HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA
CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
Abstract.We prove that stable homotopical localizations preserve ring sp*
*ec-
trum structures and module spectrum structures under suitable hypotheses,
and we use this fact to describe all possible localizations of the integ*
*ral Eilen-
berg-Mac Lane spectrum HZ . More generally, we describe the main feature*
*s of
localizations of HZ-modules (i.e., stable GEMs), motivated by similar re*
*sults
in unstable homotopy.
1.Introduction
A stable GEM is a graded Eilenberg-Mac Lane spectrum, i.e., a wedge
`
kHAk,
k2Z
where each Ak is an abelian group and HAk denotes a spectrum with ß0(HAk) ~=
Ak and ßi(HAk) = 0 if i 6= 0. A spectrum X is a stable GEM if and only if it
admits an HZ-module structure. In other words, X is a stable GEM if and only
if X is a homotopy retract of HZ ^ X. Note the analogy with unstable homotopy,
where a space X is a GEM (i.e. a weak product of abelian Eilenberg-Mac Lane
spaces) if and only if it is a homotopy retract of the infinite symmetric produ*
*ct
SP1 X, a space whose homotopy groups are the integral homology groups of X.
Farjoun and others [Bad01 ], [Far96] have shown that unstable homotopical lo-
calizations preserve GEMs. In this article we prove that the same result is tru*
*e in
stable homotopy, by developing further certain ideas used by Bousfield in [Bou9*
*6 ]
and [Bou99 ].
We work in the Bousfield-Friedlander category of spectra [BF78 ]. Localization
with respect to a map of spectra f :A -! B is a homotopy idempotent functor Lf
on the stable homotopy category taking values in the full subcategory of spectr*
*a X
such that the map of connective function spectra induced by f,
F c(B, X) -! F c(A, X),
is a homotopy equivalence. All known forms of stable localizations are f-locali*
*zations
for suitable choices of the map f. Among these, the classical Bousfield locali*
*za-
tions (that is, homological localizations) commute with suspension. Not all f-
localizations have this property; we give necessary and sufficient conditions f*
*or a
localization to commute with suspension.
We prove that if f is any map of spectra and E is a ring spectrum (in the hom*
*o-
topical sense), then LfE is a ring spectrum and the localization map E -! LfE
is a ring map, if we either assume that E is connective or that Lf commutes with
____________
1991 Mathematics Subject Classification. 55P42, 55P43, 55P60.
Key words and phrases. Localization, ring spectrum, module spectrum, stable *
*GEM.
The authors were supported by DGES grants PB97-0202 and FP98 16587447.
1
2 CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
suspension. Similarly, if M is an E-module spectrum, then the localization map
M -! LfM is an E-module map, provided that E is connective or Lf commutes
with suspension.
We believe that a stronger result holds, namely that strict (not just up to h*
*o-
motopy) ring spectrum structures and module spectrum structures are preserved
by homotopy idempotent functors, if one works in monoidal categories of spectra,
such as S-modules [EKMM97 ] or symmetric spectra [HSS00 ]. We plan to address
this question in subsequent work.
It follows from the aforementioned results that f-localizations send HZ-modul*
*es
to HZ-modules; that is, the class of stable GEMs is preserved by f-localization*
*s.
We show that, in fact, for every abelian group G,
LfHG ' HA _ HB
for certain abelian groups A and B. When G = Z, we show that B = 0, and the
group A admits a ring structure with unit, for which Hom (A, A) ~=A via evaluat*
*ion
at the unit. Rings A with this property were called "rigid" in [CRT00 ]. There *
*is a
proper class of nonisomorphic rigid rings, and for every rigid ring A there is *
*a map
f such LfHZ ' HA.
As special cases, we show that if E = K (complex K-theory) or E = E(n)
(the Johnson-Wilson spectrum) for some n, then the E*-localization of the sphere
spectrum has the following homology groups:
H0(LE S) ~=Q, Hi(LE S) = 0 for i 6=.0
A more conceptual explanation of the fact that, for every abelian group G, the
spectrum LfHG has at most two nonzero homotopy groups is that the homotopy
category of HZ-modules is equivalent to the homotopy category of (Z-graded) cha*
*in
complexes of abelian groups. We do not give a reference for this fact in the ar*
*ticle,
but give instead an argument to prove it. The appropriate context to discuss
such equivalences of categories is again the theory of structured ring spectra,*
* as
in [EKMM97 ] or [HSS00 ]. Note, however, the distinction between homotopy HR-
modules and strict HR-modules in such categories. For certain rings, including
R = Z, the corresponding homotopy categories are equivalent, but in general they
are not. This aspect will also be discussed in more detail elsewhere.
Acknowledgements. Conversations with Gustavo Granja, John Greenlees, Stefan
Schwede and Neil Strickland were greatly helpful to us.
2.Localization of spectra with respect to a map
The notion of homotopical localization with respect to a map can be formulated
in any category C with a simplicial model structure, or in even more general mo*
*del
categories; see [GJ99 ] or [Hir00] for further details about methods and termin*
*ology.
Given a map in a simplicial model category C, pick a cofibration f :A -! B
between cofibrant objects in its homotopy class. Then an object X is called f-l*
*ocal
if X is fibrant and the induced fibration of simplicial sets
HOM (B, X) -! HOM (A, X)
is a weak equivalence. We state the following standard properties for later use.
Lemma 2.1. Any homotopy retract of an f-local object is f-local.
HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA 3
Proof.If X is f-local and Y -! X has a left homotopy inverse, then the map
HOM (B, Y ) -! HOM (A, Y )
induced by f is a homotopy retract of the corresponding map for X, and hence_it
is a weak equivalence. |__|
Lemma 2.2. Every homotopy limit of f-local objects is f-local.
Proof.If I is a small category and D :I -! C is a diagram in C taking values in
f-local objects, then
HOM (B, holimID) ~=holimIHOM (B, D)
' holimIHOM (A, D) ~=HOM (A, holimID).
Details about the fact that HOM commutes with holim in any simplicial model_
category can be found in [Hir00, Ch. 19]. |__|
A map g :X -! Y is an f-equivalence if there is a cofibrant approximation
~g:~X-! ~Ysuch that the induced fibration of simplicial sets
HOM (Y~, E) -! HOM (X~, E)
is a weak equivalence for every f-local object E. An f-localization of an obje*
*ct
X is an f-equivalence l :X -! LfX where LfX is f-local. This map l is initial
in the homotopy category HoC among maps from X to f-local objects, and it is
terminal in HoC among f-equivalences with domain X. Each of these two universal
properties ensures that, if an f-localization of X exists, then it is unique up*
* to
homotopy.
The existence of f-localization for every map f and all objects X is guarante*
*ed
if the simplicial model category C satisfies certain additional assumptions. Sp*
*ecifi-
cally, the following result is proved as indicated in [Bou77 ] or in [Hir00]. A*
*n object
X is called ~-small, where ~ is an infinite cardinal, if every morphism from X *
*to
the direct limit of a sequence of cofibrations indexed by a limit ordinal great*
*er than
or equal to ~ factors through some object in the sequence.
Theorem 2.3. Let C be a cofibrantly generated simplicial model category. Suppo*
*se
that every object X of C is ~-small for some infinite cardinal ~ (which may dep*
*end
on X). Then f-localization exists for every map f in C. Moreover, Lf can be __
constructed as a functor in C which is idempotent up to homotopy. |_*
*_|
The Bousfield-Friedlander category of spectra [BF78 ] satisfies the assumptio*
*ns
stated in Theorem 2.3. In this category, HOM (X, Y ) is the simplicial set wh*
*ose
n-simplices are the maps X ^ [n]+ -! Y of spectra, and for X cofibrant and Y
fibrant, one has
ßk(HOM (X, Y )) ~=ßk(F (X, Y )) for k 0,
where F (X, Y ) denotes the function spectrum from X to Y . Therefore, the ho-
motopy groups of the simplicial set HOM (X, Y ) are isomorphic to those of the
connective cover F c(X, Y ) of the function spectrum.
Hence, for a map f :A -! B, a spectrum Y is f-local if and only if it is fibr*
*ant
and the induced map of connective covers of function spectra
F c(B, Y ) -! F c(A, Y )
induces isomorphisms of all homotopy groups.
4 CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
Proposition 2.4.Let f be any map.
(a) If E is f-local, then -kE is also f-local for k 0.
(b) If g :X ! Y is an f-equivalence, then kg is also an f-equivalence for k *
* 0.
Proof.If E is f-local, then f induces a homotopy equivalence
F c(B, -kE) ' F c(A, -kE) for k 0,
since ßi(F c(B, -kE)) ~=ßi+k(F c(B, E)) if i 0 and k 0. The proof of_part *
*(b)
is similar. |__|
Since F c(B, E) ' F c( kB, kE) for all B, E and k 2 Z, we may also infer tha*
*t if
E is f-local then kE is kf-local for every k 2 Z, and similarly for f-equival*
*ences.
From this fact we deduce the following result.
Proposition 2.5.For every map of spectra f and every spectrum X we have a
homotopy equivalence
Lf -kX ' -kL kfX for all k 2.Z
Proof.Since l :X -! L kfX is a kf-equivalence, -kl is an f-equivalence.
Moreover, -kL kfX is f-local since L kfX is kf-local, so our claim follows. *
*|___|
This is to be compared with the expression Lf kX ' kL kfX for spaces,
which was proved in [Far96].
Localization with respect to a map of the form f :A -! * is called A-nullific*
*a-
tion, and it is denoted by PA instead of Lf. The corresponding local spectra are
called A-null. Thus, a spectrum X is A-null if and only if F c(A, X) ' *.
For every map f, there is a natural transformation PC -! Lf, where C is the
cofibre of f. This follows from the fact that every f-local spectrum is C-null.
As a consequence of Proposition 2.4, for every spectrum X there is a natural
map
LfX -! Lf X.
We say that Lf commutes with suspension if this natural map is a homotopy
equivalence for all X.
Theorem 2.6. Let f :A -! B be a map of spectra. Then the following statements
are equivalent:
(i)Lf commutes with suspension.
(ii) LfX ' Lf X for every spectrum X.
(iii) kLfX ' Lf kX for every spectrum X and every k 2 Z.
(iv)If E is any f-local spectrum, then kE is also f-local for any k 2 Z.
(v) The map F (B, E) -! F (A, E) induced by f is a homotopy equivalence for
every f-local spectrum E.
(vi)If E is f-local and X is any spectrum, then F (X, E) is f-local.
(vii)If g :X -! Y and h: M -! N are arbitrary f-equivalences, then the map
g ^ h: X ^ M -! Y ^ N is also an f-equivalence.
(viii)If g is an f-equivalence, then kg is also an f-equivalence for all k 2 Z.
(ix)LfX ' L kfX for every spectrum X and every k 2 Z.
(x) If X ! Y ! Z is any cofiber sequence of spectra, then LfX ! LfY ! LfZ
is also a cofiber sequence.
HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA 5
Proof.The implications (i) ) (ii) ) (iii) ) (iv) are trivial. Statement (v) is
equivalent to the fact that f induces homotopy equivalences
F c(B, kE) ' F c(A, kE)
for all k 2 Z, and hence it follows from (iv). To prove (vi), we have to verify*
* that
F c(B, F (X, E)) ' F c(A, F (X, E)),
which is equivalent to
F c(X, F (B, E)) ' F c(X, F (A, E)),
and this follows from (v). Statement (vii) is proved from (vi) by taking any f-*
*local
spectrum E and observing that
F c(Y ^ N, E) ' F c(Y, F (N, E)) ' F c(X, F (N, E))
' F c(N, F (X, E)) ' F c(M, F (X, E)) ' F c(X ^ M, E).
Statement (viii) follows from (vii) by smashing g with the identity of kS, for
k 2 Z. The claim made in (ix) is equivalent to (viii). We next deduce (x) from *
*(ix).
Given a cofibre sequence X -! Y -! Z, let C be the cofibre of Lf -1Z -! LfX.
Thus, we have a cofibre sequence
LfX -! C -! Lf -1Z
and Lf -1Z ' L fZ ' LfZ, by Proposition 2.5 and by our assumption (ix).
As in Theorem 2.2 of [Bou96 ], there is a map Y -! C which is an f-equivalence,
and C is f-local. Hence, C ' LfY . Finally, to prove that (x) implies (i),_pic*
*k the
cofibre sequence X -! * -! X, for any spectrum X. |__|
Statement (v) tells us that f-localization commutes with suspension if and on*
*ly
if f-local spaces may be defined by means of the full function spectrum F inste*
*ad
of its connective cover F c. We also emphasize that, by the following observati*
*on,
every f-localization which commutes with suspension is a nullification.
Corollary 2.7.If a localization functor Lf commutes with suspension, then the
natural transformation PC -! Lf, where C is the cofiber of f, is a homotopy
equivalence.
Proof.There are natural transformations L f -! PC -! Lf which correspond to
inclusions of the respective classes of local spectra. Since the composite L f *
*-! Lf
is an equivalence by part (ix) of Theorem 2.6, the arrow PC - ! Lf is also_an
equivalence. |__|
3. Examples of f-localizations
We discuss three examples of localizations in the stable homotopy category,
namely Postnikov sections, homological localizations and localizations at sets *
*of
primes. These examples serve to illustrate certain features of f-localizations *
*that
we wish to emphasize.
6 CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
3.0.1. Postnikov sections. Localization of a spectrum E with respect to the map
f : k+1S -! *, where S is the sphere spectrum and k 2 Z, is homotopy equivalent
to the k-th Postnikov section of E; that is, P k+1SE ' E(k), where ßi(E(k)) = 0
for i > k and there is a map E -! E(k)that induces isomorphisms of homotopy
groups in dimension less than or equal to k.
Postnikov sections do not commute with suspension. Indeed, if ßk(E) 6= 0, then
P k+1S E 6' P k+1SE.
3.0.2. Homological localizations. Homological localizations in stable homotopy *
*were
first discussed by Bousfield in [Bou79 ]. Let E be any spectrum. A spectrum X
is called E*-acyclic if Ek(X) = 0 for all k 2 Z or, equivalently, if E ^ X ' *.*
* A
map of spectra g :X -! Y is an E*-equivalence if it induces an isomorphism in
E-homology, i.e., if the map g*: Ek(X) -! Ek(Y ) is an isomorphism for all k 2 *
*Z.
A spectrum Z is E*-local if each E*-equivalence f :X -! Y induces a homotopy
equivalence F (Y, Z) ' F (X, Z) or, equivalently, if F (A, Z) ' * for each E*-a*
*cyclic
spectrum A.
An E*-localization of a spectrum X is an E*-equivalence X -! LE X from X to
an E*-local spectrum. Each E*-localization is an f-localization for a suitable *
*map
f. In fact, it is a nullification, as shown in [Bou79 ].
Theorem 3.1. Let E be any spectrum. Then there exists an E*-acyclic spectrum __
Z such that there is a natural equivalence PZX ' LE X for every spectrum X. |_*
*_|
Homological localizations commute with suspension.
3.0.3. Localization at sets of primes. Let G be any abelian group and let MG de*
*note
its associated Moore spectrum. Thus, MG is a spectrum such that (HZ)0(MG) ~=
ß0(MG) ~=G, ßi(MG) = 0 if i < 0, and (HZ)i(MG) = 0 if i 6= 0.
Lemma 3.2. There is a natural exact sequence
0 -! Ext(G, ßk+1(X)) -! [ kMG, X] -! Hom (G, ßk(X)) -! 0
for each spectrum X and each abelian group G, where MG is the Moore spectrum
associated with G.
Proof.Pick a free abelian presentation of the group G and use the associated co*
*fibre_
sequence of Moore spectra; cf. [Bou79 , (2.2)]. |_*
*_|
Let ZP denote the integers localized at a set of primes P (possibly empty). F*
*or
every spectrum X, the P -localization of X is the map
1 ^ j :X ^ S -! X ^ MZP
where j is given by the unit in ß0(MZP ) = ZP . If we denote XP = X ^ MZP ,
then for each k 2 Z we have
ßk(XP ) ~=ßk(X) ß0(MZP ) ~=ßk(X) ZP .
From this fact it follows that, if E is any spectrum, then
Ek(XP ) ~=ßk(E ^ X ^ MZP ) ~=Ek(X) ZP .
Hence P -localization of spectra induces P -localization of their homotopy and *
*ho-
mology groups. On the other hand, about cohomology, Lemma 3.2 yields
Ek(XP ) ~=[MZP , F (X, kE)] ~=Hom (ZP , Ek(X)) Ext(ZP , Ek-1(X)).
HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA 7
It follows from the definition that P -localization is a homological localizati*
*on,
namely (MZP )*-localization. Hence, P -localization commutes with suspension.
We also emphasize that
XP ' X ^ SP ,
for all X; that is, P -localization is smashing, as defined in [Rav84 ].
An explicit map f such that LfX ' XP for all X can be displayed as follows.
Theorem 3.3. Let P be a set of primes and let g :_q62PS -! _q62PS be a wedge
of maps inducing multiplication by q in ß0(S) for each prime q not in P , and l*
*et
f = _n<0 ng. Then LfX ' X ^ MZP for all X.
Proof.The map f has been chosen so that f-local spectra are precisely those spe*
*ctra
whose homotopy groups are ZP -modules. Thus, the map j :S -! MZP is an f-
localization, since MZP is f-local and the map j is an f-equivalence because, by
Lemma 3.2, [ kMZP , Y ] -! ßk(Y ) is an isomorphism when ßk(Y ) is a ZP -module.
By part (vii) of Theorem 2.6, the map
1 ^ j :X ^ S -! X ^ MZP
is also an f-equivalence, and our claim follows. |_*
*__|
4. Localizations of Ring Spectra and Module Spectra
We recall the definition of ring spectra and module spectra in the homotopical
sense, as in [Ada74 ]. A spectrum E is called a ring spectrum if it is equipped
with two maps ~: E ^ E -! E and j :S -! E such that the following diagrams
commute up to homotopy:
~^1 j^1 1^j
E ^ E ^ E_____//E ^ E S ^ E_____//JE ^oEo__E ^ S
JJ | ttt
1^~|| |~| JJJJJ |~ tttt
fflffl| fflffl| J%%Jfflffl|yyttt
E ^ E___~_____//E E.
It is said that E is commutative if ~ O ø ' ~, where ø :E ^ E -! E ^ E is the
twist map. A spectrum M is called a module spectrum over a ring spectrum E or
an E-mo dule if it is equipped with a map m: E ^ M -! M such that the following
diagrams commute up to homotopy:
~^1 j^1
E ^ E ^ M _____//E ^ M S ^ MH_____________//E ^ M
HH vvv
1^m || |m| HHHHH vvmvv
fflffl| fflffl| H## zzvv
E ^ M ___m____//_M M.
Every ring spectrum E is an E-module spectrum with m = ~. Note also that, if
M is an E-module, then it is a homotopy retract of E ^ M.
A ring map between ring spectra (E, ~, j) and (E0, ~0, j0) is a map f :E -! E0
such that f O ~ ' ~0O (f ^ f) and f O j ' j0. An E-module map is defined simila*
*rly.
If R is an associative ring with unit and M is an R-module, then the Eilenber*
*g-
Mac Lane spectrum HR is a ring spectrum, and HM is a module spectrum over
HR. The structure maps on HR and HM come from the product R R -!
R and the unit Z -! R in the ring R, and from the structure homomorphism
8 CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
R M -! M of M as an R-module. For every ring R, the spectrum HR is an HZ-
module. Moreover, every HR-module is an HZ-module via the map HZ -! HR
corresponding to the unit Z -! R.
Remark 4.1.If M is an E-module spectrum, then, for every spectrum X, the
graded abelian group [X, M]* is a ß*(E)-module, as follows. For every map ff 2
ßi(E) and every map f 2 [X, M]j we obtain another map in [X, M]i+jby smashing
ff with f and composing with the structure map of M:
i+jX ' iS ^ jX ff^f-!E ^ M -m! M.
In particular, if M is an HR-module spectrum, then ßn(M) is an R-module for
every n.
As we next prove, in the case of f-localization functors that commute with su*
*s-
pension, the f-localizations of ring spectra or module spectra acquire a compat*
*ible
ring structure or module structure. In the rest of this chapter, we assume that*
* f is
a fixed map of spectra, and we write L instead of Lf.
Theorem 4.2. Let f :A -! B be any map of spectra. If the f-localization functor
L commutes with suspension, then the following hold:
(i)If E is a ring spectrum, then the spectrum LE has a unique ring spectrum
structure such that the localization map lE :E -! LE is a ring map. If E is
commutative, then LE is also commutative.
(ii)If M is an E-module, then the spectrum LM has a unique E-module struc-
ture such that the localization map lM :M - ! LM is an E-module map.
Moreover, LM admits a unique LE-module structure extending the E-module
structure.
Proof.For the first part we need to construct a product __~and a unit __jon LE.*
* Let
~ and j be the product and the unit of the ring spectrum E, respectively. We ha*
*ve
an equivalence F (E, LE) ' F (LE, LE) because LE is f-local and the functor L
commutes with suspension by assumption. Then,
[E ^ E, LE] ~=[E, F (E, LE)] ~=[E, F (LE, LE)] ~=[E ^ LE, LE]
~=[LE, F (E, LE)] ~=[LE, F (LE, LE)] ~=[LE ^ LE, LE].
Hence, the product ~: E ^ E -! E extends to a unique map __~:LE ^ LE -! LE
rendering homotopy commutative the diagram
~
E ^ E_______//E
lE^lE|| lE||
fflffl| fflffl|
LE ^ LE ___~//____________LE.
We define the unit __jas the composition lE O j. The commutativity of the diagr*
*ams
for __~and __jfollows from the commutativity of the diagrams for ~ and j and the
universal property of L (using part (vii) of Theorem 2.6).
The commutativity of LE when E is commutative and the statements in part_(ii)
are proved in the same way. |__|
As we next show, localization functors not commuting with suspension need not
preserve ring structures nor module structures in general. The following lemma *
*is
HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA 9
useful to prove that certain spectra fail to be ring spectra or module spectra.*
* The
idea is due to Rudyak [Rud98 , Ch. II, 4.31].
Lemma 4.3. Let E and F be ring spectra, and let M be an E-module spectrum.
If F0(E) = 0, then Fk(M) = 0 for all k 2 Z.
Proof.The diagram
ß0(E) OFk(M)O_h_1_//F0(E) Fk(M)
(jE)* 1|| |m*|
| fflffl|
ß0(S) Fk(M)____~=___//Fk(M)
is commutative, where (jE )* is induced by the unit of the ring spectrum E and *
*h is
the Hurewicz homomorphism of F . The map m* is induced by the multiplication of
M ^ F as an (E ^ F )-module. So, if F0(E) = 0, then the bottom row isomorphism_
factors through zero and hence Fk(M) = 0 for all k 2 Z. |__|
Example 4.4.Given a natural number n and a fixed prime p, let K(n) denote
the ring spectrum corresponding to n-th Morava K-theory. If we consider its
nullification P SK(n), then, according to [Rud98 ], (HZ=p)0(P SK(n)) = 0 yet
(HZ=p)k(P SK(n)) 6= 0 for some k > 0. This implies that P SK(n) cannot be
a ring spectrum, by Lemma 4.3. The same argument, now considering K(n) as
a K(n)-module and using that (HZ=p)0(K(n)) = 0 and Lemma 4.3, shows that
P SK(n) is not a K(n)-module.
This difficulty can be repaired by imposing suitable connectivity conditions.*
* The
following result extends an observation made by Bousfield in [Bou99 ].
Theorem 4.5. Let f :A -! B be any map of spectra and let L be f-localization.
Then the following hold:
(i)If E is a connective ring spectrum and LE is connective, then the spectrum
LE has a unique ring structure such that the localization map lE :E -! LE
is a ring map. If E is commutative, then LE is also commutative.
(ii)If M is an E-module, where E is a connective ring spectrum, then LM has
a unique E-module structure such that the localization map lM :M -! LM
is an E-module map. Moreover, if LE is connective, then LM also admits a
unique LE-module structure extending the E-module structure.
Proof.Using that E is a connective spectrum, we have equivalences
F c(E, F c(X, Y )) ' F c(E, F (X, Y )) ' F c(E ^ X, Y )
that give a bijection [E, F c(X, Y )] ~=[E ^ X, Y ]. Then one proceeds as_in th*
*e proof
of Theorem 4.2. |__|
5.Localization of Stable GEMs
As we next recall, the stable GEMs are precisely the HZ-modules. Thus, we
may use our results in the previous section to prove that every f-localization *
*sends
stable GEMs to stable GEMs.
10 CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
Definition 5.1.Let R be a ring. A spectrum E is called a stable R-GEM if it
is homotopy equivalent to a wedge of suspensions of Eilenberg-Mac Lane spectra
_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 stable GEMs.
For a ring spectrum E, let HosE-moddenote the subcategory of the stable homo-
topy category Hos whose objects are the E-module spectra and whose morphisms
are (ordinary) homotopy classes of E-module maps. If M and N are E-module
spectra, let [M, N]E-mod [M, N] denote the set of morphisms M -! N in this
subcategory.
If E is a ring spectrum, then for every spectrum X the smash product E ^ X
has an E-module structure given by E ^ E ^ X ~^1-!E ^ X. We are indebted to
Gustavo Granja for pointing out the following fact to us.
Lemma 5.2. Let E be any ring spectrum. Then the functor Ho s-! Ho sE-mod
assigning to every spectrum X the spectrum E ^ X is left adjoint to the forgetf*
*ul
functor HosE-mod-! Hos. That is, for every spectrum X and every E-module M,
there is a natural isomorphism
[X, M] ~=[E ^ X, M]E-mod
induced by the unit of E.
Proof.Let M be an E-module, X a spectrum, and f :X -! M any map. We are
going to show that there is a homotopy unique E-module map ~f:E ^ X -! M
such that the diagram
f
S ^ X ' X _____//M99_____
_____
j^1 || __________
fflffl|~f______
E ^ X
commutes up to homotopy. The following diagram
1^f
S ^ X_____//S ^HM
| HH
j^1|| |j^1HH'HHH
fflffl| fflffl| H##
E ^ X _1^f_//E ^ M__m__//M
commutes, so if we define ~f= m O (1 ^ f), then it satisfies ~fO (j ^ 1) ' f.
The map ~fis a map of E-modules, since the following diagram is commutative:
1^(1^f) 1^mM
E ^ (E ^ X)_____//E ^ (E ^ M)____//E ^ M
'|||| ' |||| ' ||||
|| (1^1)^f || ~E^1 ||
(E ^ E) ^ X_____//(E ^ E) ^ M____//E ^ M
~E^1|| ~E || mM ||
fflffl| fflffl| fflffl|
E ^ X ___1^f____//E ^ M__mM____//M.
HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA 11
If there exists another E-module map g satisfying g O (j ^ 1) ' f, then g ' ~*
*f,
because the following diagram also commutes:
______________________
GF E D_
|| E ^ (S ^ X)__'__//E ^ X
|| 1^(j^1)|| |1^f|
1|| fflffl|1^g fflffl|
| E ^ (E ^ X)_____//E ^ M
| | |
| 1^~| |m
@A| fflffl| fflffl|
______//E ^ X___g____//M.
The upper square commutes by hypothesis and the lower square commutes because__
g is a map of E-modules, so g ' m O (1 ^ f) ' ~f. |__|
Proposition 5.3.For every HZ-module M, there is a map of HZ-modules
HZ ^ (_k2Z kMGk) -~ff!M
which is a homotopy equivalence, where Gk = ßk(M).
Proof.The argument was sketched in [Ada74 , p. 307]. For each k 2 Z, take one
map ffk 2 [ kMGk, M] mapping to the identity in Hom (Gk, Gk). Let ff = _k2Zffk
be the wedge of all ffk constructed in this way. This yields, by Lemma 5.2, an
HZ-module map ~ff:HZ ^ (_k2Z kMGk) -! M, namely ~ff= m O (1 ^ ff). This
map ~ffinduces an isomorphism on ßk for all k 2 Z, because j ^ 1 and 1 ^ ff are_
isomorphisms on ßk. So ~ffis a homotopy equivalence. |__|
Proposition 5.3 tells us that HZ-module spectra and stable GEMs are the same
thing, because HZ ^ MG ' HG for any abelian group G and therefore, if M is an
HZ-module, then M ' _k2Z kHßk(M). Similarly, the HR-modules are precisely
the stable R-GEMs because each HR-module spectrum is an HZ-module spectrum,
and the homotopy groups of HR-module spectra are R-modules (see Remark 4.1).
Corollary 5.4.Let A and B be abelian groups. Then
[HA, HB]HZ-mod ~=Hom (A, B),
[HA, HB]HZ-mod ~=Ext(A, B), and
[HA, kHB]HZ-mod = 0 if k 6= 0,.1
Proof.As a special case of Proposition 5.3, HA ' HZ ^ MA as HZ-modules. By
Lemma 5.2, there is a natural bijection
[MA, kHB] ~=[HZ ^ MA, kHB]HZ-mod
so the result follows directly from Lemma 3.2. |___|
In the rest of this section, L denotes f-localization with respect to a fixed*
* but
arbitrary map f. Using the results on f-localizations of E-modules of Section 4,
we have the following.
Theorem 5.5. Let R be any associative ring with unit. If E is a stable R-GEM,
then LE is also a stable R-GEM and the localization map lE :E -! LE is an
HR-module map.
12 CARLES CASACUBERTA AND JAVIER J. GUTI'ERREZ
Proof.A stable R-GEM is the same as an HR-module, and HR is a connective__
spectrum. Hence we may apply Theorem 4.5. |__|
In other words, if a spectrum E is homotopy equivalent to _i2Z iHAi where
Ai is an R-module for each i 2 Z, then LE ' _i2Z iHGi where each Gi is an
R-module as well.
Next, we are going to study the case when the spectrum E is a suspension of a
single Eilenberg-Mac Lane spectrum, i.e., E ' nHG where G is an R-module. By
Theorem 5.5 we know that L nHG ' _k2Z kHGk with each Gk an R-module. In
fact, most of the R-modules Gk are zero, as we next explain. Consider the follo*
*wing
sequence of HZ-module maps, where fi is a homotopy inverse of the map given by
Proposition 5.3, and pi is the projection onto the i-th factor:
fi pi
nHG ___l_//L nHG _____//_k2Z kHGk ____//_ iHGi
By Corollary 5.4, [ nHG, iHGi]HZ-mod = 0 unless i = n or i = n + 1. The
universal property of localization and the fact that iHGi is f-local because i*
*t is
a retract of _k2Z kHGk (see Lemma 2.1) tell us that Gi= 0 if i 6= n or i 6= n +*
* 1.
What we get is that the localization of any suspension of an Eilenberg-Mac La*
*ne
spectrum has at most two nonzero homotopy groups.
Theorem 5.6. Let G be any abelian group and n 2 Z. Then L nHG ' nHG1_
n+1HG2 as HZ-modules for some abelian groups G1, G2. If G is an R-module __
for some ring R, then G1 and G2 are also R-modules. |__|
There are some special cases in which the localization of an Eilenberg-Mac La*
*ne
spectrum is a single Eilenberg-Mac Lane spectrum.
Theorem 5.7. If G is a free abelian group and n is an integer, then L nHG '
nHA for some abelian group A.
Proof.From Theorem 5.6, we know that L nHG has at most two nonzero homo-
topy groups, A and B. By Corollary 5.2,
[ nHG, n+1HB]HZ-mod ~=Ext(G, B).
If G is free, then Ext(G, B) = 0, and this tells us that the projection nHG -!
n+1HB is nullhomotopic. Moreover, n+1HB is f-local, because it is a retract __
of L nHG. The universal property of localization forces that B = 0. |_*
*_|
If we now project L nHG onto the first summand
nHG Q_l__// nHA _ n+1HB
QQQ
QQQQ |p1
p1OlQQQ((QQfflffl||
nHA
we can obtain information about the group A. This diagram yields an isomorphism
of abelian groups
[HA, HA] x [ HB, HA] ~=[HG, HA],
and [ HB, HA] ~=(HA)0( HB) ~=Hom (ß0( HB), A) = 0. Hence we get
Hom (A, A) ~=[HG, HA] ~=Hom (G, A).
In the case when G = Z, this says that Hom (A, A) ~=A. Therefore, if G = Z and
A is nonzero, then A admits a ring structure, with a multiplication coming from
HOMOTOPICAL LOCALIZATIONS OF MODULE SPECTRA 13
composition in Hom (A, A) and a unit coming from the identity homomorphism.
Moreover, the isomorphism Hom (A, A) ~=A is given by evaluation at the unit.
Definition 5.8.A ring A with unit such that Hom (A, A) ~=A via ' 7! '(1) is
called a rigid ring.
This terminology was first used in [CRT00 ]. However, rigid rings had previou*
*sly
been studied in a different context, under the name of E-rings. The most obvious
examples of such rings are Z, Q, Z=p, or the p-adics ^Zp, for any p. There are *
*many
other examples. In fact, as shown in [DMV87 ], there are rigid rings of arbitr*
*arily
large cardinality.
Rigid rings are commutative. Solid rings in the sense of [BK72] are rigid rin*
*gs,
but not conversely (in fact, solid rings are countable). Proofs of these claims*
* and
further details about rigid rings can be found in [CRT00].
All rigid rings occur as homotopy groups of localizations of HZ, since, if A *
*is
any rigid ring, then LfHZ ' HA where f is the map HZ -! HA induced by the
unit homomorphism Z -! A.
We can summarize the results obtained for f-localizations of HZ in the follow*
*ing
theorem.
Theorem 5.9. Let f be any map of spectra. Then the f-localization of the spec-
trum HZ has at most one nonzero homotopy group, i.e., LfHZ ' HA. Moreover, *
* __
the group A has a rigid ring structure if A 6= 0. All rigid rings appear this w*
*ay. |__|
We conclude with an example. As already mentioned above, a localization L is
called smashing if LX ' X ^ LS for all spectra X, where S is the sphere spec-
trum. It follows from this definition that every smashing localization is homol*
*ogical
(namely, L ' LE , where E = LS) and hence it commutes with suspension. More-
over, LS is a commutative ring spectrum, by Theorem 4.3.
As shown in [Rav84 , 1.27], a homological localization LE is smashing if and *
*only
if it commutes with direct limits. This happens, for example, if E is the spect*
*rum
K of (complex) K-theory or the Johnson-Wilson spectrum E(n) for any n.
Theorem 5.10. If L is smashing, then Hn(LS) = 0 if n 6= 0, and it is either ze*
*ro
or a rigid ring if n = 0.
Proof.We have that
Hn(LS) = ßn(HZ ^ LS) ~=ßn(LHZ) ~=ßn(HA)
for some rigid ring A, by Theorem 5.9. |___|
The ring A happens to be Q if L is localization with respect to E = K or
E = E(n) for any n. In each of these cases, the spectrum HQ is E*-local,
since it is a retract of E ^ MQ. Hence, it suffices to show that the natural
map HZ - ! HQ is an E*-equivalence. For this, we may use the fact that
Ek(HZ) = limiEk+i(K(Z, i)). Then, for E = K, our claim follows from [AH68 ].
For E = E(n), it is a consequence of [Bou82 , Example 7.5].
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Departament d'`Algebra i Geometria, Universitat de Barcelona, Gran Via, 585,
E-08007 Barcelona, Spain
E-mail address: casac@mat.ub.es
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193
Bellaterra, Spain
E-mail address: jgutierr@mat.uab.es