MODEL STRUCTURES FOR MODULES 133 Model structures for modules

3. Model structures for modules

With the symmetric monoidal smash product and a compatible model structure in place, we are ready to explore ring and module spectra. In this section we construct model structures on the category of modules over a symmetric ring spectrum. We restrict our attention to stable model structures and show that the forgetful functor to symmetric spectra ‘creates’ various such model structure. The forgetful functor also creates various level model structures, but we have no use for that and so will not discuss level model structures forR-modules.

The various stable model structures are also ‘stable’ in the technical sense that the suspension functor on the homotopy category is an equivalence of categories. As consequence of this is that stable homotopy category of modules over a ring spectrum is a triangulated category. The free module of rank one is a small generator.

We originally defined a symmetric ring spectrum in Definition I.1.3 in the ‘explicit’ form, i.e., as a family

{Rn}n≥0of pointed simplicial sets with a pointed Σn-action onRnand Σp×Σq-equivariant multiplication mapsµp,q :Rp∧Rq −→Rp+q and two unit maps subject to an associativity, unit and centrality condition. Using the internal smash product of symmetric spectra we saw in Theorem I.3.8 that a symmetric ring spectrum can equivalently be defined as a symmetric spectrumRtogether with morphismsµ:R∧R−→R

and ι : _S −→ R, called the multiplication and unit map, which satisfy certain associativity and unit conditions. In this ‘implicit’ picture a morphism of symmetric ring spectra is a morphism f :R−→S of symmetric spectra commuting with the multiplication and unit maps, i.e., such thatf◦µ=µ◦(f∧f) and

f ◦ι=ι.

Similarly, ifR is a symmetric ring spectrum, a right R-modulewas originally defined explicitly, but it can also be given in an implicit form as a symmetric spectrumM together with an action mapM∧R−→M

satisfying associativity and unit conditions. A morphism of right R-modules is a morphism of symmetric spectra commuting with the action of R. We denote the category of rightR-modules by mod-R.

The unit S of the smash product is a ring spectrum in a unique way, and S-modules are the same

as symmetric spectra. The smash product of two ring spectra is naturally a ring spectrum. For a ring spectrumR the opposite ring spectrumRopis defined by composing the multiplication with the twist map

R ∧R−→R∧R(so in terms of the bilinear mapsµp,q:Rp∧Rq−→Rp+q, a block permutation appears). The definitions of left modules and bimodules is hopefully clear; leftR-modules andR-T-bimodule can also be defined as right modules over the opposite ring spectrum Rop_{, respectively right modules over the ring} spectrum Rop _{∧ T.}

A formal consequence of having a closed symmetric monoidal smash product of symmetric spectra is that the category ofR-modules inherits a smash product and function objects. The smash productM∧RN of a right R-module M and a left R-module N can be defined as the coequalizer, in the category of symmetric spectra, of the two maps

M ∧ R ∧ N ////M ∧ N

given by the action of R on M and N respectively. Alternatively, one can characterize M ∧RN as the universal example of a symmetric spectrum which receives a bilinear map from M and N which is R- balanced, i.e., all the diagrams

(3.1) Mp∧Rq∧Nr αp,q∧Id Id∧αq,r // Mp∧Nq+r ιp,q+r Mp+q∧Nr _ι p+q,r // (M ∧N)p+q+r

commute. If M happens to be a T-R-bimodule and N an R-S-bimodule, then M ∧R N is naturally a

T-S-bimodule. If R is a commutative ring spectrum, the notions of left and right module coincide and agree with the notion of a symmetric bimodule. In this case∧R is an internal symmetric monoidal smash product for R-modules. There are also symmetric function spectra HomR(M, N) defined as the equalizer

of two maps

Hom(M, N) −→ Hom(R∧M, N).

The first map is induced by the action of R on M, the second map is the composition of R ∧ − : Hom(M, N) −→ Hom(R∧M, R∧N) followed by the map induced by the action of R on N. The in- ternal function spectra and function modules enjoy the ‘usual’ adjointness properties with respect to the various smash products. [spell out]

Theorem3.2. LetRbe a symmetric ring spectrum of topological spaces or simplicial sets. The category

of right R-modules admits the following four stable model structures in which the weak equivalences are those morphisms ofR-modules which are stable equivalences on underlying symmetric spectra.

(i) In the absolute projective stable model structurethe fibrations are those morphisms ofR-modules which are absolute projective stable fibrations on underlying symmetric spectra.

(ii) In thepositive projective stable model structurethe fibrations are those morphisms ofR-modules which are positive projective stable fibrations on underlying symmetric spectra.

(iii) In theabsolute flat stable model structurethe fibrations are those morphisms ofR-modules which are absolute flat stable fibrations on underlying symmetric spectra.

(iv) In thepositive flat stable model structure the fibrations are those morphisms ofR-modules which are positive flat stable fibrations on underlying symmetric spectra.

Moreover we have:

• All four stable model structures are proper, simplicial and cofibrantly generated.

• If R is commutative then all four stable model structures are monoidal with respect to the smash product overR.

If underlying symmetric spectrum ofRis flat, then the category of rightR-modules admits the following twoinjective stable model structuresin which the weak equivalences are those morphisms ofR-modules which are stable equivalences on underlying symmetric spectra.

(v) In the absolute injective stable model structure the fibrations are those morphisms ofR-modules which are absolute injective stable fibrations on underlying symmetric spectra.

(vi) In the positive injective stable model structurethe fibrations are those morphisms of R-modules which are positive injective stable fibrations on underlying symmetric spectra.

Moreover, both injective stable model structures are proper, simplicial and cofibrantly generated. In all six model structures, a cofibration of R-modules is a monomorphism of underlying symmetric spectra.

Proof. In the language of Definition 1.3 of Appendix A we claim that in all of the six cases the forgetful

functor from R-modules to symmetric spectra creates a model structure onR-modules. In Theorem A.1.4 we can find sufficient conditions for this, which we will now verify.

The category of R-modules is complete, cocomplete and simplicial; in fact all limits, colimits, tensors and cotensors with simplicial sets are created on underlying symmetric spectra. In particular the forgetful functor preserves filtered colimits. The forgetful functor has a left adjoint free functor, given by smashing with R. [Smallness]

It remains to check the condition which in practice is often the most difficult one, namely that every (J∧R)-cell complex is a weak equivalence. We claim that in all six cases the free functorX 7→X∧Rtakes stable acyclic cofibrations of symmetric spectra of the respective kind to stable equivalences of R-modules which are monomorphisms. In the first four cases (where we have no assumption onR) this uses that every generating acyclic cofibration i : A−→ B is in particular a flat cofibration, soi∧Id : A∧R −→ B∧R

is injective and a stable equivalence by parts (i) and (iv) of Proposition 1.11. In the ‘injective’ cases (v) and (vi) the argument is slightly different; then the assumption thatRis flat assures that for every injective stable equivalence i:A−→B the morphismi∧Id :A∧R−→B∧R is again injective (by the definition of flatness) and a stable equivalence (by Proposition II.5.14).

So in all the six cases, the free functor− ∧Rtakes the generating stable acyclic cofibrations to injective stable equivalences ofR-modules. Since colimits ofR-modules are created on underlying symmetric spectra,

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