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.
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*bielefeld.de