LOCALIZATIONS OF ABELIAN
EILENBERG-MAC LANE SPACES OF FINITE TYPE
By Carles Casacuberta, Jose L. Rodriguez and Jin-Yen Tai
Abstract. Using recent techniques of unstable localization, we extend ear*
*lier results
on homological localizations of Eilenberg-Mac Lane spaces, and show that *
*several
deep properties of such localizations can be explained by the preservatio*
*n of certain
algebraic structures under the effect of idempotent functors.
We study localizations LfK(G; n) of Eilenberg-Mac Lane spaces with resp*
*ect to
any map f, where n 1 and G is abelian. We find that, if G is finitely ge*
*nerated,
then the result is a K(A; n), where A can be computed using cohomological*
* data
derived from f. If G = Z, then A is a commutative ring which is isomorphi*
*c to the
ring End(A) of its own additive endomorphisms; such rings, which we call *
*rigid, form
a proper class which contains the set of solid rings. From this fact it f*
*ollows that there
is a proper class of distinct homotopical localizations of the circle S1.*
* Among other
applications of our results, we show that, if X is a product of abelian E*
*ilenberg-Mac
Lane spaces and f is any map, then the homotopy groups ssm (LfX) become m*
*odules
over the ring ss1(LfS1).
0. Introduction
We refer the reader to [5] and [15] for the essentials of homotopical locali*
*zation.
Given any map f: W ! V between CW-complexes, for each space X there is a map
j: X ! LfX where LfX is universal in the homotopy category with the property
that the map of function spaces
map (V; LfX) ! map (W; LfX)
induced by f is a weak homotopy equivalence. The space LfX is called the
f-localizationof X.
As explained in [15, 4.B], if f is any map, G is any abelian group, and n 1,
then the f-localization of an Eilenberg-Mac Lane space K(G; n) has at most two
nontrivial homotopy groups, and it is in fact a product
K(A; n) x K(B; n + 1):
______________
This article was begun during a stay of the authors at The Fields Institute *
*in Toronto. The
two first-named authors were partially supported by the DGR under grants 1995BE*
*AI400083 and
1996BEAI200187 and the DGICYT under grant PB94-0725.
Typeset by AM S-T*
*EX
1
2 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Earlier results on homological localizations of Eilenberg-Mac Lane spaces [4] a*
*re
special cases of this fact. More recently, the articles [21] and [31] describe*
* other
approaches related to homological localizations of Eilenberg-Mac Lane spaces.
In the present article, for any map f: W ! V , we firstly compute the homoto*
*py
groups A and B of the f-localization LfK(G; n) in terms of f when G is finitely
generated; in fact, in this case it happens that the group B is necessarily zer*
*o.
Since localization commutes with finite direct products, it suffices to cons*
*ider
the case when G is cyclic. If G = Z=pr, then we prove that either A = 0 or
A = Z=pj for some j r. This was first observed by Chacholski in the case when
V is contractible; under this assumption on V , one finds that necessarily j =*
* r.
Contrary to this fact, if no restriction is imposed on the spaces V and W , th*
*en
each j r can occur, and the value of j is determined by two (possibly infinite)
numbers np(f) and ip(f) which are defined as follows. The number np(f) is the
greatest integer n such that the homomorphism
f* : Hn (V ; Z=p) ! Hn (W ; Z=p)
is an isomorphism, and we set np(f) = 1 if no such upper bound exists. This
concept is analogous to the transitional dimension of a homology theory defined
in [4, 8.1]; it was called the mod p transitional dimension of Lf in [7, 9.14].*
* The
number ip(f), which we call the height of f at the transitional dimension, is t*
*he
greatest integer i such that the homomorphism
f* : Hnp(f)(V ; Z=pi) ! Hnp(f)(W ; Z=pi)
is an isomorphism, and we write ip(f) = 1 if no such upper bound exists. We pro*
*ve
that LfK(Z=pr; n) = K(Z=pip(f); n) if n = np(f) and r > ip(f). The remaining
cases are then straightforward.
If G = Z, then the nth homotopy group A of LfK(Z; n) is either zero or it ad*
*mits
the structure of a commutative ring with 1 having the property that evaluation *
*at 1
induces an isomorphism of rings
Hom (A; A) ~=A; ' 7! '(1): (0:*
*1)
(The elements of Hom (A; A) are endomorphisms of A as an abelian group, operati*
*ng
under composition.) Rings A satisfying (0.1) will be called rigid. More general*
*ly, if
R is any commutative ring, we say that an R-algebra A with 1 is rigid if evalua*
*tion
at 1 yields an isomorphism of R-algebras Hom R(A; A) ~=A. Some basic properties
of rigid R-algebras are described in Section 4, where we extend earlier results*
* of
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 3
Bowshell and Schultz [9], [26]. Rigid rings have previously been investigated u*
*nder
the name of E-rings. Torsion-free rigid rings of finite rank are well understoo*
*d [24].
All solid rings (meaning that the multiplication map A A ! A is an isomor-
phism) are rigid; however, the p-adics bZpor a product Z[1=p] x Z[1=q] with p 6*
*= q
are rigid yet fail to be solid. (The term "solid" was introduced in [8] to desi*
*gnate
a ring A whose core is A itself. Besides the fact that solid implies rigid, we*
* have
chosen this terminology in order to emphasize that rigid rings have few additive
endomorphisms. Further justification comes from the fact that a ring is rigid i*
*f and
only if its underlying abelian group admits only one multiplication with a fixe*
*d left
identity element; see Theorem 4.3 below.) The rigid rings turn out to be precis*
*ely
the localizations of Z in the category of groups, while the solid rings are pre*
*cisely
the Z-epimorphs.
It is known that there exist rigid rings of arbitrarily large cardinality [1*
*6]. Since
every rigid ring R occurs as the fundamental group of LfS1 for a certain map f
(namely, the map f: S1 ! K(R; 1) induced by the inclusion of 1 into R), we
infer that there is a proper class of distinct homotopy types of the form LfS1,
where f ranges over all possible maps. This contrasts greatly with the fact tha*
*t, as
proved in [22], the distinct localizations of any given space with respect to h*
*omology
theories form a set.
The knowledge of the ring R = ss1(LfS1) gives important information about
the functor Lf. Namely, as we show in Theorem 7.2, the homotopy groups of the
f-localizationof any GEM (that is, any weak product of abelian Eilenberg-Mac
Lane spaces) are then R-modules. If R is finite, then the f-localization of any*
* GEM
is a K(A; 1). On the other hand, if R is not cyclic, then the higher homotopy g*
*roups
of the f-localization of any GEM are either P -local for a certain set of prime*
*s P
when R=Z is torsion, or else they are Ext -P -complete when R=Z has elements of
infinite order. This result is derived from Lemma 5.5 in [5]; the set P consis*
*ts of
those primes p such that multiplication by p is an automorphism of R=Z.
It is crucial to observe that if G is any abelian group (not necessarily fin*
*itely
generated), then the group A = ssn(LfK(G; n)) can be described as the localizat*
*ion
of G in the category of groups with respect to a certain group homomorphism;
indeed, A = LiG where i is the homomorphism G ! A induced by the localization
map K(G; n) ! LfK(G; n). In Section 6 we use this fact to show that A inherits
much of the algebraic structure of G, and it does so uniquely. In particular, *
*if R
is any commutative ring, then the nth homotopy group of every localization of a
K(R; n) admits a unique compatible structure of a rigid R-algebra.
4 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Acknowledgements. This study originated from discussions with W. Chacholski.
Some parts of this article were included in the Ph. D. thesis of the second-nam*
*ed
author [25]. Our insight on rigid rings owes much to W. Dicks, A. Facchini, and
R. G"obel, whose advice and interest we appreciate. We also thank E. Dror Farjo*
*un
and B. Oliver for their comments.
1. Preliminaries
All spaces in this paper have the homotopy type of CW-complexes, except for
auxiliary occurrences of mapping spaces. Let f: W ! V be fixed throughout.
Recall that a space Y is called f-local if the induced map of function spaces
map (f; Y ): map (V; Y ) ! map (W; Y )
is a weak homotopy equivalence. A map g is called an f-equivalence if map (g; Y*
* )
is a weak homotopy equivalence for each f-local space Y . An f-localization of*
* a
space X is a map
jX : X ! LfX
which is an f-equivalence and where LfX is f-local. Such a map always exists
and it is unique up to homotopy; see [3] or [15]. In fact, jX is initial in t*
*he ho-
motopy category among maps from X to f-local spaces, and it is terminal among
f-equivalences going out of X. Thus, (Lf; j) is an idempotent monad in the homo-
topy category. If the induced map ss0(f) of connected components is not bijecti*
*ve,
then LfX is contractible for all X; see [29, x 3].
From general properties of idempotent monads (see [1] or [12]) it follows, a*
*mong
other things, that a map g: X ! Y is an f-equivalence if and only if it induce*
*s a
homotopy equivalence LfX ' LfY , and that every homotopy retract of an f-local
space is f-local.
If the map f is of the form W ! *, then f-local spaces are called W -null.
Thus, Y is W -null if and only if the based function space map *(W; Y ) is wea*
*kly
contractible. In this case, it is customary to use the notation PW instead of *
*Lf.
As in [7], a space X will be called f-acyclic if LfX ' *. A space A is said*
* to
be a universal f-acyclic space if the two conditions LfX ' * and PA X ' * are
equivalent for each space X. It was proved in Theorem 4.4 of [7] that universal
f-acyclic spaces exist for each map f; however, such a space A is not homotopy
unique in general with the stated property (instead, it is determined up to nul*
*lity
equivalence, in the sense of [6]).
We shall use the following version of the Zabrodsky Lemma; cf. [15, 1.H.1]
and [30].
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 5
Lemma 1.1. For any fibration F ! E ! X with E and X connected, if F is
f-acyclic, then the map E ! X is an f-equivalence.
This implies in particular that, for any connected space X, if the loop spac*
*e X
is f-acyclic then so is X. We also recall that the homotopy inverse limit of a*
*ny
diagram of f-local spaces is f-local [15, 1.A.8]; thus, any product of f-local *
*spaces
is f-local, and if F ! E ! X is a fibration where E and X are f-local, then F is
f-local as well.
For any abelian group G and any n 1, we know from [15, 4.B] that
LfK(G; n) ' K(A; n) x K(B; n + 1) (1:*
*1)
for some abelian groups A and B; see also Corollary 2.11 in [6]. Essentially t*
*he
same argument shows that, if G admits the structure of an R-module for some
ring R, then A and B are also R-modules. This fact has important consequences;
Theorem 1.3 below is one of them. Firstly we need to observe the following.
Lemma 1.2. Let f be any map and n 1. If F is a field and a K(F; n) is f-local,
then K(G; n) is f-local for every vector space G over F.
Proof. If the dimension of G is finite, then K(G; n) is a product of f-local sp*
*aces
and hence it is f-local. In the general case, let G0 be a product of copies of*
* F
indexed by a basis of G. Then K(G0; n) is f-local. Since the natural inclusion *
*of G
into G0 splits as a map of vector spaces, K(G; n) is a homotopy retract of K(G0*
*; n)
and hence it is f-local as well.
Theorem 1.3. Let f be any map and n 1. Suppose that G is a vector space
over any field. Then K(G; n) is either f-local or f-acyclic.
Proof. Let G be a nonzero vector space over a field F. We know from (1.1) that
LfK(G; n) is a product K(A; n) x K(B; n + 1) for some F-vector spaces A and B.
Then K(A; n) is a retract of an f-local space and hence it is itself f-local. I*
*f A 6= 0,
then K(F; n) is a retract of K(A; n) and therefore it is f-local too. By Lemma *
*1.2,
this implies that K(G; n) is f-local. Now suppose that A = 0 and B 6= 0. Then t*
*he
space K(B; n + 1) is f-local and so is also K(B; n), since K(B; n + 1) ' K(B; n*
*).
As above, from the fact that K(F; n) is a retract of K(B; n) we infer that K(F;*
* n)
is f-local and, by Lemma 1.2, K(G; n) is f-local. But this is incompatible with*
* the
assumption that A = 0. Therefore, if A = 0, then B = 0 as well, so K(G; n) is
f-acyclic.
6 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Lemma 1.4. For any map f and arbitrary integers n 1, r 1, we have:
(1) K(Z=pr; n) is f-acyclic if and only if K(Z=p; n) is f-acyclic.
(2) If K(Z=pr; n) is f-local, then K(Z=pj; n) is f-local for each j r.
Proof. (1) If K(Z=p; n) is f-acyclic, then we may apply Lemma 1.1 and induction
to the fibrations
K(Z=p; n) ! K(Z=pj; n) ! K(Z=pj-1; n);
where j r, to infer that K(Z=pr; n) is f-acyclic. Conversely, suppose that
K(Z=pr; n) is f-acyclic but K(Z=p; n) is not. Then, by Theorem 1.3, K(Z=p; n) is
f-local. If we apply Lemma 1.1 to the fibration
K(Z=pr; n) ! K(Z=p; n) ! K(Z=pr-1 ; n + 1); (1:*
*2)
we obtain that LfK(Z=pr-1 ; n + 1) ' K(Z=p; n). But this cannot happen by (1.1);
thus, K(Z=p; n) is also f-acyclic.
(2) Suppose that K(Z=pr; n) is f-local. Then it follows from part (1) that
K(Z=p; n) is not f-acyclic, and hence it is f-local by Theorem 1.3. In order to
prove that K(Z=pj; n) is f-local for each j r, argue by downward induction usi*
*ng
the fibrations
K(Z=pj-1; n) ! K(Z=pj; n) ! K(Z=p; n);
together with the fact that the homotopy fibre of any map between f-local spaces
is f-local.
2. Transitional dimensions and heights
For each prime p, let np(f) denote the supremum of all positive integers n s*
*uch
that K(Z=p; n) is f-local. If no such integer exists, then we set np(f) = 0. *
*If
all integers n fulfill this condition, then we write np(f) = 1. This is called*
* the
mod p transitional dimension of f. Thus, for any map f, we have np(f) = n if
and only if the homomorphism Hi(f; Z=p) is an isomorphism for i n but not for
i = n + 1. Likewise, np(f) = 1 if and only if f is a mod p equivalence. Note th*
*at
np(f) = np(f) + 1 for every map f.
For a space W , we denote by np(W ) the dimension np(f) where f: W ! *.
Using the natural isomorphism Hj(W ; Z=p) ~= Hom (Hj(W ; Z=p); Z=p) for all j
(cf. [27, Ch. 5]), we see that, for a space W , the following statements are eq*
*uivalent:
(1) np(W ) = n.
(2) "Hj(W ; Z=p) = 0 for j n and Hn+1 (W ; Z=p) 6= 0.
(3) "Hj(W ; Z=p) = 0 for j n and Hn+1 (W ; Z=p) 6= 0.
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 7
Lemma 2.1. Let f and g be two maps for which there is a natural transformation
of functors Lf ! Lg. Then np(f) np(g).
Proof. The assumption made means precisely that every g-local space is f-local;
cf. [1] and also [29, x 2]. Therefore, if a space K(Z=p; n) is g-local then it*
* is also
f-local.
Corollary 2.2. For any map f: W ! V , if C denotes the homotopy cofibre of f
and A is a universal f-acyclic space, then
np(f) = np(A) np(C) np(f) + 1:
Moreover, np(f) = np(C) if and only if Hn+1 (C; Z=p) 6= 0, where n = np(f).
Proof. This follows from the definitions, by resorting to the natural transform*
*ations
Lf ! PC ! PA ! Lf;
together with the mod p homology long exact sequence associated to the cofibre
sequence W ! V ! C.
Lemma 2.3. For any map f, if n < np(f), then K(Z=pr; n) is f-local for every
integer r.
Proof. By assumption, both K(Z=p; n) and K(Z=p; n + 1) are f-local. Hence we
may argue by induction using the fibrations
K(Z=pj; n) ! K(Z=pj-1; n) ! K(Z=p; n + 1):
Now we can associate another number to each map f. For any prime p, let ip(f)
be the supremum of all integers i such that the space K(Z=pi; np(f)) is f-local.
If all integers i fulfill this condition, then we write ip(f) = 1. Thus, if n =
np(f) (implying that Hn (f; Z=p) is an isomorphism) then ip(f) = i if and only *
*if
Hn (f; Z=pj) is an isomorphism for j i but not for j = i + 1. We call this num*
*ber
ip(f) the height of f at the mod p transitional dimension.
3. Localizing Eilenberg-Mac Lane spaces
Let G be any abelian group and n 1. Let
j: K(G; n) ! K(A; n) x K(B; n + 1)
be the f-localization map, as in (1.1). From the fact that j is an f-equivalenc*
*e we
derive the following algebraic relations.
8 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Theorem 3.1. Let f be any map, G any abelian group, and n 1. For the abelian
groups A = ssn(LfK(G; n)) and B = ssn+1 (LfK(G; n)), the following hold:
(1) Hom (A; A) ~=Hom (G; A).
(2) Hom (A; B) ~=Hom (G; B).
(3) Hom (B; B) Ext(A; B) ~=Ext (G; B) if n 2, or else
(4) Hom (H2(K(A; 1)); B) Hom (B; B) Ext(A; B) ~=
Hom (H2(K(G; 1)); B) Ext(G; B) if n = 1.
Proof. The fact that the space K(A; n) is f-local yields an isomorphism
Hn (K(A; n) x K(B; n + 1); A) ~=Hn (K(G; n); A);
which implies the isomorphism (1). Claim (2) is deduced in the same way from the
fact that K(B; n) is f-local, and claims (3) and (4) hold because K(B; n + 1) is
f-local. Here we need to recall from Theorem V.7.8 in [32] that
Hn+1 (K(A; n)) = 0
for any abelian group A if n 2, and also for n = 1 if A is cyclic.
If G = Z, then we infer from (3) or (4) that B = 0, for any n. Furthermore, *
*we
can consider the homomorphism ssn(j): Z ! A induced by j on the nth homotopy
group, and let e be its value on 1. If we identify Hom (Z; A) with A in the ob*
*vious
way, then the isomorphism Hom (A; A) ~= A in Theorem 3.1 sends each endomor-
phism ' 2 Hom (A; A) to '(e). Composition in Hom (A; A) defines a multiplicati*
*on
in A for which e is the identity element (unless A = 0). Therefore, if A is non*
*zero,
then A admits a ring structure, and this ring structure is of a very special ki*
*nd.
Here we give it a name and postpone its study until the next section.
Definition 3.2. A ring R with 1 is rigid if the evaluation map Hom (R; R) ! R
given by ' 7! '(1) is bijective.
Theorem 3.3. For any map f and any integer n 1,
LfK(Z; n) ' K(A; n);
where A is either zero or it admits the structure of a rigid ring.
From this fact it follows, for example, that ssn(LfK(Z; n)) cannot be isomor*
*phic
to Z=p1 nor to Z[1=p] x Z[1=p]. However, it can be isomorphic to Z[1=p] x Z[1=*
*q]
if p and q are distinct primes.
We next address the case when G = Z=pr, where p is any prime and r 1.
First of all, if n > np(f) then K(Z=p; n) is f-acyclic by Theorem 1.3 and hence*
* so
is K(Z=pr; n), by Lemma 1.4. If n < np(f), then it follows from Lemma 2.3 that
K(Z=pr; n) is f-local. The general result, including the case n = np(f), reads*
* as
follows.
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 9
Theorem 3.4. For any map f and arbitrary integers n; r 1, we have
8
>< * if n > np(f);
LfK(Z=pr; n) ' > K(Z=pip(f); n) if n = np(f) and r > ip(f);
: K(Z=pr; n) otherwise.
Proof. After our previous remarks, only the case n = np(f) requires a proof. As
above, let A and B be the homotopy groups of LfK(Z=pr; n). Since B is a Z=pr-
module and hence of bounded exponent, B is either zero or a direct sum of groups
Z=pj with 1 j r; cf. Theorem 6 in [17]. If B 6= 0, then K(Z=pj; n + 1) is a
retract of K(B; n + 1) for some 1 j r and hence f-local. But then it follows
from Lemma 1.4 that K(Z=p; n + 1) is f-local, and this contradicts our choice o*
*f n.
This shows that B = 0. On the other hand, if A = 0 then part (1) of Lemma 1.4
tells us that K(Z=p; n) is f-acyclic, contradicting again our choice of n. Thus*
* A is
a nonzero Z=pr-module and part (1) of Theorem 3.1 shows that A = Z=pj for some
j r. Finally, we want to prove that j = ip(f). Suppose instead that j < ip(f).
Then K(Z=pj+1; n) is f-local. This yields an isomorphism
Hom (Z=pj; Z=pj+1) ~=Hom (Z=pr; Z=pj+1);
where the left-hand side equals Z=pj and the right hand side equals Z=pj+1. This
contradiction completes the argument.
Moreover, when n = np(f) and r > i = ip(f), then the localization map
j: K(Z=pr; n) ! K(Z=pi; n) coincides, up to a homotopy equivalence, with the
map induced by the natural projection Z=pr ! Z=pi. To see this, observe that the
map K(Z=pr; n) ! K(Z=pi; n) induced by the projection must factor through j up
to homotopy, and this forces j*(1) to be a unit in Z=pi.
Example 3.5. If the map f is of the form W ! *, then ip(f) = 1. Therefore,
for any space W , we have
ae* if n > n (W );
PW K(Z=pr; n) ' p
K(Z=pr; n) otherwise.
This result was communicated to us by Chacholski and was in fact one of the
motivations of our work.
Example 3.6. Let f: K(Z; n) ! K(Z=pi; n) be the map induced by the projection
of Z onto Z=pi, where n 1. Then K(Z=p; n) is f-local but K(Z=p; n + 1) is not.
Hence, np(f) = n. Likewise, K(Z=pi; n) is f-local but K(Z=pi+1; n) is not, which
implies that ip(f) = i. This shows that all heights can occur in practice. No*
*w it
follows from Theorem 3.4 that LfK(Z=pr; n) ' K(Z=pi; n) for r i.
Since f-localization functors preserve finite products, Theorems 3.3 and 3.4*
* yield:
10 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Corollary 3.7. For any map f, any integer n 1, and any finitely generated
abelian group G, we have LfK(G; n) = K(A; n) for some abelian group A. More-
over, if G is torsion, then A is a quotient of G.
This is of course not true in general if G is not assumed to be finitely gen*
*erated.
For example, if f is any map for which Lf is ordinary homological localization *
*with
Z=p coefficients, then it follows from the exact sequence
0 ! Z ! Z[1=p] ! Z=p1 ! 0
that
LfK(Z=p1 ; n) ' LfK(Z; n + 1) ' K(bZp; n + 1):
4. Rigid rings and algebras
In this section, all rings are assumed to be associative and have an identity
element, which we denote by 1 if no confusion can arise. For any two abelian
groups A and B, we abbreviate Hom Z(A; B) to Hom (A; B) and A Z B to A B.
Recall from Definition 3.2 that a ring A is called rigid if the evaluation m*
*ap
Hom (A; A) ! A given by ' 7! '(1) is bijective. Such rings were first conside*
*red
in [26] and [9] under the name of E-rings. The basic examples are the rings Z=m,
the subrings of Q, and the ring bZpof p-adic integers, for any p. If A, B are r*
*igid
rings and Hom (A; B) = Hom (B; A) = 0, then the product A x B is rigid. Other
less obvious examples of rigid rings are the products
Y Y Y
Z=p; Z[1=p]; bZp;
p2P p2P p2P
where P is an arbitrary set of primes, possibly infinite. A classification of*
* rigid
rings which are torsion-free of finite rank was achieved in [24].
As a preparation for the findings in Section 6, we shall now discuss a more
general notion, namely rigid algebras. Most of the following results generalize*
* basic
properties of rigid rings that can be found in [26] or [9]. Some observations a*
*re new,
notably Theorem 4.3. In the rest of this section, R will be a fixed commutative
ring with 1. By an R-algebra we mean a ring A equipped with a central ring
homomorphism R ! A.
Definition 4.1. An R-algebra A will be called rigid if the evaluation map
Hom R (A; A) ! A
given by ' 7! '(1) is bijective.
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 11
Theorem 4.2. If an R-algebra A is rigid, then A is commutative.
Proof. Fix any element a 2 A. Then the R-endomorphisms '1, '2 of A given by
'1(x) = ax; '2(x) = xa
satisfy '1(1) = '2(1) and hence coincide.
If A is any R-algebra, then left multiplication defines a map : A ! Hom R (A*
*; A).
Both and the evaluation map ": Hom R (A; A) ! A are R-module homomorphisms
and the composition " O is the identity map. Therefore, " is surjective and *
*is
injective for every R-algebra A. It follows that an R-algebra is rigid if and o*
*nly if
the evaluation map " is injective.
Theorem 4.3. An R-algebra A is rigid if and only if the underlying R-module
admits only one compatible multiplication where 1 acts as a left identity.
Proof. Suppose firstly that A is rigid, and denote by O an arbitrary multiplica*
*tion
in A which is compatible with the R-module structure and where 1 O a = a for
all a. Then, for any fixed element a 2 A, the R-endomorphisms '1, '2 given by
'1(x) = xa and '2(x) = xOa satisfy '1(1) = '2(1) and hence coincide. This proves
one implication.
Conversely, suppose that the multiplication in A is unique with the prescrib*
*ed
conditions. If is an R-endomorphism of A such that (1) = 1, then the multipl*
*i-
cation defined by a O b = (a)b endows A with an R-algebra structure where 1 is*
* a
left identity. By assumption, a O b = ab for all a; b 2 A, which implies that *
* = id.
Now, if '1 and '2 are two R-endomorphisms of A such that '1(1) = '2(1), then
= id- '1 + '2 satisfies (1) = 1, and hence '1 = '2. This proves that A is ri*
*gid,
as claimed.
Example 4.4. The abelian group ZZ admits precisely a two-parameter family of
distinct multiplications for which (1; 1) is the identity. Each of these is det*
*ermined
by a 2x2 matrix with integer entries, representing multiplication by (1; 0) in *
*ZZ.
Thus, if we impose the condition that the product of this matrix with (1; 1) eq*
*uals
(1; 0), we obtain the family of solutions
(x; y)O(z; t) = (xz +(1-)xt+(1-)yz -(1-)yt; xz -xt-yz +(1+)yt);
where and are arbitrary integers. These multiplications are all associative a*
*nd
commutative.
12 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Theorem 4.5. For an R-algebra A, the following statements are equivalent:
(1) A is rigid.
(2) The map : A ! Hom R (A; A) given by (a)(x) = ax is bijective.
(3) Hom R (A=<1>; A) = 0, where <1> is the R-submodule of A generated by 1.
(4) Every ' 2 Hom R (A; A) is an A-module endomorphism.
(5) The evaluation map ": Hom R (A; A) ! A is an isomorphism of R-algebras.
(6) The endomorphism ring Hom R(A; A) is commutative.
Proof. The equivalence of (1) and (2) follows from the fact that is right-inve*
*rse
to ". Next, observe that the inclusion of the submodule <1> into A gives rise t*
*o a
short exact sequence of R-modules
0 ! Hom R (A=<1>; A) ! Hom R (A; A) ! Hom R (<1>; A) ! 0;
where the third arrow coincides with the evaluation map " and hence it is surje*
*ctive.
This proves that (1) and (3) are equivalent. Next we prove that (1) ) (4). Let '
be any R-endomorphism of A. Fix any element a 2 A. Then the endomorphisms
'1, '2 given by
'1(x) = x'(a); '2(x) = '(xa)
satisfy '1(1) = '2(1) and hence coincide. This shows that ' is an A-module en-
domorphism, as required. The implication (4) ) (1) is immediate, since under (4)
any ' 2 Hom R (A; A) is completely determined by its value on 1. We can now inf*
*er
that (4) ) (5), since
"( O ') = ('(1)) = '(1) (1) = (1) '(1) = "( ) "('):
The fact that (5) ) (6) follows from Theorem 4.2. We conclude by showing that
(6) ) (4). Thus, assume that Hom R(A; A) is a commutative ring, and pick any
' 2 Hom R(A; A). Then, by assumption, ' commutes with (a) for any a 2 A,
which yields
'(ax) = [' O (a)](x) = [(a) O '](x) = a'(x)
for all x 2 A, as we wanted to prove.
Recall from [8] that a ring A with 1 is called solid if the multiplication m*
*ap
m: A A ! A; m(a b) = ab;
is bijective. Such rings were called T -rings in [9] and Z-epimorphs in [14]. I*
*ndeed,
by [28, XI.1.2], a ring A is solid if and only if the unit map Z ! A is an epim*
*orphism
of rings.
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 13
More generally, if R is any commutative ring, an R-algebra A will be called *
*solid
if the multiplication map m: A R A ! A is bijective or, equivalently, if the un*
*it
map R ! A is an epimorphism of rings.
The fact that an R-algebra A is solid forces that a b = ab 1 = 1 ab in
A R A, for all a and b. Therefore, if A is solid, then, for every ' 2 Hom R(A;*
* A),
we can consider the homomorphism : A R A ! A given by (a b) = a'(b) and
infer that
'(ax) = (1 ax) = (a x) = a'(x):
Hence, every ' 2 Hom R(A; A) is an A-module endomorphism, and Theorem 4.5
yields the following result, which generalizes Corollary 1.8 in [9].
Theorem 4.6. Every solid R-algebra is rigid.
Note that the p-adic integers are rigid as a Z-algebra, but not solid. Solid*
* rings
have been classified; see [8], [9], [14]. We warn the reader that, while the c*
*lass of
solid rings is closed under quotients, the class of rigid rings is not. For ex*
*ample,
the quotient of A = Z[1=2] x Z[1=3] by the ideal 5A is isomorphic to Z=5 x Z=5.
5. Localizing groups
In this section we deal with localization in the category of groups with res*
*pect
to any group homomorphism ': H ! K, as in [11, x3] or [13, x1]. A group L is
said to be '-local if the induced map
Hom ('; L): Hom (K; L) ! Hom (H; L)
is a bijection of sets. A '-equivalence of groups is a homomorphism such that
Hom ( ; L) is a bijection for every '-local group L. For every group G there *
*is a
'-equivalence jG : G ! L' G into a '-local group L' G, with universal properties
analogous to those written down in Section 1; thus, (L' ; j) is an idempotent m*
*onad
in the category of groups. We call L' G the '-localization of G.
Let G be any abelian group and n 1. Then, as we know, for any map f
between connected spaces, the f-localization of a K(G; n) takes the form
j: K(G; n) ! K(A; n) x K(B; n + 1):
If we denote by i the homomorphism G ! A induced by j at the nth homotopy
group, then part 1 of Theorem 3.1 says precisely that the group A is i-local. S*
*ince
i is of course a i-equivalence, the group A is the i-localization of G. Moreov*
*er,
part 2 of Theorem 3.1 tells us that B is i-local as well. Therefore, we have:
14 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Theorem 5.1. Given any abelian group G, any n 1, and any map f between
connected spaces, there exists a group homomorphism i such that
LfK(G; n) ' K(LiG; n) x K(B; n + 1);
and the group B is i-local.
Theorem 5.1 can be improved if the source and target of f are assumed to
be (n - 1)-connected spaces. In that case, as stated in Theorem 5.4 below, the
homomorphism i can be chosen to be ssn(f).
First of all observe that, if f is a map between (n-1)-connected spaces, the*
*n the
localization PSn with respect to Sn ! * (i.e., the (n-1)-th Postnikov section) *
*turns
f trivially into a homotopy equivalence. This implies that, for all spaces X, *
*the
f-localization map X ! LfX induces a homotopy equivalence PSn X ' PSn LfX;
cf. [29, x 2]. From this fact we derive the following generalization of Corolla*
*ry 4.4
in [5]; cf. also [29, x 7]. We are thankful to Jeff Smith for making this resul*
*t evident
to us.
Theorem 5.2. Let f: W ! V be any map where W and V are (n - 1)-connected.
Then, for all connected spaces X, the natural map of (n - 1)-connected covers
X ! (LfX) is an f-localization, that is, it induces a homotopy
equivalence
Lf(X) ' (LfX):
Proof. Apply fibrewise f-localization to the homotopy fibration
X ! X ! PSn X;
yielding a homotopy fibration
Lf(X) ! Y ! PSn X;
together with a map h: X ! Y which is an f-equivalence; cf. [15, 1.F.1]. Sinc*
*e,
by our assumption, the mapping spaces map *(V; PSn X) and map *(W; PSn X) are
weakly contractible, we infer that map *(V; Y ) ! map *(W; Y ) is a weak homoto*
*py
equivalence, and hence Y is f-local. This means of course that Y ' LfX. Sin*
*ce
PSn X ' PSn LfX, our claim follows.
Using this observation and the same arguments as in Proposition 3.3 of [11],*
* we
find that for an arbitrary map f: W ! V between (n - 1)-connected spaces, if we
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 15
denote by ': ssn(W ) ! ssn(V ) the induced homomorphism of nth homotopy groups,
then the following hold:
(1) A group G is '-local if and only if K(G; n) is f-local.
(2) If g is any f-equivalence of connected spaces, then the homomorphism ss*
*n(g)
is a '-equivalence of groups.
(To prove (2), notice that if g is an f-equivalence then so is the lifting o*
*f g to
the (n - 1)-connected covers, by Theorem 5.2.)
In particular, since jX : X ! LfX is an f-equivalence, it follows from (2) t*
*hat
there is a natural homomorphism
ssn(LfX) ! L' ssn(X)
which is a '-equivalence and therefore it is an isomorphism if and only if ssn(*
*LfX)
is '-local. This leads to the following improvement of Theorem 2.1 in [13].
Theorem 5.3. Let f: W ! V be a map where W is a wedge of copies of Sn with
n 1, and V has cells in dimensions n and n + 1 only. Let ' = ssn(f). Then
ssn(LfX) ~=L' ssn(X) for all connected spaces X.
Proof. We only need to prove that ssn(LfX) is '-local. The assumption made on
W ensures that, given any group homomorphism : ssn(W ) ! ssn(LfX), there ex-
ists a map g: W ! LfX inducing on the nth homotopy group. Since LfX is
f-local, there is a map g0: V ! Lf(X) such that g0 O f ' g, yielding a homomor-
phism 0: ssn(V ) ! ssn(LfX) such that 0O ' = , as desired. If 00is any other
homomorphism with this property, then it is induced by some map g00: V ! LfX.
Then g00O f and g induce the same homomorphism on the nth homotopy group
and hence they are homotopic, since W is a wedge of copies of Sn . It follows t*
*hat
g00' g0 and therefore 00= 0, as needed.
Theorem 5.4. For any abelian group G and any map f between (n - 1)-connected
spaces, where n 1, we have
LfK(G; n) ' K(L' G; n) x K(B; n + 1);
where ' = ssn(f). Moreover, the group B is '-local.
Proof. Let A and B be the homotopy groups of LfK(G; n). Then the localization
map j: K(G; n) ! LfK(G; n) induces a homomorphism ssn(j): G ! A. Since the
map j is an f-equivalence, the homomorphism ssn(j) is a '-equivalence. Moreover,
the space K(A; n) is f-local, and hence the group A is '-local. This proves th*
*at
A ~=L' G. From the fact that K(B; n + 1) is f-local it follows that K(B; n) is *
*also
f-local and therefore the group B is '-local.
16 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Corollary 5.5. Suppose that f is a map between (n - 1)-connected spaces such
that the homomorphism ssn(f) is surjective. Then, for any abelian group G, the
natural homomorphism G ! ssn(LfK(G; n)) is surjective.
Proof. This follows from the fact that if ' is an epimorphism, then the localiz*
*ation
map j: G ! L' G is an epimorphism for all groups G. To prove this claim, check
directly that the image of j is '-local and its inclusion into L' G is a '-equi*
*valence
and hence an isomorphism.
The following example shows that the assumption that f is a map of (n - 1)-
connected spaces cannot be removed from Theorem 5.4 and Corollary 5.5. Consider
the map f: M(Z[1=p]; 1) ! *, where the letter M stands for a Moore space. Then
ssn(LfK(Z; n)) is the ring bZp of p-adic integers if n 2; cf. [13]. However, *
*any
homomorphism induced by f on homotopy groups will be surjective and bZpcannot
be obtained by localizing Z with respect to any epimorphism.
From Theorem 5.4 it follows that the localization of any abelian group with
respect to any group homomorphism is abelian. In fact, a more general result is
true. The following purely algebraic argument was suggested to us by Dror Farjo*
*un.
Proposition 5.6. Let (L; j) be any idempotent monad in the category of groups.
If A is any abelian group, then LA is also abelian.
Proof. For any element a 2 A, conjugation by j(a) is the identity homomorphism
on j(A) and hence it is the identity homomorphism on LA. In particular, for each
x 2 LA, conjugation by x is the identity on j(A) and hence it is the identity o*
*n LA.
This shows that LA is indeed abelian.
Proposition 5.7. Let ': K ! M be an arbitrary group homomorphism, and de-
note by OE: H1(K) ! H1(M) its abelianization. Then for any abelian group G we
have a natural isomorphism L' G ~=LOEG.
Proof. Since every localization of an abelian group is abelian, the functors LO*
*Eand
L' can both be restricted to the full subcategory of abelian groups. Moreover, *
*an
abelian group is OE-local if and only if it is '-local. This says that the rest*
*rictions of
LOEand L' have the same image class in the category of abelian groups, and hence
they are naturally isomorphic.
As a consequence, we can replace L' G with LOEG in Theorem 5.4, where OE = Hn(f*
*),
for any n 1.
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 17
Example 5.8. Let f: X ! Y be any map inducing the projection OE: Z ! Z=m
on the first homology group, where m is any integer. Then an abelian group A is
OE-local if and only if mA = 0. Therefore,
ss1(LfK(G; 1)) ~=LOEG ~=G=mG
for every abelian group G. In fact, it follows from Corollary 7.3 that ss2(LfK(*
*G; 1))
vanishes and hence LfK(G; 1) ' K(G=mG; 1).
Similarly, if g: X ! Y is any map where H1(X) = 0 and H1(Y ) ~=Z=pr, where
p is a prime and r 1, then ss1(LgK(G; 1)) ~= G=TpG; where TpG denotes the
p-torsion subgroup of G, for any abelian group G.
We next specialize to the case G = Z. Let ' = ss1(f) be the homomorphism
induced by f on fundamental groups, and denote by OE = H1(f) its abelianization.
The following result follows from Theorem 3.3, Theorem 5.4, and Proposition 5.7.
Theorem 5.9. For any given map f between connected spaces, we have
LfS1 ' K(L' Z; 1) ' K(LOEZ; 1);
where ' = ss1(f) and OE = H1(f).
Corollary 5.10. Suppose that H1(f) is surjective. Then ss1(LfS1) is cyclic.
Theorem 5.11. For a nonzero abelian group A, the following statements are equi*
*v-
alent:
(1) A admits the structure of a rigid ring.
(2) There is a group homomorphism OE such that LOEZ ~=A.
(3) There is a map f such that LfS1 ' K(A; 1).
Proof. We first prove that (1) ) (2). If A is any rigid ring, then it follows d*
*irectly
from the definition (Definition 3.2) that A is OE-local, where OE: Z ! A is the*
* only
ring homomorphism with OE(1) = 1. Since OE is obviously a OE-equivalence, we ob*
*tain
that A ~= LOEZ. The implication (2) ) (3) is a consequence of Theorem 5.9, and
the implication (3) ) (1) has been proved in Theorem 3.3.
Corollary 5.12. The collection of homotopy types of the form LfS1, where f
ranges over all maps, is a proper class (i.e., it is not a set).
Proof. This follows from Theorem 5.11 since, by Corollary 4.10 in [16], there a*
*re
rigid rings of arbitrarily large cardinality.
This result is striking, since the distinct homological localizations of S1 *
*are listed
in [4] and certainly form a set. Furthermore, Ohkawa proved in [22] that the st*
*able
18 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Bousfield equivalence classes of spectra form a set. This implies that there is*
* only
a set of nonisomorphic homological localization functors, both in the stable an*
*d in
the unstable homotopy categories. We are indebted to Neil Strickland for drawing
Ohkawa's article to our attention and showing us his own simpler argument.
We close this section with the following curious consequences of Theorem 5.9.
Recall that PW denotes localization with respect to W ! *.
Theorem 5.13. If W is any space and X is a (possibly infinite) wedge of circle*
*s,
then ae
X if H1(W ) = 0;
PW X '
* if H1(W ) 6= 0:
Proof. If ss1(W ) ! ss1(X) is a nontrivial homomorphism, then its image is a no*
*n-
trivial free group and hence there is a nonzero homomorphism ss1(W ) ! Z. Hence,
if H1(W ) = 0, then Hom (ss1(W ); ss1(X)) is trivial. This implies that ss0 map*
* *(W; X)
is trivial and, since X is one-dimensional, map *(W; X) is weakly contractible,*
* i.e.,
X is W -null. On the other hand, if H1(W ) 6= 0, then it follows from Theorem 5*
*.9
that PW S1 ' * and therefore PW X ' * as well.
Corollary 5.14. Suppose that a space W satisfies H1(W ) 6= 0. Then for every
noncontractible CW-complex X there exists at least one essential map rW ! X
for some r 0.
Proof. Since PW S1 ' *, there is a natural transformation PS1 ! PW , and hence
PW X ' PW PS1X ' * for all connected spaces X; that is, no space is W -null, un*
*less
it is contractible. This implies that, if X is not contractible, then the mapp*
*ing
space map *(W; X) is not weakly contractible. Hence, [rW; X] is nontrivial for
some value of r.
6. Some algebraic structures preserved under localization
Let f be any map such that Lf is isomorphic to ordinary homological localiza*
*tion
with mod p coefficients, and let g be the wedge of all prime power maps of S1 (*
*thus,
Lg is rationalization). Then
LgLfS1 ' K(bZp Q; 1):
Since the p-adic field bZp Q does not admit a rigid ring structure, this is an *
*easy
example where the composite of two localization functors fails to be a localiza*
*tion.
However, bZp Q is rigid as a bZp-algebra.
In this section we generalize this example by showing that the fundamental g*
*roup
of any iterated localization of S1 has a commutative ring structure. In fact, *
*the
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 19
same is true for the nth homotopy group of any iterated localization of a K(R; *
*n)
where R is a commutative ring with 1.
From now on we fix a commutative ring R with 1 and an arbitrary idempotent
monad (L; j) in the category of groups. If A is a ring or a module over some ri*
*ng,
we denote by LA the localization of the underlying abelian group, which is again
abelian, according to Proposition 5.6.
From the fact that the functor L is left adjoint to the identity it follows *
*that L
is additive; that is, the natural map
Hom (A; B) ! Hom (LA; LB)
is a group homomorphism for all abelian groups A and B; see [18, p. 83]. In the*
* case
when A = B, this map is in fact a ring homomorphism (under composition). Thus,
if M is any R-module with structure map R ! Hom (M; M), then LM inherits the
R-module structure. More precisely, we have
Theorem 6.1. If M is any R-module, then LM admits a unique R-module struc-
ture such that the localization map j: M ! LM is an R-module map.
Lemma 6.2. If M is any R-module, then the natural map Hom R(LR; LM) ! LM
induced by the localization map j: R ! LR is an isomorphism.
Proof. The universal property of j gives rise to an isomorphism of abelian grou*
*ps
Hom (LR; LM) ~=Hom (R; LM), which restricts to a monomorphism
Hom R (LR; LM) ! Hom R (R; LM) ~=LM: (6:*
*1)
Now, given an R-module map : R ! LM, it follows again from the universal
property of j that the induced homomorphism ": LR ! LM is an R-module map,
since
"(rj(s)) = "(j(rs)) = (rs) = r (s) = r "(j(s))
for all r; s 2 R. This shows that (6.1) is in fact bijective.
Theorem 6.3. Let R be any commutative ring with 1. Then LR is either zero or
it admits a unique ring structure such that j: R ! LR is a ring homomorphism.
Moreover, LR is rigid as an R-algebra. If R is a field and LR is nonzero, then j
is an isomorphism.
Proof. We can use (6.1) with M = R to endow LR with a ring structure, where the
multiplication is induced by composition in Hom R(LR; LR). It follows from th*
*is
definition that j is a ring homomorphism and that LR is rigid as an R-algebra. *
*As
such, the multiplication in LR is unique, by Theorem 4.3. If R is a field, then*
* LR
is free as an R-module. Therefore, since a retract of any L-local group is L-lo*
*cal,
we infer that R is already L-local if LR 6= 0.
20 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Lemma 6.4. Let R and S be commutative rings with 1 and let f: R ! S be a
ring homomorphism. Then the induced map Lf: LR ! LS is also a ring homo-
morphism.
Proof. We write g instead of Lf for convenience. For any fixed element a 2 R, t*
*he
group homomorphisms '1(x) = g(j(a)x) and '2(x) = g(j(a)) g(x) from LR to LS
coincide on the image of j and hence '1 = '2. Now, for a fixed element b 2 LR,
the group homomorphisms 1(x) = g(xb) and 2(x) = g(x)g(b) coincide on the
image of j and hence 1 = 2, as claimed.
Lemma 6.5. If B is any L-local abelian group, then Hom (A; B) is L-local for *
*all
abelian groups A.
Proof. Let ': C ! D be an arbitrary L-equivalence. Then Hom (D; Hom (A; B)) ~=
Hom (A; Hom (D; B)) ~=Hom (A; Hom (C; B)) ~=Hom (C; Hom (A; B)), which proves
our claim.
Theorem 6.6. If M is any R-module, then the R-module structure of LM can be
extended uniquely to an LR-module structure.
Proof. Use Lemma 6.2 or Lemma 6.5 to endow LM with an LR-module structure
and Lemma 6.5 to guarantee its uniqueness.
Now we apply the above results to the case of f-localizations of Eilenberg-M*
*ac
Lane spaces.
Theorem 6.7. Given any commutative ring R with 1, an arbitrary map f, and
an integer n 1, we have
LfK(R; n) ' K(LiR; n) x K(B; n + 1);
for a certain homomorphism i of abelian groups. The group LiR is either zero or
a rigid R-algebra. Moreover, B is i-local and an LiR-module.
Proof. In view of Theorem 5.1, only the last statement requires a proof. We know
that B is an R-module and B ~= LiB. Therefore, by Theorem 6.6, the R-module
structure of B can be extended uniquely to an LiR-module structure.
Corollary 6.8. If f1; : :;:fk is any finite collection of maps and R is a comm*
*utative
ring with 1, then ssn(Lfk . .L.f1K(R; n)) is either zero or a rigid algebra ove*
*r the
ring ssn(Lfk-1 . .L.f1K(R; n)), for any n 1.
Proof. It is enough to prove it for two maps f1, f2. Firstly write
Lf1K(R; n) ' K(LiR; n) x K(B; n + 1);
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 21
for a certain homomorphism i. Thus, LiR is either zero or a rigid R-algebra. Si*
*nce
localization commutes with finite products and ssn(Lf2K(B; n + 1)) = 0, we have
ssn(Lf2Lf1K(R; n)) ~= ssn(Lf2K(LiR; n)). By Theorem 6.7, this group is either
zero or a rigid LiR-algebra.
Of course, this implies in particular that the nth homotopy group of every i*
*ter-
ated localization Lfk . .L.f1K(R; n) is a commutative R-algebra with 1.
7. Effect on the higher homotopy groups
In this last section, we explain how the knowledge of LfK(Z; n) or LfK(Z=pr;*
* n)
gives very relevant information about the homotopy groups of LfX for other
spaces X. Fix a map f and let R = ssn(LfK(Z; n)) with n 1, which is either zero
or a commutative ring with 1, according to Theorem 3.3. Since the localization
map K(Z; n) ! K(R; n) is an f-equivalence, we obtain a bijection
[K(R; n); Y ] ~=[K(Z; n); Y ] (7:*
*1)
for every f-local space Y . From this fact we infer the next results.
Theorem 7.1. If R denotes the ring ss1(LfS1) for a certain map f, then every
f-local space Y satisfies
ssm (Y ) ~=[m-1 K(R; 1); Y ] for m 1.
Proof. Use (7.1) with n = 1, together with the fact that if Y is f-local, then *
*so are
all the spaces iY for i 1.
As customary, a space which is homotopy equivalent to a weak product of abel*
*ian
Eilenberg-Mac Lane spaces will be called a generalized Eilenberg-Mac Lane space,
shortly a GEM.
Recall from [15, 4.B] that each localization of a GEM is a GEM. Thus, if X is
a GEM and A = ssm (LfX) where m 1, then the space K(A; m) is a homotopy
retract of LfX and hence it is f-local. If m n, then K(A; n) ' m-n K(A; m)
and hence K(A; n) is f-local too. Thus, we may infer from (7.1) that the unit m*
*ap
Z ! R induces an isomorphism
Hom (R; A) ~=Hom (Z; A): (7:*
*2)
This says precisely that A is local with respect to Z ! R, from which it follows
that A admits a unique R-module structure; cf. Theorem 6.6.
22 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
Modules A satisfying (7.2) were called E(R)-modules by Pierce in [23] and R-
groups by Bowshell and Schultz in [9, x 2]. Of course, there is no restriction *
*on A
if R = Z or more generally if Z ! R is a ring epimorphism (i.e., if the ring R
is solid). Indeed, if Z ! R is a ring epimorphism, then by [28, XI.1.2] we have
canonical isomorphisms
Hom (R; A) ~=Hom R(R; A) ~=Hom (Z; A)
for every R-module A. However, if Z ! R is not a ring epimorphism, then there
is at least one ring A and two distinct ring homomorphisms R ! A such that the
composites Z ! A coincide. Then A becomes an R-module which violates (7.2).
Hence, in our context, condition (7.2) imposes a nonvoid restriction on the R-
module A precisely when the ring R is rigid but not solid. For example, if R = *
*bZp
and A = bZp Q, then (7.2) does not hold.
If we assume, in addition, that m > n, then K(A; n + 1) will be f-local and *
*from
(7.1) it follows that Ext(R; A) = 0. We summarize our conclusions in the follow*
*ing
theorem.
Theorem 7.2. Let f be any map and let R = ssn(LfK(Z; n)), where n 1. Let X
be a GEM. For m n, consider the group A = ssm (LfX). Then A admits a unique
R-module structure satisfying Hom (R; A) ~= Hom (Z; A) if R 6= 0, and A = 0 if
R = 0. Moreover, if m > n then in addition Ext (R; A) = 0.
Corollary 7.3. Suppose that LfK(Z; n) ' K(Z=t; n), where t is any positive in-
teger and n 1. If X is a GEM, then ssm (LfX) = 0 for m > n.
Proof. From Theorem 7.2 we know that each of the homotopy groups ssm (LfX)
is a Z=t-module for m n. But if an abelian group A satisfies tA = 0 and
Ext (Z=t; A) = 0, then A = 0.
The conclusion of Corollary 7.3 seems to hold for a much broader class of sp*
*aces,
not necessarily products of Eilenberg-Mac Lane spaces. Perhaps even the answer
to the following question is affirmative.
Question: Let f: S1 ! K(Z=t; 1) induce the projection Z ! Z=t on the funda-
mental group, where t is any positive integer. Is it true that ssm (LfX) = 0 if*
* m 2,
for all spaces X?
Observe that, if R is a rigid ring and the unit map Z ! R is not injective, *
*then
R ~= Z=t for some integer t. Indeed, if the identity element of R has finite o*
*rder,
then tR = 0 for some integer t and, for a rigid ring, this implies that R is cy*
*clic;
this fact was already noted in [26]. Therefore, if LfK(Z; n) ' K(R; n), then ei*
*ther
LOCALIZATIONS OF ABELIAN EILENBERG-MAC LANE SPACES 23
R is cyclic or the induced map Z ! R is a proper monomorphism. We next address
the latter case.
For a set of primes P , an abelian group A is said to be P -cotorsion or Ext*
* -
P -complete if Hom (Z[P -1]; A) = Ext (Z[P -1]; A) = 0. We recall from [2] and
Lemma 5.5 in [5] that, given an abelian group G, if P denotes the set of primes*
* p
for which the map x 7! px is an automorphism of G, then the class of abelian gr*
*oups
A such that Hom (G; A) = 0 = Ext(G; A) consists precisely of the P -local grou*
*ps if
G is torsion, and it consists of the Ext -P -complete abelian groups otherwise.
Theorem 7.4. Suppose that LfK(Z; n) ' K(R; n), where n 1 and R is not
cyclic. Let X be any GEM, and let P be the set of primes p such that multiplica*
*tion
by p is an automorphism of R=Z. If R=Z is torsion, then the groups ssm (LfX) are
P -local for m > n; if R=Z has elements of infinite order, then the groups ssm *
*(LfX)
are Ext -P -complete for m > n.
Proof. Let A = ssm (LfX) with m > n. By Theorem 7.2 we have Ext (R; A) = 0
and Hom (R; A) ~= Hom (Z; A). Hence, by applying the functor Hom (-; A) to t*
*he
short exact sequence 0 ! Z ! R ! R=Z ! 0 we infer that
Hom (R=Z; A) = Ext(R=Z; A) = 0;
so that our claim follows from Lemma 5.5 in [5].
Theorem 7.4 is conveniently illustrated by ordinary homological localization*
* with
coefficients in Z(p)or Z=p, and even better by localization with respect to Mor*
*ava
K-theories; see Examples 7.4 and 7.5 in [4].
Theorem 7.5. Let f be any map and p a prime. Suppose that the transitional
dimension np(f) is finite. If X is any GEM, then:
(1) The groups ssm (LfX) are Z[1=p]-modules for m np(f) + 2 and ssm (LfX)
is p-torsion free if m = np(f) + 1.
(2) If the height ip(f) is finite, then the groups ssm (LfX) are Z[1=p]-mod*
*ules
for m np(f) + 1 and the p-torsion subgroup of ssm (LfX) is annihilated*
* by
pip(f)for m = np(f).
Proof. If m np(f) + 1 and we write A = ssm (LfX), then K(Z=p; m) is f-acyclic
and K(A; m) is f-local. It follows that Hom (Z=p; A) = 0 and hence A is p-tors*
*ion
free. If m np(f) + 2, then we also have Ext (Z=p; A) = 0, which, together with
the fact that A is p-torsion free, guarantees that A is a Z[1=p]-module.
If i = ip(f) is finite, then it follows from Theorem 3.4 that the natural map
K(Z=pr+1; np(f)) ! K(Z=pr; np(f)) is an f-equivalence for r i. If m = np(f)+1,
24 BY CARLES CASACUBERTA, JOSE L. RODRIGUEZ AND JIN-YEN TAI
we obtain that Ext (Z=pr; A) ~=Ext (Z=pr+1; A) for r i. Hence, Ext (Z=p; A) = 0
and we may infer again that A is a Z[1=p]-module. Finally, if m = np(f), then we
can deduce that Hom (Z=pi; A) ~= Hom (Z=pr; A) for r i, from which it follows
that the p-torsion subgroup of A is a Z=pi-module.
Example 7.6. For the map f: K(Z=p; 1) ! * we have np(f) = 0, which implies,
by Theorem 7.5, that the homotopy groups of any f-local GEM are Z[1=p]-modules
in dimensions higher than 1. Indeed, from the fibration
K(Z=p1 ; n - 1) ! K(Z; n) ! K(Z[1=p]; n)
it follows that LfK(Z; n) ' K(Z[1=p]; n) for n 2; cf. [10, x7]. A similar argu*
*ment
shows that LfK(G; n) ' K(GZ[1=p]; n) for every abelian group G and each n 2.
On the other hand, all finite-dimensional CW-complexes are f-local by Miller's *
*main
theorem in [19], yet their homotopy groups need not be Z[1=p]-modules. This sho*
*ws
that the above theorems are false if we omit the assumption that X be a GEM.
Example 7.7. Let f be any map such that Lf is localization with respect to
complex K-homology. Since K(Z=p; 1) is K-local and K(Z=p; 2) is K-acyclic for a*
*ll
primes p (see [4] or [20]), it follows that np(f) = 1 for every p. Thus, Theore*
*m 7.5
tells us that if X is any GEM, then the homotopy groups ssm (XK ) of the K-
localization of X are Q-vector spaces if m 3, and ss2(XK ) is torsion-free. T*
*his
observation enlightens Theorem 2.2 in [20]. Indeed, if X is any 2-connected GEM
then XK is a 2-connected rational GEM. Since the class of K-equivalences with
rational coefficients coincides with the class of ordinary homology equivalence*
*s with
rational coefficients (see Lemma 1.8 in [20]), the rationalization X0 is K-loca*
*l. From
this fact it follows directly that XK ' X0 if X is any 2-connected GEM.
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Departament de Matematiques, Universitat Autonoma de Barcelona, E-08193
Bellaterra, Spain
E-mail address: casac@mat.uab.es, jlrodri@mat.uab.es
Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA
E-mail address: Jin-Yen.Tai@Dartmouth.edu