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STRICT MODULES AND HOMOTOPY MODULES

IN STABLE HOMOTOPY

JAVIER J. GUTI ´ERREZ

(communicated by Johannes Huebschmann)

Abstract

LetR be any associative ring with unit and letHRdenote the corresponding Eilenberg–Mac Lane spectrum. We show that the category of algebras over the monadX 7→HR∧X on the homotopy category of spectra is equivalent to the homo-topy category associated to a model category of HR-module spectra, if the ringR is a field or a subring of the rationals, but not for all rings.

1.

Introduction

Classically, ring spectra and module spectra were defined as objects of the stable homotopy category equipped with suitable structure maps. The stable homotopy category, as described by Adams in [1], has a smash product which is associative and commutative up to homotopy. The structure maps that define ring spectra and module spectra give rise to diagrams that commute up to homotopy. For a given ring spectrumE, the E-modules in this sense, together with the E-module maps, form a subcategory which can be seen as the Eilenberg–Moore category associated with the monad defined by X 7→ E ∧X, i.e., the category of algebras over this monad. We call this category thecategory of homotopy E-modules.

The recent discovery of new structured model categories for stable homotopy, such as the categories of S-modules [7] or symmetric spectra [9], equipped with a strictly associative and commutative smash product, allows one to define strict ring spectra (the monoids in the category) and strict module spectra (modules over monoids). The structure maps for these objects give rise to diagrams that truly commute in the model category. Thus, for a strict ring spectrumE, we can as well consider the homotopy category of strictE-modules, by endowing the category of strictE-modules with a model structure as in [7] or in [9].

The categories of strict modules have better properties than the categories of homotopy modules. The fibre of anE-module map of strict E-modules is a strict

E-module, yet this need not be true for homotopyE-modules. The model category of strict HR-modules is Quillen equivalent to the category Ch(R) of unbounded chain complexes ofR-modules; see [7], [12], [13].

The author was supported by MCyT grants BFM2001–2031 and FP98 16587447. Received January 12, 2005; published on May 16, 2005.

2000 Mathematics Subject Classification: 55P43, 18E30, 55P42.

Key words and phrases: Ring spectrum, module spectrum, derived category. c

The homotopy category of strictHR-modules is not equivalent to the category of homotopyHR-modules in general. However, they are equivalent in some special cases, for example when R=Z (the homotopy HZ-modules are also called stable GEMs). We give a sufficient condition for a ring Rin order that there is an equiv-alence between the homotopy category of strict HR-modules and the category of homotopyHR-modules. This condition is fulfilled by fields and by subrings ofQ.

Acknowledgements.I would like to thank my advisor Carles Casacuberta for his support and encouragement in writing this paper, and also to John Greenlees and Stephan Schwede for many helpful discussions and suggestions during my stay at the Isaac Newton Institute in Cambridge in 2002.

2.

Monads and Eilenberg–Moore categories

In this section we recall the definition of a monad on a category, and collect some basic results about Eilenberg–Moore categories. These and other facts about monads can be found in [3, Ch. 3], [4, Ch. 4] or [11, Ch. VI].

If (N, n) is another T-algebra, then a morphism of T-algebras or a T-morphism

f: (M, m) −→ (N, n) is a morphism f: M −→ N in C such that the following whose objects are theT-algebras and whose morphisms are theT-morphisms. The categoryCT

There is a forgetful functorU:CT of monads S−→Tis a natural transformationλ:S −→T such that the following diagrams commute:

Remark 2.1. Any morphism of monadsλ:S −→T yields a faithful functor between the categories of algebras Q: CT

−→ CS

, since any T-algebra has an S-algebra structure via the morphismλ. Thus, Qis defined asQ(M, m) = (M, m◦λM) and

whereU is the forgetful functor. This shows that the functorQis faithful.

Example 2.2. Let Ab be the category of abelian groups and let R be a ring with unit. The functor R⊗ −: Ab −→ Ab together with the product and the unit of

R is a monad on the category of abelian groups. The Eilenberg–Moore category associated with this monad is the category of leftR-modules.

3.

Stable categories of

E

-modules

In this section we describe, for a ring spectrum E, the category of strict E -modules and the category of homotopyE-modules as particular cases of Eilenberg– Moore categories associated with monads. We will work in the category SpΣ

It is said thatE iscommutative ifµ◦τ =µwhereτ:E∧E−→E∧E is the twist

Example 3.1. IfR is an associative ring with unit andM is a leftR-module, then the Eilenberg–Mac Lane spectrum HR is a ring spectrum and the spectrumHM

is an HR-module spectrum. The structure maps of HR and HM come from the product and the unit of R, and from the structure homomorphism of M as an

R-module.

A map of E-modules or an E-module map f: (M, m) −→ (N, n) is a map

f:M −→N such that the following diagram commutes:

E∧M

. This functor sends any X to an E-module spectrum E∧X. The natural transformationsη∧Id: IdSpΣ −→E∧ − andµ∧Id:E∧E∧ − −→E∧ − form

a monad on the category of symmetric spectra, by the commutativity of (3.1). The Eilenberg–Moore category associated with the monad (E∧ −, η∧Id, µ∧Id) will be denoted byE-modand called thecategory of strictE-modules. By [13], this category admits a model category structure. IfHo(E-mod) is the corresponding homotopy category, andM and N are objects in this category, we denote by [M, N]E-modthe

group of morphisms betweenM andN in Ho(E-mod). Thus, (1) yields a bijection

E-mod(E∧M, N)∼=SpΣ (M, N)

for anyE-module spectrumNand anyM. This bijection does not induce a bijection of homotopy classes of maps in general, as shown in Corollaries 4.5 and 4.6.

Now, for a ring spectrumE∈SpΣ_{, consider the monad (E∧ −, η∧Id, µ∧Id) on} the homotopy category Ho(SpΣ_{). The Eilenberg–Moore category associated with} this monad will be called the category of homotopy E-modules, and denoted by

E-hmod. IfM andN are objects inE-hmod, we denote by [M, N]E-hmodthe group

of morphisms between them in the Eilenberg–Moore category. IfN is a homotopy

E-module andM is any spectrum, then the bijection (1) gives an isomorphism

[E∧M, N]E-hmod ∼= [M, N].

4.

Homotopy modules and derived categories

The categories E-hmod and Ho(E-mod), defined in the previous section, are very different in general. In this section we compare these two categories in the case where E is the ring spectrumHR, for some associative ring R with unit. In what followsR-modules will be left modules.

The derived categoryD(R) of the ringRis defined as the homotopy category of Ch(R), the model category of unbounded chain complexes ofR-modules; see [8]. The weak equivalences are the quasi-isomorphisms, i.e., the maps inducing isomorphisms in homology. IfE is anyR-module, we will denote byE[k] the chain complex

· · · −→0−→0−→E−→0−→0−→ · · ·

whereEis located in dimensionk. IfAandBare twoR-modules, then the following holds:

D(R)(A[0], B[k]) = ExtkR(A, B);

see [14, Ch. 10] for a useful description of the derived category.

Theprojective dimension pd(A) of anR-moduleAis the minimum integern(if it exists) such that there is a projective resolution ofAof lengthn,

0−→Pn−→Pn−1−→ · · · −→P0−→A−→0.

If no such integer exists, we say that pd(A) =∞. Theglobal dimension of a ringR

is defined as gd(R) = sup_A∈R-mod{pd(A)}. For example, gd(Z) = 1, gd(Z/p

2_{) =∞} ifpis a prime, gd(R[x1, . . . , xn]) = gd(R) +n.

The rings R with gd(R) = 0 are called semisimple. All fields and finite direct products of fields are semisimple rings. In general, gd(R) = 0 if and only ifR is a finite direct product of matrix rings over division rings, by the Wedderburn–Artin Theorem; see [2,§13], for example. Thus, if R is commutative, then gd(R) = 0 if and only ifRis a finite direct product of fields.

The groups ExtkR are closely related with the global dimension of the ring R.

A classical theorem in homological algebra states that gd(R) = k if and only if ExtiR(A, B) = 0 fori > k and allR-modulesAandB; see [14].

Proposition 4.1. If gd(R)61, then any chain complex ofR-modules

C: · · · −→Cn d

n

−→Cn−1

dn−1

−→ · · · −→C0

d0

−→C−1−→ · · ·

is weakly equivalent, and hence isomorphic inD(R), to⊕k∈_ZHk(C)[k].

Proof. If gd(R) = 0, then there exists a mapp_ek:Hk(C)−→kerdk⊂Ck such that π◦p_ek =id, whereπdenotes the projection kerdk−→Hk(C), since everyR-module

is projective. Hence, for eachk∈Zwe have a map

· · · //_Ck //C_k−1 //· · ·

Hk(C)

e

pk

O

O

/

/₀

O

and this yields a map of chain complexes φ: ⊕k∈_ZHk(C)[k] −→ C inducing an

projective resolution of thek-th homology groupHk(C). TakeAkto be the complex · · · →0→Rk→Fk→0→ · · · withFk in dimensionkandRk in dimensionk+ 1.

Now we construct a map from⊕k∈_ZAk toCinducing an isomorphism in homology.

For eachk∈Z, we have the following diagram:

Ck+1

Remark 4.2. Note that Proposition 4.1 does not hold if gd(R)>1. Suppose that gd(R) =k >1 and consider a nonzero elementξ in Extk

R(M, N), where M andN

areR-modules. This elementξ can be represented by an extension of modules

0−→N−→Ek−→ · · · −→E1−→M −→0,

whereE1, . . . , Ek are free [10, Corollary III.6.5]. Take now the chain complex

E: · · · −→0−→Ek −→Ek−1−→ · · · −→E1−→0−→ · · ·

whereE1is in dimension 0. This complex has homology only in dimensions 0 andk− 1, namelyH0(E) =M andHk−1(E) =N. But if this complex is quasi-isomorphic

to M[0]⊕N[k−1] thenξ= 0, becauseE1, . . . , Ek are free and hence there exists

commutative diagram

The following result was first proved in [12]. A recent generalization can be found in [13].

Theorem 4.3. For any ringRthere is a Quillen equivalence between the model cat-egory of unbounded chain complexes ofR-modules and the model category of (strict)

HR-modules. This equivalence induces an equivalence between the homotopy cate-gories D(R)andHo(HR-mod)that sends eachHR-moduleM to a chain complex

C such thatHk(C)∼=πk(M)for everyk∈Z.

The objects of the categoryHR-hmodare precisely the stableR-GEMs and have been studied in [6, Section 5]. Recall that a spectrum E∈Ho(SpΣ

particular case. Thus, the study of morphisms inHZ-hmodamounts to the study of [HA,Σk_HB]

HZ-hmod. These abelian groups have already been described in [6,

Section 5], as follows:

Proposition 4.4. For all abelian groupsA andB, the following holds: [HA, HB]H_Z-hmod

Proof. The left adjoint of the monad given by HZ∧ − yields an isomorphism [M A,Σk_{HB]∼= [HA,Σ}k_HB]

HZ-hmod because HA≃HZ∧M A whereM A denotes

a Moore spectrum for the abelian groupA; see [1]. Now use the exact sequence

0→Ext(A, πk+1X)→[ΣkM A, X]→Hom(A, πkX)→0

in the caseX =HB.

Corollary 4.5. Given any ring R and R-modules A and B, if k 6= 0,1, then [HA,Σk_HB]

H R-hmod= 0.

Proof. The inclusionZ֒→Rprovides a natural transformationHZ∧− −→HR∧−

Corollary 4.6. There is no equivalence of categories between the categories

Ho(HR-mod)andHR-hmodif gd(R)>1.

Proof. Suppose that there exists an equivalence between the categoriesHR-hmod

andHo(HR-mod). Then for anyR-modulesA,B and anyk∈Z, we have that

[HA,ΣkHB]H R-hmod∼= [HA,Σ

k_HB]

HR-mod= Ext

k R(A, B)

by Theorem 4.3. But [HA,Σk_HB]

H R-hmod = 0 for k 6= 0,1 by Corollary 4.5, and

this is a contradiction since gd(R)>1.

We will now discuss [HA,Σk_HB]

HR-hmod in the cases k= 0 andk = 1, for any

ringR. The following proposition generalizes Proposition 4.4 for any ringR in the casek= 0.

Proposition 4.7. For any ringRand allR-modulesAandB, the correspondence

f 7→π0(f)yields a natural isomorphism [HA, HB]HR-hmod

∼

= HomR(A, B).

Proof. Letf:HA−→HBbe any map. Recall thatHAandHB areHR-modules because A and B are R-modules. The map f will be a map in HR-hmod if the diagram

HR∧HA IdH R∧f //

mHA

²

²

HR∧HB

mH B

²

²

HA _f //_HB

commutes up to homotopy. We can define a map Φ : [HA, HB]−→[HR∧HA, HB] by Φ(f) =f◦mHA−mHB◦(IdHR∧f). Thenf is a map inHR-hmodif and only

iff ∈ker Φ. But [HA, HB]∼= Hom(A, B) and [HR∧HA, HB]∼= Hom(R⊗A, B). The mapf is in ker Φ if and only iff(ra) =rf(a) for allr∈Randa∈A, and this is the same as stating thatf ∈HomR(A, B).

The case k = 1 is more involved. Although we can give a description of [HA,ΣHB]H R-hmod as the kernel of a map Φ : [HA,ΣHB]−→ [HR∧HA,ΣHB],

as in the proof of Proposition 4.7, and

[HA,ΣHB]∼= Ext(A, B) and [HR∧HA,ΣHB]∼= Ext(R⊗A, B),

it turns out that [HA,ΣHB]H R-hmod6∼= ExtR(A, B) in general. Indeed, suppose that

[HA,ΣHB]H R-hmod∼= ExtR(A, B). Then, by Remark 2.1 and Proposition 4.4, there

would be an injective map

ExtR(A, B)// //Ext(A, B)

Example 4.8. Let R = Q[x], A = Q[x]/(xn_{) and B = Q. Then Ext}

R(A, B) 6= 0

because the exact sequence

Q// //_Q[x]/(xn+1₎ ////Q[x]/(xn₎

does not split. If this splitting did exist, then xn _{= 0 in Q[x]/(x}n+1_{), which is a} contradiction. On the other hand, Ext(A, B) = 0 as abelian groups because Q is divisible.

Note that this example shows that, for the ring R = Q[x], which has global dimension 1, there is no possible equivalence of categories between Ho(HR-mod) andHR-hmod.

5.

An equivalence of categories

In this section we study for which rings R there is an equivalence of categories betweenHo(HR-mod) and HR-hmod. As we have seen, Corollary 4.6 states that there is no possible equivalence if gd(R)>1. But not all rings of global dimension one yield such an equivalence, as illustrated by Example 4.8. However, as we next show, the equivalence holds if the ringRis a field orRis a subring of the rationals. Proposition 5.3 of [6] can be extended to the case ofHR-modules when R is a field orR is a torsion free solid ring. For these rings, equality (5) also holds if one replacesZbyR. IfRis a field, then everyR-module splits as a direct sum of copies ofR, and hence

[HA,ΣHB]H R-hmod∼= [∨iHR,ΣHB]H R-hmod∼= Y

i

[S,ΣHB] = 0.

A ringR issolid if the multiplication induces an isomorphism R⊗R∼=R, where the tensor product is overZ. Solid rings were introduced in [5]. IfRis solid, then in particularR⊗A∼=AwheneverAis anR-module. This implies that, ifR is solid, then

HomR(A, B)∼= Hom(A, B) and ExtR(A, B)∼= Ext(A, B).

IfR is torsion-free, thenHR∧M A≃H(R⊗A) for anyR-moduleA. If a ring R

satisfies these two conditions, then [HA,ΣHB]HR-hmod= ExtR(A, B).

Lemma 5.1. If R is a torsion-free solid ring of global dimension one, thenRis a subring of the rationals.

Proof. This follows from the classification of solid rings (see [5]).

Theorem 5.2. If R is a field or R is a subring of the rationals, then there is an equivalence of categories betweenHo(HR-mod)andHR-hmod.

Proof. We construct a functor Φ : HR-hmod −→ D(R) that is an equivalence of categories. It will be enough to define the functor on objects of the form Σi_{HA, since}

anyM ∈HR-hmodis isomorphic to∨k∈_ZΣkHAk inHR-hmod, and on morphisms

φis an equivalence of categories,Ho(HR-mod)≃HR-hmodby Theorem 4.3. We consider separately the case of a field and of a subring of the rationals.

IfRis a field, then gd(R) = 0 and hence ExtR(A, B) = 0. We define Φ(ΣkHA) = A[k] and, thus, if M ∈ HR-hmod is such that M ≃ ∨k∈_ZΣkHAk, then Φ(M) = ⊕k∈_ZAk[k]. Thus, for a map f:HA −→ HB we define Φ(f) = π0(f), the

cor-responding map between A[0] and B[0]. Now, Φ is a functor and it is full and faithful. Moreover, every object inD(R) lies in the image of Φ up to isomorphism by Proposition 4.1, so it is an equivalence of categories.

IfRis a subring ofQ, we define Φ(Σk_{HA) =P}

k(A) wherePk(A) is the complex

· · · −→0−→Rk −→Fk−→0−→ · · ·

withFk in dimensionk, andRk →Fk→A is a projective resolution ofA. IfM ≃ ∨k∈_ZΣkHAk, then Φ(M) =⊕k∈_ZPk(Ak). A mapf ∈[HA, HB]HR-hmodcorresponds

to a morphism of R-modules fromA toB and hence lifts to a mapfebetween the projective resolutions ofAand B

RA //

²

²

FA

²

²

RB //FB.

This yields a map from the complex P0(A) = Φ(HA) to P0(B) = Φ(HB). We define Φ(f) =fe.

Similarly, a mapg∈[HA,ΣHB]HR-hmod lifts to a mapegbetween the complexes P0(A) = Φ(HA) andP1(B) = Φ(ΣHB),

RA //

²

²

FA

RB //FB.

We define Φ(g) =eg. The Yoneda product

Exti

R(B, C)⊗Ext j R(A, B)

Y

−→Exti_R+j(A, C)

makes Φ a functor. This functor is full and faithful by Proposition 4.7 and because [HA,ΣHB]H R-hmod= ExtR(A, B). By Proposition 4.1, every object inD(R) lies in

the image of Φ up to isomorphism, so it is an equivalence of categories.

References

[1] J. F. Adams,Stable Homotopy and Generalised Homology.Chicago Lectures in Mathematics. University of Chicago Press, Chicago, London, 1974.

[2] F. W. Anderson and K. R. Fuller,Rings and Categories of Modules.Graduate Texts in Mathematics, vol. 13. Springer-Verlag, New York, 1974.

[4] F. Borceux, Handbook of Categorical Algebra. Basic Category Theory. En-cyclopedia of Mathematics and its Applications, 50. Cambridge University Press, Cambridge, 1994.

[5] A. K. Bousfield and D. M. Kan, The core of a ring. J. Pure Appl. Algebra2

(1972), pp. 73–81.

[6] C. Casacuberta J. J. Guti´errez, Homotopical localizations of module spectra. To appear inTrans. Amer. Math. Soc..

[7] A. D. Elmendorf, I. Kriz, M. A. Mandell J. P. May,Rings, Modules, and Al-gebras in Stable Homotopy Theory. Mathematical Surveys and Monographs, 47. Amer. Math. Soc., Providence, 1997.

[8] M. Hovey,Model Categories.Mathematical Surveys and Monographs, vol. 63. Amer. Math. Soc., Providence, 1999.

[9] M. Hovey, B. Shipley J. Smith, Symmetric spectra. J. Amer. Math. Soc.13

(2000), no. 1, pp. 149–208.

[10] S. Mac Lane, Homology. Die Grundlehren der mathematischen Wis-senschaften, Band 114. Springer-Verlag, New York, 1971

[11] S. Mac Lane,Categories for the Working Mathematician.Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York, 1971.

[12] A. Robinson, The extraordinary derived category.Math. Z.96(1987), no. 2, pp. 231–238.

[13] S. Schwede B. Shipley, Stable model categories are categories of modules. Topology42(2003), no. 1, pp. 103–153.

[14] C. A. Weibel, An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge, 1994.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2005/n1a3/v7n1a3.(dvi,ps,pdf)

Javier J. Guti´errez javier.gutierrez@ub.edu

Departament d’ `Algebra i Geometria Universitat de Barcelona