MODEL STRUCTURES FOR MODULES 135 the class of injective stable equivalences is closed under wedges, cobase change and transfinite composition.

So every (J ∧R)-cell complex is a stable equivalence. So we have verified the hypothesis of Theorem 1.4, which thus shows that the forgetful functor creates the six model structure. It also shows that the model structures are simplicial and right proper.

[left proper] [monoidal ifR commutative.] [preservation of cofibrations] [Is there an ‘strongly injective’ stable model structure in which cofibrations are the monomorphisms of

R-modules ? make exercise?]

Proposition 3.3. A morphismf :M −→N of right R-modules is a flat cofibration if and only if for every morphism g:V −→W of leftR-modules the pushout product map

f∧Rg : M ∧RW∪M∧RV N∧RW −→ N∧RW

is an injective morphism of symmetric spectra.

There are also characterizations of flat and projective cofibrations in terms of ‘R-module latching objects’, see Exercise 4.1.

As we just proved, cofibrations of R-modules are always monomorphisms of underlying symmetric spectra, but sometimes more is true. As the special case S=Sof Theorem 3.4 (iii) below we will see that

if R is flat as a symmetric spectrum, then every flat cofibration ofR-modules is also a flat cofibration on underlying symmetric spectra. Similarly, ifRis projective as a symmetric spectrum, then every projective cofibration ofR-modules is also a projective cofibration on underlying symmetric spectra.

For a morphismf :S−→Rof symmetric ring spectra, there is are two adjoint functor pairs relating the modules over S and R. The functors are analogous to restriction and extension respectively coextension of scalars. Every R-module becomes an S-module if we let S act through the homomorphism f; more precisely, given an R-moduleM we define anS-modulef∗M as the same underlying symmetric spectrum as M and withS-action given by the composite

(f∗M)∧S = M ∧S −−−→Id∧f M∧R −−→α M .

We call the resulting functorf∗: mod-R−→mod-Srestriction of scalarsalongf and note that it has both a left and right adjoint. We call the left adjointextension of scalars and denote it byf∗: mod-S−→mod-R.

The left adjoint takes an S-moduleN to the R-modulef∗N =N∧SR, whereS is a leftR-module via f, and with right R-action through the right multiplication action of R on itself. We call the right adjoint of

f∗thecoextension of scalars and denote it byf!: mod-S−→mod-R. The right adjoint takes anS-module

N to theR-modulef!N = Hommod-S(R, N), where S is a right R-module viaf, and with rightR-action through the left multiplication action of Ron itself.

Theorem 3.4. Letf :S−→R be a homomorphism of symmetric ring spectra.

(i) The functor pair

mod-S f∗ // mod-R f∗ o o

is a Quillen adjoint functor pair with respect to the absolute projective, the positive projective, the absolute flat and the positive flat stable model structures on both sides.

(ii) IfS andRare flat as symmetric spectra then(f∗, f∗)is a Quillen adjoint functor pair with respect

to the absolute injective and the positive injective stable model structures on both sides.

(iii) Suppose that the morphism f : S −→ R makes R into a flat (respectively projective) right S- module. Then the functor pair

mod-R f∗ _// mod-S f! o o

is a Quillen adjoint functor pair with respect to the absolute and positive flat stable (respectively absolute and positive projective stable) model structures on both sides. In particular, the restriction

of scalars f∗ then takes flat (respectively projective) cofibrations ofR-modules to flat (respectively projective) cofibrations ofS-modules.

(iv) If the homomorphism f :S −→R is a stable equivalence, then the adjoint functor pairs (f∗, f∗)

and(f∗, f!)are a Quillen equivalences in all the cases when they are Quillen adjoint functors.

Proof. (i) In each case, the weak (i.e., stable) equivalences and the various kinds of fibrations are

defined on underlying symmetric spectra, hence the restriction functor preserves fibrations and acyclic fibrations. By adjointness, the extension functor preserves cofibrations and trivial cofibrations.

(iv) Iff :S −→Ris a stable equivalence, then for every flat rightS-moduleN the morphism

N ∼= N∧SS −→ N∧SR = f∗N

is a stable equivalence. Thus if Y is a fibrant left R-module, an S-module map N −→ Y is a weak equivalence if and only if the adjointR-module mapf∗N−→Y is a weak equivalence. Example 3.5 (Modules over Eilenberg-Mac Lane spectra). For every ring A we have an associated

Eilenberg-Mac Lane ring spectrum, see Example I.2.7. This symmetric spectrum arises from a Γ-space by evaluation on spheres, so it is flat as a symmetric spectrum (Proposition II.5.18). Hence all six model structure of Theorem 3.2 are defined on the category of HA-modules, and they are Quillen-equivalent to each other.

The homotopy category of HA-modules can be described purely algebraically in terms ofA-modules. More precisely, the stable model structures ofHA-modules are Quillen equivalent to the category of chain complexes ofA-modules in any of the model structures which have the quasi-isomorphisms as weak equiv- alences. In particular, we get an equivalence of triangulated categories

Ho(mod-HA) ∼= D(A) to the unbounded derived category of the ring A.

Exercises

Exercise 4.1. LetRby a symmetric ring spectrum. We define anR-bimodule ¯R by

¯

Rn =

(

∗ forn= 0

Rn forn≥1.

We define then-latching objectLR_nM of a rightR-moduleM byLR_nM = (M∧RR¯)n. The latching object has a left action of the symmetric group Σn and a right action of the pointed monoid R0. The inclusion

¯

R−→R is a morphism ofR-bimodules and thus induces a morphism of Σn-R0 simplicial bisets

νn : LRnM = (M ∧RR¯)n −→ (M ∧RR)n∼=Mn . Show:

(i) A morphism f : M −→ N is a flat cofibration of R-modules if and only if the maps νn(f) :

LR nN∪LR

nM Mn−→Nn are cofibrations of rightR0-simplicial sets.

(ii) A morphism f : M −→ N is a projective cofibration of R-modules if and only if the maps

νn(f) :LR_nN∪LR

nM Mn−→Nn are cofibrations of Σn-R0-simplicial bisets.

(Hint: define a suitable R-module analog of the filtration FmA of a symmetric spectrum A so that the proof of Proposition II.5.10 can be adapted.)

History and credits

The projective and injective level and stable model structures for symmetric spectra are constructed in the original paper [25] of Hovey, Shipley and Smith. The flat model structures show up in the literature under the name ofS-model structure. (the ‘S’ refers to the sphere spectrum). The cofibrant objects in this model structure (which we call ‘flat’ and Hovey, Shipley and Smith call ‘S-cofibrant’) and parts of the model structures show up in [25] and in [52], but the first verification of the full model axioms appears in Shipley’s paper [58]. I prefer the term ‘flat’ model structure because the cofibrant objects are very analogous to flat

HISTORY AND CREDITS 137 modules in algebra and because we can then also use the term ‘flat model structure’ for modules over a symmetric ring spectrum. Shipley [58] calls the flat model structure for modules over a symmetric ring spectrum Rthe ‘R-model structure’.

APPENDIX A

1. Tools from model category theory

1.1. Cofibrantly generated model categories and a lifting theorem. In this section we review cofibrantly generated model categories and a general method for creating model category structures. If a model category is cofibrantly generated, its model category structure is completely determined by a set of cofibrations and a set of acyclic cofibrations. The transfinite version of Quillen’s small object argument allows functorial factorization of maps as cofibrations followed by acyclic fibrations and as acyclic cofibra- tions followed by fibrations. Most of the model categories in the literature are cofibrantly generated, e.g. topological spaces and simplicial sets, as are all model structures involving symmetric spectra which we discuss in this book.

The only complicated part of the definition of a cofibrantly generated model category is formulating the definition of relative smallness. For this we need to consider the following set theoretic concepts. The reader might keep in mind the example of a compact topological space which isℵ0-small relative to closed inclusions.

Ordinals and cardinals. Anordinal γ is an ordered isomorphism class of well ordered sets; it can be identified with the well ordered set of all preceding ordinals. For an ordinalγ, the same symbol will denote the associated poset category. The latter has an initial object ∅, the empty ordinal. An ordinal κ is a

cardinal if its cardinality is larger than that of any preceding ordinal. A cardinalκ is calledregular if for every set of sets {Xj}j∈J indexed by a setJ of cardinality less thanκsuch that the cardinality of eachXj is less than that of κ, then the cardinality of the union S

JXj is also less than that ofκ. The successor cardinal (the smallest cardinal of larger cardinality) of every cardinal is regular.

Transfinite composition. LetC be a cocomplete category and γ a well ordered set which we identify with its poset category. A functor V :γ −→ C is called a γ-sequence if for every limit ordinal β < γ the natural map colimV|β −→V(β) is an isomorphism. The map V(∅)−→ colimγV is called the transfinite composition of the maps of V. A subcategory C1 ⊂ C is said to be closed under transfinite composition if for every ordinal γ and every γ-sequence V:γ−→ C with the map V(α)−→ V(α+ 1) inC1 for every ordinal α < γ, the induced mapV(∅)−→colimγV is also inC1. Examples of such subcategories are the cofibrations or the acyclic cofibrations in a closed model category.

Relatively small objects. Consider a cocomplete category C and a subcategory C1 ⊂ C closed under transfinite composition. If κ is a regular cardinal, an object C ∈ C is called κ-small relative to C1 if for every regular cardinalλ≥κand every functor V:λ−→ C1 which is aλ-sequence inC, the map

colimλHomC(C, V) −→HomC(C,colimλV)

is an isomorphism. An object C ∈ C is calledsmall relative toC1 if there exists a regular cardinalκsuch that C isκ-small relative toC1.

I-injectives,I-cofibrations andI-cell complexes. Given a cocomplete categoryCand a classI of maps, we denote

• by I-inj the class of maps which have the right lifting property with respect to the maps in I. Maps inI-inj are referred to asI-injectives.

• by I-cof the class of maps which have the left lifting property with respect to the maps in I-inj. Maps inI-cof are referred to asI-cofibrations.

• byI-cell⊂I-cof the class of the (possibly transfinite) compositions of pushouts (cobase changes) of maps inI. Maps inI-cell are referred to asI-cell complexes.

In [46, p. II 3.4] Quillen formulates hissmall object argument, which immediately became a standard tool in model category theory. In our context we will need a transfinite version of the small object argument, so we work with the ‘cofibrantly generated model category’, which we now recall. Note that here I has to be aset, not just a class of maps. The obvious analogue of Quillen’s small object argument would seem to require that coproducts are included in theI-cell complexes. In fact, any coproduct of anI-cell complex is already anI-cell complex, see [24, 2.1.6].

Lemma 1.1. LetC be a cocomplete category andI a set of maps inCwhose domains are small relative

toI-cell. Then

• there is a functorial factorization of any map f in C asf =qi with q∈I-injand i∈I-celland thus

• every I-cofibration is a retract of anI-cell complex.

Definition 1.2. A model category C is called cofibrantly generatedif it is complete and cocomplete

and there exists a set of cofibrationsI and a set of acyclic cofibrationsJ such that

• the fibrations are precisely the J-injectives;

• the acyclic fibrations are precisely theI-injectives;

• the domain of each map in I(resp. inJ) is small relative toI-cell (resp.J-cell). Moreover, here the (acyclic) cofibrations are theI (J)-cofibrations.

For a specific choice of I and J as in the definition of a cofibrantly generated model category, the maps in I (resp. J) will be referred to as generating cofibrations (resp. generating acyclic cofibrations). In cofibrantly generated model categories, a map may be functorially factored as an acyclic cofibration followed by a fibration and as a cofibration followed by an acyclic fibration.

Definition 1.3. LetC be a model category

R : D −→ C

a functor. We say that R creates a model structureon the category Dif the following definitions make D

into a model category: a morphismf inDis a

• weak equivalence if the morphismR(f) is a weak equivalence inC,

• fibration if the morphismR(f) is a fibration inC,

• cofibration if it has the left lifting property with respect to all morphisms in D which are both fibrations and weak equivalences.

Theorem 1.4. LetC be a model category,D a category which is complete and cocomplete and let

R : D −→ C : L

be a pair of adjoint functors such that R commutes with filtered colimits. LetI (J) be a set of generating cofibrations (resp. acyclic cofibrations) for the cofibrantly generated model categoryC. LetLI (resp.LJ) be the image of these sets under the left adjointL. Assume that the domains ofLI (LJ) are small relative to LI-cell(LJ-cell). Finally, suppose everyLJ-cell complex is a weak equivalence. ThenR:D −→ Ccreates a model structure on Dwhich is cofibrantly generated withLI (LJ) a generating set of (acyclic) cofibrations.

If the model category C is right proper, then so is the model structure onD.

If C andD are simplicially enriched, the adjunction(L, R) is simplicial, and the model structure ofC is simplicial, then the model structure on Dis again simplicial.

If C andD are topologically enriched, the adjunction (L, R) is continuous, and the model structure of C is topological, then the model structure onDis again topological.