊗ Le H = k but Proposition 4.13 shows that L ( e G ⊗ e H ) is freely generated by the setWide( G, H ) / conjugacy.We now start to prove that various categories are colimit-exact. Our first example is easy: Proposition 9.4. If U ≤ G is a groupoid, then it is colimit-exact.Proof. Choose a family ( G i ) i ∈ I containing precisely one representative of each isomorphism class in U . If X ∈ AU then the group Out( G i ) acts on X ( G i ), and we write X ( G i ) Out( G i ) for the module of coinvariants.As we work over a field of characteristic zero and Out( G i ) is finite, this is an exact functor of X . It is easyto identify lim −→ X with L i X ( G i ) Out( G i ) , and this makes it clear that the colimit functor is exact as well. (cid:3) For other examples we will use the following notion: efinition 9.5. A colimit tower for U is a diagram G ǫ ←− G ǫ ←− G ←− · · · in U such that(a) For every H ∈ U there is a pair ( i, α ) with i ∈ N and α ∈ U ( G i , H ).(b) For every diagram G i α −→ H β ←− G i in U there exists γ ∈ U ( G i +1 , G i +1 ) making the following diagramcommute: G i +1 G i +1 G i H G i . ǫ i γ ǫ i α β (c) For every diagram G i +1 α −→ H φ ←− K in U with U ( G i , K ) = ∅ , there exists β ∈ U ( G i +1 , K ) such that φ ◦ β = α . Construction 9.6. Suppose we have a colimit tower as above. For any X ∈ AU we define Λ i X to bethe group of coinvariants X ( G i ) Out( G i ) , and we let ρ i : X ( G i ) → Λ i X be the obvious reduction map. Bytaking H = G i and β = 1 in condition (b), we see that every automorphism of G i can be covered by anautomorphism of G i +1 . It follows that there is a unique map Λ i X → Λ i +1 X making the following diagramcommute: X ( G i ) X ( G i +1 )Λ i X Λ i +1 X ǫ ∗ i ρ i ρ i +1 We will just write ǫ ∗ i for this map. We define Λ ∞ X to be the colimit of the sequenceΛ X ǫ ∗ −→ Λ X ǫ ∗ −→ Λ X ǫ ∗ −→ · · · , and we let σ i denote the canonical map Λ i X → Λ ∞ X . As we are working over a field of characteristic zeroand Out( G n ) is finite, we see that Λ n is an exact functor. As sequential colimits are exact, we see thatΛ ∞ : AU → Vect k is also an exact functor. Proposition 9.7. For any colimit tower, there is a natural isomorphism Λ ∞ X → LX . Thus, if U has acolimit tower, then it is colimit-exact.Proof. Let θ H : X ( H ) → LX be the canonical morphism. It is formal that there is a unique map φ : Λ ∞ X → LX with φσ i ρ i = θ G i for all i . In the opposite direction, suppose we have H ∈ U . We can choose ( i, α ) asin condition (a) and define ψ H,i,α = σ i ρ i α ∗ : X ( H ) → Λ ∞ X. Using the obvious cone properties we see that this is the same as ψ H,i +1 ,αǫ i , or as ψ H,i,αµ for any µ ∈ Out( G i ).By using these rules together with condition (b), we see that ψ H,i,α is independent of the choice of ( i, α ), so wecan just denote it by ψ H . It is now easy to see that for any ζ : H → K we have ψ H ζ ∗ = ψ K : X ( K ) → Λ ∞ X .This means that there is a unique ψ : LX → Λ ∞ X with ψθ H = ψ H for all H . This is clearly inverse to φ . (cid:3) Remark 9.8. So far we only used conditions (a) and (b) in the definition of colimit tower. Condition (c)will play an important role in Section 12. Example 9.9. Let C be the category of cyclic groups. The morphisms can be described as follows:(a) If | G | = n then the group Aut( G ) = Out( G ) is canonically identified with ( Z /n ) × .(b) If | H | divides | G | then C ( G, H ) is a torsor for Aut( H ). Moreover, for any α : G → H and φ ∈ Aut( H ) = ( Z / | H | ) × there exists ψ ∈ Aut( G ) = ( Z / | G | ) × that reduces to φ , and any such ψ satisfies αψ = φα .(c) On the other hand, if | H | does not divide | G | then C ( G, H ) = ∅ . rom these observations it follows easily that the groups G n = Z /n ! form a colimit tower, and so C iscolimit-exact. Similarly, the groups Z /p n form a colimit tower in the category C [ p ∞ ] of cyclic p -groups, so C [ p ∞ ] is also colimit exact. For a more degenerate example, we can fix a positive integer d and consider thecategory C [ d ] of cyclic groups of order dividing d . We find that the constant sequence with value Z /d is acolimit sequence for C [ d ], so this category is again colimit-exact.Recall the category Z [ p r ] of finitely generated Z /p r -modules and its subcategory F [ p r ] of free Z /p r -modules. Lemma 9.10. Consider a diagram of epimorphisms A BC α γ β with A ∈ F [ p r ] , B ∈ Z [ p r ] and rk( A ) ≥ rk( B ) . Then the dotted arrow can be filled by another epimorphism.Proof. Put rk( A ) = n , rk( B ) = m and rk( C ) = l so that n ≥ m ≥ l . Choose elements c , . . . , c l ∈ C that project to a basis of C/pC over Z /p (so they form a minimal generating set for C ). Choose elements b , . . . , b l ∈ B with β ( b i ) = c i . The images of b , . . . , b l in B/pB will then be linearly independent. Chooseadditional elements b l +1 , . . . , b m ∈ B so that b , . . . , b m gives a basis for B/pB . After adding multiples of b , . . . , b l to b l +1 , . . . , b m if necessary, we can ensure that β ( b i ) = 0 for i > l . In the same way, we can findelements a , . . . , a n ∈ A such that α ( a i ) = c i for i ≤ l , and α ( a i ) = 0 for i > l , and a , . . . , a n gives a basisfor A/pA over Z /p . As A is free of rank n over Z /p r , it follows that the same elements give a basis over Z /p r . Thus, there is a unique morphism γ : A → B with γ ( a i ) = ( b i if 0 ≤ i ≤ m m < i ≤ n. As all the generators b i lie in the image of γ , we see that γ is surjective. It also satisfies βγ = α byconstruction. (cid:3) Example 9.11. Consider the category Z [ p r ] of finite abelian p -groups of exponent dividing p r , which isequivalent to the category of finitely generated Z /p r -modules and linear maps. Using Lemma 9.10 one seesthat the groups ( Z /p r ) n form a colimit tower. It follows that Z [ p r ] is colimit-exact. As these groups lie inthe subcategory F [ p r ] ≤ Z [ p r ], it is clear that they form a colimit tower for that subcategory as well.Most of the rest of this section is devoted to the proof of the following result: Theorem 9.12. If U ≤ G is submultiplicative then it is colimit-exact. We will prove this by giving a less explicit but much more general construction of colimit towers. For this,we will need a bit of preparation. Recall the functor T from Example 5.10. Lemma 9.13. Let X be a finite set and consider a diagram of epimorphisms between groups in U GT X H αλµ in which | G | ≤ | X | . Then the dotted arrow can be filled in by another epimorphism.Proof. Put L = ker( α ), so | L || H | = | G | ≤ | X | . Let i : X → T X be the canonical inclusion, and put X h = ( λi ) − { h } ⊆ X for each h ∈ H . We then have P h | X h | = | X | ≥ | H || L | , so we can choose h with | X h | ≥ | L | . Let µ h : X h → α − { h } be chosen arbitrarily, except that we choose µ h to be surjective. Bycombining these maps, we get µ ′ : X → G such that αµ ′ = λi . By the defining properties of T X , we see thatthere is a unique homomorphism µ : T X → G with µi = µ ′ . This satisfies αµi = λi and i ( X ) generates T X so αµ = λ . Now note that the restriction of α to the image of µ is an epimorphism since αµ is surjective.Also, the image of µ contains L as µ h is surjective. It follows that | Im( µ ) | = | L || H | = | G | so µ is surjectiveas required. (cid:3) emma 9.14. If G = 1 then ǫ : T G → G is not injective, so | T G | ≥ | G | .Proof. Choose any nontrivial g ∈ G and let τ : G → G be the transposition that exchanges 1 and g . Let e and e g denote the corresponding generators of F G or T G . The map τ induces an automorphism α of T G which exchanges e and e g . The homomorphism ǫα sends e to g = 1, so e N , so e gives a nontrivialelement of T G . However, this lies in the kernel of ǫ , so ǫ is not injective, and | T G | = | G || ker( ǫ ) | ≥ | G | . (cid:3) Remark 9.15. This lower bound is pitifully weak; in practice T G is enormously larger than G . Lemma 9.16. Suppose that α, β : G → H are surjective homomorphisms in U . Then there is an automor-phism γ of T G making the following diagram commute: T G T GG H G. γǫ ǫα β Proof. Put m = | G | / | H | = | ker( α ) | = | ker( β ) | . For each h ∈ H we have | α − { h }| = m = | β − { h }| , so wecan choose a bijection α − { h } → β − { h } . By combining these choices, we obtain a bijection σ : G → G such that βσ = α . This gives an automorphism γ = T σ of T G . We claim that βǫγ = αǫ : T G → H . It willsuffices to check this on the generating set G ⊂ T G , and that reduces to the relation βσ = α , which holdsby construction. (cid:3) Proof of Theorem 9.12. The claim is clear if U = { } . Suppose instead that U contains a nontrivial group G . Put G n = T n G , so we have a tower G ǫ ←− G ǫ ←− G ǫ ←− · · · . We claim that this is a colimit tower for U . Using Remark 9.15 we see that | G n | → ∞ as n → ∞ . For fixed H ∈ U we can therefore choose a surjective function G n → H for some n , and this will induce a surjectivehomomorphism G n +1 → H giving condition (a) of the colimit tower. Condition (b) holds by Lemma 9.16and (c) follows from Lemma 9.13, so U is colimit-exact. (cid:3) Proposition 9.17. Suppose that V ⊆ U ⊆ G , that U is colimit-exact and that V is closed upwards in U .Then V is also colimit-exact.Proof. Let i : V → U be the inclusion, and let c be the functor U → 1. Note that i ! : AV → AU is justextension by zero, as proved in Lemma 5.3, and so is exact. We are given that the functor L U = c ! is exact,so the composite L V = ( ci ) ! = c ! i ! is exact as well. (cid:3) We conclude with a counterexample. Example 9.18. The category G ≤ is not colimit-exact. Proof. The subcategory U ′ = { , C , C } is a skeleton of G ≤ , which makes it easy to calculate colimits. Let X < be given by X ( G ) = 0 when | G | = 1 and X ( G ) = ( G ) = k when | G | > 1. We find that LX = k but L = k , so L does not send the monomorphism X → to a monomorphism, so L is not exact. (cid:3) Complete subcategories In this section we introduce a well-behaved type of subcategory and present some examples. Definition 10.1. Let U be a subcategory of G .(a) For T ∈ G , we denote by δ ( T ) the minimum possible size of a generating set for T .(b) For m ∈ N , we put R m = { T ∈ U | δ ( T ) ≥ m } . (c) We say that U is expansive if for all G ∈ U and m ∈ N we have R m ↓ G = ∅ . d) Let U be expansive. For X ∈ AU and n > ω U n ( X ) = lim sup m →∞ { dim( X ( T )) /n δ ( T ) | T ∈ R m } ∈ [0 , ∞ ] . and W ( U ) n = { X ∈ AU | ω U n ( X ) < ∞} . It is easy to see that if ω U n ( X ) > ω U m ( X ) = ∞ for m < n . Similarly, if ω U n ( X ) < ∞ then ω U m ( X ) = 0 for m > n . Thus, there is at most one n such that 0 < ω U n ( X ) < ∞ . If such an n exists,we call it the order of X . Remark 10.2. We will often drop the superscript and just write ω n ( X ).Using the properties of the limsup we obtain the following result. Lemma 10.3. For any short exact sequence X → Y → Z in AU we have max( ω n ( X ) , ω n ( Z )) ≤ ω n ( Y ) ≤ ω n ( X ) + ω n ( Z ) . In particular, for any X and Z we have max( ω n ( X ) , ω n ( Z )) ≤ ω n ( X ⊕ Z ) ≤ ω n ( X ) + ω n ( Z ) . (cid:3) Corollary 10.4. The category W ( U ) n is closed under finite direct sums, subobjects, quotients, extensionsand retracts. It also contains e G for all G ∈ U ≤ n .Proof. The closure properties easily follow from Lemma 10.3. For the second claim, note that if A ⊂ T is agenerating set for T ∈ U , then the restriction map Hom( T, G ) → Map( A, G ) is injective, so | Hom( T, G ) | ≤| G | | A | . It follows that |U ( T, G ) | = | Epi( T, G ) | / | Inn( G ) | ≤ | Hom( T, G ) | / | Inn( G ) | ≤ | G | δ ( T ) / | Inn( G ) | = | G | δ ( T ) − | ZG | . From this it is easy to see that ω n ( e G ) ≤ | Inn( G ) | − if | G | = n , and ω n ( e G ) = 0 if | G | < n . (cid:3) We are now ready to introduce an important family of subcategories. Definition 10.5. A subcategory U of G is complete if the following conditions are satisfied: • U is expansive, i.e., for all G ∈ U and n > T ∈ U with δ ( T ) ≥ n and U ( T, G ) = ∅ ; • For all n > G ∈ U n , we have 0 < ω U n ( e G ) < ∞ . In other words, e G has order exactly | G | . Example 10.6. Recall that we always have ω n ( e G ) ≤ | Inn( G ) | − if | G | = n . • The category C [ p ∞ ] of cyclic p -groups is not complete, as it is not expansive. • The category E [ p ] of elementary abelian p -groups is complete. Indeed we have ω p n ( e C np ) = lim m →∞ | Epi( C mp , C np ) | p nm = lim m →∞ ( p m − p m − p ) · · · ( p m − p n − ) p nm = 1 . Let us produce more examples of complete subcategories. Proposition 10.7. If U ≤ G is nontrivial and submultiplicative, then it is complete.Proof. As U is nontrivial and subgroup-closed, it must contain C p for some p . Then for G ∈ U we have G × C np ∈ U with δ ( G × C np ) ≥ n , showing that U is expansive. We now need to show that ω | G | ( e G ) > G ∈ U . Without loss of generality we can assume that G = 1. For X m a set with m elements, considerthe group T X m ∈ U as defined in Example 5.10. By definition, there is a natural bijection Hom( T X m , G ) =Hom( F X m , G ) ≃ G m for all the groups G ∈ U ≤ m . Since by [15, Theorem 1] we havelim m →∞ | Epi( F X m , G ) | / | G | m = 1we deduce that lim m →∞ | Epi( T X m , G ) | / | G | m = 1 . t only remains to notice that δ ( T X m ) ≤ m so ω | G | ( e G ) ≥ lim m →∞ |U ( T X m , G ) || G | m = lim m →∞ | Epi( T X m , G ) || Inn( G ) || G | m = 1 | Inn( G ) | > . (cid:3) The completeness assumption give us information on the growth of the indecomposable projectives. Lemma 10.8. Let U be a complete subcategory of G . For G ∈ U and V an Out( G ) -representation, we have < ω | G | ( e G,V ) < ∞ .Proof. We show that dim( e G,V ( T )) = dim( V ) | Out( G ) | − | dim( e G ( T )) | , and so the claim follows by com-pleteness. It is easy to see that Out( G ) acts freely on U ( T, G ). Choose a subset M ⊂ U ( T, G ) containing onerepresentative of every orbit, so that | M | = | Out( G ) | − |U ( T, G ) | . We also see that M is a basis for e G ( T )as a module over the ring R = k [Out( G )], so e G,V ( T ) = V ⊗ R e G ( T ) ≃ V | M | . This gives dim( e G,V ( T )) = dim( V ) | M | = dim( V ) | Out( G ) | − dim( e G ( T ))as claimed. (cid:3) Proposition 10.9. Let U be complete subcategory of G . Then any monomorphism between projective objectsof AU is split.Proof. Let u : P → Q be a monomorphism between projective objects. By Proposition 8.4, we can write P = L n P n and Q = L n Q n , where P n and Q n are in the image of ( i n ) ! : AU n → AU , so AU ( P n , Q m ) = 0when n < m . We put P ≤ m = L k ≤ m P k = L ≤ m P , and similarly for Q . It is then clear that u restricts togive a monomorphism u ≤ m : P ≤ m → Q ≤ m . We will prove by induction on m that u ≤ m splits. The claim istrivial if m = 0. Let m > s 1, whereas if K m is nonzero, it musthave order m . It follows that K m must actually be zero, so u m is a monomorphism in AU m , so there is asplitting v : Q m → P m . Let s ≤ m : Q ≤ m → P ≤ m be given by s We introduce various finiteness conditions on objects of A and prove some implications amongst them. Werefer the reader to Remarks 11.9 and 13.2 for a summary. Definition 11.1. Consider a subcategory U ≤ G and an object X ∈ AU .(a) We say that X has finite type if dim( X ( G )) < ∞ for all G ∈ U .(b) We say that X is finitely projective if it can be expressed as the direct sum of a finite family ofindecomposable projectives.(c) We say that X is finitely generated if there is an epimorphism P → X , for some finitely projectiveobject P (or equivalently, for some object P of the form L ni =1 e G i ).(d) We say that X is finitely presented if there is a right exact sequence P → P → X , where P and P are finitely projective.(e) We say that X is finitely resolved if there is a resolution P ∗ → X , where each P i is finitely projective.(f) We say that X is perfect if there is a resolution P ∗ → X , where P i is finitely projective for all i , and P i = 0 for i ≫ g) We say that X has finite order if there exists n > ω n ( X ) < ∞ . (This is only meaningfulin the case where U is expansive.) Lemma 11.2. Let i : U → V be the inclusion of a subcategory. (a) The functor i ∗ always preserves objects of finite type. If U is closed downwards, then i ∗ preserves allfiniteness conditions from Definition 11.1 excluding that of finite order. (b) The functor i ! always preserves finitely presented and finitely generated objects. If U is closed upwards(and therefore expansive), then i ! preserves all finiteness conditions. (c) If U is closed downwards, then i ∗ preserves objects of finite type.Proof. Clearly, i ∗ preserves objects of finite type. If U is closed downwards, then i ∗ ( e G ) is either e G (if G ∈ U ) or 0 (if G 6∈ U ). It follows that i ∗ preserves (finitely) projective objects. Since i ∗ is also exact byLemma 5.3(e), it follows that i ∗ preserves conditions (a) to (f) in Definition 11.1.By Lemma 5.3(e) and (i), the functor i ! preserves colimits and preserves (finitely) projective objects. Itfollows that i ! preserves finitely presented and finitely generated objects. If U is closed upwards, then i ! isextension by zero by Lemma 5.3(f) so it preserves objects of finite type and finite order (if U expansive). Itis also exact so it preserves all the other finiteness conditions.Finally, part (c) follows from Lemma 5.3(g) as i ∗ is extension by zero. (cid:3) Remark 11.3. We have seen that the restriction functor i ∗ preserves projectives if U is closed downwards.This is no longer true if we relax the conditions on U as the following example shows. Choose a group G ∈ G ,and consider U = { H ∈ G | U ( H, G ) = ∅ , U ( G, H ) = ∅} = { H ∈ G ≥ G | H G } . Note that U is complete as it is closed upwards in G . Let i : U → G denote the inclusion functor. We claimthat i ∗ e G is not projective in U . Suppose that i ∗ e G was projective, so we could write i ∗ e G = L i e H i ,V i forsome groups H i ∈ U . Note that we must have | H i | > | G | for all i . If we calculate the order of these objectswe see that ω U| G | ( i ∗ e G ) = ω G| G | ( e G ) and so i ∗ e G has order | G | by completeness of G . On the other hand, for n = max i | H i | we have 0 < ω U n ( L i e H i ,V i ) < ∞ so this has order n . We have found a contradiction since n > | G | so i ∗ e G cannot be projective.It is useful to have a criterion to detect objects which are not finitely generated. Recall the notion of supportfrom Definition 2.6. Lemma 11.4. If X is finitely generated, then min(supp( X )) is finite.Proof. If X is finitely generated, we can find an epimorphism ϕ : L ri =1 e G i → X . Without loss of generalitywe can assume that each component e G i → X is nonzero so that X ( G i ) = 0 for all i . We claim thatmin(supp( X )) ⊆ { [ G ] , . . . , [ G r ] } which will prove the lemma. If [ H ] ∈ min(supp( X )), then X ( H ) = 0, so L i e G i ( H ) = 0, so we can choose an index i with e G i ( H ) = 0, so we can choose a morphism α : H → G i in U . Now both [ H ] and [ G i ] lie in supp( X ), and [ H ] is assumed to be minimal, so α must be an isomorphism,so [ H ] = [ G i ] as required. (cid:3) Proposition 11.5. Let U be a complete subcategory of G . Then any object of AU with a finite projectiveresolution is projective. In particular any perfect object is finitely projective.Proof. Let P ∗ → X be a projective resolution and suppose that P i = 0 for all i > n . If n > d n : P n → P n − must be a monomorphism, so Proposition 10.9 tells us that it is split. Nowlet Q ∗ be the same as P ∗ except that Q n = 0 and Q n − = cok( d n ). We find that this is again a projectiveresolution of X . By repeating this construction, we eventually obtain a projective resolution of length one,showing that X itself is projective. (cid:3) Remark 11.6. The Proposition above is not true if we drop the completeness condition. For example let C [ p ∞ ] be the subcategory of cyclic p -groups. Then there is a short exact sequence 0 → c C p → c C p → t C p ,k → t C p ,k is perfect. On the other hand, we have t C p ,k ( C p r ) = 0 for all r > 1, and itfollows easily from this that t C p ,k is not projective. roposition 11.7. Let U be a complete subcategory of G . Then any finitely projective object in AU hasfinite order.Proof. The zero object has by definition finite order. For r ≥ 1, we have0 < ω n r M i =1 e G i ,S i ! < ∞ if n = max i ( | G i | )by Lemma 10.8. (cid:3) Lemma 11.8. Let U ≤ G be finite (meaning that it has only finitely many isomorphism classes). Then thefollowing are equivalent for an object X ∈ AU : (a) X has finite type; (b) X is finitely generated; (c) X is perfect.Proof. Recall the explicit projective resolution from Construction 5.6. We have P = l ! l ∗ ( X ) = L G ∈U ′ e G,X ( G ) .If X has finite type, then P is a finitely generated projective object since U is finite. This gives (a) ⇒ (b).Clearly, (b) ⇒ (a) so (a) is equivalent to (b).Now suppose that X is finitely generated (and hence of finite type) and consider the canonical projectiveresolution P • → X . The explicit formula for P i tells us that P i has finite type, and it follows from theprevious paragraph that P i is finitely generated too. To prove (b) ⇒ (c), we need to show that P n = 0for n ≫ 0. Recall from Remark 5.7 that base( P n ) ≥ base( X ) + n . Now note that any object in AU withsufficiently large base is zero as U is finite. Hence P n = 0 for n ≫ ⇒ (a) is clear. (cid:3) Remark 11.9. So far we have the following implications:finitely resolved finitely presented finitely generated finite typeperfect finitely projective finite order. completeness completeness Torsion and torsion-free objects In this section we introduce the notions of torsion, absolutely torsion and torsion-free object. We study theirformal properties and give some examples. Definition 12.1. Consider an object X of AU . • We say that x ∈ X ( G ) is torsion if there exists H ∈ U and f ∈ U ( H, G ) such that f ∗ ( x ) = 0. • We say that x ∈ X ( G ) is absolutely torsion if there exists m ∈ N such that for all f ∈ U ( H, G ) with | H | ≥ m we have f ∗ ( x ) = 0. • We say that X is torsion (resp., absolutely torsion ) if it consists entirely of torsion (resp., absolutelytorsion) elements. • We say that X is torsion-free if it does not contain any nonzero torsion element. Equivalently, X istorsion-free if and only if all the maps α ∗ : X ( G ) → X ( H ) are injective. • We write tors( X )( G ) for the subset of torsion elements in X ( G ). Hypothesis 12.2. Throughout we will assume that U ≤ G has a colimit tower as in Definition 9.5. This isnot a very restrictive assumption as we have shown in Section 9 that most natural examples satisfy this. Lemma 12.3. For an element x ∈ X ( H ) , the following are equivalent: (a) x is torsion. (b) There exists α ∈ U ( G n , H ) for some n such that α ∗ ( x ) = 0 in X ( G n ) . (c) There exists n such that for all n ≥ n and all α ∈ U ( G n , H ) we have α ∗ ( x ) = 0 in X ( G n ) . roof. By condition (a) of the colimit tower, we see that U ( G n , H ) = ∅ for large n . It follows that ( c ) ⇒ ( b ) ⇒ ( a ). Now suppose that ( a ) holds, so we can choose β ∈ U ( G, H ) for some G with β ∗ ( x ) = 0. Now let n be least such that U ( G n − , G ) = ∅ . Suppose that n ≥ n , so U ( G n − , G ) = ∅ . If α ∈ U ( G n , H ), thencondition (c) of the colimit tower gives us a morphism γ ∈ U ( G n , G ) with α = β ◦ γ , and it follows that α ∗ ( x ) = 0. Thus, part ( c ) holds. (cid:3) Recall the colimit functor L : AU → Vect k from Section 9. Lemma 12.4. Consider an object X ∈ AU and an element x ∈ X ( G ) . Then x is torsion if and only if theelement G ⊗ x ∈ ( e G ⊗ X )( G ) maps to zero in L ( e G ⊗ X ) .Proof. Suppose that x is torsion, so we can choose α : G ′ → G with α ∗ ( x ) = 0. This means that α ∗ (1 G ⊗ x ) = α ⊗ α ∗ ( x ) = 0. The description L ( e G ⊗ X ) = lim −→ H ( e G ⊗ X )( H ) shows that 1 G ⊗ x is sent to zero in L ( e G ⊗ X ).For the converse, suppose we have an integer n and a morphism α ∈ U ( G n , G ). Put Γ = Out( G n ) and∆ = { δ | αδ = α } . Define a map ξ : ( e G ⊗ X )( G n ) → X ( G n ) , ξ ( π ⊗ m ) = X { γ ∗ m | γ ∈ Γ , πγ = α } One checks that ξθ ∗ = ξ for all θ ∈ Γ, so there is an induced map from the coinvariants ξ : ( e G ⊗ X )( G n ) Γ → X ( G n ). One also checks that ξ ( α ∗ (1 G ⊗ x )) = | ∆ | α ∗ ( x ) for all x ∈ X ( G ). The condition that α ∗ (1 G ⊗ x )maps to zero in L ( e G ⊗ X ) is equivalent to α ∗ (1 G ⊗ x ) mapping to zero in ( e G ⊗ X )( G n ) Γ for some n ≥ ξ (0) = ξ ( α ∗ (1 G ⊗ x )) = | ∆ | α ∗ ( x ) so x is torsion. (cid:3) Corollary 12.5. If x ∈ X ( G ) is not torsion, then there is a morphism u : e G ⊗ X → such that u (1 G ⊗ x ) = 1 .Proof. As the image of 1 G ⊗ x is nonzero in L ( e G ⊗ X ), we can choose a k -linear map u : L ( e G ⊗ X ) → k sending this image to 1. Then u is adjoin to a morphism u : e G ⊗ X → as claimed. (cid:3) Lemma 12.6. For any finite dimensional subspace V ≤ tors( X )( G ) , there is a map α : H → G in U with α ∗ ( V ) = 0 . Moreover, tors( X ) defines a subobject of X in AU which is the largest torsion subobject of X .The assignment tors is functorial in X so we have a functor tors: AU → AU .Proof. Suppose we have torsion elements x , . . . , x s ∈ tors( X )( G ). By Lemma 12.3(c), we can choose n largeso that α ∗ ( x i ) = 0 for all α ∈ U ( G n , G ) and all 1 ≤ i ≤ s . Thus, if V is the span of { x , . . . , x n } , we have α ∗ ( V ) = 0, so V ≤ tors( X )( G ). This proves in particular that tors( X )( G ) is a vector subspace of X ( G ).Now suppose we have α ∗ ( x ) = 0, and we also have another morphism β : G ′ → G in U . By condition (b) ofthe colimit tower, we can fill the dotted arrow in the diagram G ′ H G. βαγ We have γ ∗ β ∗ ( x ) = α ∗ ( x ) = 0, so β ∗ ( x ) is a torsion element. This shows that tors( X ) is a subobject of X .All remaining claims are now clear. (cid:3) The following example illustrates the fact that many things can go wrong if we do not assume that U has acolimit tower. Example 12.7. Consider the following object of AG ≤ X = ( k x pr x ←−− k x ⊕ k y pr y −−→ k y ) . Note that x, y ∈ X (1) are torsion since pr x ( y ) = 0 = pr y ( x ). On the other hand, x + y ∈ X (1) is not torsionsince pr x ( x + y ) = x and pr y ( x + y ) = y . In particular tors( X )(1) is not a vector subspace of X (1). emark 12.8. The sum of two torsion-free subobjects need not be torsion-free. To see this, consider atorsion-free object Y , a nonzero torsion object Z and an epimorphism f : Y → Z . In Y ⊕ Z we have a copyof Y , and the graph of f is another subobject Y ′ ≤ Y ⊕ Z which is also isomorphic to Y and so is torsion-free.However, Y + Y ′ is all Y ⊕ Z and so is not torsion-free. Lemma 12.9. For any object X of AU , the quotient X/ tors( X ) is torsion-free.Proof. Consider an element x ∈ ( X/ tors( X ))( G ), so x is represented by some element x ∈ X ( G ). If x is atorsion element, then we have α ∗ ( x ) = 0 for some α ∈ U ( H, G ), or equivalently α ∗ ( x ) ∈ tors( X )( H ). Thismeans that there exists β ∈ U ( K, H ) with ( αβ ) ∗ ( x ) = β ∗ ( α ∗ ( x )) = 0. Thus x is a torsion element and x = 0as required. (cid:3) Recall the objects e G,V and t G,V from Definition 2.7. Lemma 12.10. For all G ∈ U , we have that e G,V is torsion-free and t G,V is absolutely torsion. Thus, anyprojective object is torsion-free.Proof. It is clear that t G,V is absolutely torsion as t G,V ( K ) is zero as soon as | K | > | G | . It is enough toshow that e G is torsion-free as e G,V is a retract of a direct sum of e G ’s. Thus, we need to show that forany epimorphism ϕ : H → K the linear map ϕ ∗ : k [ U ( K, G )] → k [ U ( H, G )] is injective. This is equivalent toproving that the map ϕ ∗ : U ( K, G ) → U ( H, G ) is injective, or in other words that ϕ is an epimorphism inthe category U . This is the content of Lemma 2.2. (cid:3) We write AU t and AU f for the subcategories of torsion and torsion-free objects. Lemma 12.11. (a) For an object X ∈ AU , we have X ∈ AU t if and only if AU ( X, Y ) = 0 for all Y ∈ AU f . (b) For an object Y ∈ AU , we have Y ∈ AU f if and only if AU ( X, Y ) = 0 for all X ∈ AU t .Proof. If f : X → Y then f (tors( X )) ≤ tors( Y ). If X ∈ AU t and Y ∈ AU f then tors( X ) = X andtors( Y ) = 0 so this becomes f ( X ) = 0 and f = 0. Thus, for X ∈ AU t and Y ∈ AU f we have A ( X, Y ) = 0.Now suppose that X is such that AU ( X, Y ) = 0 for all Y ∈ AU t . In particular, this means that the projection X → X/ tors( X ) is zero, so tors( X ) = X and X ∈ AU t .Suppose instead that Y is such that AU ( X, Y ) = 0 for all X ∈ AU t . In particular, this means that theinclusion tors( Y ) → Y is zero, so tors( Y ) = 0 and Y ∈ AU f . (cid:3) Lemma 12.12. Consider objects X ∈ AU t and Y ∈ AU f . Then for all Z ∈ AU , we have (a) X ⊗ Z is torsion; (b) Hom( X ⊗ Z, Y ) = 0 .Proof. Any element of ( X ⊗ Z )( G ) can be written as a finite linear combination of homogeneous terms x i ⊗ z i . For each of such term we can find α i : H i → G such that α ∗ i ( x i ) = 0. Thus we have α ∗ ( x ⊗ z ) = α ∗ ( x ) ⊗ α ∗ ( z ) = 0. As a finite linear combination of torsion elements is again torsion by Lemma 12.6, wededuce that X ⊗ Z is torsion. For all G ∈ U , we haveHom( X ⊗ Z, Y )( G ) = AU ( e G ⊗ X ⊗ Z, Y ) = 0by part (a) and Lemma 12.11. (cid:3) Lemma 12.13. The subcategory AU t is localizing that is, it is closed under arbitrary sums, subobjects,extensions and quotients.Proof. Consider an exact sequence X i −→ Y p −→ Z in which X and Z are torsion objects. Consider an element y ∈ Y ( G ). As Z is a torsion object, we can choose α : H → G with α ∗ ( p ( y )) = 0. This means that ( α ∗ ( y )) = 0, so α ∗ ( y ) = i ( x ) for some x ∈ X ( H ). As X is a torsion object, we can choose β : K → H with β ∗ ( x ) = 0, and it follows that ( αβ ) ∗ ( y ) = β ∗ i ( x ) = i ( β ∗ ( x )) = i (0) = 0 . This shows that Y is also a torsion object so AU t is closed under extensions.Now let X be a sum of torsion objects X i and consider an element x ∈ X ( G ). By definition, we can write x = x i + . . . + x i n for torsion elements x i k ∈ X i k ( G ). By Lemma 12.3(c), we can find large n such that α ∗ ( x i k ) = 0 for all i k and α ∈ U ( G n , H i k ). Thus, we have α ∗ ( x ) = α ∗ i ( x i ) + . . . α ∗ i n ( x i n ) = 0so x is torsion. This shows that AU t is closed under arbitrary sums. All the other claims are clear. (cid:3) Lemma 12.14. The subcategory AU f is closed under subobjects, extensions, arbitrary sums and arbitraryproducts.Proof. From Lemma 12.11(b) it is clear that AU f is closed under products and subobjects. As products andsums are computed objectwise, we see that every sum injects in the corresponding product, so AU f is alsoclosed under coproducts. Now consider a short exact sequence as follows, in which X and Z are torsion-free X Y Z. f g If T is a torsion object, this gives a left exact sequence0 = AU ( T, X ) AU ( T, Y ) AU ( T, Z ) = 0 , f ∗ g ∗ proving that AU ( T, Y ) = 0. It follows that AU f is also closed under extensions. (cid:3) We will now give another characterization of torsion-free objects under some mild conditions on U . But firstwe need a little bit of preparation. Construction 12.15. Recall the inclusion functor l : U × → U and the functor l ! l ∗ from Construction 5.6.For any object X ∈ AU , we set SX = D ( l ! l ∗ ( DX )) which is injective by Proposition 9.1. Adjoint to thecounit map l ! l ∗ ( DX ) → DX , we have a map X ⊗ l ! l ∗ ( DX ) → which is itself adjoint to a map ξ : X → SX .If we fix a skeleton U ′ for U , we have the explicit formula SX = Y G ∈U ′ De G,DX ( G ) , and the map ξ has G -component which is adjoint to the evaluation map ev : X ⊗ e G,DX ( G ) → . Moreexplicitly, we haveev : X ( H ) ⊗ e G ( H ) ⊗ Out( G ) AU ( e G ⊗ X, ) → k, x ⊗ [ α ] ⊗ f f ([ α ] ⊗ x )for all H ∈ U . Proposition 12.16. Suppose that U is a multiplicative global family and consider an object X ∈ AU . Then SX is projective and we have an exact sequence → tors( X ) → X ξ −→ SX. In particular, X is torsion-free if and only if it can be embedded in a projective object.Proof. We first show that SX is projective. By Proposition 8.6, it is enough to show that De G,DX ( G ) is projective. Note that Out( G ) acts freely on DX ( G ) = AU ( e G ⊗ X, ). Choose u , . . . , u r ∈ DX ( G )containing precisely one element from each Out( G )-orbit so that DX ( G ) = L ri =1 k [Out( G )] and hence e G,DX ( G ) = L ri =1 e G . Therefore we have reduced the problem to showing that De G is projective. This nowfollows from Theorem 4.20. f SX is projective, then it is also torsion-free by Lemma 12.10, so we get tors( X ) ⊆ ker( ξ ). If x ∈ X ( G ) isnot torsion, then by Corollary 12.5 we can find u ∈ DX ( G ) such that u (1 G ⊗ x ) = 0. In particular, we haveev(1 G ⊗ x ⊗ u ) = 0 and hence ker( ξ ) ⊆ tors( X ). This shows that the sequence is exact. (cid:3) Let X be a finitely generated torsion object. It is tempting to conclude that X ( G ) should be zero when G is sufficiently large, in some sense. However, the following example shows that this is not correct. Example 12.17. Let θ : P → Q be a non-split epimorphism between groups in U . This gives a map θ ∗ : e P → e Q , and we define X to be the cokernel (so X is finitely presented). The obvious generator x ∈ X ( Q ) satisfies θ ∗ ( x ) = 0 by construction, so x is torsion. As x generates X , it follows that X is a torsionobject. Note that X ( G ) is the quotient of k [ U ( G, Q )] in which we kill every basis element [ α ] for which thehomomorphism α : G → Q can be lifted to P . Note that no split epimorphism α : G → Q can be lifted to P ,because that would give rise to a splitting of θ . In particular, if H admits a split epimorphism to Q , then X ( H ) = 0. Thus, we have X ( H × Q ) = 0 for all H ∈ U .It is true, however, that if X is a finitely generated torsion object, and G is both sufficiently large andsufficiently free, then X ( G ) = 0. We now proceed to make a precise version of this statement. Definition 12.18. We say that an object X ∈ AU is G ∗ - null if X ( G n ) = 0 for large n . Lemma 12.19. If X is G ∗ -null, then it is torsion. The converse holds if X is finitely generated.Proof. First suppose that X is G ∗ -null. Consider an element x ∈ X ( H ). Choose n large enough that X ( G n ) = 0 and U ( G n , H ) = ∅ . Then for α ∈ U ( G n , H ) we have α ∗ ( X ) = 0, as required.Conversely, suppose that X is finitely generated, with generators x i ∈ X ( H i ) for i = 1 , . . . , d say. ByLemma 12.3 we can choose n i such that α ∗ ( x i ) = 0 in X ( G m ) for all m ≥ n i , and all α ∈ U ( G m , H i ). Put n = max( n , . . . , n d ); then we find that X ( G n ) = 0 for all n ≥ m . (cid:3) We finish this section by giving some examples of torsion objects. Example 12.20. Let G be cyclic of order p , so Aut( G ) is cyclic of order p − 1, and let ψ ∈ Aut( G ) be agenerator. Let X be the cokernel of ψ ∗ − e G → e G . By definition X ( H ) is the quotient of k [ U ( H, G )] bythe subspace generated by the elements [ ψα ] − [ α ] for all α . As ψ is a generator of Aut( G ), we can identify X with c G from Definition 2.7. In particular, X is projective and torsion-free. This illustrates the fact thatwe can introduce quite a lot of relations without creating torsion. Example 12.21. Take U = Z [ p ∞ ] and let C be cyclic of order p . Let λ, ρ : C → C be the two projections,and let X be the cokernel of λ ∗ − ρ ∗ : e C → e C . This means that X ( G ) = k [ T ( G )], where T G is thecoequaliser of the maps λ ∗ , ρ ∗ : k [ U ( G, C )] → k [ U ( G, C )]. Let Q ( G ) be the Frattini quotient of G , so Q ( G ) ≃ C d ( G ) for some d ( G ) ≥ 0. If d ( G ) = 0 then G = 1 and T ( G ) = ∅ and X ( G ) = 0. If d ( G ) = 1 then G is cyclic and U ( G, C ) = ∅ so T ( G ) = U ( G, C ) = U ( Q ( G ) , C ) (which is a set of size p − 1) so X ( G ) ≃ k p − .Now suppose that d ( G ) ≥ 2. If α and β are epimorphisms from G to C with different kernels then thecombined map φ = ( α, β ) : G → C is again surjective with λφ = α and ρφ = β so [ α ] = [ β ] in T ( G ). Even if α and β have the same kernel, we can choose a third epimorphism γ : G → C with different kernel (becauseof the fact that d ( G ) ≥ α ] = [ γ ] = [ β ]. From this we see that T ( G ) is a singleton and so X ( G ) = k . To summarize X ( G ) = k [ T ( G )] = d ( G ) = 0 U ( G, C ) ≃ k p − if d ( G ) = 1 k if d ( G ) ≥ . From our discussion we also see thattors( X )( G ) ≃ ( k p − if G is nontrivial and cyclic0 otherwise( X/ tors( X ))( G ) ≃ ( G = 1 k if G = 1 . xample 12.22. Take U = Z [2 ∞ ]. There are then three morphisms λ, ρ, σ ∈ U ( C , C ), and we define X to be the cokernel of λ ∗ + ρ ∗ + σ ∗ : e C → e C . We claim that X is a torsion object. To see this, we put u = λ + ρ + σ ∈ e C ( C ) so that X ( G ) is the quotient of k [ U ( G, C )] by all elements of the form φ ∗ ( r ) as φ runs over U ( G, C ). If d ( G ) = 1 then U ( G, C ) is a singleton and U ( G, C ) = ∅ and X ( G ) = k . If d ( G ) = 2then k [ U ( G, C )] has three elements, say α, β, γ , and X ( G ) = k { α, β, γ } / ( α + β + γ ) ≃ k . Now consider X ( C ). This is spanned by the seven nonzero homomorphisms C → C . There are sevensubgroups of order 4 in Hom( C , C ) ≃ C : A = { , e ∗ , e ∗ , ( e + e ) ∗ } A = { , e ∗ , e ∗ , ( e + e ) ∗ } A = { , e ∗ , e ∗ , ( e + e ) ∗ } A = { , e ∗ , ( e + e ) ∗ , ( e + e + e ) ∗ } A = { , e ∗ , ( e + e ) ∗ , ( e + e + e ) ∗ } A = { , e ∗ , ( e + e ) ∗ , ( e + e + e ) ∗ } A = { , ( e + e ) ∗ , ( e + e ) ∗ , ( e + e ) ∗ } where e ∗ , e ∗ and e ∗ denote the canonical generators. For each of these A i we have a relation, saying that thesum of the three nonzero homomorphisms in that subgroup is zero. For example, the relation attached to A tells us that e ∗ + e ∗ + ( e + e ) ∗ = 0. Let u be the sum of all these relations, and let v α be the sum ofthe subset that involve a particular morphism α . A calculation shows that (3 v α − u ) / α . It follows thatthe resulting quotient X ( C ) is zero. If d ( G ) ≥ α ∈ U ( G, C ) can be factored through C , and itfollows from this that X ( G ) = 0. Thus X is a torsion object as claimed.13. Noetherian abelian categories The goal of this section is to study when the category AU is locally noetherian. Definition 13.1. Let U be a subcategory of G . • An object X ∈ AU is noetherian if every subobject of X is finitely generated. • The category AU is locally noetherian if e G is noetherian for all G ∈ U . Remark 13.2. Suppose that U is locally noetherian. After adding the obvious consequences of the noether-ian property to Remark 11.9 we get the following diagram of finiteness conditions for objects in AU :finitely resolved finitely presented finitely generated finite typeperfect finitely projective finite order. completeness completeness It is not difficult to find subcategories of G for which AU is not locally noetherian. Proposition 13.3. Let U be a full subcategory containing the trivial group and infinitely many cyclic groupsof prime order. Then AU is not locally noetherian.Proof. Let χ + ∈ AU be the subobject of given by χ + ( T ) = ( | T | = 1 k if | T | > . Note that min(supp( χ + )) contains the isomorphism classes of all cyclic groups of prime orders. ApplyLemma 11.4 to see that χ + cannot be finitely generated. (cid:3) The rest of this section will be devoted to prove the following theorem. Theorem 13.4. Fix a prime number p and a positive integer n . The abelian category AU is locally noetherianfor the following choices of subcategories U : (a) F [ p n ] = { free Z /p n - modules } . (b) C [ p ∞ ] = { cyclic p - groups } . c) Z [ p n ] = { fin . gen . Z /p n - modules } . (d) Z [ p ∞ ] = { finite abelian p - groups } .Proof. Sam and Snowden proved part (a) [19, 8.3.1]. The proofs of part (b),(c),(d) will be given in the nextsubsections. (cid:3) Part (b). We start by introducing the criterion for noetherianity developed in [6] which applies to aspecial type of subcategories. Definition 13.5 ([6, 2.2]) . Let U be a subcategory of G and fix a skeleton U ′ for U . If G, H ∈ U we write G ≫ H to mean that U ( G, H ) = ∅ . We say that U has type A ∞ if there exists an isomorphism of posets( U ′ , ≫ ) ≃ ( N , ≥ ). Example 13.6. The subcategory C of all cyclic groups is not of type A ∞ as there are no epimorphisms C → C or C → C . However if we fix a prime number p , then the subcategory C [ p ∞ ] of cyclic p -groupshas type A ∞ . Recall that F [ p n ] is (equivalent to) the category of finitely generated free modules over Z /p n ;this also has type A ∞ . The same is true of the category E [ p ] of elementary abelian p -groups (because it isthe same as F [ p ]).For compatibility with our work, we reformulate [6, 3.1] for contravariant diagrams. Definition 13.7. We say that the category U has the transitivity property if the action of Out( G ) on U ( G, H )is transitive whenever G ≫ H . Definition 13.8. Suppose that U has the transitivity property. For any pair ( G, H ) with G ≫ H we letOut( G ) act diagonally on U ( G, H ) and put U ( G, H ) = U ( G, H ) / Out( G ). Lemma 13.9. Suppose we fix α ∈ U ( G, H ) and put Φ( α ) = { φ ∈ Out( G ) | αφ = α } . Then there is anatural bijection ζ : U ( G, H ) / Φ( α ) → U ( G, H ) .Proof. We have a map U ( G, H ) → U ( G, H ) given by γ ( α, γ ), and this induces a map ζ : U ( G, H ) / Φ( α ) → U ( G, H ) . If ( β, γ ) ∈ U ( G, H ) then the transitivity property gives θ ∈ Out( G ) with βθ = α and it follows that[ β, γ ] = [ βθ, γθ ] = ζ ( γθ ) in U ( G, H ). This shows that ζ is surjective.On the other hand, if ζ [ β ] = ζ [ β ] then there exists φ ∈ U ( G ) with ( αφ, β φ ) = ( α, β ). This means that αφ = α (so φ ∈ Φ( α )) and β φ = β (so [ β ] = [ β ] in U ( G, H ) / Φ( α )). This shows that ζ is also injective. (cid:3) Lemma 13.10. Suppose that G ′ ≫ G and u ∈ U ( G, H ) , so u ⊆ U ( G, H ) . Put λ ( u ) = λ GG ′ ( u ) = { ( αφ, βφ ) | ( α, β ) ∈ u, φ ∈ U ( G ′ , G ) } ⊆ U ( G ′ , H ) . Then λ ( u ) is a Out( G ′ ) -orbit, or in other words an element of U ( G ′ , H ) . The map λ can also be characterisedby λ [ α, β ] = [ αφ, βφ ] for any φ ∈ U ( G ′ , G ) .Proof. A typical element of λ ( u ) has the form x = ( αφ, βφ ) with ( α, β ) ∈ u and φ ∈ U ( G, H ). If θ ∈ Out( G )then the map φ ′ = φθ also lies in U ( G, H ) and θ ∗ x = ( αφ ′ , βφ ′ ); this shows that λ ( u ) is preserved by Out( G ).Now suppose we fix an element x = ( α, β ) ∈ u and a map φ ∈ U ( G, H ) and put x ′ = ( αφ, βφ ) ∈ λ ( u ).Any element of u has the form ( αζ, βζ ) for some ζ ∈ Out( G ). Thus, any element y ∈ λ ( u ) has the form y = ( αζψ, βζψ ) for some ζ ∈ Out( G ) and ψ ∈ U ( G ′ , G ). By the transitivity property we can find ξ ∈ Out( G ′ )with ζψ = φξ , so y = ( αφξ, βφξ ) = ξ ∗ ( x ′ ). It follows that λ [ x ] = [ x ′ ], so in particular λ [ x ] is an orbit asclaimed. (cid:3) Definition 13.11. We say that U has the bijectivity property if for all H there exists G ≫ H such that forall G ′ ≫ G the map λ : U ( G, H ) → U ( G ′ , H )is bijective. emark 13.12. Our bijectivity property is not visibly the same as that of [6, 3.2]. However, Lemma 13.9shows that they are equivalent (and we consider that our version is more transparent).We are finally ready to state the criterion. Theorem 13.13 ([6, 3.7]) . Let U be a subcategory of G of type A ∞ . Suppose that U satisfies the transitivityand bijectivity properties. Then AU is locally noetherian. We now apply the criterion to our case of interest. Theorem 13.14. Fix a prime number p and let C [ p ∞ ] be the family of cyclic p -groups. Then the category AC [ p ∞ ] is locally noetherian.Proof. We have already seen that C [ p ∞ ] has type A ∞ so it is enough to check that it satisfies the transitivityand bijectivity property. Recall the discussion on the morphisms of C [ p ∞ ] from Example 9.9.Consider cyclic groups G and H and suppose that | H | divides | G | so that U ( G, H ) = ∅ . We know that forany α ∈ U ( G, H ) and φ ∈ Aut( H ) there exists ψ ∈ Aut( G ) such that αψ = φα . Combining this with thefact that U ( G, H ) is a torsor for Aut( H ), we find that U ( G, H ) is a single orbit for Aut( G ). Thus C [ p ∞ ]satisfies the transitivity condition.If ( α, β ) ∈ U ( G, H ) then there is a unique element φ ∈ Aut( H ) with β = φ ◦ α . This is unchanged if wecompose α and β with any surjective homomorphism ǫ : G ′ → G . It follows that the rule [ α, β ] φ givesa well-defined bijections ξ = ξ GH : U ( G, H ) → Aut( H ). This also satisfies ξ G ′ H λ = ξ GH , so all the maps λ are bijective, and so C [ p ∞ ] satisfies the bijectivity condition. (cid:3) Part (c) and (d). The rest of this section will be devoted to proving the following result. Theorem 13.15. Fix a prime number p . Recall that Z [ p ∞ ] is the category of finite abelian p -groups, andthat Z [ p n ] is the subcategory where the exponent divides p n . Then the categories AZ [ p ∞ ] and AZ [ p n ] arelocally noetherian. We will apply a different criterion due to Sam and Snowden that we shall now recall [19]. The basic outline isas follows. One way to prove that polynomial rings are noetherian is to use the technology of Gröbner bases.If C is a category satisfying appropriate combinatorial and order-theoretic conditions, we can use similartechniques to prove that [ C , Vect k ] is locally noetherian. If U ≤ G and we have a functor C → U op withappropriate finiteness properties, we can then deduce that AU is locally noetherian. In the case U = Z [ p ∞ ]we will take C to be something like the category of finite abelian p -groups with a specified presentation,although the precise details are somewhat complex. Remark 13.16. Some of the definitions and constructions below can be done for preordered sets or forsmall categories. We regard a preordered set P as a small category with one morphism a → b whenever a ≤ b , and no morphisms a → b if a b . We regard a small category C as a preordered set by declaring that a ≤ b if and only if C ( a, b ) = ∅ .The first combinatorial condition that we need to use is as follows: Definition 13.17. Let C be a small category. • A sequence in C means a map u : N → obj( C ) • A subsequence of u is a map of the form u ◦ f , where f : N → N is strictly increasing. • We say that u is good if there exists i < j such that u ( i ) ≤ u ( j ) (meaning that C ( u ( i ) , u ( j )) = ∅ , asin Remark 13.16). • We say that u is very good if u ( i ) ≤ u ( j ) for all i ≤ j . • We say that C is well-quasi-ordered (or wqo ) if every sequence in C is good. • We say that C is cowqo if C op is wqo. • We say that C is slice-wqo if the slice category X ↓ C is wqo for all objects X . Remark 13.18. It is clear that the definition of wqo is compatible with the identifications in Remark 13.16. emark 13.19. If C is finite then any sequence u : N → obj( C ) is non-injective and therefore good. Remark 13.20. Now let P be a well-ordered set. For any sequence u : N → P , the set u ( N ) must have asmallest element, say u ( k ), and then we have u ( k ) ≤ u ( k + 1), showing that u is good. It follows that P iswqo.The following lemma is a basic ingredient. Lemma 13.21. Suppose that C is wqo. Then any sequence in C has a very good subsequence.Proof. Given any sequence u : N → obj( C ) and i ∈ N , put I ( u, i ) = { j > i | u ( i ) ≤ u ( j ) } . Then put J ( u ) = { i | | I ( u, i ) | = ∞} . Suppose that J ( u ) is empty, so I ( u, i ) is finite for all i . Define f : N → N recursively by f (0) = 0 and f ( i + 1) = min { j | j > f ( i ) and j > k for all k ∈ I ( u, f ( i )) } . It is then not hard to see that u ◦ f is bad, contradicting the assumption that C is wqo. It follows that J ( u )must actually be nonempty. Put j ( u ) = min( J ( u )), so I ( u, j ( u )) is infinite. Put T ( u ) = u ◦ f , where f : N → N is the unique strictly increasing map with image I ( u, j ( u )). Now define R ( u ) : N → obj( C ) recursively by R ( u )(0) = u ( j (0)) and R ( u )( i + 1) = R ( T ( u ))( i ). We find that R ( u ) is a very good subsequence of u . (cid:3) Definition 13.22. Let C be a small category. • We say that C is rigid if every endomorphism is an identity. • A hom-ordering on C consists of a system of well-orderings of the hom sets C ( X, Y ) such that for all α : Y → Z , the induced map α ∗ : C ( X, Y ) → C ( X, Z ) is monotone. Definition 13.23. Let C be a small category and let D be essentially small. • We say that C is Gröbner if it is rigid, slice-wqo and it admits a hom-ordering. • We say that D is quasi-Gröbner if there is a Gröbner category C and an essentially surjective functor M : C → D such that each comma category ( x ↓ M ) has a finite weakly initial set. In more detail,the condition is as follows: for each x ∈ D there must exist a finite list of objects y , . . . , y n ∈ C and morphisms f i : x → M ( y i ), such that for any y ∈ C and any f : x → M ( y ) there exists i and g : y i → y with f = M ( g ) ◦ f i . This is known as Condition (F) .We are finally ready to state the criterion. Theorem 13.24. [19, 4.3.2] Let D be a quasi-Gröbner category. Then the category [ D , Vect k ] is locallynoetherian. Remark 13.25. Here and elsewhere we have used terminology and notation that seems clear to us andcompatible with the rest of our work, but which differs from that in [19] and related references. In particular,our “rigid” (as in Definition 13.22) is their “direct”, and our “wqo” is their “noetherian”. Our “hom-ordering”is their condition (G1), and our “slice-wqo” condition is their (G2).Before proving Theorem 13.15 we need to introduce more notation and prove some technical results. Well-quasi orders.Remark 13.26. To deal with some set-theoretic issues, we let X denote the set of hereditarily finite sets, so X is countable and closed under taking subsets, products and quotients, and contains sets of all finite orders.When we discuss categories of finite sets with extra structure, we will implicitly assume that the underlyingsets are in X , so that the category will be small. Definition 13.27. Let C and D be preordered sets, and let f : C → D be a function.(a) We say that f is monotone if p ≤ p ′ implies f ( p ) ≤ f ( p ′ ).(b) We say that f is comonotone if f ( p ) ≤ f ( p ′ ) implies p ≤ p ′ . emark 13.28. Here C is and D might be small categories, regarded as preordered sets as in Remark 13.16.In that case, any functor f : C → D gives a monotone map. Proposition 13.29. If f : C → D is comonotone and D is wqo then C is wqo.Proof. If u : N → C is a sequence, then f ◦ u must be good, so there exists i ≤ j with f u ( i ) ≤ f u ( j ), butthat implies u ( i ) ≤ u ( j ) by the comonotone property. (cid:3) Proposition 13.30. Any finite product of wqo preordered sets is again wqo.Proof. It suffices to show that if P and Q are wqo, then so is P × Q . Let u : N → P × Q be a sequence. As P is wqo, we can find a subsequence v such that π P ◦ v is nondecreasing. As Q is wqo, we can then find asubsequence w of v such that π Q ◦ w is nondecreasing. Now w is nondecreasing subsequence of u . (cid:3) We now recall the Nash-Williams theory of minimal bad sequences [14]. Definition 13.31. Let P be a preordered set. We say that a finite list u ∈ P n is bad if there is no pair ( i, j )with 0 ≤ i < j < n and u ( i ) ≤ u ( j ). We say that such a finite list u is very bad if there is an infinite badsequence extending it. If so, the set E ( u ) = { u ′ ∈ P | ( u (0) , . . . , u ( n − , u ′ ) is very bad } is nonempty. Now suppose we have a well-ordered set W and a function λ : P → W . Put EM ( u ) = { u ′ ∈ E ( u ) | λ ( u ′ ) = min( λ ( E ( u ))) } 6 = ∅ . We say that a very bad list u ∈ P n is λ - minimal if for all k < n we have u ( k ) ∈ EM ( u Proposition 13.33. Let P and λ be as above. Let P be a subset of P , and let χ : P → P be a map suchthat (a) For all x ∈ P we have χ ( x ) ≤ x and λ ( χ ( x )) < λ ( x ) . (b) Every bad sequence u : N → P has a subsequence v contained in P with the following property: if i < j with χ ( v ( i )) ≤ χ ( v ( j )) , then v ( i ) ≤ v ( j ) .Then P is wqo.Proof. Suppose not, so there exists a minimal bad sequence u . Let v be a subsequence as in (b), so v ( n ) = u ( f ( n )) for some strictly increasing map f : N → N . Define w ( n ) = u ( n ) for n < f (0) and w ( f (0) + k ) = χ ( v ( k )). We claim that w is bad. If not, we have i < j with w ( i ) ≤ w ( j ). If j < f (0) this gives u ( i ) ≤ u ( j ),contradicting the badness of u . Suppose instead that i < f (0) ≤ j , so w ( i ) = u ( i ) and w ( j ) = χ ( v ( j ′ )) = χ ( u ( j ′′ )) for some j ′ ≥ j ′′ ≥ f (0). We now have u ( i ) ≤ χ ( u ( j ′′ )) ≤ u ( j ′′ ), again contradictingthe badness of u . This just leaves the possibility that f (0) ≤ i < j , so w ( i ) = χ ( v ( i ′ )) = χ ( u ( i ′′ )) and w ( j ) = χ ( v ( j ′ )) = χ ( u ( j ′′ )) for some i ′ , j ′ , i ′′ , j ′′ with i ′ < j ′ and i ′′ < j ′′ . We now have χ ( v ( i ′ )) ≤ χ ( v ( j ′ ))so v ( i ′ ) ≤ v ( j ′ ) b y condition (b), so u ( i ′′ ) ≤ u ( j ′′ ), yet again contradicting the badness of u . It follows that w must be bad after all. However, this contradicts the λ -minimality of u ( f (0)) in E ( u Let C be a wqo category. We define SC to be the category of pairs ( X, p ), where X is afinite, totally ordered set, and p : X → C . A morphism from ( X, p ) to ( Y, q ) consists of a strictly monotonemap φ : X → Y together with a family of morphisms φ x : p ( x ) → q ( φ ( x )) for each x ∈ X . These arecomposed in the obvious way. We put λ ( X, p ) = | X | . Remark 13.35. If C is just a preordered set, then a morphism from ( X, p ) to ( Y, q ) is just a strictly monotonemap φ : X → Y such that p ( x ) ≤ q ( φ ( x )) for all x . he following result is standard (although typically formulated a little differently). We give the proof toillustrate the use of Proposition 13.33. Proposition 13.36 (Higman’s Lemma) . SC is wqo.Proof. For ( X, p ) with X = ∅ we define x = min( X ) and ǫ ( X, p ) = p ( x ) ∈ C and χ ( X, p ) = ( X ′ , p ′ ),where X ′ = X \ { x } and p ′ = p | X ′ . This clearly satisfies condition (a) of Proposition 13.33. If u : N → SC is bad then u ( n ) can never be empty (otherwise we would have u ( n ) ≤ u ( n + 1)), so we have a sequence u = ǫ ◦ u : N → C . As C is wqo, we can choose a strictly increasing map f : N → N such that u ◦ f : N → P is very good. Now put v = u ◦ f . If i < j and χ ( v ( i )) ≤ χ ( v ( j )) then we also have ǫ ( v ( i )) ≤ ǫ ( v ( j )) and itfollows easily that v ( i ) ≤ v ( j ). Using Proposition 13.33 we can now see that SC is wqo. (cid:3) Definition 13.37. Let X and Y be nonempty finite totally ordered sets. Let φ : X → Y be a surjectivemap, which need not preserve the order. We define an φ † : Y → X by φ † ( y ) = min( φ − { y } ). We say that φ is † -monotone if φ † is monotone. Lemma 13.38. For any φ we have φφ † ( y ) = y for all y ∈ Y , and φ † φ ( x ) ≤ x for all x ∈ X . If φ is † -monotone then we have φ ( x ) < y whenever x < φ † ( y ) . In particular, if x and y are the smallest elementsof X and Y , then φ ( x ) = y and φ † ( y ) = x .Proof. It is clear by definition that φφ † ( y ) = y . Next, if x ∈ X then x is a preimage of φ ( x ), whereas φ † φ ( x ) is the smallest preimage, so φ † φ ( x ) ≤ x . Now suppose that φ is † -monotone. If y ≤ φ ( x ) then φ † ( y ) ≤ φ † φ ( x ) ≤ x . By the contrapositive, if x < φ † ( y ) we must have φ ( x ) < y , as claimed. We now claimthat x = φ † ( y ). Indeed, if not then x < φ † ( y ) so φ ( x ) < y , contradicting the definition of y . We musttherefore have x = φ † ( y ) after all, and it follows that φ ( x ) = φφ † ( y ) = y . (cid:3) Corollary 13.39. Suppose we have † -monotone maps X φ −→ Y ψ −→ Z. Then ( ψφ ) † = φ † ψ † , and so ψφ is also † -monotone.Proof. Given z ∈ Z put y = ψ † ( z ) and x = φ † ( y ) = φ † ψ † ( z ). Using the Lemma we get ψφ ( x ) = z . We alsosee that if x ′ < x = φ † ( y ) then φ ( x ′ ) < y = ψ † ( z ) and thus ψ ( φ ( x ′ )) < z . This means that x has the definingproperty of ( ψφ ) † ( z ). We therefore have ( ψφ ) † = φ † ψ † . This is the composite of two increasing maps, so itis again increasing, so ψφ is † -monotone. (cid:3) Definition 13.40. We define a category L † as follows. The objects are finite nonempty sets equipped with amap e X : X → N , together with a total order on X . The morphisms from X to Y are † -monotone surjectivemaps φ : X → Y such that e Y ( φ ( x )) ≤ e X ( x ) for all x ∈ X . Definition 13.41. We define α, β : L † → N by α ( X ) = e X (min( X )) and β ( X ) = min( e X ( X )). Next, for x ∈ X \ { min( X ) } we define e ′ X ( x ) = min { e X ( x ′ ) | x ′ < x } ∈ N , and e ∗ X ( x ) = ( e X ( x ) , e ′ X ( x )) ∈ N . The set X \ { min( X ) } together with the map e ∗ X define an object γ ( X ) ∈ S ( N ). Proposition 13.42. The map ( α, β, γ ) : L op † → N × S ( N ) is comonotone, so L † is cowqo.Proof. Suppose that α ( X ) ≤ α ( Y ) and β ( X ) ≤ β ( Y ) and γ ( X ) ≤ γ ( Y ); we need to construct a morphismfrom Y to X . As β ( X ) ≤ β ( Y ), we can choose a strictly increasing map ψ : X \ { min( X ) } → Y \ { min( Y ) } with e X ( x ) ≤ e Y ( ψ ( x )) and e ′ X ( x ) ≤ e ′ X ( ψ ( x )) for all x . We extend ψ over all of X by putting ψ (min( X )) =min( Y ), and note that the relation e X ( x ) ≤ e Y ( ψ ( x )) remains true. We define φ : ψ ( X ) → X by φ ( ψ ( x )) = x .Now consider an element y ∈ Y \ ψ ( X ), so y = min( Y ). If y > max( ψ ( X )) we choose x with e X ( x ) = β ( X )and define φ ( y ) = x , noting that e Y ( y ) ≥ β ( Y ) ≥ β ( X ) = e X ( x ). Otherwise, we let x ′ be least such that ψ ( x ′ ) > y , then choose x < x ′ with e X ( x ) = e ′ X ( x ′ ). This gives e Y ( y ) ≥ e ′ Y ( ψ ( x ′ )) ≥ e ′ X ( x ′ ) = e X ( x ) , nd we define φ ( y ) = x . We now have a surjective map φ : Y → X with e Y ( y ) ≥ e X ( φ ( y )) for all y . We alsohave φ ( ψ ( x )) = x , and φ ( y ) < x whenever y < ψ ( x ), so that ψ = φ † . This means that φ is a morphism in L † , as required. (cid:3) Corollary 13.43. L † is slice-cowqoProof. The construction ( X p ←− U ) ( p − { x } ) x ∈ X gives a full and faithful embedding L † ↓ X → Q x ∈ X L † .Finally apply Proposition 13.30. (cid:3) Hom-orderings.Remark 13.44. In Definition 13.22 we defined the notion of a hom-ordering on C . We can spell out thedual notion as follows: a hom-ordering of C op consists of a system of well-orderings of the hom sets C ( X, Y )such that for all β : W → X , the induced map β ∗ : C ( X, Y ) → C ( W, Y ) is monotone. Remark 13.45. If F : C → D is a faithful functor and we have a hom-ordering on D then we can define ahom-ordering on C by declaring that φ ≤ ψ if and only if F φ ≤ F ψ . Definition 13.46. Let F † be the category of finite totally ordered sets and † -monotone surjections. Weorder F † ( X, Y ) lexicographically, so φ < ψ if and only if there exists x ∈ X with φ ( x ) < ψ ( x ) and φ ( x ) = ψ ( x ) for all x < x . Proposition 13.47. This gives a hom-ordering on F op † .Proof. It is standard and easy that the above rule gives a total order on the finite set of surjections from X to Y . Now suppose we have θ : W → X and φ, ψ : X → Y with φ ≤ ψ ; we must show that φθ ≤ ψθ .By assumption there exists x ∈ X with φ ( x ) < ψ ( x ) and φ ( x ) = ψ ( x ) for all x < x . Put w = θ † ( x ) = min( θ − { x } ). Then ( φθ )( w ) = φ ( x ) < ψ ( x ) = ( ψθ )( w ). On the other hand, if w < w thenLemma 13.38 tells us that θ ( w ) < x and so ( φθ )( w ) = ( ψθ )( w ). (cid:3) Corollary 13.48. The faithful forgetful functor L op † → F op † gives a hom-ordering to L op † . (cid:3) Proof of Theorem 13.15. For the duration of this proof we put P = Z [ p ∞ ] = { finite abelian p -groups } and C [ k ] = Z /p k ∈ P . If k ≥ m , we write π for the standard surjective homomorphism C [ k ] → C [ m ]. For A ∈ P and a ∈ A , we let η a be the natural number such that a has order p η a By combining Corollaries 13.43 and 13.48, we see that L op † is Gröbner.We define an essentially surjective functor M : L op † → P op as follows. For an object X ∈ L † , we set M X = Q x ∈ X C [ e X ( x )]. Given a morphism φ : X → Y in L † , we define φ ∗ : M X → M Y by( φ ∗ m ) y = Y φ ( x )= y π ( m x ) . Let us introduce some terminology before proceeding with the proof. A framing of A ∈ P is a surjectivehomomorphism M X → A for some X ∈ L † . This corresponds to a map α : X → A such that η ( α ( x )) ≤ e X ( x ) for all x , and α ( X ) generates A . We say that the framing is tautological if X is a subset of A and α is just the inclusion and e X ( x ) = max { η ( w ) | w ∈ X, w ≤ x } . It is clear from the definition that there are only finitely many tautological framings. Unravelling thedefinitions, we see that M satisfies condition (F) if any framing α : X → A factors as X → X → A where the first arrow is in L † and the second one is a tautological framing. So if α : X → A is an arbitraryframing, we define X = α ( X ) ⊂ A and e X = η | X and set α : X → A to be the inclusion. We also define α † : A → X by α † ( a ) = min( α − ( a )) and order X by declaring that a < b iff α † ( a ) < α † ( b ). This makes α nto a tautological framing and gives the required factorization. Therefore P op is quasi-Gröbner and so part(a) holds.For part (b), we put Ω = { η a | A ∈ U , a ∈ A } ⊂ N . Define L U† to be the full subcategory of L † consisting of objects X with image( e X ) ⊂ Ω. This is still Gröbnerby [19, 4.4.2]. It is now easy to check that the functor M : ( L U† ) op → U op defined as above is essentiallysurjective and satisfies property ( F ). Thus U op is quasi-Gröbner and AU is locally noetherian.14. Representation stability In this section we show that any finitely presented object can be recovered by a finite amount of data via astabilization recipe. This phenomenon is called central stability and it was first introduced by Putman [16].We also show that under the noetherian assumption, any finitely generated object satisfies the analogue of theinjectivity and surjectivity conditions in the definition of representation stability due to Church–Farb [4, 1.1]. Definition 14.1. Let U be a subcategory of G . For X ∈ AU , we put τ n ( X ) = i ≤ n ! i ∗≤ n ( X ) ∈ AU , and note that there is a counit map τ n ( X ) → X . We also define natural maps τ n ( X ) → τ n +1 ( X ) as follows.Let j denote the inclusion U ≤ n → U ≤ ( n +1) , so we have a counit map j ! j ∗ ( Y ) → Y for all Y ∈ AU ≤ ( n +1) .Taking Y = i ∗≤ ( n +1) ( X ) for some X ∈ AU , we get a map j ! i ∗≤ ( n +1) X → i ∗≤ ( n +1) X . Applying the functor i ≤ ( n +1)! to this gives the required map τ n ( X ) → τ n +1 ( X ).We list a few important properties of the truncation functor. Proposition 14.2. Consider an object X ∈ AU . (a) Then X is the colimit of the objects τ n ( X ) . (b) We have τ n ( e G ) = e G if G ∈ U ≤ n and τ n ( e G ) = 0 otherwise. (c) For all G ∈ U and n ≥ , we have τ n ( X )( G ) = lim −→ H ∈ N ( G,n ) X ( G/H ) where N ( G, n ) = { H ⊳ G | | G/H | ≤ n } .Proof. For part (a) it is enough to notice that τ n ( X )( G ) = X ( G ) for | G | ≤ n . Part (b) follows fromLemma 5.3(i). Using the formula for Kan extensions, we see that τ n ( X )( G ) can be written as a colimitover the comma category ( G ↓ U ≤ n ). Suppose we have objects ( G α −→ A ) and ( G β −→ B ) in the commacategory so A, B ∈ U ≤ n . As α and β are surjective, we find that there is a unique morphism from α to β if ker( α ) ≤ ker( β ), and no morphisms otherwise. This shows that the comma category is equivalent to theposet N ( G, n ) so part (c) follows. (cid:3) The following is a characterization of finitely generated and finite presented objects. Proposition 14.3. Consider an object X ∈ AU . (a) X is finitely generated if and only if X has finite type and there exists N ∈ N such that the canonicalmap τ n ( X ) → X is an epimorphism for all n ≥ N . (b) X is finitely presented if and only if X has finite type and there exists N ∈ N such that the canonicalmap τ n ( X ) → X is an isomorphism for all n ≥ N .Proof. For part (a), assume that the map τ n ( X ) → X is an epimorphism for all n ≥ N . Note that we canconstruct an epimorphism M G ∈U ≤ n dim( X ( G )) e G → i ∗≤ n ( X ) s X has finite type. We apply i ≤ n ! to get an epimorphism M G ∈U ≤ n dim( X ( G )) e G → τ n ( X )since i ≤ n ! preserves all colimits by Lemma 5.3(f). Post-composition with τ n ( X ) → X gives the desiredepimorphism. Conversely, assume that X is finitely generated so that we have a short exact sequence0 → K → P → X → P finitely projective. Note that by Proposition 14.2(b), there must exist N ∈ N such that τ n ( P ) ≃ P for all n ≥ N . The commutativity of the diagram P X τ n ( P ) τ n ( X ) ≃ implies that the map τ n ( X ) → X is an epimorphism for all n ≥ N .For part (b), assume that X is finitely presented. Then there exists a short exact sequence 0 → K → P → X → P finitely projective and K finitely generated. By Part (a), it is enough to show that thecanonical map τ n ( X ) → X is eventually monic. Note that for large n , we have a diagramker( i nK ) 0 ker( i nX ) τ n ( K ) τ n ( P ) τ n ( X ) 00 K P X i nK ) 0 cok( i nX ) i nK ≃ i nX where the bottom row is exact and the top is only right exact. By assumption both K and X are finitelygenerated, so the maps i nK and i nX are epimorphisms by part (a). Thus, the Snake Lemma tell us thatker( i nX ) = 0. Conversely, assume that the natural map is an isomorphism. By part (a), X is finitelygenerated so we have a short exact sequence 0 → K → P → X → P finitely projective. By applyingthe Snake Lemma to the diagram above, we see that cok( i nK ) = 0 for large n , so K is finitely generated and X is finitely presented. (cid:3) We note that by combining Propositions 14.2 and 14.3 we obtain Theorem B from the introduction. Remark 14.4. Recall the functor q ≤ n from Example 5.9. We have seen that q ≤ n is left adjoint to theinclusion U ⋆ ≤ n → G . If U is closed downwards, then q ≤ n is also the left adjoint to the inclusion U ≤ n → U . Proposition 14.5. Let U be multiplicative and closed under passage to subgroups, and consider a finitelypresented object X ∈ AU . Then there exists n ∈ N such that X ( G ) = X ( q ≤ n G ) for all G ∈ U .Proof. Choose a finite presentation r M i =1 e G i f −→ s M j =1 e H j → X → . Choose n large enough so that G i , H j ∈ U ∗≤ n for all i and j . Let Y be cokernel of f in AU ∗≤ n . We claim that X = q ∗≤ n ( Y ). As the functor q ∗ n preserves all colimits it is enough to show that q ∗≤ n e G = e G for all G ∈ U ∗≤ n .Using that q ≤ n is left adjoint to the inclusion U ∗≤ n → U we see that( q ∗≤ n e G )( H ) = k [ U ( q ≤ n H, G )] = k [ U ( H, G )] = e G ( H )which concludes the proof. (cid:3) e now restrict to the locally noetherian case. Recall the definition of eventually torsion-free and stablysurjective object from the introduction, see Definition A. Theorem 14.6. Let X ∈ AZ [ p ∞ ] be a finitely generated object. Then the restriction of X to AC [ p ∞ ] and AF [ p n ] for n ≥ , is eventually torsion-free and stably surjective.Proof. The restriction of X to AC [ p ∞ ] is eventually torsion-free and stably surjective by [7, 5.1, 5.2].Let Z [ p n ] denote the subfamily of abelian p -groups of exponent dividing p n . Write Y n for the restriction of X to AZ [ p n ] and note that this is still finitely generated by Lemma 11.2. Put q = p n . Note that tors( Y n ) isfinitely generated so G ∗ -null by Lemma 12.19. This means that tors( Y n )( C rq ) = 0 for r ≫ 0. Note that anyelements in the kernel of α ∗ : Y n ( A ) → Y n ( B ) lies in tors( Y n ). This shows that the restriction of Y n to F [ q ]is eventually torsion-free.For the surjectivity condition, choose an epimorphism P ։ Y n in AZ [ q ] from a finitely projective object. Bythe commutativity of the diagram P ( C rq ) ⊗ k [ G ( C r +1 q , C rq )] P ( C r +1 q ) Y n ( C rq ) ⊗ k [ G ( C r +1 q , C rq )] Y n ( C r +1 q ) θ P θ Yn it is enough to show that θ P is surjective. Equivalently, we need to show that the map G ( C rq , A ) × G ( C r +1 q , C rq ) → G ( C r +1 q , A ) , ( α, β ) → β ◦ α is surjective, for all A ∈ Z [ q ] and r ≫ 0. This now follows from Lemma 9.10. (cid:3) We conclude this section by proving Theorem D from the introduction. Proof of Theorem D. First of all note that the equivalence (1.0.1) in the introduction descents to an equiv-alence between the full subcategories of compact objects (Sp Q U ) ω ≃ D ( AU ) ω for any family U ≤ G . We canapply [9, 2.3.12] to deduce that D ( AZ [ p ∞ ]) ω = thick( e G | G ∈ Z [ p ∞ ])where the right hand side denotes the smallest thick (=closed under retracts) triangulated subcategorycontaining the generators e G for G ∈ Z [ p ∞ ].Consider the full subcategory T = { X | H ∗ ( X ) is finitely generated } ⊂ D ( AZ [ p ∞ ]) ω . Since AZ [ p ∞ ] is locally noetherian one easily checks that T is a thick triangulated subcategory. Clearly e G ∈ T for all G ∈ Z [ p ∞ ] so by the discussion in the previous paragraph we see that any compact objectlies in T . Finally apply Theorems B and D. (cid:3) Injectives We now turn to study the injective objects of AU . Unlike in the projective case, a complete classificationof the indecomposable injective objects seems at the moment far out of reach. The main difficulty arisesfrom the fact that any projective object is necessarily torsion-free whereas an injective object can be torsion,absolutely torsion or torsion-free.Recall that if U has a colimit tower then the dual of any object is injective by Proposition 9.1. Let us producemore examples of injective objects. Proposition 15.1. Let U be a multiplicative global family. Then the torsion-free injective objects coincidewith the projective objects. roof. Suppose that U is a multiplicative global family and consider a projective object P . We will show that P is injective giving one of the implications in the proposition. We can write P = Q n P n by Proposition 8.6,so it will suffice to show that P n is injective. We have P n = ( i n ) ! ( i ∗ n P n ) and i ∗ n P n is projective in AU n . Wecan write i ∗ n P n as a retract of an object Q = L t e G t with G t ∈ U n . This embeds in the product R = Q t e G t ,and all monomorphisms in AU n are split, so i ∗ n P n is a retract of R . We know that ( i n ) ! preserves productsby Proposition 7.3, so P n = ( i n ) ! ( i ∗ n P n ) is a retract of Q t ( i n ) ! ( e G t ) = Q t e G t . Therefore, it is enough toshow that e G t is injective. This now follows from the fact that De G t is injective and that e G t is a summandof De G t by Theorem 4.20. Therefore P is injective as claimed. Conversely, let I be a torsion-free injective.By Proposition 12.16, we can embed I into a projective object SI . Since I is injective, the inclusion I → SI splits showing that I is projective as required. (cid:3) Remark 15.2. Let C [2 ∞ ] be the family of cyclic 2-groups. Then we have a short exact sequence0 → e C → → t ,k → is torsion-free and t ,k is torsion. Hence e C is not injective in AC [2 ∞ ].The following structural result, classically due to Matlis [12], suggests that we can restrict our attention toindecomposable injectives. Theorem 15.3 ([5, Chaper IV]) . Any injective object in a locally noetherian abelian category is a sum ofindecomposable injectives. Lemma 15.4. Let U be multiplicative global family of V . (a) For any G ∈ V and V irreducible Out( G ) -representation, the object t G,V is indecomposable andinjective in AV . Furthermore, t G,V is the injective envelope of s G,V . (b) For any G ∈ U and V irreducible Out( G ) -representation, the object χ U ⊗ e G,V is indecomposableand injective in AV .Proof. We have seen that t G,V is injective and it is indecomposable by Lemma 5.3(e). If U is a multiplicativeglobal family, then e G,V is injective and so combining part (e) and (i) of Lemma 5.3 we see that i ∗ ( e G,V ) = χ U ⊗ e G,V is an indecomposable injective. Finally note that there is a canonical monomorphism s G,V → t G,V ,so the injective hull of s G,V is a direct summand of t G,V so the claim follows by indecomposability (cid:3) The next result classifies the indecomposable injective objects which are absolutely torsion. 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MR0389953(Pol) Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, Regensburg 93040, Deutschland Email address : [email protected] (Strickland) School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK