ENDOFINITENESS IN STABLE HOMOTOPY THEORY HENNING KRAUSE AND ULRIKE REICHENBACH Abstract.We study endofinite objects in a compactly generated triangulat* *ed cat- egory in terms of ideals in the category of compact objects. Our result* *s apply in particular to the stable homotopy category. This leads, for example, to * *a new inter- pretation of stable splittings for classifying spaces of finite groups. Introduction A CW-complex is said to be endofinite if all stable homotopy groups sssn(X) =* * {Sn; X} are finite length modules over the ring {X; X} of stable self-maps of X. In thi* *s paper we study CW-complexes having this finiteness property. One motivation for this * *work is to understand stable splittings of certain spaces, for instance the classify* *ing space BG of a finite group G. We shall give an algebraic interpretation of such splittin* *gs in terms of certain ideals in the category of finite spectra. A natural framework for the study of endofinite objects are compactly generat* *ed triangulated categories. Thus we consider a triangulated category C, for examp* *le the stable homotopy category, which has arbitrary coproducts and is generated by a * *set of compact objects (an object X in C is compact if the representable functor Hom (* *X; -) preserves coproducts). An object X in C is said to be endofinite if the group H* *om (C; X) is a finite length module over the endomorphism ring End (X) for all compact ob* *jects C in C. This finiteness condition has a number of interesting consequences. For e* *xample, in [15] it is shown that every endofinite object decomposes essentially uniquel* *y into indecomposable objects with local endomorphism rings. Here, we shall study endo* *finite objects in terms of certain ideals in the full subcategory C0 of compact object* *s in C. Viewing C0 as a ring with several objects, we consider for every X in C the ann* *ihilator Ann X of X which is the ideal of maps OE in C0 such that Hom (OE; X) = 0. Two t* *ypes of ideals I in C0 are of particular importance: o I is cohomological if for every sequence of maps in C0 0 OE OE00 X0 OE-!X -! Y - ! Y 00 such that OE0 is a fibre of OE00OE, the map OE belongs to I if OEOE0 and OE00OE* * belong to I. We shall see that the cohomological ideals in C0 are precisely the annihilator ide* *als of objects in C. In fact, for every cohomological ideal I there exists a canonical object* * EI such that I = Ann EI. o I is cofinite if for every X in C0 there exists n 2 N such that for every s* *equence of maps in C0 X -OE0!X0 -OE1!X1 -OE2!: :O:En-!Xn the composition of OEi-1: :O:E0 with a fibre of OEi: :O:E0 belongs to I for som* *e i n. Let us denote by Max C0 the set of maximal elements among the cofinite cohomological i* *deals I 6= C0 in C0. ____________ Date: November 1998. 1991 Mathematics Subject Classification. 55P42, 55U35. 1 2 HENNING KRAUSE AND ULRIKE REICHENBACH Theorem A. An object X in C is endofinite if and only if the annihilator of * *X is cofinite. Moreover the maps X 7! Ann X and I 7! EI induce mutually inverse bije* *ctions between the set of isomorphism classes of indecomposable endofinite objects in * *C and Max C0. ` Any endofinite object X has an essentially unique decomposition X ~= i2IXi i* *nto indecomposable endofinite objects, and the annihilator of X controls this decom* *position. More precisely, an indecomposable object Y arises as a direct factor of an end* *ofinite object X if and only if Ann X Ann Y . This is a particular consequence of the * *following result: Theorem B. Let X be an endofinite object in C with I = Ann X. Then " I = J IJ2Max C0 and X has a unique decomposition a X ~= XJ IJ2Max C0 such that each XJ is a non-empty coproduct of copies of EJ. Moreover, Ann X An* *n Y holds for any object Y if and only if Y is isomorphic to a direct factor of a (* *co)product of copies of X. Let us now explain some special results for the stable homotopy category of s* *pectra. A fairly immediate consequence of Theorem B is the fact that we can work one pr* *ime at a time. Corollary. Let X be an endofinite spectrum. Then the p-localization map X ! Xp * *is a split epimorphism for every prime ideal p in Z. If XtordenotesWthe fibre of the* * rational localization X ! X(0), then X ~=X(0)_Xtorand Xtor~= p(Xtor)p where p runs thro* *ugh all non-zero prime ideals in Z. Having developed a general theory of endofinite spectra, one may ask for cond* *itions on a CW-complex X such that its suspension spectrum is endofinite. It turns out* * that this happens if and only if all stable homotopy groups of X are finite length m* *odules over the ring of stable self-maps of X, hence if the CW-complex X is endofinite* *. Here is a useful criterion: Theorem C. Let X be a connected CW-complex such that the singular homology gr* *oups Hn(X; Z) are finite for all n > 0. Then X is an endofinite CW-complex, and ther* *efore its suspension spectrum is also endofinite. Important examples of CW-complexes with finite homology are the classifying s* *paces of finite groups. Corollary. The classifying space BG of a finite group G is endofinite. It is a well-known consequence of Carlsson's solution of the Segal conjecture* * [2] that BG splits stably into indecomposable spectra; see [9] for an alternative proof.* * We recover this result, and our ideal theoretic approach reduces the problem of fi* *nding such splittings to the category of finite spectra. The concept of endofiniteness has its origin in representation theory of fini* *te dimen- sional algebras. Following Crawley-Boevey, a module X over some associative rin* *g is endofinite if it has finite length when regarded as a module over its endomorph* *ism ring ENDOFINITENESS IN STABLE HOMOTOPY THEORY 3 End (X). Such modules have nice decomposition properties and they play a deci* *sive role for the representation type of a finite dimensional algebra; see [3] for a* *n excellent survey. Comparing the category of modules over some associative ring and the st* *able ho- motopy category of spectra, one finds that a number of results about endofinite* * objects carry over from one setting to the other. However, the additional triangulated * *structure of the stable homotopy category led to the discovery of some genuine new phenom* *ena. In fact, there seems to be no analogue of Theorem A and Theorem B for the categ* *ory of modules over a ring. This paper is organized as follows: In Section 1 we introduce cohomological * *and cofinite ideals for arbitrary triangulated categories and study their basic pro* *perties. For a compactly generated triangulated category C, the restricted Yoneda functor C -! Mod C0; X 7! HX = Hom (-; X)|C0 is an important tool which translates the triangulated structure of C into the * *abelian structure of the category Mod C0 of C0-modules. A detailed treatment of this * *functor can be found in [14], but we recall in Section 2 the basic results and prove th* *at any cohomological ideal I in C0 is the annihilator of some canonical object EI in C* *. The correspondence between endofinite objects and cofinite ideals is the main theme* * of Sec- tion 3 which contains the proofs of Theorem A and Theorem B. In the following t* *wo sections we apply the results from the previous ones to the stable homotopy cat* *egory of spectra. In Section 4 we investigate p-localizations of endofinite spectra and * *introduce the endolength of a spectrum X which is a numerical invariant, i.e. a family (`* *n(X))n2Z of numbers `n(X) 2 N [ {1} such that X is endofinite if and only if `n(X) is fi* *nite for all n 2 Z. The final Section 5 is devoted to the criterion for a CW-complex* * to be endofinite which is formulated in Theorem C. 1. Cohomological ideals In this section we introduce cohomological and cofinite ideals for arbitrary * *triangu- lated categories and study their basic properties. We use the Yoneda functor w* *hich embeds a triangulated category into an abelian category. 1.1. The Yoneda functor. Let C be any additive category. A C-module is by defin* *ition an additive functor Cop ! Ab into the category Ab of abelian groups, and for C-* *modules M and N we denote by Hom (M; N) the class of natural transformations M ! N. A sequence L ! M ! N of maps between C-modules is exact if the sequence L(X) ! M(X) ! N(X) is exact for all X in C. A C-module M is finitely generated if the* *re exists an exact sequence Hom (-; X) ! M ! 0 for some X in C, and M is finitely presented if there exists an exact sequence Hom (-; X) ! Hom (-; Y ) ! M ! 0 wi* *th X and Y in C. Note that Hom (M; N) is a set for every finitely generated C-mo* *dule M by Yoneda's lemma. The finitely presented C-modules form an additive category with cokernels which we denote by mod C. It is well-known that mod C is abelia* *n if and only if every map Y ! Z in C has a weak kernel X ! Y , i.e. the sequence Hom (-; X) ! Hom (-; Y ) ! Hom (-; Z) is exact. In particular, mod C is abelian* * if C is triangulated. Suppose now that C is a triangulated category [18]. Any map :Y ! Z in C can* * be completed to a triangle (*) X -OE!Y - ! Z -O! X[1] 4 HENNING KRAUSE AND ULRIKE REICHENBACH and we call the map OE a fibre of and O a cofibre of . Recall that a functor* * F :C ! A from C into an abelian category A is cohomological if for every triangle (*) th* *e sequence F (X) ! F (Y ) ! F (Z) ! F (X[1]) is exact. Examples of cohomological functors* * are the representable functors Hom (X; -): C ! Ab and Hom (-; X): Cop ! Ab for any X in C. Another important example is the fully faithful Yoneda functor H :C -! mod C; X 7! HX = Hom (-; X): A map OE: X ! Y is sent to the natural transformation HOE:Hom (-; X) ! Hom (-; * *Y ). The Yoneda functor has the following well-known universal property: Freyd's Lemma. Every additive functor F :C ! A into an abelian category A ex* *tends, up to isomorphism, uniquely to a right exact functor F 0:mod C ! A such that F = F 0OH. The functor F 0is exact if and only if F is a cohomological functor. * * __ Proof.See [6, Theorem 3.1]. * *|__| In order to study the category mod C it is important to observe that any obj* *ect F in mod C is of the form F = Im HOEfor some map OE: Y ! Z in C. In fact, * *if HX ! HY ! F ! 0 is a presentation of F , then F = Im HOEwhere OE: Y ! Z denotes a cofibre of the map X ! Y . Moreover, the representable functors HX are proje* *ctive objects in mod C by Yoneda's lemma, and they are also injective since C is tria* *ngulated. Our next aim is a description of the exact sequences in mod C. To this end we m* *ake the following definition: 0 OE OE00 Definition 1.1. A sequence X0 OE!X ! Y ! Y 00of maps in C is called cohomolog* *ical exact if OE0is a fibre of the composition OE00OE. The following two lemmas explain the relevance of the cohomological exact seq* *uences. 0 OE OE00 Lemma 1.2. Let X0 OE!X ! Y ! Y 00be a cohomological exact sequence in C. Th* *en the induced sequence 0 -! Im HOEOE0-! Im HOE-! Im HOE00OE-! 0 is exact in mod C. Proof.Clearly, the map Im HOE! Im HOE00OEis an epimorphism, and Im HOEOE0is the* * kernel __ of this map since OE0is a fibre of OE00OE. * * |__| Lemma 1.3. Let ": 0 ! F 0! F ! F 00! 0 be an exact sequence in mod C. Then th* *ere 0 OE OE00 exists a cohomological exact sequence X0 OE!X ! Y ! Y 00in C such that the in* *duced sequence 0 -! Im HOEOE0-! Im HOE-! Im HOE00OE-! 0 is isomorphic to ". Proof.Let F = Im HOEfor some map OE: X ! Y . Taking a monomorphism F 00! HY 00, the composition F ! F 00! HY 00extends to a map HY ! HY 00since HY 00is injecti* *ve. By Yoneda's lemma, this map is induced by a map OE00:Y ! Y 00and it follows th* *at F 00= Im HOE00OE. Taking for OE0:X0 ! X a fibre of the composition OE00OE, it f* *ollows from the preceding lemma that the induced sequence 0 -! Im HOEOE0-! Im HOE-! Im HOE00OE-! 0 * * __ is isomorphic to ". * *|__| ENDOFINITENESS IN STABLE HOMOTOPY THEORY 5 1.2. Cohomological ideals. Let C be a triangulated category. An ideal I in C co* *nsists of subgroups I(X; Y ) in Hom (X; Y ) for every pair of objects X; Y in C such * *that for all OE in I(X; Y ) and all maps ff: X0 ! X and fi :Y ! Y 0in C the composition * *fiOEff belongs to I(X0; Y 0). Given an additive functor F :C ! D, the ideal of maps O* *E in C satisfying F (OE) = 0 is called the annihilator of F and is denoted by Ann F . * *Note that any ideal I in C is of the form I = Ann F for some functor F :C ! D since we ca* *n take for F the projection C ! C=I onto the additive quotient category C=I. Definition 1.4. An ideal in C is said to be cohomological if it is the annihila* *tor of some cohomological functor C ! A. For example, if F :C ! D is an exact functor between triangulated categories,* * then Ann F is a cohomological ideal since Ann F = Ann (H OF ) where H :D ! mod D de- notes the Yoneda functor which is faithful. Recall that a full subcategory S of an abelian category A is a Serre subcateg* *ory provided that for every exact sequence 0 ! X0 ! X ! X00! 0 in A the object X belongs to S if and only if X0 and X00belong to S. The corresponding quotient c* *ategory A=S is constructed as follows [7]: The objects of A=S are those of A and Hom A=S(X; Y ) = lim-!HomA(X0; Y=Y 0) with X0 X, Y 0 Y and X=X0; Y 02 S. Again the category A=S is abelian and there is canonically defined the quotient functor Q: A ! A=S such that Q(X) = X; it i* *s exact with Ker Q = S. Here the kernel Ker F of a functor F :A ! B is, by definition, * *the full subcategory of all objects X in A such that F (X) = 0. Now suppose that F :A ! B is an exact functor. Then Ker F contains S if and only if F induces a (unique * *and exact) functor G: A=S ! B such that F = G OQ. Moreover, G is faithful if and on* *ly if S = KerF . Proposition 1.5. The assignments I 7! SI = {Im HOE| OE 2 I} and S 7! IS = {OE 2 C | Im HOE2 S} induce mutually inverse bijections between the class of cohomological ideals in* * C and the class of Serre subcategories in mod C. Proof.Let A = mod C. Suppose that I = Ann F for some cohomological functor F :C* * ! A0 and denote by F 0:A ! A0 the unique exact functor extending F which exists by Freyd's lemma. Clearly, Ker F 0= SI, and this is a Serre subcategory of A s* *ince F 0is exact. Conversely, if S is a Serre subcategory of A, then IS = Ann F whe* *re F :C ! A=S denotes the composition of the Yoneda functor C ! A with the quotien* *t __ functor A ! A=S. It is easily checked that these constructions are mutually inv* *erse. |__| As a consequence of this proposition, we obtain an internal characterization of* * cohomo- logical ideals in terms of maps in C. Corollary 1.6. An ideal I in C is cohomological if and only if for every cohomo* *logical exact sequence in C 0 OE OE00 X0 OE-!X -! Y - ! Y 00 the map OE belongs to I provided that OEOE0 and OE00OE are in I. Proof.Combine the description of exact sequences in mod C given in Lemma 1.2 an* *d_ Lemma 1.3 with Proposition 1.5. |* *__| 6 HENNING KRAUSE AND ULRIKE REICHENBACH 1.3. Cofinite ideals. Let C be a triangulated category and let A = mod C. Recal* *l that an abelian category is a length category if every object has finite composition* * length. Definition 1.7. Let I be a cohomological ideal in C and SI be the corresponding* * Serre subcategory of A. Then I is called cofinite if the quotient category A=SI is a* * length category. In order to give an internal characterization of cofinite ideals in C we need* * another definition. Definition 1.8. Let I be a cohomological ideal in C. (1) An I-sequence is a sequence of maps X OE0!X0 OE1!: :O:En!Xn in C such tha* *t for every i 2 {1; : :;:n} the composition of OEi-1: :O:E0 with a fibre of OEi* *: :O:E0 does not belong to I. (2) The I-length of an object X in C is `I(X) = sup{n 2 N | there exists an I-sequence X OE0!X0 OE1!:O:E:n!Xn}: We are now in a position to state a characterization of cofinite ideals in C. Proposition 1.9. A cohomological ideal I in C is cofinite if and only if the I* *-length of every object in C is finite. The proof of this proposition requires two lemmas. Lemma 1.10. Let 0 = F0 F1 : : :Fn = HX be a chain of subobjects in A. (1) There exists a sequence X !OE0X0 OE1!: :O:En!Xn of maps in C such that Fi* * = Ker HOEi:::OE0for all i. (2) Denote for every i by i a fibre of OEi: :O:E0. Then Fi=Fi-1 ~=Im HOEi-1:* *::OE0fior all i. Proof.(1) Put X0 = X and OE0 = idX . Choose for every i > 0 a copresentation 0 ! Fi! HX ! HXi. The inclusion Fi-1! Fi induces a commutative diagram 0 -! Fi-1? -! HXfl -! HXi-1? ?y flfl ?y 0 -! Fi -! HX -! HXi and the map HXi-1 ! HXi is induced by a map OEi:Xi-1! Xi. This gives the desired sequence X OE0!X0 OE1!:O:E:n!Xn. (2) An application of the snake lemma shows that Fi=Fi-1 is isomorphic to the* * kernel __ of the map Im HOEi-1:::OE0! Im HOEi:::OE0which is Im HOEi-1:::OE0biy Lemma 1.2.* * |__| Lemma 1.11. Let I be a cohomological ideal and X 2 C. Then `I(X) is the lengt* *h of HX in A=SI. Proof.Let Q: A ! A=SI be the quotient functor. Any chain of subobjects of Q(HX * *) is the image of a chain 0 = F0 F1 : : :Fn = HX in A, and we denote by X OE0!X0 O* *E1! : :O:En!Xn the sequence in C with Fi = Ker HOEi:::OE0which exists by Lemma 1.10* *. By construction, Q(Fi)=Q(Fi-1) 6= 0 if and only if Fi=Fi-1 62 SI. Furthermore, Fi=* *Fi-1 62 SI if and only if the composition of OEi-1: :O:E0 with a fibre of OEi: :O:E0 do* *es_not_belong to I by Lemma 1.10. Thus `I(X) is the length of Q(HX ). * * |__| ENDOFINITENESS IN STABLE HOMOTOPY THEORY 7 Proof of Proposition 1.9.Any object in A=SI is the quotient of HX for some X in C. Therefore A=SI is a length category if and only if `I(X) is finite for all_* *X by Lemma 1.11. |__| Now suppose that I is a cofinite cohomological ideal in C and denote by Q: A * *! A=SI the corresponding quotient functor. Then the assignment X 7! SX = {A 2 A | X does not occur as a composition factor of Q(A)} induces a bijection between the isomorphism classes of simple objects in A=SI a* *nd the maximal Serre subcategories S 6= A of A containing SI. We include the following consequence for later reference. Lemma 1.12. Let I be a cofinite cohomological ideal in C. The map X 7! ISX in* *duces a bijection between the isomorphism classes of simple objects in A=SI and the m* *aximal cohomological ideals J 6= C in C containing I. Proof.Apply the inclusion preserving bijection from Proposition 1.5 between Ser* *re sub-_ categories of A and cohomological ideals in C. * * |__| 2. Annihilator ideals Let C be a compactly generated triangulated category [17]. More precisely, C* * is a triangulated category and has arbitrary coproducts.`An object X in C is called`* *compact if for every family (Yi)i2I in C the canonical map iHom (X; Yi) ! Hom (X; i* *Yi) is an isomorphism. We denote by C0 the full subcategory of compact objects in C a* *nd observe that C0 is a triangulated subcategory of C. For C being compactly gene* *rated the isomorphism classes of objects in C0 need to form a set, and Hom (C; X) = 0* * for all C in C0 implies X = 0 for every object X in C. 2.1. The restricted Yoneda functor. We denote by Mod C0 the category of C0- modules, and our main tool for studying the category C is the restricted Yoneda* * functor C -! Mod C0; X 7! HX = Hom (-; X)|C0: We need to recall the concept of purity for a compactly generated triangulated * *category which has been introduced in [14]. A map X ! Y in C is a pure monomorphism if t* *he induced map Hom (C; X) ! Hom (C; Y ) is a monomorphism for all C in C0. An obje* *ct X in C is pure-injective if every pure monomorphism X ! Y is a split monomorphism* *. The following lemma summarizes some essential properties of the restricted Yoneda f* *unctor. Lemma 2.1. The restricted Yoneda functor C ! Mod C0 identifies the pure-injec* *tives in C with the injectives in Mod C0. Moreover, an object X in C is pure-injectiv* *e if and only if the map Hom (Y; X) ! Hom (HY ; HX ), OE 7! HOE, is bijective for every * *Y in C. A map OE in C is a pure monomorphism if and only if HOEis a monomorphism in Mod C* *0. * * __ Proof.See Corollary 1.9 and Lemma 1.7 in [14]. * * |__| We denote by Sp C the set of isomorphism classes of indecomposable pure-injecti* *ves in C, and Spec C0 denotes the set of isomorphism classes of indecomposable injecti* *ves in Mod C0. 8 HENNING KRAUSE AND ULRIKE REICHENBACH 2.2. The annihilator. We define the annihilator of an object in C as follows: Definition 2.2. The annihilator Ann X of an object X in C is the annihilator of* * the functor Hom (-; X)|C0. Note that Ann X is a cohomological ideal in C0 since Hom (-; X)|C0 is a cohom* *ological functor. Given a cohomological ideal I in C0, we denote by EI the product of a* *ll X in Sp C with I Ann X. The following result shows that the indecomposable pure- injectives in C control the cohomological ideals in C0. Theorem 2.3. Let I be a cohomological ideal in C0 and X be an object in C. (1) I = Ann EI. Q (2) I Ann X if and only if there exists a pure monomorphism X ! IEI for so* *me set I. The theorem is a consequence of a result about the category of C0-modules whi* *ch we take from [12]. Lemma 2.4. Let S be a Serre subcategory of mod C0 and let ES be the product o* *f all X 2 SpecC0 with Hom (S; X) = 0. Also, let X be an object in Mod C0. (1) If X 2 mod C0, then X 2 S if and only if Hom (X; ES) = 0. Q (2) Hom (S; X) = 0 if and only if there exists a monomorphism X ! IES for s* *ome set I. * * __ Proof.See [12, Proposition 3.2]. * * |__| Proof of Theorem 2.3.Let S = SI be the Serre subcategory of mod C0 corresponding to I. Given a map OE in C0 and an object X in C, we have OE 2 Ann X if and only if Hom (Im HOE; HX ) = 0; in particular, I Ann X if and only if Hom (S; HX ) =* * 0. Therefore the restricted Yoneda functor C ! Mod C0 identifies EI with ES and in* *duces a bijection Hom (X; EI) ! Hom (HX ; ES) for every X in C by Lemma 2.1. The asse* *rtion_ of the theorem now follows from these observations and the preceding lemma. * * |__| 3. Endofiniteness Let C be a compactly generated triangulated category. The notion of an endofi* *nite object has been introduced by Crawley-Boevey for so-called locally finitely pre* *sented additive categories [4]. An analogue of this concept for compactly generated tr* *iangulated categories has been studied in [15]. Let us recall this definition. Definition 3.1. An object X in C is called endofinite if for every compact obje* *ct C in C the End (X)-module Hom (C; X) has finite composition length. 3.1. A characterization. We have the following characterization of endofinite o* *bjects in terms of cohomological ideals. Theorem 3.2. An object X in C is endofinite if and only if the annihilator of* * X is a cofinite ideal in C0. Proof.Let X = End (X) and X = End (HX ). We define A = mod C0 and observe that the map Hom (-; C) 7! Hom (C; -) induces an equivalence Aop ! mod (Cop0). Thus * *we can view the C0-module HX :Cop0! Ab as an exact functor FX :Aop ! Ab by Freyd's lemma. In fact, FX is an exact functor Aop ! Mod X in the obvious way, and * *the Serre subcategory S = SAnnX of A corresponding to Ann X is the kernel of FX . * * In [13, Theorem 10.2] it is shown that FX (A) is a finite length X -module for eve* *ry A in ENDOFINITENESS IN STABLE HOMOTOPY THEORY 9 A if and only if A=S is a length category. Suppose now that X is endofinite. * *Then FX (A) is for every A in A a finite length X -module, and therefore also a fini* *te length X -module since the X -action is induced by the canonical map X ! X . Thus A=S is a length category and Ann X is cofinite. Conversely, if Ann X is cofinite, t* *hen FX (A) is a finite length X -module for every A in A. It follows from [13, Theorem 9.6* *] and the characterization of pure-injectives in [14, Theorem 1.8] that X is pure-injecti* *ve. Thus the canonical map X ! X is an isomorphism by Lemma 2.1, and therefore FX (A) is of finite length over X for all A. We conclude that X is endofinite, and this * *finishes_ the proof. |* *__| 3.2. Decomposing endofinite objects. Endofinite objects have nice decomposition properties. We recall the basic result which has been established in [15]. Proposition 3.3. An endofinite`object X is pure-injective and has, up to isomo* *rphism, a unique decomposition X ~= i2IXi into indecomposable objects with local endom* *or- phism ring. * * __ Proof.See [15, Theorem 1.2]. * *|__| The indecomposable endofinite objects can be described in terms of certain co* *homo- logical ideals. In order to state this result we denote by Max C0 the set of m* *aximal elements among the cofinite cohomological ideals I 6= C0 in C0. Theorem 3.4. The assignments I 7! EI and X 7! Ann X induce mutually inverse b* *i- jections between Max C0 and the set of isomorphism classes of indecomposable en* *dofinite objects in C. Proof.Suppose that I is a cofinite cohomological ideal in C0 and let S = SI be * *the corresponding Serre subcategory of A = mod C0. Then A=S is a length category. We denote by T the localizing subcategory of Mod C0 which is generated by S, i.e* *. T is the smallest Serre subcategory of Mod C0 containing S which is closed under ta* *king coproducts. The quotient category Mod C0=T is locally finite in the sense of [7* *, II.4] since the quotient functor Q: Mod C0 ! Mod C0=T sends A to a full subcategory of Mod* * C0=T which is equivalent to A=S and generates Mod C0=T ; see [12, Theorem 2.6] for d* *etails. In fact, A=S is equivalent to the full subcategory of finite length objects in * *Mod C0=T . The quotient functor Q identifies the indecomposable injective objects X in Mod* * C0 satisfying Hom (S; X) = 0 with the indecomposable injective objects in Mod C0=* *T [7, III.3]. Therefore the indecomposable pure-injective objects X in C satisfying I* * Ann X are identified via X 7! Q(HX ) with the indecomposable injective objects in Mod* * C0=T by Lemma 2.1. Every indecomposable injective object in a locally finite catego* *ry is the injective envelope of a simple object [7, IV.1]. Therefore card{X 2 Sp C |* * I Ann X} = 1 if and only if A=S has a unique simple object. The cofinite cohomolo* *gical ideal I is maximal if and only if A=S has a unique simple object by Lemma 1.12,* * and therefore EI is indecomposable if and only if I is a maximal cofinite cohomolog* *ical ideal in C0. Moreover, EI is endofinite by Theorem 3.2 since I = Ann EI which holds * *by Theorem 2.3. Now suppose that X 2 Sp C is endofinite and let I = Ann X which is cofinite by Theorem 3.2. Let S be a simple object in A=SI which we view as a simple object* * in Mod C0=T . By construction, Hom (S; Q(HX )) 6= 0 and therefore Q(HX ) is isomor* *phic to an injective envelope of S. It follows that S is the unique simple object in A=* *SI. Thus I is a maximal cofinite cohomological ideal in C0; in particular X = EI. This f* *inishes_ the proof. |* *__| 10 HENNING KRAUSE AND ULRIKE REICHENBACH We discuss now a number of consequences of Theorem 3.2 and Theorem 3.4. The following lemma will be useful. Lemma 3.5. Let (Xi)i2Ibe a family of objects in C. Then a " Y Ann Xi= Ann Xi= Ann Xi: i2I i2I i2I ` ` Q Proof.UseQthe isomorphisms iHom (C; Xi) ~=Hom (C; iXi) and Hom (C; iXi) * *~=_ iHom (C; Xi) for C in C0. * *|__| Corollary 3.6. Let X be an endofinite object in C with I = Ann X. Then " I = J IJ2Max C0 and X has a unique decomposition a X ~= XJ IJ2Max C0 such that each XJ is a non-empty coproduct of copies of EJ. ` Proof.Let X ~= i2IXibe the decomposition into indecomposable objects which exi* *sts by Proposition 3.3. The annihilator Ann Xi contains Ann X and is therefore cofi* *nite by Theorem 3.2. Thus Xi is endofinite for all i. Applying Theorem 3.4 and Lemma 3.* *5 we obtain " " I = Ann X = Ann Xi= J: i2I IJ2Max C0 Now define for every J 2 Max C0 containing I a XJ = Xi: AnnXi=J Clearly, this gives the desired decomposition a X ~= XJ: IJ2Max C0 It remains to show that XJ 6= 0 for every J 2 Max C0 containing I. We apply the* * ideas from the proof of Theorem 3.4 and keep the same notation. We fix J. The ideal* * J corresponds to a simple object S in A=SI by Lemma 1.12 which we view as a simple object in Mod C0=T . The injective envelope of S is Q(HEJ) which is isomorphi* *c to a direct factor of Q(HX ) since Hom (S; Q(HX )) 6= 0. It follows that EJ is isomo* *rphic to a direct factor of X`since Hom (EJ; X) ~=Hom (Q(HEJ); Q(HX )). The uniqueness of* * the decomposition X ~= i2IXiimplies that Xi~= EJ for some i 2 I, and therefore XJ * *6=_0. This completes the proof. * *|__| Corollary 3.7. Let X and Y be objects in C and suppose that X is endofinite. T* *hen Ann X Ann Y if and only if Y is isomorphic to a direct factor of a (co)produ* *ct of copies of X. Proof.If Y is isomorphic to a direct factor of a (co)product of copies of X, th* *en Ann X Ann Y by Lemma 3.5. To prove the converse suppose that Ann X Ann Y .`Observe that`Y is endofinite by Theorem 3.2 since Ann Y is cofinite. Let X ~= i2IXi an* *d Y ~= j2JYj be the decompositions into indecomposables which exist by Proposition 3* *.3. It follows from Corollary 3.6 that every factor Yj is isomorphic to Xi for some i * *2 I since Yj ~=EJ for some J 2 Max C0 containing Ann Y and therefore containing Ann X. The ENDOFINITENESS IN STABLE HOMOTOPY THEORY 11 ` family of split monomorphisms Yj ! X induces`aQsplit monomorphism Y ! JX and the composition with the canonical map JX ! JX is again a split monomorphism since it is a pure monomorphism and Y is pure-injective by Proposition 3.3. We * *conclude_ that Y is isomorphic to a direct factor of a (co)product of copies of X. * * |__| Corollary 3.8. An indecomposable object X is endofinite if and only if every pr* *oduct of copies of X is a coproduct of copies of X. Q Proof.Suppose first that X is indecomposableQand endofinite. Any product IX * *is again endofiniteQby Theorem 3.2, since Ann IX = Ann X by Lemma 3.5. The decom- position of IX as a coproduct of copies of X then follows directly from Corol* *lary 3.6. Now suppose that X is any indecomposable object such that every product of copi* *es of X is a coproduct of copies of X. We consider the corresponding C0-module M = HX* * , and it is clear that every product of copies of M is a coproduct of copies of M* *. It follows from [8, Theoreme] that M is a pure-injective C0-module. In fact, M is injectiv* *e since M is fp-injective by [14, Lemma 1.6] and therefore every monomorphism M ! N in Mo* *d C0 is a pure monomorphism, hence a split monomorphism. It follows from Lemma 2.1 t* *hat X is a pure-injective object in C. Viewing X as an exact functor FX :Aop ! Ab * * for A = mod C0 as in the proof of Theorem 3.2, it follows from the characterizatio* *n of __ endofinite objects in [13, Corollary 10.5] that X is also an endofinite object * *in C. |__| 4.Endofinite spectra In the following we apply the results of the previous sections to the stable * *homotopy category C in the sense of [1]. The objects are CW-spectra, and the morphisms * *X ! Y between two spectra X and Y are denoted by {X; Y }. We identify each CW-complex with its suspension spectrum. The stable homotopy category is a triangulated ca* *tegory where the suspension X 7! X = X ^ S1 acts as a translation functor. In particul* *ar, Sn = n(S0) is defined for all n 2 Z, and for a spectrum X we denote by sssn(X) * *the n-th stable homotopy group {Sn; X}. Coproducts are given by one point unions ca* *lled wedge, and arbitrary coproducts do exist in C. Recall that a spectrum X is fin* *ite if for some n 2 N the spectrum nX is the suspension spectrum of a finite CW-comple* *x. Note that the finite spectra are precisely the compact objects in C. Therefore* * C is a compactly generated triangulated category since sssn(X) = 0 for all n 2 Z impli* *es X = 0. These facts are well known, see for instance [16]. 4.1. p-localization. We denote for every prime ideal p in Z by X ! Xp the p- localization map for a spectrum X. The fibre of the rational localization X ! * *X(0) is denoted by Xtor. Theorem 4.1. Let X be an endofinite spectrum. (1) The p-localization map X ! Xp is a split epimorphism for every prime idea* *l p in Z. (2) X ~=Xtor_ X(0). W (3) Let Y = Xtor. Then Y ~= p Yp where p runs through all non-zero prime ide* *als in Z. Proof.(1) Fix a prime ideal p. The localization map X ! Xp induces, by definiti* *on, for every n 2 Z an isomorphism sssn(X) Z Zp ~=sssn(Xp) which extends to an isomorph* *ism {C; X} Z Zp ~={C; Xp} 12 HENNING KRAUSE AND ULRIKE REICHENBACH for all C in C0. Therefore Ann X Ann Xp. If X is indecomposable endofinite, th* *en Xp is a direct factor of a wedge of copies of X by Corollary 3.7. Thus Xp = X or X* *p = 0, and therefore X ! Xp is a split epimorphism. An arbitrary endofinite spectrum X* * is a wedge of indecomposable endofinite spectra by Proposition 3.3, and therefore X * *! Xp is a split epimorphism since p-localization preserves wedges; e.g. see [16, Cor* *ollary 8.3]. (2) The sequence 0 ! Xtor! X ! X(0)! 0 is split exact by (1). (3) Suppose first that Y is indecomposable. There exists a prime ideal q 6=* * 0 such that {C; Y } Z Zq 6= 0 for some C in C0. Thus Ann Yq 6= C0 and thereforeWY = Yq* * by (1). Moreover, Yp = 0 for all p 6= q since Y is torsion, and Y ~= p Yp follows* *. Finally,_ the assertion holds for arbitrary Y since p-localization preserves wedges. * * |__| 4.2. The endolength. Let X be any spectrum. We denote for every n 2 Z by `n(X) the length of the stable homotopy group sssn(X) as {X; X}-module, and call `(X) = (`n(X))n2Z the endolength of X. The endolength leads to a nice characterisation of endofi* *nite spectra: Proposition 4.2. A spectrum X is endofinite if and only if `n(X) is finite for* * all n 2 Z. Proof.Consider the full subcategory CX of spectra Y such that {nY; X} is of f* *inite length over {X; X} for all n 2 Z. This is easily seen to be closed under cofibr* *ations and retracts, and therefore CX is a thick subcategory of C. On the other hand, C0 i* *s the small- est thick subcategory containing the sphere spectrum S0; e.g. see [10, Theorem * *2.1.3]. Consequently, if S0 is an object of CX then C0 is contained in CX . In particu* *lar,_X is endofinite. * *|__| We continue with another interpretation of the length `n(X). Let I = Ann X a* *nd denote by S = SI the corresponding Serre subcategory of A = mod C0. The locali* *z- ing subcategory of Mod C0 generated by S is denoted by T , and the quotient fu* *nctor Q: Mod C0 ! Mod C0=T identifies A=S with a full subcategory of Mod C0=T ; see* * [12, Theorem 2.6]. Lemma 4.3. Let X be a pure-injective spectrum. Then `n(X) equals the length o* *f HSn in Mod C0=T which is also the length of HSn in A=S; in particular `n(X) = `I(S* *n). The proof of this lemma is essentially an application of the following easy l* *emma: Lemma 4.4. Let M and N be objects in an abelian category and suppose that N is an injective cogenerator. Then the length of M equals the length of the End (N)* *-module Hom (M; N). * *__ Proof.Straightforward. |* *__| Proof of Lemma 4.3. Put M = Q(HSn) and N = Q(HX ), and observe that {Y; X} ~=Hom (HY ; HX ) ~=Hom (Q(HY ); Q(HX )) for all Y in C by Lemma 2.1. Also, Q(HX ) is injective by the same lemma since * *X is pure- injective. The assertion now follows from the preceding lemma since Hom (M0; N)* * 6= 0 for all M0 = Q(A) 6= 0 with A 2 A. Moreover, M0 is a simple object in Mod C0=T* * if and only if M0 is simple when regarded as an object in A=S. Finally, `n(X) = `I* *(Sn)_ follows from Lemma 1.11. |* *__| We are now inWa position to compute the endolength of a spectrum X from a dec* *om- position X ~= iXi. We proceed in two steps. ENDOFINITENESS IN STABLE HOMOTOPY THEORY 13 W Proposition 4.5. Let X be a spectrum and I be a non-empty set. Suppose that * *IX is pure-injective. Then for all n 2 Z _ Y `n( X) = `n(X) = `n( X): I I W Q Proof.The assertion follows from Lemma 4.3 since Ann IX = Ann X = Ann IX by_ Lemma 3.5. |__| W Proposition 4.6. Let X ~= i2IXi be a decomposition of a pure-injective spectr* *um. Suppose that any indecomposable spectrum arises as a wedge summand of Xi for at most one i 2 I. Then for all n 2 Z _ X `n( Xi) = `n(Xi): i2I i2I Proof.We keep the above notation and denote for every i 2 I by Qi: Mod C0=T ! Mod C0=Ti the quotient functor with respect to the localizing subcategory Ti wh* *ich is generated by SAnnXi. Now suppose that A 6= 0 is any subquotient of Q(HSn) in A=* *S. Then Qi(A) 6= 0 for some i 2 I since a a 0 6= Hom (A; Q(HX )) ~= Hom (A; Q(HXi)) ~= Hom (Qi(A); QiOQ(HXi)): i i If A is simple then Qi(A) 6= 0 for precisely one i 2 I by our assumption on the* * family (Xi)i2I, since the injective envelope of A in Mod C0=T is isomorphic to Q(HY ) * *for some indecomposableWsummand Y of X. Using the exactness of the Qi, the formula for `n( i2IXi) follows from the fact that `n computes the length of Q(HSn) by Lemm* *a_4.3. |_* *_| Corollary 4.7. Let X be an endofinite spectrum with I = Ann X. Then for all n 2* * Z X `n(X) = `n(EJ): IJ2Max C0 W Proof.Combine Proposition 4.5 and Proposition 4.6 with the decomposition X ~= * *JXJ_ from Corollary 3.6. * *|__| 4.3. Subgroups of finite definition. Let X be any spectrum. Given a finite spec* *trum F and an element OE 2 sssn(F ) for some n 2 Z, we denote by sssn;OE(X) the imag* *e of the induced map {OE; X}: {F; X} -! {Sn; X} = sssn(X): A subgroup of sssn(X) which is of the form sssn;OE(X) for some OE 2 sssn(F ) is* * said to be of finite definition. Note that any subgroup of finite definition is automa* *tically a {X; X}-submodule of sssn(X). Lemma 4.8. Let OE 2 sssn(F ) for some finite spectrum F . (1) The subgroups of finite definition form a latticeWin sssn(X)`for every sp* *ectrum X. Q (2) IfQ(Xi)i2Iis a family of spectra, then sssn;OE( iXi) ~= isssn;OE(Xi) an* *d sssn;OE( iXi) ~= s issn;OE(Xi). Proof.(1) Let OEi 2 sssn(Fi) (i = 1; 2). Then sssn;OE1(X) + sssn;OE2(X) = sssn* *;OE(X) where OE: Sn ! F1 _ F2 has components OEi. Let :F1 _ F2 ! G be a cofibre of OE, wi* *th components i, and set O = 1OE1. The sequence {G; X} ! {F1 _ F2; X} ! {Sn; X} * *is exact, and it follows that sssn;OE1(X) \ sssn;OE2(X) = sssn;O(X). * * __ (2) Clear, since {OE; -} preserves wedges and products. * * |__| 14 HENNING KRAUSE AND ULRIKE REICHENBACH The endofiniteness of a spectrum can be characterized in terms of the lattice* * of sub- groups of finite definition. Proposition 4.9. A spectrum X is endofinite if and only if sssn(X) has the asc* *ending and descending chain conditions on subgroups of finite definition for all n 2 Z* *. If this holds, then the {X; X}-submodules of sssn(X) are precisely the subgroups o* *f finite definition. * * __ Proof.As for modules, see [3, Proposition 4.1]. * * |__| For the sake of completeness we include the following characterization of the* * spectra having the descending chain condition on subgroups of finite definition: Theorem 4.10. The following conditions are equivalent for a spectrum X: W (1) X is -pure-injective,Wi.e.Q IX is pure-injective for every set I; (2) the canonical map NX ! NX is a split monomorphism; (3) the descending chain condition holds for the subgroups of finite definiti* *on of sssn(X) for every n 2 Z; (4) every product of copies of X is a wedge of indecomposable spectra with lo* *cal en- domorphism rings. Proof.Adapt the argument for modules [11], using the restricted Yoneda functor_* *C_! Mod C0. |_* *_| Remark 4.11. Concepts and results from this section carry over to any compact* *ly generated triangulated category C if one replaces the sphere spectrum S with an* *y set of compact objects which generates C. 5. A criterion for endofiniteness In this section we give a criterion for the suspension spectrum of a CW-compl* *ex to be endofinite in terms of its singular homology groups. Following the definiti* *on of an endofinite module [3], we say that a CW-complex is endofinite if the stable hom* *otopy groups sssn(X) are modules of finite length over {X; X} for all n 2 Z. Theorem 5.1. Let X be a connected CW-complex such that the singular homology groups Hn(X; Z) are finite for all n > 0. Then X is an endofinite CW-complex, * *and therefore its suspension spectrum is also endofinite. Proof.We prove that a CW-complex X satisfying the above assumptions has finite stable homotopy groups, i.e. sssn(X) is finite for all n 2 Z. Clearly, this imp* *lies that X is an endofinite CW-complex, and therefore the suspension spectrum of X is endo* *finite by Proposition 4.2. To prove the finiteness of the stable homotopy groups, we a* *pply the Atiyah-Hirzebruch-Whitehead spectral sequence for the (unreduced) homology theo* *ry hn(-) = sssn(- +). The subscript + means adding of a disjoint basepoint. The co* *efficient groups hn(pt:) are the stable homotopy groups of S0 and are known to be trivial* * for n < 0, isomorphic to Z for n = 0, and finite for n > 0. Therefore, the E2-term* * of the spectral sequence, E2p;q= Hp(X; hq(pt:)), has non-trivial values only for p* *; q 0 and these are finite with the only exception E20;0= Z. This form is preserved * *for all Ek-terms, k 2. The spectral sequence converges to h*(X). This means that hn(X) is the limit* * of groups F0;n F1;n-1 : : :which are inductively calculated from F0;n = E10;nand the extension problem 0 ! Fp-1;n-p+1! Fp;n-p! E1p;n-p! 0. From this inductive ENDOFINITENESS IN STABLE HOMOTOPY THEORY 15 construction it follows easily that Fp;n-p= 0 for n < 0 and hn(X) = Fn;0is fini* *te for n > 0. For the respective reduced homology theory "hn(-) = sssn(-) we get ( 0 for n 0; sssn(X) = sssn(X+ )=sssn(S0) = finite for n > 0. __ |_* *_| Important examples of CW-complexes with finite homology are the classifying s* *paces of finite groups. Corollary 5.2. The classifying space BG of a finite group G is endofinite. Proof.Since the homology groups Hn(BG; Z) are finite in all positive degrees we* * may_ apply the preceding theorem. * *|__| In Theorem 5.1, the condition on X to be connected cannot be omitted. For exa* *mple, X = S0 has finite homology in all positive degrees but the {X; X}-module sss0(X* *) ~=Z is not of finite length. It has been pointed out by Stefan Schwede that Theorem 5.1 can be generalized* * as follows. Theorem 5.3. A connected CW-spectrum X is endofinite if and only if the homol* *ogy groups Hn(X; Z) with coefficients in Z are modules of finite length over {X; X}* * for all n 2 Z. Proof.The argument is essentialy that given in the proof of Theorem 5.1. To pr* *ove endofiniteness of X one uses the Atiyah-Hirzebruch-Whitehead spectral sequence H*(X; sss*) =) sss*(X) as above. To prove finiteness of the homology of X one uses the less well-known* * spectral sequence s Torss*(sss*(X); Z) =) H*(X; Z): * * __ Both spectral sequences are natural in X and can be found in [5]. * * |__| Acknowledgement. Both authors would like to thank Stefan Bauer and Hans-Werner Henn for a number of helpful conversations concerning the material of this pape* *r. The second named author was supported by the SFB 343. References [1]J.F. Adams, Stable homotopy and generalised homology, The University of Chi* *cago Press (1974). [2]G. Carlsson, Equivariant stable homotopy and Segal's Burnside ring conjectu* *re, Ann. 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Soc. 9 (1996), 205-236. [18]J.L. Verdier, Categories derivees,etat 0, Springer Lec. Notes 569 (1977), 2* *62-311. Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33501 Bielefeld, Germany E-mail address: henning@mathematik.uni-bielefeld.de, reichenb@mathematik.uni-* *bielefeld.de