HOMOLOGICAL LOCALIZATIONS OF EILENBERG-MAC LANE SPECTRA JAVIER J. GUTI'ERREZ Abstract. We discuss the Bousfield localization LEX for any spectrum E and any HR-module X, where R is a ring with unit. Due to the splitting property of HR-modules, it is enough to study the localization of Eilenb* *erg- Mac Lane spectra. Using general results about stable f-localizations, we* * give a method to compute the localization of an Eilenberg-Mac Lane spectrum LEHG for any spectrum E and any abelian group G. We describe LEHG explicitly when G is one of the following: finitely generated abelian gr* *oups, p- adic integers, Pr"ufer groups, and subrings of the rationals. The result* *s depend basically on the E-acyclicity patterns of the spectrum HQ and the spectr* *um HZ=p for each prime p. 1. Introduction Homological localizations were first defined by Adams [Ada73 ]. Bousfield de* *vel- oped the theory further by proving the existence of homological localizations i* *n the category of spaces [Bou75 ] and in the category of spectra [Bou79b ]. Given any spectrum E, a homological localization functor with respect to E i* *s a homotopy idempotent transformation LE :Hos -! Hos, where Hos is the stable homotopy category, that turns E-homology equivalences into homotopy equiva- lences in a universal way. Homological localizations are special cases of f-loc* *aliza- tions in the sense of [Dro96] and commute with the suspension operator. In [CG05 ], we presented a general study of f-localizations of HR-module spe* *ctra and discussed the preservation of several structures under the effect of these * *func- tors. In this paper, we restrict our attention to homological localizations in * *order to obtain more explicit results. In fact, we translate to spectra some of the r* *esults of [Bou82 ], by using ideas of [CG05 ] to simplify the arguments. In [Bou79b ], Bousfield determined the homological localizations of connecti* *ve spectra with respect to connective homology theories. A spectrum X is connective if ssk(X) = 0 for k < 0. If either E or X fail to be connective, then LE X is s* *omehow unpredictable. For example, the spectrum LK S, where K denotes complex K- theory and S is the sphere spectrum, has infinitely many nonzero homotopy groups in both positive an negative dimensions (see [Rav84 , Theorem 8.10] or [CG05 , Corollary 5.15]). We study LE X where E is any homology theory (not necessarily connective) and X is any HR-module spectrum for a ring R with unit. Since any HR-module splits as a wedge of suspensions of Eilenberg-Mac Lane spectra, we focus on the study of LE HG for any homology theory E and any abelian group G, where HG denotes the Eilenberg-Mac Lane spectrum associated to G. We describe all possib* *le homological localizations in the case of finitely generated abelian groups and * *other groups, including the p-adic integers, the Pr"ufer groups Z=p1 , and subrings o* *f the rationals. For example, in the case of the spectrum HZ, by the general approach ____________ 2000 Mathematics Subject Classification. 54P60, 54P42. Key words and phrases. Homological localization, Eilenberg-Mac Lane spectrum. The author was supported by MEC-FEDER grant MTM2004-03629. 1 2 JAVIER J. GUTI'ERREZ of [CG05 ] we know any of its localization has at most one nonzero homotopy gro* *up and that this group has the structure of a rigid ring in the sense of [CRT00 ].* * We prove that, for homological localizations of HZ, the only rigid rings that appe* *ar are subrings of the rationals or products of p-adic integers for diferent prime* *s. The computations of these localizations depend on the E-acyclicity patterns * *of the spectra HZ=p and HQ, and on the set of primes p such that G is uniquely p-divisible, similarly as in [Bou82 ]. Acknowledgements. I am especially indebted to A. K. Bousfield for ecouraging me to write this paper and for sharing his insight. I would also like to thank * *Carles Casacuberta and Mark Hovey for many useful conversations. 2. Homological localization of spectra We will work in the stable homotopy category of spectra Hos (see [Ada74 ]). * *Any spectrum E in Hos gives rise to a homology theory defined as Ek(X) = ssk(E ^ X) for any spectrum X and any k 2 Z. Homological localization with respect to the homology theory E is a functor that transforms homology equivalences with respe* *ct to this theory into homotopy equivalences in a universal way. It is unique up * *to homotopy and idempotent. A map of spectra f :X -! Y is an E-equivalence if the map f*: Ek(X) -! Ek(Y ) is an isomorphism for all k 2 Z. A spectrum X 2 Hos is called E-acyclic if Ek(X) = 0 for all k 2 Z, that is, if E ^ X is contractible. A spectrum Z is E-local if each E-equivalence X -! Y induces induces a homotopy equivalence F (Y, Z) ' F (X, Z), or equivalently if F (W, Z) = 0 for each E-acyclic spectru* *m W , where F (X, Y ) denotes the function spectrum from X to Y . An E-localization of a spectrum X is a map jX :X -! LE X, where X is an E-local spectrum and jX is an E-equivalence. Homological localization is universal in the following se* *nse: the localization map jX is initial among maps from X to E-local spectra and it * *is terminal among all E-equivalences with domain X. The class of all the E-acyclic spectra for a given spectrum E is denoted by * * and called the Bousfield class or the acyclicity class of E. Given two spectra * *E and F , the E-localization functor and the F -localization functor are equivalent i* *f and only if = . By Ohkawa's theorem, there is only a set of Bousfield class* *es [DP01 ], and therefore a set of non-equivalent homological localization functor* *s. 3. Acyclicity patterns of HZ=p and localization of HR-modules In this section we study how the E-acyclicity patterns of HZ=p determine the localization LEZ=pX for any spectrum E and any HR-module spectrum X. For any spectrum E and any abelian group G, let EG = E ^ MG, where MG is the Moore spectrum associated to G. A spectrum E is called a stable R-GEM if it is homotopy equivalent to a wedge of suspensions of Eilenberg-Mac Lane spectra, i.e., E ' _k2Z kHAk, where each Ak is an R-module (hence, each HAk is an HR-module spectrum). If R = Z, then stable Z-GEMs are called simply stable GEMs. The Eilenberg-Mac Lane spectrum HG is an R-GEM if G is an R-module. The stable R-GEMs are precisely the spectra that admit a module structure over the ring spectrum HR (see for example [CG05 , Proposition 4.4]). The splitting property of HR-modules allows us to describe their localization easily. Note that every HR-module is an HZ-module trivially via the morphism Z ! R that sends the unit of Z to the unit of the ring R. And also that homolog* *ical localizations commute with suspension, i.e., LE kHG ' kLE HG for all k 2 HOMOLOGICAL LOCALIZATIONS OF EILENBERG-MAC LANE SPECTRA 3 Z, since the desuspension of an E-equivalence is again an E-equivalence [CG05 , Proposition 2.4]. Proposition 3.1. LE (_k2Z kHAk) ' _k2Z( kLE HAk) for any spectrum E. Proof. The spectrum _k2Z( kLE HAk) is E-local, since the natural map ` Y kLE HAk -! kLE HAk k2Z k2Z is a homotopy equivalence, because by [CG05 , Theorem 5.6] for each value of k,* * at most two nonzero homotopy groups appear in LE HAk. Now, the map ` ` kHAk -! kLE HAk k2Z k2Z is an E-equivalence, because it is a wedge of E-equivalences. In [Bou79a ], Bousfield showed that ` (3.1) = _ , p2P for any spectrum E, where P is the set of all primes. In fact, what this means essentially is that we can recover LE X for any E and X from information on what happens rationally, LEQ X, and at each prime, LEZ=pX. The following result of Bousfield [Bou79b , Proposition 2.9] ilustrates this fact. Recall that a commut* *ative diagram of spectra f X _____//Y h || g|| fflffl| fflffl| Z ___i_//W is an arithmetic square if there is a map j :W -! X such that (f,h) (g,-i) j X _____//Y ^ Z____//W_____// X is a cofiber sequence of spectra. Proposition 3.2. For all spectra E and X, there is an arithmetic square Q (3.2) LE X ________//_p2PLEZ=pX | | | | fflffl| Q |fflffl LEQ X _____//LEQ ( p2P LEZ=pX), where P is the set of all primes. The EQ-localizations were completely determined in [Bou79b ]. For any spec- trum E, all these localizations are equivalent to rationalization. In fact, LEQ* * X = LMQ X = X ^ MQ for all E and X. The computation of LEZ=pX for any E and any HR-module spectrum X depends on the E-acyclicity types of the spectrum HZ=p for each prime p. Note that if iHZ=p is E-acyclic for some i 2 Z, then kHZ=p is E-acyclic for all k 2 Z, sin* *ce homological localizations commute with suspension. Proposition 3.3. If HZ=p is E-acyclic, then LEZ=pX = 0 for any HR-module spectrum X. 4 JAVIER J. GUTI'ERREZ Proof. It is enough to check that MZ=p is E-acyclic, because in this case EZ=p ^ X ' E ^ MZ=p ^ X = 0. The spectrum MZ=p ^ X is obviously EQ-acyclic and EZ=q-acyclic for q 6= p. In the case q = p, we have that MZ=p ^ X ^ EZ=p ' X0^ HZ ^ MZ=p ^ EZ=p = 0 since X is an HZ-module and therefore splits as HZ ^ X0 for some spectrum X0, and HZ ^ EZ=p ' E ^ HZ=p = 0. Now using the decomposition (3.1), we have that X is EZ=p-acyclic. Lemma 3.4. If HZ=p is not E-acyclic and f :X -! Y is an EZ=p-equivalence, then it is an HZ=p-equivalence. Proof. Since homological localizations commute with suspension, if we smash f with any spectrum the resulting map is an EZ=p-equivalence. In particular, if we smash with the spectrum HZ, the map f ^ HZ induces an equivalence EZ=p ^ HZ ^ X ' EZ=p ^ HZ ^ Y. The spectrum EZ=p ^ HZ ' E ^ HZ=p is an HZ=p-module, so its homotopy gropus are Z=p-vector spaces and it splits as a wedge _k2I kHZ=p (there may be repetitions in the index set I) and this wedge is non-trivial since by hypothes* *is E ^ HZ=p 6= 0. Hence, f induces an equivalence ` ` kHZ=p ^ X ' kHZ=p ^ Y k2I k2I which turns f into an HZ=p-equivalence. The following theorem allows us to compute the localization LEZ=p of connect* *ive spectra or HR-modules when the spectrum HZ=p is not E-acyclic. Theorem 3.5. If HZ=p is not E-acyclic, then LEZ=pX ' LMZ=p X for every spec- trum X that is connective or an HR-module. Proof. If X is a connective spectrum, then LMZ=p X ' LHZ=pX (see [Bou79b , Theorem 3.1]). The localization map X - ! LMZ=p X ' LHZ=pX is an MZ=p- equivalence and therefore and EZ=p-equivalence. Moreover, the spectrum LHZ=pX is EZ=p-local since by Lemma 3.4 every HZ=p-local spectrum is EZ=p-local. If X is an HR-module, the result follows from the above and Proposition 3.1. The case LMZ=p X can be computed using [Bou79b , Proposition 2.5]: Proposition 3.6. For any spectrum X, we have that LMZ=p X ' F ( -1MZ=p1 , X), and there is a splittable exact sequence 0 -! Ext(Z=p1 , ssk(X)) -! ssk(LMZ=p X) -! Hom (Z=p1 , ssk-1(X)) -! 0 for any k 2 Z. In the particular case when X is an Eilenberg-Mac Lane spectrum HG, we have that LMZ=p HG ' HA _ HB where A ~=Ext(Z=p1 , G) and B ~=Hom (Z=p1 , G). HOMOLOGICAL LOCALIZATIONS OF EILENBERG-MAC LANE SPECTRA 5 4. Localizations of Eilenberg-Mac Lane spectra In the study of homological localizations of HR-module spectra, we can focus* * our attention on the particular case of homological localizations of Eilenberg-Mac * *Lane spectra LE HG, by Proposition 3.1. Homological localizations are a particular e* *x- ample of homotopical localizations or f-localizations. These localizations in * *the stable homotopy category have been studied in [CG05 ]. In that paper, we proved that the localization of any Eilenberg-Mac Lane spectrum has at most two nonzero homotopy groups in dimensions zero and one (see [CG05 , Theorem 5.6]). Thanks to the Bousfield arithmetic square, to compute LE HG it is enough to determine LEZ=pHG for every prime p, since in the rational case LEQ HG = H(Q G) for any spectrum E. An abelian group G is called uniquely p-divisible if for every g 2 G there e* *xists a unique h 2 G such that g = ph. This condition is equivalent to saying that Z=p G = 0 and Tor(Z=p, G) = 0. Lemma 4.1. For any spectrum E, the group ssk(E) is uniquely p-divisible for all k 2 Z if and only if EZ=p = 0. Proof. The result follows using the exact sequence Z=p ssk(E) -! ssk(EZ=p) -! Tor(Z=p, ssk-1(E)), which is valid for every k 2 Z. As a particular case, we have that the abelian group (HZ)k(E) is uniquely p- divisible for all k 2 Z if and only if HZ=p is E-acyclic. Note also that if ssk* *(E) is uniquely p-divisible, then (HZ)k(E) is uniquely p-divisible. Proposition 4.2. If HZ=p is not E-acyclic, then LEZ=pHG = 0 if and only if G is uniquely p-divisible. Proof. If LEZ=pHG = 0, then E ^ HZ=p ^ MG = 0. Since E ^ HZ=p is an HZ=p- module spectrum, we have that E ^ HZ=p ^ MG = _k2I kHZ=p ^ MG = 0. Therefore, HZ=p ^ MG = MZ=p ^ HG = 0 and thus G is uniquely p-divisible by Lemma 4.1. On the other hand, if G is uniquely p-divisible, then by Lemma 4.1 we have t* *hat HG ^ MZ=p = 0 and hence LEZ=pHG = 0. By means of Theorem 3.5 and Proposition 4.2, one can now compute the local- ization LEZ=pHG depending on the E-acylicity patterns of HZ=p. If HZ=p is E- acyclic, then LEZ=pHG = 0. If HZ=p is not E-acyclic, then LEZ=pHG ' LMZ=p HG if G is not uniquely p-divisible and zero otherwise. The arithmetic square (3.2* *) in the case X = HG is the following: Q (4.1) LE HG __________//_p2PLEZ=pHG | | | | fflffl| Q fflffl| H(G Q) _____//MQ ^ ( p2P LEZ=pHG), where P is the set of all primes p such that HZ=p is not E-acyclic and G is not uniquely p-divisible. Theorem 4.3. Let Ap = Ext(Z=p1 , G), Bp = Hom (Z=p1 , G), and let P be the set of primes such that HZ=p is not E-acyclic and G is not uniquely p-divisible. For any spectrum E and any abelian group G, we have the following: 6 JAVIER J. GUTI'ERREZ (i)If HQ is E-acyclic, then Y LE HG = (HAp _ HBp). p2P (ii)If HQ is not E-acyclic, then there is a cofiber sequence of spectra Y Y LE HG -! H(Q G) _ (HAp _ HBp) -! MQ ^ (HAp _ HBp). p2P p2P Proof. The result follows from Proposition 3.3, Proposition 4.2, Theorem 3.5 and the arithmetic square (4.1). 5. Some examples In this section, we compute homological localizations of Eilenberg-Mac Lane spectra and HR-module spectra in some concrete examples. First, we compute LE X for some non-connective homology theories E and any HR-module spec- trum X. 5.1. Localization with respect to n-th Morava K-theory K(n). Let n 0, p a fixed prime and K(n) the spectrum of the n-th Morava K-theory at p. Recall that ss*K(n) ~=Z=p [v-1n, vn] where |vn| = 2(pn - 1) for n 1. If n = 0, then K(0) = HQ = MQ and so LK(0)HG = H(G Q). In the case n 1 we know that K(n) ^ MQ = 0 and HZ=p is K(n)-acyclic for every prime p, because K(n) ^ HZ=p = 0 for all primes p (see for example [Rav84, Theorem 2.1]). Thus LK(n)HG = 0 for n 1. Hence, Proposition 5.1. For any HR-module X, its localization with respect to K(n) is either zero if n 1, or rationalization if n = 0, i.e., LK(0)X = X ^ MQ. 5.2. Localization with respect to Johnson-Wilson spectra E(n). The Bous- field class of E(n) splits as a wedge of Morava K-theories, = (see [Rav84, Theorem 2.1]); therefore LE(n)HG = LK(0)HG = H(G Q) since LK(i)HG = 0 if i 0. 5.3. Localization with respect to complex K-theory. The spectrum HZ=p is K-acyclic for every prime p and KQ 6= 0, so LK HG = H(G Q). Therefore, we infer the following: Proposition 5.2. For any HR-module spectrum X, its localization with respect to E(n) or K-theory is rationalization. In the next examples, we use Theorem 4.3 to compute all the possible homolog- ical localizations of the spectrum HG with respect to any E for some families of abelian groups. Given any spectrum E and any abelian group G, we have the fol- lowing acyclicity patterns that determine the localization LE HG completely. Th* *ese patterns are the stable analogues of Condition I and Condition II of [Bou82 , S* *ection 4]: o Pattern I: EQ = 0 and E ^ HZ=p = 0 for all primes p. o Pattern II: EQ 6= 0 and E ^ HZ=p = 0 for all primes p. o Pattern III: EQ = 0 and E ^HZ=p 6= 0 for all primes p in a set of primes* * P. o Pattern IV: EQ 6= 0 and E ^HZ=p 6= 0 for all primes p in a set of primes* * P. Note that if Pattern I holds, we have that LE HG = 0 for any abelian group G. HOMOLOGICAL LOCALIZATIONS OF EILENBERG-MAC LANE SPECTRA 7 5.4. Localizations of HZ. The abelian group of the integers is not uniquely p- divisible for any prime p. If Pattern II holds, then LE HZ = HQ. We have that Hom (Z=p1 , Z) = 0 and Ext(Z=p1 , Z) = bZp, whereQbZp is the ring of p-adic integers. If Pattern III holds, then LE HZ = H( p2P bZp). And if Pattern IV holds, then taking ss0 in the square (4.1) we have the following pullback diagram of abelian groups: Q ss0(LE HZ)_______//_p2PbZp | | | | | fflffl| fflffl|Q Q _________//Q p2PbZp, where P is the set of all primes p such that E ^ HZ=p 6= 0. So LE HZ = HZP . In [CG05 , Theorem 5.12] we proved that every f-localization of the spectrum HZ has at most one nonzero homotopy group, which aquires the structure of a rigid ring in the sense of [CRT00 ]. A ring A with unit is rigid if evaluation* * a 1 induces an isomorphism of abelian groups Hom (A, A) ~=A. In the special case of homological localizations we get the following: Proposition 5.3. For any spectrum E, we have that LE HZ is either zero or HA, where the rigid ring A is a subring of Q or a product of p-adic integers for di* *fferent primes. 5.5. Localizations of HZ=pk for a prime p. The group Z=pk is uniquely q- divisible for every q 6= p and moreover LEQ HZ=pk ' H(Q Z=pk) = 0 for all p. We have that Hom (Z=p1 , Z=pk) = 0 and Ext(Z=p1 , Z=pk) = Z=pk, hence LE HZ=pk = 0 under Pattern II and LE HZ=pk = HZ=pk under Pattern III or Pattern IV. 5.6. Localizations of HQ. The group Q is uniquely p-divisible for every prime p, so LEZ=pHQ = 0 for all p. If Pattern III holds, then LE HQ = 0 and LE HQ = HQ under Pattern II or Pattern IV. 5.7. Localization of HZR for a set of primes R. For every prime p 2 R, we have that Hom (Z=p1 , ZR ) = 0 and Ext(Z=p1 , ZR ) = bZp. In fact, Hom (Z=p1 , G) = 0 if G is a torsion-free abelian group and Ext(Z=p1 ,* * G) = 0 if and only if G is p-divisible. If Pattern II holds,Qthen LE HZR = HQ because Q ZR ~=Q. If Pattern III holds, then LE HZR = H( p2R\P bZp). And if Pattern IV holds, then LE HZR = HZR\P , where P is the set of all primes p such that HZ=p is not E-acyclic. Note that this case generalizes the cases of the localiz* *ation of HZ (when R = ;) and HQ (when R is the set of all primes). 5.8. Localizations of HZ=p1 . The group Z=p1 is uniquely p-divisible for every prime q 6= p. In this case, LEQ HZ=p1 = 0 since Q Z=p1 = 0 for all p. We have that Hom (Z=p1 , Z=p1 ) = bZp and Ext(Z=p1 , Z=p1 ) = 0. Thus, under Pattern II, LE HZ=p1 = 0. If Pattern III holds, then LE HZ=p1 ' HZbp. And if Pattern IV holds, then by Theorem 4.3 we have a cofiber sequence of spectra LE HZ=p1 - ! HZbp-! HQb- ! H(bQ=bZp), 8 JAVIER J. GUTI'ERREZ where bQ ~=bZp Q are the p-adic rationals. Hence, LE HZ=p1 = H(bQ=bZp) ' HZ=p1 . 5.9. Localization of HZbp. We only have to focus on the prime p, because bZpis uniquely q divisible for all primes q 6= p. In this case, we have that Hom (Z=p1 , bZp) = 0 and Ext(Z=p1 , bZp) = bZp. If Pattern II holds, then LE HZbp= HQbp. And if Pattern III or Pattern IV hold, then LE HZbp= HZbp. The following table summarizes the results obtained for the homological loca* *l- izations of Eilenberg-Mac Lane spectra for different groups. The set P is the s* *et of primes p such that HZ=p is not E-acyclic. __________________________________________________________ | |Pattern A|Pattern B| Pattern C |Pattern D | |____________|________|__________|_____________|__________| | LE HZ | 0 | HQ | Q HZbp | HZP | |___________|_________|_________|____p2P______|__________|_ | LE HZ=pk | 0 | 0 | HZ=pk | HZ=pk | |____________|________|__________|____________|__________|_ | LE HQ | 0 | HQ | 0 | HQ | |___________|_________|_________|______________|________|_ | LE HZR | 0 | HQ | Q HZbp | HZP\R | |___________|_________|_________|___p2P\R______|_________|_ | LE HZ=p1 | 0 | 0 | HZbp | HZ=p1 | |___________|_________|__________|____________|__________|_ | LE HZbp | 0 | HQbp | HZbp | HZbp | |____________|________|_________|_____________|___________| 5.10. Localization of HG where G is a finitely generated abelian group. Every finitely generated abelian group splits as a direct sum G = ni=1Ci where each Ci is either Z or Z=pk for some prime p and k 1. Since HG ' _ni=1HCi, then LE HG = _ni=1LE HCi and the localization of each HCi is determined using the results of sections 5.4 and 5.5. 5.11. Localization of HG where G is a divisible abelian group. If G is a divisible abelian group, then G ~=R T , where R = iQ and TQ= p( jpZ=p1 ). In this case LE HG ' LE HR _ LE HT . Since R is a retract of iQ, we have that LE HR = HR or LE HR = 0 depending on whether HQ is E-local or E-acyclic. The localization LE HT can be determined using the exact sequence of abelian groups 0 -! ZP -! Q -! p2PZ=p1 - ! 0 together with the results of sections 5.6 and 5.8, and the fact that homological localizations preserve cofiber sequences. 6. Localization of reduced Eilenberg-Mac Lane spectra In all the examples we have studied, except in the case of HZ=p1 , all the h* *omo- logical localizations of HG have at most one nonzero homotopy group in dimension zero. This property also holds when the group G is abelian reduced. An abelian group is reduced if it does not have nontrivial divisible subgroups. We say tha* *t the Eilenberg-Mac Lane spectrum HG is reduced if the group G is reduced. Theorem 6.1. If HG is reduced, then LE HG is either zero or HA for some abelian group A and for any spectrum E. Proof. If G is reduced, then Hom (Z=p1 , G) = 0. The result follows now from Theorem 4.3. HOMOLOGICAL LOCALIZATIONS OF EILENBERG-MAC LANE SPECTRA 9 Any abelian group G splits as a direct sum G ~= G1 G2, where G1 is the maximal divisible subgroup of G and G2 is reduced. Morover, G1 splits as a dire* *ct sum of Q's and Z=p1 for several primes p. Hence, by Theorem 4.3 and the results in the previous section, the only possibility for the homological localization * *of an Eilenberg-Mac Lane spectrum HG to have a nonzero homotopy group in dimension one, is that some Z=p1 appears as a factor of the decomposition of G and that LE HZ=p1 = HZbp. Corollary 6.2. If Z=p1 does not occur as a direct summand of G for any prime p, then LE HG is either zero or HA for some abelian group A and any spectrum E. References [Ada73]J. F. Adams. Mathematical Lectures. University of Chicago, 1973. [Ada74]J. F. Adams. Stable Homotopy and Generalised Homology. Chicago Lectures * *in Mathe- matics. University of Chicago Press, 1974. [Bou74]A. K. Bousfield. Types of acyclicity. J. Pure Appl. Algebra, 4:293-298, * *1974. [Bou75]A. K. Bousfield. The localization of spaces with respect to homology. To* *pology, 14:133- 150, 1975. [Bou79a]A. K. Bousfield. The Boolean algebra of spectra. Comment. Math. Helv., * *54(3):368-377, 1979. [Bou79b]A. K. Bousfield. 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Departament d'`Algebra i Geometria, Universitat de Barcelona, Gran Via, 585, E-08007 Barcelona, Spain E-mail address: javier.gutierrez@ub.edu