MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY

A. D. ELMENDORF, I. KRIZ, M. A. MANDELL, AND J. P. MAY

Contents

Introduction 1 1. Spectra and the stable homotopy category 6 2. Smash products and twisted half-smash products 11 3. The category of 𝕃-spectra 13 4. The smash product of 𝕃-spectra and function 𝕃-spectra 15 5. The category of S-modules 19 6. S-algebras and their categories of modules 21 7. The smash product of R-modules and function R-modules 24 8. Tor and Ext in topology and algebra 28 9. Universal coefficient and Künneth spectral sequences 31 10. Algebraic constructions in the derived category of R-modules 33 11. R-ring structures on localizations and on quotients by ideals 36 12. The specialization to MU-modules and MU-ring spectra 38 References 40

Introduction

It is a truism that algebraic topology is a very young subject. In some of its most fundamental branches, the foundations have not yet reached a state of shared consensus. Our theme will be stable homotopy theory and an emerging consensus on what its foundations should be. The consensus is different than would have been the case as recently as a decade ago. We shall illustrate the force of the change of paradigm with new constructions of some of the most basic objects in modern algebraic topology, namely the various spectra and cohomology theories that can be derived from complex cobordism. The two following articles will give introductions to completions in stable homotopy theory and to equivariant stable homotopy theory. The three papers have a common theme: the relationship between commutative algebra and stable homotopy theory, both relations of analogy and relations of application.

Stable homotopy theory began around 1937 with the Freudenthal suspension theorem. In simplest terms, it states that, if q is small relative to n, then π_{n+q}(S^{n}) is independent of n. Stable phenomena had of course appeared earlier, at least implicitly: reduced homology and cohomology are examples of functors that are invariant under suspension without limitation on dimension. Stable homotopy theory emerged as a distinct branch of algebraic topology with Adams’ introduction of his eponymous spectral sequence and his spectacular conceptual use of the notion of stable phenomena in his solution to the Hopf invariant one problem. Its centrality was reinforced by two related developments that occurred at very nearly the same time, in the late 1950’s. One was the introduction of generalized homology and cohomology theories and especially K-theory, by Atiyah and Hirzebruch. The other was the work of Thom which showed how to reduce the problem of classifying manifolds up to cobordism to a problem, more importantly, a solvable problem, in stable homotopy theory.

The reduction of geometric phenomena to solvable problems in stable homotopy theory has remained an important mathematical theme, the most recent major success being Stolz’s use of Spin cobordism to study the classification of manifolds with positive scalar curvature. In an entirely different direction, the early 1970’s saw Quillen’s introduction of higher algebraic K-theory and the recognition by Segal and others that it could be viewed as a construction in stable homotopy theory. With algebraic K-theory as an intermediary, there has been a growing volume of work that relates algebraic geometry to stable homotopy theory. With Waldhausen’s introduction of the algebraic K-theory of spaces in the late 1970’s, stable homotopy became a bridge between algebraic K-theory and the study of diffeomorphisms of manifolds. Within algebraic topology, the study of stable homotopy theory has been and remains the focus of much of the best work in the subject. The study of nilpotence and periodic phenomena by Hopkins, Mahowald, Ravenel, and many others has been especially successful.

We shall focus on the study of structured ring, module, and algebra spectra. This study plays a significant role in all of the directions of work that we have just mentioned and would have been technically impossible within the foundational consensus that existed a decade ago.

Stable homotopy theory demands a category in which to work. One could set up the ordinary Adams spectral sequence ad hoc, as Adams did, but it would be ugly at best to set up the Adams spectral sequence based on a generalized homology theory that way. One wants objects – called spectra – that play the role of spaces in unstable homotopy theory, and one wants a category in which all of the usual constructions on spaces are present and, up to homotopy, the suspension functor is an equivalence. At this point, we introduce a sharp distinction: there is a category of point-set level objects, and there is an associated derived category. There has been consensus on what the latter should be, up to equivalence of categories, since the fundamental work of Boardman in the 1960’s. The change in paradigm concerns the point-set level category that underlies the stable homotopy category. There is a growing recognition that one needs a good point-set level category in order to study stable topological algebra seriously.

There is an analogy with algebra that is fundamental to an understanding of this area of mathematics. Suppose given a (discrete) commutative ring R. It has an associated category ℳ_{R} of (ℤ-graded) chain complexes, there is a notion of homotopy between maps of chain complexes, and there is a resulting homotopy category hℳ_{R}. However, this is not the category that algebraists are interested in. For example, if R-modules M and N are regarded as chain complexes concentrated in degree zero, then, in the derived category, the homology of their tensor product should be their torsion product Tor_{∗}^{R}(M, N). Formally, the fundamental invariants of chain complexes are their homology groups, and one constructs a category that reflects this. A map of chain complexes is said to be a quasi-isomorphism if it induces an isomorphism of homology groups. The derived category 𝒟_{R} is obtained by adjoining formal inverses to the quasi-isomorphisms. The best way to make this rigorous is to introduce a notion of cell R-module such that every quasi-isomorphism between cell R-modules is a chain homotopy equivalence (Whitehead theorem) and every chain complex is quasi-isomorphic to a cell R-module. Then 𝒟_{R} is equivalent to the ordinary homotopy category of cell R-modules. See [15, 21]. This is a topologist’s way of thinking about the appropriate generalization to chain complexes of projective resolutions of modules.

We think of the sphere spectrum S as the analog of R. We think of spectra as analogs of chain complexes, or rather as a first approximation to the definitive analogs, which will be S-modules. We let 𝒮 denote the category of spectra. There is a notion of homotopy of maps between spectra, and there is a resulting homotopy category h𝒮. The fundamental invariants of spectra are their homotopy groups, and a map of spectra is a weak equivalence if it induces an isomorphism of homotopy groups. The stable homotopy category, which we denote by h̄𝒮, is obtained by formally inverting the weak equivalences. This is made rigorous by introducing CW spectra. A weak equivalence between CW spectra is a homotopy equivalence and every spectrum is weakly equivalent to a CW spectrum. Then h̄𝒮 is equivalent to the ordinary homotopy category of CW spectra.

Now the category ℳ_{R} has an associative and commutative tensor product. If we regard R as a chain complex concentrated in degree zero, then R is a unit for the tensor product. A differential R-algebra A is a chain complex with a unit R ⟶ A and product A ⊗_{R} A ⟶ A such that the evident associativity and unity diagrams commute. It is commutative if the evident commutativity diagram also commutes. These are, obviously enough, point-set level structures. Algebraists would have trouble taking seriously the idea of an algebra defined in 𝒟_{R}, with unit and product only defined in that category.

The category 𝒮 has a smash product but, in contrast with the tensor product, it is not associative, commutative, or unital. The induced smash product on the stable homotopy category h̄𝒮 is associative and commutative, and it has S as unit. Topologists routinely study ring spectra, which are objects E of h̄𝒮 with a unit η : S ⟶ E and product ϕ : E ∧ E ⟶ E such that the evident unit diagrams commute; that is, ϕ∘(η ∧id) = id = ϕ∘(id ∧η) in h̄𝒮. Similarly, E is associative or commutative if the appropriate diagrams commute in h̄𝒮. Given that the point-set level smash product is not associative or commutative, it would seem at first sight that these up to homotopy notions are the only ones possible.

It is a recent discovery that there is a category ℳ_{S} of S-modules that has an associative, commutative, and unital smash product ∧_{S} [11]. Its objects are spectra with additional structure, and we say that a map of S-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category 𝒟_{S} is obtained from ℳ_{S} by formally inverting the weak equivalences, and 𝒟_{S} is equivalent to the stable homotopy category h̄𝒮. Again, this is made rigorous by a theory of CW S-modules that is just like the theory of CW spectra.

In the category ℳ_{S}, we have a point-set level notion of an S-algebra R that is defined in terms of maps η : S ⟶ R and ϕ : R ∧_{S} R ⟶ R in ℳ_{S} such that the standard unit and associativity diagrams commute on the point-set level; we say that R is commutative if the standard commutativity diagram also commutes. There were earlier notions with a similar flavor, namely the A_{∞} and E_{∞} ring spectra introduced in [19, 20]. Here “A_{∞}” stands historically for “associative up to an infinite sequence of higher homotopies”; similarly, “E_{∞}” stands for “homotopy everything”, meaning that the product is associative and commutative up to all higher coherence homotopies. With the definitions just given, the higher homotopies are hidden in the definition of the associative and commutative smash product in ℳ_{S}, but these definitions are essentially equivalent to the earlier ones, in which the higher homotopies were exhibited in terms of an “operad action”. It is tempting to simply call these objects associative and commutative ring spectra, but that would be a mistake. These terms have long established meanings, as associative and commutative rings in the stable homotopy category, and the more precise point-set level notions do not make the older notions obsolete: there are plenty of examples of associative or commutative ring spectra that do not admit structures of A_{∞} or E_{∞} ring spectra. It is part of the new paradigm that one must always be aware of when one is working in the derived category and when one is working on the point-set level.

Now fix an S-algebra R. An R-module M is an S-module together with a map μ : R ∧_{S} M ⟶ M such that the evident unit and transitivity diagrams commute. Let ℳ_{R} be the category of R-modules. Again we have a homotopy category hℳ_{R} and a derived category 𝒟_{R} that is obtained from it by inverting the weak equivalences, by which we mean the maps of R-modules that are weak equivalences of underlying spectra. The construction of 𝒟_{R} is made rigorous by a theory of cell R-modules, the one slight catch being that, unless R is connective, in the sense that its homotopy groups are zero in negative degrees, we cannot insist that cells be attached only to cells of lower dimension, so that our cell Rmodules cannot be restricted to be CW R-modules. These categories enjoy all of the good properties that we have described in the special case R = S. There is an associative, commutative, and unital smash product over R. We can therefore go on to define R-algebras and commutative R-algebras A in terms of point-set level associative, unital, and commutative multiplications A ∧_{R} A ⟶ A. We can also define derived category level associative and commutative R-ring spectra A, exactly like the classical associative and commutative ring spectra in the stable homotopy category.

It is the derived category 𝒟_{R} that we wish to focus on in describing the current state of the art in stable homotopy theory. We can mimic classical commutative algebra in this category. In particular, for an ideal I and multiplicatively closed subset Y in the coefficient ring R_{∗} = π_{∗}(R), we will show how to construct quotients M/IM and localizations M[Y^{−1}]. When applied with R taken to be the representing spectrum MU for complex cobordism, these constructions specialize to give simple constructions of various spectra that are central to modern stable homotopy theory, such as the Morava K-theory spectra. Moreover, we shall see that these spectra are MU-ring spectra.

This account is largely a summary of parts of the more complete and technical paper [11], to which the reader is referred for further background, detailed proofs, and many more applications.

1. Spectra and the stable homotopy category

We here give a bare bones summary of the construction of the stable homotopy category, referring to [16] and [11] for details and to [22] for a more leisurely exposition. We aim to give just enough of the basic definitional framework that the reader can feel comfortable with the ideas.

By Brown’s representability theorem [6], if E^{∗} is a reduced cohomology theory on based spaces, then there are CW complexes E_{n} such that, for CW complexes X, E^{n}(X) is naturally isomorphic to the set [X, E_{n}] of homotopy classes of based maps X ⟶ E_{n}. The suspension isomorphism E^{n}(X) ≅ E^{n+1}(ΣX) gives rise to a homotopy equivalence σ̃_{n} : E_{n} ⟶ ΩE_{n+1}. The object E = {E_{n}, σ̃_{n}} is called an Ω-spectrum. A map f : E ⟶ E′ of Ω-spectra is a sequence of maps f_{n} : E_{n} ⟶ E′_{n} that are compatible up to homotopy with the equivalences σ̃_{n} and σ̃′_{n}. The category of Ω-spectra is equivalent to the category of cohomology theories on based spaces and can be thought of as an intuitive first approximation to the stable homotopy category. However, this category does not have a usable theory of cofibration sequences and is not suitable for either point-set level or homotopical work. For that, one needs more precise objects and morphisms that are defined without use of homotopies but that still represent cohomology theories and their maps. More subtly, one needs a coordinate-free setting in order to define smash products sensibly. The nth space E_{n} relates to the n-sphere and thus to ℝ^{n}. Restricting to spaces E_{n} is very much like restricting to the standard basis of ℝ^{∞} when doing linear algebra.

A coordinate-free spectrum is indexed on the set of finite dimensional subspaces V of a “universe” U, namely a real inner product space isomorphic to the sum ℝ^{∞} of countably many copies of ℝ. In detail, writing W − V for the orthogonal complement of V in W, a spectrum E assigns a based space EV to each finite dimensional subspace V of U, with (adjoint) structure maps

σ̃_{V,W} : EV ⟶^{≅}Ω^{W −V} EW

when V ⊂ W, where Ω^{W} X is the function space F (S^{W} , X) of based maps S^{W} ⟶ X and S^{W} is the one-point compactification of W. The structure maps are required to satisfy an evident transitivity relation when V ⊂ W ⊂ Z, and they are required to be homeomorphisms. A map of spectra f : E → E′ is a collection of maps of based spaces f_{V} : EV → E′V for which each of the following diagrams commutes:

\xymatrix{EV \ar[r]^{f_{V}} \ar[d]_{σ̃_{V,W}} & E′V \ar[d]^{σ̃′_{V,W}} \\ Ω^{W −V} EW \ar[r]^{Ω^{W −V} f_{W}} & Ω^{W −V} E′W.}

We obtain the category 𝒮 = 𝒮 U of spectra indexed on U. We obtain an equivalent category if we restrict to any cofinal family of indexing spaces. If we drop the requirement that the maps σ̃_{V,W} be homeomorphisms, we obtain the notion of a prespectrum and the category 𝒫 = 𝒫U of prespectra indexed on U. The forgetful functor ℓ : 𝒮 ⟶ 𝒫 has a left adjoint L. When the structure maps σ̃ are inclusions, (LE)(V) is just the union of the spaces Ω^{W −V} EW for V ⊂ W. We write σ : Σ^{W −V} EV ⟶ EW for the adjoints of the maps σ̃, where Σ^{V} X = X ∧ S^{V}.

Examples 1.1. Let X be a based space. The suspension prespectrum Π^{∞}X is the prespectrum whose V th space is Σ^{V} X; the structure maps σ are the evident identifications Σ^{W −V} Σ^{V} X ≅ Σ^{W} X. The suspension spectrum of X is Σ^{∞}X = LΠ^{∞}X. Let QX = ∪Ω^{V} Σ^{V} X, where the union is taken over the inclusions obtained from the adjoints of the cited identifications. Then (Σ^{∞}X)(V) = Q(Σ^{V} X). The functor Σ^{∞} from based spaces to spectra is left adjoint to the functor that assigns the zeroth space E_{0} = E({0}) to a spectrum E. More generally, for a fixed subspace Z ⊂ U, define Π_{Z}^{∞}X to be the analogous prespectrum whose V th space is Σ^{V −Z}X if Z ⊂ V and a point otherwise and define Σ_{Z}^{∞}X = LΠ_{Z}^{∞}X. Then Σ_{Z}^{∞} is left adjoint to the functor that sends a spectrum to its Zth space EZ; these functors are generally called “shift desuspensions”.

Functors on prespectra that do not preserve spectra are extended to spectra by applying the functor L. For example, for a based space X and a prespectrum E, we have the prespectrum E ∧ X specified by (E ∧ X)(V) = EV ∧ X. When E is a spectrum, the structure maps for this prespectrum level smash product are not homeomorphisms, and we understand the smash product E ∧X to be the spectrum L(ℓE ∧ X). Function spectra are easier. We set F (X, E)(V) = F (X, EV) and find that this functor on prespectra preserves spectra. If we topologize the set 𝒮 (E, E′) as a subspace of the product over V of the function spaces F (EV, E′V) and let 𝒯 be the category of based spaces with sets of maps topologized as function spaces, then there result homeomorphisms

𝒮 (E ∧ X, E′) ≅ 𝒯 (X, 𝒮 (E, E′)) ≅ 𝒮 (E, F (X, E′)).

Recall that a category is said to be cocomplete if it has all colimits and complete if it has all limits.

Proposition 1.2. The category 𝒮 is complete and cocomplete.

Proof. Limits and colimits are defined on prespectra spacewise. Limits preserve spectra, and colimits of spectra are obtained by use of the left adjoint L. □

We write Y_{+} for the union of a space Y and a disjoint basepoint. A homotopy in the category of spectra is a map E ∧ I_{+} ⟶ E′. We have cofibration and fibration sequences that are defined exactly as on the space level (e.g. [29]) and enjoy the same homotopical properties. Let [E, E′] denote the set of homotopy classes of maps E ⟶ E′; we shall later understand that, when using this notation, E must be of the homotopy type of a CW spectrum. For based spaces X and Y with X compact, we have

[Σ^{∞}X, Σ^{∞}Y] ≅ colim_{n} [Σ^{n}X, Σ^{n}Y].

Fix a copy of ℝ^{∞} in U. In the equivariant generalization of the present theory, it is essential not to insist that ℝ^{∞} be all of U, but the reader may take U = ℝ^{∞} here. We write Σ_{n}^{∞} = Σ_{ℝ^{n}}^{∞}. For n ≥ 0, the sphere spectrum S^{n} is Σ^{∞}S^{n}. For n > 0, the sphere spectrum S^{−n} is Σ_{n}^{∞}S^{0}. We write S for the zero sphere spectrum. The nth homotopy group of a spectrum E is the set [S^{n}, E] of homotopy classes of maps S^{n} ⟶ E, and this fixes the notion of a weak equivalence of spectra. The adjunctions of Examples 1.1 make it clear that a map f of spectra is a weak equivalence if and only if each of its component maps f_{Z} is a weak equivalence of spaces. The stable homotopy category h̄𝒮 is constructed from the homotopy category of spectra by adjoining formal inverses to the weak equivalences, a process that is made rigorous by CW approximation.

The theory of CW spectra is developed by taking sphere spectra as the domains of attaching maps of cells CS^{n} = S^{n} ∧ I [16, I§5]. The one major difference from the space level theory of CW complexes is that we have to construct CW spectra as unions E = ∪E_{n}, where E_{0} is the trivial spectrum and where we are allowed to attach cells of arbitrary dimension when constructing E_{n+1} from E_{n}. There results a notion of a cell spectrum. We define a CW spectrum to be a cell spectrum whose cells are attached only to cells of lower dimension. Thus CW spectra have two filtrations, the sequential filtration {E_{n}} that gives the order in which cells are attached, and the skeletal filtration {E^{q}}, where E^{q} is the union of the cells of dimension at most q. We say that a map between CW spectra is cellular if it preserves both filtrations. In fact, by redefining the sequential filtration appropriately, we can always arrange that the sequential filtration is preserved. We have three basic results, whose proofs are very little different from their space level counterparts.

Theorem 1.3 (Whitehead). If E is a CW spectrum and f : F ⟶ F′ is a weak equivalence of spectra, then f_{∗} : [E, F] ⟶ [E, F′] is an isomorphism. Therefore a weak equivalence between CW spectra is a homotopy equivalence.

Theorem 1.4 (Cellular approximation). Let A be a subcomplex of a CW spectrum E, let F be a CW spectrum, and let f : E ⟶ F be a map whose restriction to A is cellular. Then f is homotopic relative to A to a cellular map. Therefore any map E ⟶ F is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic.

Theorem 1.5 (Approximation by CW spectra). For a spectrum E, there is a CW spectrum ΓE and a weak equivalence γ : ΓE ⟶ E. On the homotopy category h𝒮, Γ is a functor such that γ is natural.

It follows that the stable category h̄𝒮 is equivalent to the homotopy category of CW spectra. Homotopy-preserving functors on spectra that do not preserve weak equivalences are transported to the stable category by first replacing their variables by weakly equivalent CW spectra.

Observe that there has been no mention of space level CW complexes in our development so far. The total lack of hypotheses on the spaces and structural maps of our prespectra allows considerable point-set level pathology, even if, as usual in modern algebraic topology, we restrict attention to compactly generated weak Hausdorff spaces. Recall that a space X is weak Hausdorff if the diagonal subspace is closed in the compactly generated product X × X. More restrictively, a space X is said to be LEC (locally equiconnected) if the inclusion of the diagonal subspace is a cofibration. We record the following list of special kinds of prespectra both to prepare for our discussion of smash products and to compare our definitions with those adopted in the original treatments of the stable homotopy category.

Definition 1.6. A prespectrum D is said to be

(i) Σ-cofibrant if each σ : Σ^{W −V} DV → DW is a based cofibration (that is, satisfies the based homotopy extension property).

(ii) CW if it is Σ-cofibrant and each DV is LEC and has the homotopy type of

a CW complex. (iii) strictly CW if each DV is a based CW complex and the structure maps σ

are the inclusions of subcomplexes. A spectrum E is said to be Σ-cofibrant if it is isomorphic to LD for some Σ-cofibrant prespectrum D; E is said to be tame if it is of the homotopy type of a Σ-cofibrant spectrum.

If E is a spectrum, then the maps σ̃ are homeomorphisms. Therefore the underlying prespectrum ℓE is not Σ-cofibrant unless it is trivial. However, it is a very weak condition on a spectrum that it be tame. We shall see that this weak condition is enough to avoid serious point-set topological problems. If D is a Σ-cofibrant prespectrum, then the maps σ̃ are inclusions and therefore LD(V) is just the union of the spaces Ω^{W −V} DW. We have the following relations between CW prespectra and CW spectra. Remember that CW spectra are defined in terms of spectrum level attaching maps.

Theorem 1.7. If D is a CW prespectrum, then LD has the homotopy type of a CW spectrum. If E is a CW spectrum, then each space EV has the homotopy type of a CW complex and E is homotopy equivalent to LD for some CW prespectrum D. Thus a spectrum has the homotopy type of a CW spectrum if and only if it has the homotopy type of LD for some CW prespectrum D.

In particular, spectra of the homotopy types of CW spectra are tame.

Implicitly or explicitly, early constructions of the stable homotopy category restricted attention to the spectra arising from strict CW prespectra. This is far too restrictive for serious point-set level work, and it is also too restrictive to admit a sensible equivariant analogue. Note that such a category cannot possibly be complete or have well-behaved point-set level function spectra.

One reason for focusing on Σ-cofibrant spectra is that they are built up out of their component spaces in a simple fashion.

Proposition 1.8. If E = LD, where D is a Σ-cofibrant prespectrum, then

E ≅ colim_{V} Σ_{V}^{∞}DV,

where the colimit is computed as the prespectrum level colimit of the maps

Σ_{W}^{∞}σ : Σ_{V}^{∞}DV ≅ Σ_{W}^{∞}Σ^{W −V} DV ⟶ Σ_{W}^{∞}DW.

That is, the prespectrum level colimit is a spectrum that is isomorphic to E. The maps of the colimit system are shift desuspensions of based cofibrations.

Another reason is that general spectra can be replaced functorially by weakly equivalent Σ-cofibrant spectra.

Proposition 1.9. There is a functor K : 𝒫U ⟶ 𝒫U, called the cylinder functor, such that KD is Σ-cofibrant for any prespectrum D, and there is a natural spacewise weak equivalence of prespectra KD ⟶ D. On spectra E, define KE = LKℓE. Then there is a natural weak equivalence of spectra KE ⟶ E.

In practice, if one is given a prespectrum D, perhaps indexed only on integers, and one wishes to construct a spectrum from it that retains homotopical information, one forms E = LKD. Then

π_{n}(E) = colim_{q} π_{n+q}D_{q}.

If D is an Ω-spectrum that represents a given cohomology theory on spaces, then E = LKD is a genuine spectrum that represents the same theory.

2. Smash products and twisted half-smash products

The construction of the smash product of spectra proceeds by internalization of an “external smash product”. The latter is an associative and commutative pairing

𝒮 U × 𝒮 U′ → 𝒮 (U ⊕ U′)

for any pair of universes U and U′. It is constructed by starting with the prespectrum level definition

(E ∧ E′)(V ⊕ V′) = EV ∧ E′V′.

The structure maps fail to be homeomorphisms when E and E′ are spectra, and we apply the spectrification functor L to obtain the desired spectrum level smash product.

In order to obtain smash products internal to a single universe U, we exploit the “twisted half-smash product”. The input data for this functor consist of two universes U and U′, an unbased space A with a given map α : A → ℐ (U, U′), and a spectrum E indexed on U. The output is the spectrum A ⋉ E, which is indexed on U′. It must be remembered that the construction depends on α and not just on A, although different choices of α lead to equivalent functors on the level of stable categories. When A is a point, α is a choice of a linear isometry f : U ⟶ U′ and we write f_{∗} for the twisted half-smash product. For a prespectrum D,

(f_{∗}D)(V′) = D(V) ∧ S^{V′−f(V)}, where V = f^{−1}(V′ ∩ im f).

For a spectrum E, f_{∗}E is obtained by application of L to the prespectrum level construction. The functor f_{∗} is left adjoint to the more elementary functor f^{∗} specified by (f^{∗}E′)(V) = E′(f(V)). For general A and α, the intuition is that A ⋉ E is obtained by suitably topologizing the union of the α(a)_{∗}(E). Another intuition is that the twisted half-smash product is a generalization to spectra of the “untwisted” functor A_{+} ∧ X on based spaces X. This intuition is made precise by the following “untwisting formula” relating twisted half-smash products and shift desuspensions.

Proposition 2.1. For a map A ⟶ ℐ (U, U′) and an isomorphism V ≅ V′, where V ⊂ U and V′ ⊂ U′, there is an isomorphism of spectra

A ⋉ Σ_{V}^{∞}X ≅ A_{+} ∧ Σ_{V′}^{∞}X

that is natural in spaces A over ℐ (U, U′) and based spaces X.

The twisted-half smash product functor enjoys essentially the same formal properties as the space level functor A_{+} ∧X. The functor A⋉E is homotopy-preserving in E, and it therefore preserves homotopy equivalences in the variable E. However, it only preserves homotopies over ℐ (U, U′) in A. Nevertheless, it very often preserves homotopy equivalences in the variable A. The following central technical result is an easy consequence of Propositions 1.8 and 2.1.

Theorem 2.2. Let E ∈ 𝒮 U be tame and let A be a space over ℐ (U, U′). If ϕ : A′ ⟶ A is a homotopy equivalence, then ϕ ⋉ id : A′ ⋉ E ⟶ A ⋉ E is a homotopy equivalence.

Since A ⋉ E is a CW spectrum if A is a CW complex and E is a CW spectrum, this has the following consequence.

Corollary 2.3. Let E ∈ 𝒮 U be a spectrum that has the homotopy type of a CW spectrum and let A be a space over ℐ (U, U′) that has the homotopy type of a CW complex. Then A ⋉ E has the homotopy type of a CW spectrum.

Now, as before, restrict attention to a particular universe U and write 𝒮 = 𝒮 U; again, the reader may think of U as ℝ^{∞}. We are especially interested in twisted half-smash products defined in terms of the following spaces of linear isometries.

Notations 2.4. Let U^{j} be the direct sum of j copies of U and let ℒ (j) = ℐ (U^{j}, U). The space ℒ (0) is the point i, where i : {0} → U, and ℒ (1) contains the identity map 1 = id_{U} : U → U. The left action of Σ_{j} on U^{j} by permutations induces a free right action of Σ_{j} on the contractible space ℒ (j). Define maps

γ : ℒ (k) × ℒ (j_{1}) × ⋯ × ℒ (j_{k}) ⟶ ℒ (j_{1} + ⋯ + j_{k})

by

γ(g; f_{1}, …, f_{k}) = g ∘ (f_{1} ⊕ ⋯ ⊕ f_{k}).

The spaces ℒ (j) form an operad [18, p.1] with structural maps γ, called the linear isometries operad. Points f ∈ ℒ (j) give functors f_{∗} that send spectra indexed on U^{j} to spectra indexed on U. Applied to a j-fold external smash product E_{1} ∧ ⋯ ∧ E_{j}, there results an internal smash product f_{∗}(E_{1} ∧ ⋯ ∧ E_{j}). All of these smash products become equivalent in the stable category h̄𝒮, but none of them are associative or commutative on the point set level. More precisely, the following result holds.

Theorem 2.5. Let 𝒮_{t} ⊂ 𝒮 be the full subcategory of tame spectra and let h𝒮_{t} be its homotopy category. On 𝒮_{t}, the internal smash products f_{∗}(E ∧ E′) determined by varying f ∈ ℒ (2) are canonically homotopy equivalent, and h𝒮_{t} is symmetric monoidal under the internal smash product. For based spaces X and tame spectra E, there is a natural homotopy equivalence E ∧ X ≃ f_{∗}(E ∧ Σ^{∞}X).

This implies formally that we have arrived at a stable situation. As for spaces, the suspension functor Σ is given by ΣE = E ∧ S^{1} and is left adjoint to the loop functor Ω given by ΩE = F (S^{1}, E). The cofibre Cf of a map f : E ⟶ E′ of spectra is the pushout E′ ∪_{f} CE.

Theorem 2.6. The suspension functor Σ : h𝒮_{t} ⟶ h𝒮_{t} is an equivalence of categories. A cofibre sequence E⟶^{f}E′ ⟶ Cf in 𝒮_{t} gives rise to a long exact sequence of homotopy groups

⋯ ⟶ π_{q}(E) ⟶ π_{q}(E′) ⟶ π_{q}(Cf) ⟶ π_{q−1}(E) ⟶ · · · .

Proof. For based spaces X, Σ^{∞}X is naturally isomorphic to (Σ_{1}^{∞}X) ∧ S^{1} because the structural homeomorphisms σ̃ : E_{0} ⟶ ΩE_{1} on spectra give an isomorphism between their right adjoints. Thus, for tame spectra E, the previous theorem gives a natural homotopy equivalence

E = E ∧ S^{0} ≃ f_{∗}(E ∧ Σ^{∞}S^{0}) ≅ f_{∗}(E ∧ Σ_{1}^{∞}S^{0}) ∧ S^{1}.

Therefore Σ is an equivalence of categories with inverse obtained by smashing with the (−1)-sphere spectrum S^{−1} = Σ_{1}^{∞}S^{0}. It follows categorically that ΩE ≃ f_{∗}(E ∧ S^{−1}) and that the unit and counit

η : E ⟶ ΩΣE and ε : ΣΩE ⟶ E

of the adjunction are homotopy equivalences. The last statement is a standard consequence of the fact that maps can now be desuspended. □

Note that only actual homotopy equivalences, not weak ones, are relevant to the last two results. For this reason among others, h𝒮_{t} is a technically convenient halfway house between the homotopy category of spectra and the stable homotopy category.

3. The category of 𝕃-spectra

We need a category of spectra with a canonical smash product. The category of 𝕃-spectra that we introduce here will be shown in the next section to have an associative and commutative smash product ∧_{ℒ} . This product is not quite unital, but there is a natural unit weak equivalence λ : S ∧_{ℒ} M ⟶ M. The S-modules will be the 𝕃-spectra such that λ is an isomorphism.

For f ∈ ℒ (j) and E_{i} ∈ 𝒮_{t}, Theorem 2.2 implies that the inclusion {f} ⊂ ℒ (j) induces a homotopy equivalence

f_{∗}(E_{1} ∧ ⋯ ∧ E_{j}) ⟶ ℒ (j) ⋉ (E_{1} ∧ ⋯ ∧ E_{j}).

The proof of Theorem 2.5 above is entirely based on the use of such equivalences. It therefore seems natural to think of

ℒ (j) ⋉ (E_{1} ∧ ⋯ ∧ E_{j})

as a canonical j-fold smash product. It is still not associative, but it seems closer to being so. However, to take this idea seriously, we must take note of the difference between E and its “1-fold smash product” ℒ (1) ⋉ E. The space ℒ (1) is a monoid under composition, and the formal properties of twisted half-smash products imply a natural isomorphism

ℒ (1) ⋉ (ℒ (1) ⋉ E) ≅ (ℒ (1) × ℒ (1)) ⋉ E,

where, on the right, ℒ (1) × ℒ (1) is regarded as a space over ℒ (1) via the composition product. This product induces a map μ : (ℒ (1) × ℒ (1)) ⋉ E ⟶ ℒ (1) ⋉ E, and the inclusion {1} ⟶ ℒ (1) induces a map η : E ⟶ ℒ (1) ⋉ E. Thus it makes sense to consider spectra E with an action ξ : ℒ (1)⋉E ⟶ E of the monoid ℒ (1). It is required that the following diagrams commute:

[figure: (ℒ (1) × ℒ (1)) ⋉ E; μ; ℒ (1)⋉ξ; ℒ (1) ⋉ E; ξ; and; E; =; η; ℒ (1) ⋉ E; ξ; ℒ (1) ⋉ E; ξ; E; E.]

Definition 3.1. An 𝕃-spectrum is a spectrum E together with an action ξ of ℒ (1). A map f : E → E′ of 𝕃-spectra is a map of spectra such that the following diagram commutes:

\xymatrix{ℒ (1) ⋉ E \ar[r]^{ℒ (1)⋉f} \ar[d]_{ξ_{E}} & ℒ (1) ⋉ E′ \ar[d]^{ξ_{E′}} \\ E \ar[r]_{f} & E′.}

We let 𝒮 [𝕃] denote the category of 𝕃-spectra.

A number of basic properties of the category of spectra are directly inherited by the category of 𝕃-spectra.

Theorem 3.2. The category of 𝕃-spectra is complete and cocomplete, with both limits and colimits created in the underlying category 𝒮. If X is a based space and M is an 𝕃-spectrum, then M ∧ X and F (X, M) are 𝕃-spectra, and the spectrum level fibre and cofibre of a map of 𝕃-spectra are 𝕃-spectra.

A homotopy in the category of 𝕃-spectra is a map M ∧ I_{+} ⟶ M′. A map of 𝕃-spectra is a weak equivalence if it is a weak equivalence as a map of spectra. The stable homotopy category h̄𝒮 [𝕃] is constructed from the homotopy category of 𝕃-spectra by adjoining formal inverses to the weak equivalences. There is a theory of CW 𝕃-spectra that is exactly like the theory of CW spectra, and, again, the construction of h̄𝒮 [𝕃] is made rigorous by CW approximation. We have a free functor 𝕃 from spectra to 𝕃-spectra specified by 𝕃E = ℒ (1) ⋉ E. The “sphere 𝕃-spectra” that we take as the domains of attaching maps when defining CW 𝕃spectra are the free 𝕃-spectra 𝕃S^{n}. Using the freeness adjunction

𝒮 [𝕃](𝕃E, M) ≅ 𝒮 (E, M), it is easy to prove Whitehead, cellular approximation, and approximation by CW 𝕃-spectra theorems exactly like those stated for spectra in Section 1, and h̄𝒮 [𝕃] is equivalent to the homotopy category of CW 𝕃-spectra. There is one catch: although S and all other suspension spectra are 𝕃-spectra in a natural way, using the untwisting isomorphism of Proposition 2.1 and the projection ℒ (1) ⟶ {∗}, S does not have the homotopy type of a CW 𝕃-spectrum. However, it is not hard to see that the categories h̄𝒮 and h̄𝒮 [𝕃] are equivalent.

Theorem 3.3. The following conclusions hold.

(i) The free functor 𝕃 : 𝒮 ⟶ 𝒮 [𝕃] carries CW spectra to CW 𝕃-spectra. (ii) The forgetful functor 𝒮 [𝕃] ⟶ 𝒮 carries 𝕃-spectra of the homotopy types of CW 𝕃-spectra to spectra of the homotopy types of CW spectra.

(iii) Every CW 𝕃-spectrum M is homotopy equivalent as an 𝕃-spectrum to 𝕃E

for some CW spectrum E. (iv) If E ∈ 𝒮_{t}, for example if E is a CW spectrum, then η : E ⟶ 𝕃E is a

homotopy equivalence of spectra.

(v) If M has the homotopy type of a CW 𝕃-spectrum, then ξ : 𝕃M ⟶ M is a

homotopy equivalence of 𝕃-spectra. Therefore the free and forgetful functors establish an adjoint equivalence between the stable homotopy categories h̄𝒮 and h̄𝒮 [𝕃].

4. The smash product of 𝕃-spectra and function 𝕃-spectra

One of the most surprising developments of recent years is the discovery of an associative and commutative smash product ∧_{ℒ} in the category of 𝕃-spectra. We proceed to define it. To begin with, observe that the monoid ℒ (1) × ℒ (1) acts from the right on ℒ (2) and acts from the left on ℒ (i) × ℒ (j), via instances of the structural maps γ of the linear isometries operad. Another instance of γ gives rise to a map

(1)  γ̄ : ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (i) × ℒ (j) ⟶ ℒ (i + j).

The space on the left is the balanced product (formally a coequalizer) of the two specified actions by ℒ (1) × ℒ (1). The essential, elementary, point is that this map is a homeomorphism if i ≥ 1 and j ≥ 1. To see this, choose linear isometric isomorphisms s : U ⟶ U^{i} and t : U ⟶ U^{j}. Composition on the right with s ⊕ t gives vertical homeomorphisms in the commutative diagram

[figure: ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (i) × ℒ (j); γ̄; ℒ (i + j); ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (1) × ℒ (1); γ̄; ℒ (2),]

and the lower map γ̄ is clearly a homeomorphism. Note also that ℒ (1) acts from the left on ℒ (2) and that this action commutes with the right action of ℒ (1) × ℒ (1).

Regard ℒ (1) × ℒ (1) as a space over ℐ (U^{2}, U^{2}) via the direct sum of isometries map. If M and N are 𝕃-spectra, then ℒ (1) × ℒ (1) acts from the left on the external smash product M ∧ N via the map

[figure: ξ : (ℒ (1) × ℒ (1)) ⋉ (M ∧ N) ≅ (ℒ (1) ⋉ M) ∧ (ℒ (1) ⋉ N); ξ∧ξ; M ∧ N.]

The operadic smash product of M and N is simply the balanced product (again, formally a coequalizer)

(2)  M ∧_{ℒ} N = ℒ (2) ⋉_{ℒ (1)×ℒ (1)} (M ∧ N).

The left action of ℒ (1) on ℒ (2) induces a left action of ℒ (1) on M ∧_{ℒ} N that gives it a structure of 𝕃-spectrum. Use of the transposition σ ∈ Σ_{2} and the commutativity of the external smash product easily gives a commutativity isomorphism

τ : M ∧_{ℒ} N ⟶ N ∧_{ℒ} M.

More substantially, there is a natural associativity isomorphism

(M ∧_{ℒ} N) ∧_{ℒ} P ≅ M ∧_{S} (N ∧_{ℒ} P).

In fact, using the case i = 2 and j = 1 of the homeomorphism γ̄, we obtain isomorphisms

(M ∧_{ℒ} N) ∧_{ℒ} P ≅ ℒ (2) ⋉_{ℒ (1)^{2}} (ℒ (2) ⋉_{ℒ (1)^{2}} (M ∧ N)) ∧ (ℒ (1) ⋉_{ℒ (1)} P)
≅ (ℒ (2) ×_{ℒ (1)^{2}} ℒ (2) × ℒ (1)) ⋉_{ℒ (1)^{3}} (M ∧ N ∧ P)
≅ ℒ (3) ⋉_{ℒ (1)^{3}} M ∧ N ∧ P.

The symmetric argument shows that this is also isomorphic to M ∧_{ℒ} (N ∧_{ℒ} P). In view of the generality of the homeomorphisms (4.1), the argument iterates to give

(3)  M_{1} ∧_{ℒ} ⋯ ∧_{ℒ} M_{j} ≅ ℒ (j) ⋉_{ℒ (1)^{j}} (M_{1} ∧ ⋯ ∧ M_{j}),

where the iterated smash product on the left is associated in any fashion.

On passage to the derived category h̄𝒮 [𝕃], the smash product of 𝕃-spectra just constructed can be used interchangeably with the internal smash product on the stable category h̄𝒮. To see this, one defines the latter by use of a linear isometric isomorphism f : U^{2} ⟶ U (not just an isometry). With this choice, it is not hard to check the following result.

Proposition 4.1. For spectra E and F, there are isomorphisms of 𝕃-spectra

𝕃E ∧_{ℒ} 𝕃F ≅ ℒ (2) ⋉ E ∧ F ≅ 𝕃f_{∗}(E ∧ F).

For CW 𝕃-spectra M and N, M ∧_{ℒ} N is a CW 𝕃-spectrum with one (p + q)-cell for each p-cell of M and q-cell of N.

However, we need a deeper result, one that depends on the fine structure of the linear isometries operad, to complete the comparison of smash products. By arguments like those in the proof of Theorem 2.6, its first statement implies its second statement.

Proposition 4.2. For 𝕃-spectra N, there is a natural weak equivalence of 𝕃-spectra ω : 𝕃S ∧_{ℒ} N ⟶ N, and Σ : π_{n}(N) ⟶ π_{n+1}(ΣN) is an isomorphism for all integers n. Therefore the unit η : N ⟶ ΩΣN and counit ε : ΣΩN ⟶ N of the (Σ, Ω)-adjunction are weak equivalences and any cofibre sequence N⟶^{f}N′ ⟶ Cf of 𝕃-spectra gives rise to a long exact sequence of homotopy groups

⋯ ⟶ π_{q}(N) ⟶ π_{q}(N′) ⟶ π_{q}(Cf) ⟶ π_{q−1}(N) ⟶ · · · .

It is a pleasant technical feature of the theory that this result holds whether or not the given ℒ -spectra are tame. In particular, we have the following consequence, which is the 𝕃-spectrum analog of the algebraic statement that, when computing torsion products, one need only resolve one of the tensor factors by a projective resolution.

Proposition 4.3. If M is a CW 𝕃-spectrum and ϕ : N ⟶ N′ is a weak equivalence of 𝕃-spectra, then id ∧_{ℒ} ϕ : M ∧_{ℒ} N ⟶ M ∧_{ℒ} N′ is a weak equivalence of 𝕃-spectra.

The previous results lead easily to the promised comparison between the internal smash product of spectra and the operadic smash product of 𝕃-spectra.

Theorem 4.4. For 𝕃-spectra M and N, there is a natural map of spectra

α : f_{∗}(M ∧ N) ⟶ M ∧_{ℒ} N,

and α is a weak equivalence when M is a CW 𝕃-spectrum and N is a tame spectrum. For any 𝕃-spectrum N, the functors (?) ∧_{ℒ} N and f_{∗}(? ∧ N) from h̄𝒮 [𝕃] to h̄𝒮 are naturally isomorphic.

Thus, under the forgetful functor, the operadic smash product in h̄𝒮 [𝕃] agrees with the internal smash product in h̄𝒮.

There is a function 𝕃-spectrum functor to go with the operad smash product. The twisted half-smash product functor A ⋉ E has a right adjoint twisted function spectrum functor F [A, E′) and the external smash product has a right adjoint function spectrum functor. Using these functors and appropriate equalizer diagrams, dual to the coequalizer diagrams that were implicit in the definition of ∧_{ℒ} , we obtain the following result.

Theorem 4.5. There is a function 𝕃-spectrum functor F_{ℒ} (M, N) such that 𝒮 [𝕃](M ∧_{ℒ} N, P) ≅ 𝒮 [𝕃](M, F_{ℒ} (N, P)) for 𝕃-spectra M, N, and P.

Given the adjunction, we can deduce the homotopical behavior of F_{ℒ} from that of ∧_{ℒ} . There is an internal function spectrum functor F that is induced from the external spectrum functor by use of our chosen linear isometric isomorphism f : U^{2} ⟶ U. Our function ℒ -spectrum functor gives a canonical substitute.

Proposition 4.6. If M is a CW 𝕃-spectrum and ϕ : N → N′ is a weak equivalence of 𝕃-spectra, then

F_{ℒ} (id, ϕ) : F_{ℒ} (M, N) ⟶ F_{ℒ} (M, N′)

is a weak equivalence of 𝕃-spectra.

Theorem 4.7. For 𝕃-spectra M and N, there is a natural map of spectra

α̃ : F_{ℒ} (M, N) ⟶ F (M, f^{∗}N),

and α̃ is a weak equivalence when M is a CW 𝕃-spectrum. The forgetful functor h̄𝒮 [𝕃] ⟶ h̄𝒮 carries the function 𝕃-spectrum functor F_{ℒ} to the internal function spectrum functor F.

We must still address the question of units.

Proposition 4.8. For 𝕃-spectra N, there is a natural unit map of 𝕃-spectra λ : S∧_{ℒ} N ⟶ N. It is a weak equivalence for any N, and it is a homotopy equivalence of 𝕃-spectra if N is a CW 𝕃-spectrum.

Proof. Consider the map γ̄ of (4.1). It is a non-trivial property of the linear isometries operad that γ̄, although not a homeomorphism, is a homotopy equivalence when i = 0 and j > 0. When N is the free S-module 𝕃E = ℒ (1) ⋉ E generated by a spectrum E, λ is given by the map

S ∧_{S} 𝕃E = ℒ (2) ⋉_{ℒ (1)×ℒ (1)} (ℒ (0) ⋉ S^{0}) ∧ (ℒ (1) ⋉ E)
≅ (ℒ (2) ×_{ℒ (1)×ℒ (1)} ℒ (0) × ℒ (1)) ⋉ (S^{0} ∧ E)

⟶^{γ̄⋉id}ℒ (1) ⋉ E = 𝕃E.

Since γ̄ is a homotopy equivalence, Theorem 2.2 implies that λ is a homotopy equivalence when E ∈ 𝒮_{t}. For general N, the map just constructed for 𝕃N induces the required map for N by a comparison of coequalizer diagrams. Although the arguments are not transparent, the rest can be deduced from this. □

There is one important case when λ is an isomorphism. It turns out that the map γ̄ of (4.1) is a homeomorphism when i = j = 0; that is, non-obviously since ℒ (1) is a monoid but not a group, the domain ℒ (2)/ℒ (1) × ℒ (1) of (4.1) is then a point. This implies that S ∧_{ℒ} S ≅ S. More generally, it implies that the smash product over ℒ precisely generalizes the smash product of based spaces, in the sense that

Σ^{∞}X ∧_{ℒ} Σ^{∞}Y ≅ Σ^{∞}(X ∧ Y).

5. The category of S-modules

Here, finally, is the promised definition of S-modules.

Definition 5.1. Define an S-module to be an 𝕃-spectrum M which is unital in the sense that λ : S ∧_{ℒ} M ⟶ M is an isomorphism. Let ℳ_{S} denote the full subcategory of 𝒮 [𝕃] whose objects are the S-modules. For S-modules M and N, define

M ∧_{S} N = M ∧_{ℒ} N and F_{S}(M, N) = S ∧_{ℒ} F_{ℒ} (M, N).

The justification for the name “S-module” is given by the commutative diagrams

\xymatrix{S ∧_{S} S ∧_{S} M \ar[r]^{λ∧id} \ar[d]_{id ∧λ} & S ∧_{S} M \ar[d]^{λ} & and & M \ar[r]^{λ^{−1}} \ar[dr]_{id} & S ∧_{S} M \ar[d]^{λ} \\ S ∧_{S} M \ar[r]_{λ} & M &  &  & M.}

We consistently retain the notation M ∧_{ℒ} N when the given 𝕃-spectra M and N are not restricted to be S-modules. We have the following examples of S-modules.

Proposition 5.2. For any based space X, Σ^{∞}X is an S-module, and

Σ^{∞}X ∧_{S} Σ^{∞}Y ≅ Σ^{∞}(X ∧ Y).

For any S-module M and any 𝕃-spectrum N, M ∧_{ℒ} N is an S-module. In particular, S ∧_{ℒ} N is an S-module for any 𝕃-spectrum N.

We have the following categorical relationship between 𝒮 [𝕃] and ℳ_{S}.

Lemma 5.3. The functor S ∧_{ℒ} (?) : 𝒮 [𝕃] ⟶ ℳ_{S} is left adjoint to the functor F_{ℒ} (S,?) : ℳ_{S} ⟶ 𝒮 [𝕃] and right adjoint to the inclusion ℓ : ℳ_{S} ⟶ 𝒮 [𝕃].

This implies that to lift right adjoint functors from 𝒮 [𝕃] to ℳ_{S}, we must first forget down to 𝒮 [𝕃], next apply the given functor, and then apply the functor S ∧_{ℒ} (?). For example, limits in ℳ_{S} are created in this fashion.

Proposition 5.4. The category of S-modules is complete and cocomplete. Its colimits are created in 𝒮 [𝕃]. Its limits are created by applying the functor S ∧_{S} (?) to limits in 𝒮 [𝕃]. If X is a based space and M is an S-module, then M ∧ X is an S-module, and the spectrum level cofibre of a map of S-modules is an S-module. For a based space X and S-modules M and N,

ℳ_{S}(M ∧ X, N) ≅ ℳ_{S}(M, S ∧_{ℒ} F (X, N)).

Moreover,

M ∧ X ≅ M ∧_{S} Σ^{∞}X and S ∧_{ℒ} F (X, M) ≅ F_{S}(Σ^{∞}X, M).

Lemma 5.3 also explains our definition of function S-modules. Its second adjunction and the adjunction of Theorem 4.8 compose to give the adjunction displayed in the following theorem.

Theorem 5.5. The category ℳ_{S} is symmetric monoidal under ∧_{S}, and

ℳ_{S}(M ∧_{S} N, P) ≅ ℳ_{S}(M, F_{S}(N, P))

for S-modules M, N, and P.

A homotopy in the category of S-modules is a map M ∧ I_{+} ⟶ N. A map of S-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category 𝒟_{S} of S-modules is constructed from the homotopy category hℳ_{S} by adjoining formal inverses to the weak equivalences; again, the process is made rigorous by CW approximation. We define sphere S-modules

(4)  S_{S}^{n} ≡ S ∧_{ℒ} 𝕃S^{n}

and use them as the domains of attaching maps when defining cell and CW Smodules. From here, the theory of cell and CW S-modules is exactly like the theory of cell and CW spectra and is obtained by specialization of the theory of cell R-modules to be discussed shortly. A weak equivalence of cell S-modules is a homotopy equivalence, any S-module is weakly equivalent to a CW S-module, and 𝒟_{S} is equivalent to the homotopy category of CW S-modules. Again, the Smodule S does not have the homotopy type of a CW S-module. When working homotopically, we replace it with S_{S} ≡ S_{S}^{0}.

The following comparison between CW S-modules and CW 𝕃-spectra establishes an equivalence between 𝒟_{S} and h̄𝒮 [𝕃] and thus between 𝒟_{S} and h̄𝒮. It is largely a recapitulation of results already discussed.

Theorem 5.6. The following conclusions hold.

(i) The functor S ∧_{ℒ} (?) : 𝒮 [𝕃] ⟶ ℳ_{S} carries CW 𝕃-spectra to CW Smodules.

(ii) The forgetful functor ℳ_{S} ⟶ 𝒮 [𝕃] carries S-modules of the homotopy

types of CW S-modules to 𝕃-spectra of the homotopy types of CW 𝕃-spectra. (iii) Every CW S-module M is homotopy equivalent as an S-module to S ∧_{S} N

for some CW 𝕃-spectrum N. (iv) The unit λ : S ∧_{ℒ} M ⟶ M is a weak equivalence for all 𝕃-spectra M and

is a homotopy equivalence of 𝕃-spectra if M has the homotopy type of a CW 𝕃-spectrum.

The functors S ∧_{ℒ} (?) and the forgetful functor establish an adjoint equivalence between the stable homotopy category h̄𝒮 [𝕃] and the derived category 𝒟_{S}. This equivalence of categories preserves smash products and function spectra.

When doing classical homotopy theory, we can work interchangeably in any of the categories h̄𝒮, h̄𝒮 [𝕃], or 𝒟_{S}. These three categories are equivalent, and the equivalences preserve all structure in sight. When working on the point set level, we have reached a nearly ideal situation with our construction of ℳ_{S}, and the rest of the article will describe how to exploit this.

6. S-algebras and their categories of modules

Intuitively, S-algebras are as near to associative rings with unit as one can get in stable homotopy theory, and commutative S-algebras are as near as one can get to commutative rings.

Definition 6.1. An S-algebra is an S-module R together with maps of S-modules η : S ⟶ R and ϕ : R ∧_{S} R → R such that the following diagrams of S-modules commute:

\xymatrix{S ∧_{S} R \ar[r]^{η∧_{S}id} \ar[dr]_{λ} & R ∧_{S} R \ar[d]^{ϕ} & R ∧_{S} S \ar[l]_{id ∧_{S}η} \ar[dl]^{λτ} & and & R ∧_{S} R ∧_{S} R \ar[d]_{ϕ∧_{S}id} \ar[r]^{id ∧_{S}ϕ} & R ∧_{S} R \ar[d]^{ϕ} \\  & R &  &  & R ∧_{S} R \ar[r]^{ϕ} & R;}

R is commutative if the following diagram also commutes:

\xymatrix{R ∧_{S} R \ar[dr]_{ϕ} \ar[rr]^{τ} &  & R ∧_{S} R \ar[dl]^{ϕ} \\  & R.}

We shall not review the older definitions of A_{∞} and E_{∞} ring spectra. It turns out that they are equivalent to the structures that are given by the definition above, with the single exception that the unit map λ of an A_{∞} or E_{∞} ring spectrum need not be an isomorphism. In other words, the natural ground category for A_{∞} and E_{∞} ring spectra is the category of 𝕃-spectra rather than the category of S-modules. We state this formally.

Theorem 6.2. An S-algebra or commutative S-algebra is an A_{∞} or E_{∞} ring spectrum which is also an S-module. If A is an A_{∞} ring spectrum, then S ∧_{ℒ} A is a weakly equivalent S-algebra. If A is an E_{∞} ring spectrum, then S ∧_{ℒ} A is a weakly equivalent commutative S-algebra.

This means that we can use the older theory to construct examples. For example, the classical Thom spectra occur in nature as E_{∞} ring spectra, and [20] gives a machine for manufacturing A_{∞} and E_{∞} ring spectra from space level data. It shows that the Eilenberg-MacLane spectrum Hk of a ring k is an A_{∞} ring spectrum and is an E_{∞} ring spectrum if k is commutative and that the algebraic K-theory spectrum Kk of a commutative ring k is an E_{∞} ring spectrum. Similarly, the spectra ko and ku that represent real and complex connective K-theory are E_{∞} ring spectra.

Since it is very convenient to have strict units, we shall always work with Salgebras.

Definition 6.3. Let R be an S-algebra. A (left) R-module M is an S-module together with a map μ : R ∧_{S} M → M of S-modules such that the following diagrams commute:

\xymatrix{S ∧_{S} M \ar[r]^{η∧_{S}id} \ar[dr]_{λ} & R ∧_{S} M \ar[d]^{μ} & and & R ∧_{S} R ∧_{S} M \ar[d]_{ϕ∧_{S}id} \ar[r]^{id ∧_{S}μ} & R ∧_{S} M \ar[d]^{μ} \\  & M &  & R ∧_{S} M \ar[r]^{μ} & M.}

A map f : M ⟶ M′ of R-modules is a map of S-modules such that the following diagram commutes:

\xymatrix{R ∧_{S} M \ar[r]^{id ∧f} \ar[d]_{μ} & R ∧_{S} M′ \ar[d]^{μ′} \\ M \ar[r]_{f} & M′.}

We let ℳ_{R} denote the category of R-modules.

If R is commutative, then an R-module is the same thing as a left module over R regarded as an S-algebra, exactly as in algebra. From here, we can mimic vast areas of algebra, one particularly striking direction being the development of topological Hochschild homology. However, we shall concentrate on the generalized analog of stable homotopy theory that we obtain by studying the homotopy theory of Rmodules for a fixed commutative S-algebra R. Everything that makes sense is also true in the non-commutative case.

Theorem 6.4. The category of R-modules is complete and cocomplete, with both limits and colimits created in the underlying category ℳ_{S}. Let X be a based space, K be an S-module, and M and N be R-modules. Then the following conclusions hold, where the displayed isomorphisms are obtained by restriction of the corresponding isomorphisms for S-modules.

(i) M ∧X is an R-module and the spectrum level cofibre of a map of R-modules is an R-module.

(ii) S ∧_{ℒ} F (X, N) is an R-module and

ℳ_{R}(M ∧ X, N) ≅ ℳ_{R}(M, S ∧_{ℒ} F (X, N)).

(iii) M ∧_{S} K and F_{S}(K, N) are R-modules and

ℳ_{R}(M ∧_{S} K, N) ≅ ℳ_{R}(M, F_{S}(K, N)).

(iv) F_{S}(M, K) is an R-module.

(v) As R-modules,

M ∧ X ≅ M ∧_{S} Σ^{∞}X and S ∧_{ℒ} F (X, N) ≅ F_{S}(Σ^{∞}X, N).

A homotopy in the category of R-modules is a map M ∧ I_{+} ⟶ M′. A map of R-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category 𝒟_{R} is constructed from the homotopy category hℳ_{R} by adjoining formal inverses to the weak equivalences; again, the process is made rigorous by the approximation of general R-modules by cell R-modules.

Cell theory is based on the free R-module functor 𝔽_{R} from spectra to R-modules that is specified by 𝔽_{R}X = R ∧_{S} 𝔽_{S}X, where 𝔽_{S}X = S ∧_{ℒ} 𝕃X. The term “free” is a slight misnomer, in view of the following result.

Proposition 6.5. The functor 𝔽_{R} : 𝒮 ⟶ ℳ_{R} is left adjoint to the functor that sends an R-module M to the spectrum F_{ℒ} (S, M), and there is a natural map of R-modules ξ : 𝔽_{R}M ⟶ M whose adjoint M ⟶ F_{ℒ} (S, M) is a weak equivalence of spectra. Therefore

π_{n}(M) ≅ hℳ_{R}(𝔽_{R}S^{n}, M).

In the stable homotopy category h̄𝒮, 𝔽_{R}X is naturally isomorphic to the internal smash product R ∧ X when X is tame.

Thus 𝔽_{R} is left adjoint to a functor that is weakly equivalent to the obvious forgetful functor. This is the price to be paid for insisting on strict units, and it introduces no serious complications in the theory. Homotopically, the functor 𝔽_{R} behaves as one would expect. Generalizing (5.6), we define sphere R-modules by

(5)  S_{R}^{n} = 𝔽_{R}S^{n},

and we use them as the domains of attaching maps when developing the cell theory of R-modules. For cells, we note that the cone functor CE = E ∧ I commutes with 𝔽_{R}, so that CS_{R}^{n} ≅ 𝔽_{R}CS^{n}. Thus, via the adjunction, maps out of sphere R-modules and their cones are induced by maps on the spectrum level. Using this, we can simply parrot the theory of cell spectra in the context of R-modules, reducing proofs to the spectrum level via adjunction. We easily obtain the Whitehead theorem for cell R-modules, and the approximation theorem to the effect that any R-module is weakly equivalent to a cell R-module. The category 𝒟_{R} is equivalent to the homotopy category of cell R-modules. If R is connective, but not otherwise, we obtain the cellular approximation theorem when we restrict attention to CW R-modules, namely cell R-modules such that cells are only attached to cells of lower dimension.

The category 𝒟_{R} has all homotopy limits and colimits; the former are created as the corresponding constructions on the underlying diagrams of spectra; the latter require application of the functor S ∧_{ℒ} (?). Thus we have enough information to quote the categorical form of Brown’s representability theorem given in [6]. Adams’ analog [3] for functors defined only on finite CW spectra also applies in our context, with the same proof.

Theorem 6.6 (Brown). A contravariant functor k : 𝒟_{R} → Sets is representable in the form k(M) ≅ 𝒟_{R}(M, N) for some R-module N if and only if k converts wedges to products and converts homotopy pushouts to weak pullbacks.

Theorem 6.7 (Adams). A contravariant group-valued functor k on the homotopy category of finite cell R-modules is representable in the form k(M) ≅ 𝒟_{R}(M, N) for some R-module N if and only if k converts finite wedges to direct products and converts homotopy pushouts to weak pullbacks of underlying sets.

In fact, Brown’s theorem is the kind of formal result that can be derived in any (closed) model category in the sense of Quillen (see [8] for a good exposition), and we have the following result. Serre fibrations of spectra are maps that satisfy the covering homotopy property with respect to the set of cone spectra

{Σ_{q}^{∞}CS^{n}|q ≥ 0 and n ≥ 0}.

Relative cell R-modules M ⟶ N are constructed exactly like cell R-modules, except that one starts the inductive construction of N = ∪N_{n} with N_{0} = M.

We write q-cofibrations and q-fibrations here to avoid confusion with cofibrations (HEP) and fibrations (CHP); the ambiguous use of the same term for both the classical and the model theoretic concepts is one of the banes of the literature.

Theorem 6.8. The category of R-modules is a model category. Its weak equivalences are the maps of R-modules that are weak equivalences of spectra. Its qcofibrations are the retracts of relative cell R-modules. Its q-fibrations are the maps M ⟶ N such that F_{ℒ} (S, M) ⟶ F_{ℒ} (S, N) is a Serre fibration of spectra.

7. The smash product of R-modules and function R-modules

Continuing to work with our fixed commutative S-algebra R, we mimic the definition of tensor products of modules over algebras.

Definition 7.1. For R-modules M and N, define M ∧_{R} N to be the coequalizer displayed in the following diagram of S-modules:

\xymatrix{M ∧_{S} R ∧_{S} N \ar[r]^{μ∧_{S}id} \ar[r]_{id ∧_{S}ν} & M ∧_{S} N \ar[r] & M ∧_{R} N.}

where μ and ν are the given actions of R on M and N. Then M ∧_{R}N has a canonical R-module structure induced from the R-module structure of M or, equivalently, N.

Of course, S is a commutative S-algebra and our new M ∧_{S} N coincides with our old M ∧_{S} N. The functor ∧_{R} preserves colimits in each of its variables, and smash products with spaces commute with ∧_{R}, in the sense that

X ∧ (M ∧_{R} N) ≅ (X ∧ M) ∧_{R} N.

Therefore the functor ∧_{R} commutes with cofibre sequences in each of its variables. We have analogous relations with smash products over S and an adjunction that can be thought of as completing Proposition 5.4.

Proposition 7.2. For an S-module K,

K ∧_{S} (M ∧_{R} N) ≅ (K ∧_{S} M) ∧_{R} N

and

ℳ_{S}(M ∧_{R} N, K) ≅ ℳ_{R}(N, F_{S}(M, K)).

The associativity, commutativity, and unity of the smash product over S is inherited by the smash product over R.

Theorem 7.3. Under the smash product over R, the category of R-modules is symmetric monoidal with unit R.

We can deduce not only formal but also homotopical properties of ∧_{R} from corresponding properties of ∧_{S}. As in Section 4, we use an isomorphism of universes f : U ⊕ U → U to define the internal smash product f_{∗}(E ∧ F).

Proposition 7.4. Let X and Y be spectra and let N be an R-module. There is a natural isomorphism of R-modules

𝔽_{R}X ∧_{R} N ≅ 𝔽_{S}X ∧_{S} N.

There is also a natural isomorphism of R-modules

𝔽_{R}X ∧_{R} 𝔽_{R}Y ≅ 𝔽_{R}f_{∗}(X ∧ Y).

If M and N are cell R-modules, then M ∧_{R}N is a cell R-module with one (p+q)-cell for each p-cell of M and q-cell of N.

Theorem 7.5. If M is a cell R-module and ϕ : N ⟶ N′ is a weak equivalence of R-modules, then

id ∧_{R}ϕ : M ∧_{R} N ⟶ M ∧_{R} N′

is a weak equivalence of R-modules.

We construct ∧_{R} as a functor 𝒟_{R} × 𝒟_{R} → 𝒟_{R} by approximating one of the variables by a cell R-module.

We have a function spectrum functor F_{R} to go with our smash product. It is defined as the equalizer of a certain pair of maps F_{S}(M, N) ⟶ F_{S}(R ∧_{S} M, N). The details are dictated by the expected adjunction. Again, F_{R}(M, N) inherits a structure of R-module from M or, equivalently, N.

Proposition 7.6. For R-modules N and P and an S-module K,

ℳ_{R}(K ∧_{S} N, P) ≅ ℳ_{S}(K, F_{R}(N, P)).

If M is an R-module, then

ℳ_{R}(M ∧_{R} N, P) ≅ ℳ_{R}(M, F_{R}(N, P)).

Therefore

F_{R}(M ∧_{R} N, P) ≅ F_{R}(M, F_{R}(N, P)).

Formal arguments from the adjunction, as in algebra, give a natural associative and unital composition pairing

(6)  π : F_{R}(N, P) ∧_{R} F_{R}(M, N) ⟶ F_{R}(M, P).

Parenthetically, we note that this gives rise to a host of examples of S-algebras; in fact, R itself need not be commutative in the following result.

Proposition 7.7. For an R-module M, F_{R}(M, M) is an S-algebra; For R-modules M and N, F_{R}(M, N) is an (F_{R}(N, N), F_{R}(M, M))-bimodule.

Proposition 7.8. Let X be a spectrum and M be an R-module. There is a natural isomorphism of R-modules

F_{R}(𝔽_{R}X, M) ≅ F_{S}(𝔽_{S}X, M).

The functor F_{R}(M, N) converts colimits and cofibre sequences in M to limits and fibre sequences. It preserves limits and fibre sequences in N. Using the previous result to deal with sphere R-modules, we obtain the analog of Theorem 7.5.

Proposition 7.9. If M is a cell R-module and ϕ : N ⟶ N′ is a weak equivalence of R-modules, then

F_{R}(id, ϕ) : F_{R}(M, N) ⟶ F_{R}(M, N′)

is a weak equivalence of R-modules.

In the derived category 𝒟_{R}, F_{R}(M, N) means F_{R}(ΓM, N), where ΓM is a cell approximation of M.

Summarizing, we obtain the following derived category level conclusion.

Theorem 7.10. The derived category 𝒟_{R} is symmetric monoidal under the product derived from ∧_{R}, and

𝒟_{R}(M ∧_{R} N, P) ≅ 𝒟_{R}(M, F_{R}(N, P)).

There is a formal theory of duality (explained in [16, Ch. III]) that now applies to 𝒟_{R}. We define the dual of an R-module M to be D_{R}M = F_{R}(M, R). We have an evaluation map ε : D_{R}M ∧_{R} M ⟶ R and a map η : R → F_{R}(M, M), namely the adjoint of λ : R ∧_{R} M ⟶ M. There is also a natural map

(7)  ν : F_{R}(L, M) ∧_{R} N ⟶ F_{R}(L, M ∧_{R} N).

By composition with F_{R}(id, λ), ν specializes to a map

(8)  ν : D_{R}M ∧_{R} M ⟶ F_{R}(M, M).

We say that M is “strongly dualizable” if it has a coevaluation map η̄ : R ⟶ M ∧_{R} D_{R}M such that the following diagram commutes in 𝒟_{R}:

(9)  \xymatrix{R \ar[r]^{η̄} \ar[d]_{η} & M ∧_{R} D_{R}M \ar[d]^{τ} \\ F_{R}(M, M) & D_{R}M ∧_{R} M. \ar[l]^{ν}}

The definition has many purely formal implications. The map ν of (7.12) is an isomorphism in 𝒟_{R} if either L or N is strongly dualizable. The map ν of (7.13) is an isomorphism if and only if M is strongly dualizable, and the coevaluation map η̄ is then the composite τν^{−1}η in (7.14). The natural map

ρ : M ⟶ D_{R}D_{R}M

is an isomorphism if M is strongly dualizable. The natural map

∧ : F_{R}(M, N) ∧_{R} F_{R}(M′, N′) ⟶ F_{R}(M ∧_{R} M′, N ∧_{R} N′)

is an isomorphism if M and M′ are strongly dualizable or if M is strongly dualizable and N = R.

Say that a cell R-module N is a wedge summand up to homotopy of a cell Rmodule M if there is a homotopy equivalence of R-modules between M and N ∨N′ for some cell R-module N′. We say that N is semi-finite if it is a wedge summand up to homotopy of a finite cell R-module. In contrast with the usual stable homotopy category, a semi-finite R-module need not have the homotopy type of a finite cell R-module.

Theorem 7.11. A cell R-module is strongly dualizable if and only if it is semifinite.

The analogy with finitely generated projective modules in algebra should be clear.

8. Tor and Ext in topology and algebra

Still restricting for definiteness to a commutative S-algebra R and its modules, we define Tor and Ext groups as the homotopy groups of derived smash product and function modules.

Definition 8.1. For R-modules M and N, define

Tor_{n}^{R}(M, N) = π_{n}(M ∧_{R} N)

and

Ext_{R}^{n}(M, N) = π_{−n}(F_{R}(M, N)).

Note that Tor_{∗}^{R}(M, N) and Ext_{R}^{∗}(M, N) are π_{∗}(R)-modules.

We emphasize that the smash product and function spectra are understood to be taken in the derived category 𝒟_{R}. At this point in our exposition, we act as traditional topologists, taking it for granted that all spectra and modules are to be approximated as cell modules, without change of notation, whenever necessary. Various properties reminiscent of those of the classical Tor and Ext functors follow directly from the definition and the results of the previous sections. The intuition is that the definition gives an analogue of the differential Tor and Ext functors (alias hyperhomology and cohomology functors) in the context of differential graded modules over differential graded algebras. In particular, the grading should not be thought of as the resolution grading of the classical torsion product, but rather as a total grading that sums a resolution degree and an internal degree; this idea will be made precise by the grading of the spectral sequences that we shall describe for the calculation of these functors.

Proposition 8.2. Tor_{∗}^{R}(M, N) satisfies the following properties.

(i) If R, M, and N are connective, then Tor_{n}^{R}(M, N) = 0 for n < 0. (ii) A cofibre sequence N′ → N → N′′ gives rise to a long exact sequence ⋯ → Tor_{n}^{R}(M, N′) → Tor_{n}^{R}(M, N) → Tor_{n}^{R}(M, N′′) → Tor_{n−1}^{R}(M, N′) → · · · . (iii) Tor_{∗}^{R}(M, R) ≅ π_{∗}(M) and, for a spectrum X,

Tor_{∗}^{R}(M, 𝔽X) ≅ π_{∗}(M ∧ X).

(iv) The functor Tor_{∗}^{R}(M,?) carries wedges to direct sums.

The commutativity and associativity relations for the smash product imply various further properties. We content ourselves with the following examples:

Tor_{∗}^{R}(M, N) ≅ Tor_{∗}^{R}(N, M)

and

Tor_{∗}^{R}(M ∧_{R} N, P) ≅ Tor_{∗}^{R}(M, N ∧_{R} P).

Say that a spectrum N is coconnective if π_{n}(N) = 0 for n > 0.

Proposition 8.3. Ext_{R}^{∗}(M, N) satisfies the following properties.

(i) If R and M are connective and N is coconnective, then Ext_{R}^{n}(M, N) = 0 for n < 0.

(ii) Fibre sequences N′ → N → N′′ and cofibre sequences M′ → M → M′′ give

rise to long exact sequences ⋯ → Ext_{R}^{n}(M, N′) → Ext_{R}^{n}(M, N) → Ext_{R}^{n}(M, N′′) → Ext_{R}^{n+1}(M, N′) → ⋯

and

⋯ → Ext_{R}^{n}(M′′, N) → Ext_{R}^{n}(M, N) → Ext_{R}^{n}(M′, N) → Ext_{R}^{n+1}(M′′, N) → · · · .

(iii) Ext_{R}^{∗}(R, N) ≅ π_{−∗}(N) and, for a spectrum X,

Ext_{R}^{∗}(𝔽X, N) ≅ π_{−∗}(F (X, N)).

(iv) The functor Ext_{R}^{∗}(?, N) carries wedges to products.

Passing to homotopy groups from the pairings (7.7), we obtain the following further property. As usual, for a spectrum E, abbreviate

E_{n} = π_{n}(E) = E^{−n}.

Proposition 8.4. There is a natural, associative, and unital system of pairings of R^{∗}-modules

π^{∗} : Ext_{R}^{∗}(M, N) ⊗_{R^{∗}} Ext_{R}^{∗}(L, M) ⟶ Ext_{R}^{∗}(L, N).

The formal duality theory of the previous section implies the following result, together with various other such isomorphisms.

Proposition 8.5. For a finite cell R-module M and any R-module N,

T or_{n}^{R}(D_{R}M, N) ≅ Ext_{R}^{−n}(M, N).

Thinking of the derived category 𝒟_{R} as a stable homotopy category, we may change notations and reinterpret the functors Tor and Ext as prescribing homology and cohomology theories in this category.

Definition 8.6. Let M and E be R-modules. Define

E_{n}^{R}(M) = π_{n}(E ∧_{R} M) and E_{R}^{n}(M) = π_{−n}(F_{R}(M, E)).

The properties of Tor and Ext translate directly to statements about homology and cohomology. All of the standard homotopical machinery is available to us, and the previous result now takes the form of Spanier-Whitehead duality.

Corollary 8.7. For a finite cell R-module M and any R-module E,

E_{n}^{R}(D_{R}M) ≅ E_{R}^{−n}(M).

Since the equivalence between the classical stable homotopy category and the derived category of S-modules preserves smash products and function spectra, we obtain all of the usual homology and cohomology theories by taking R = S.

We also obtain the classical algebraic Tor and Ext groups as special cases, by specialization to Eilenberg-Mac Lane spectra. Thus let R be a discrete commutative ring for a moment. Recall that HR denotes a spectrum whose zeroth homotopy group is R and whose remaining homotopy groups are zero. It follows from multiplicative infinite loop space theory [20] that the Eilenberg-Mac Lane spectrum HR = K(R, 0) is an E_{∞} ring spectrum. Analogously, if M is an R-module, then HM can be constructed as an HR-module. We shall see a quick and easy construction shortly. Granting this, we have the following result.

Theorem 8.8. For a discrete commutative ring R and R-modules M and N,

Tor_{∗}^{R}(M, N) ≅ Tor_{∗}^{HR}(HM, HN)

and

Ext_{R}^{∗}(M, N) ≅ Ext_{HR}^{∗}(HM, HN).

Under the second isomorphism, the topologically defined pairing

Ext_{HR}^{∗}(HM, HN) ⊗_{R} Ext_{HR}^{∗}(HL, HM) ⟶ Ext_{HR}^{∗}(HL, HN)

coincides with the algebraic Yoneda product.

The proof is clear enough: we just check the axioms for Tor and Ext.

We can elaborate this result to an equivalence of derived categories. Recall from [28] or [15, Ch.III] that the derived category 𝒟_{R} is obtained from the homotopy category of chain complexes over R by formally inverting the quasi-isomorphisms, exactly as we obtained the category 𝒟_{HR} from the homotopy category of HRmodules by inverting the weak equivalences. The algebraic theory of cell and CW chain complexes over R in the latter source makes the analogy precise and gives a treatment of tensor products and Hom functors in 𝒟_{R} that exactly parallels our treatment of ∧_{HR} and F_{HR}. The proof of the equivalence is quite easy. The category 𝒟_{HR} is equivalent to the homotopy category of CW HR-modules and cellular maps. It is a simple matter to see that CW HR-modules have associated chain complexes. This gives a functor 𝒟_{HR} ⟶ 𝒟_{R}. An inverse functor Φ is obtained by applying Brown’s representability theorem. Indeed, for a given chain complex X, the functor k on 𝒟_{HR} specified by k(M) = 𝒟_{R}(C_{∗}(M), X) satisfies the hypotheses of that result, and we let Φ(X) represent this functor. Specialization to R-modules regarded as chain complexes concentrated in degree zero gives the promised construction of Eilenberg-Mac Lane HR-modules from R-modules.

Theorem 8.9. The cellular chain functor C_{∗} on HR-modules induces an equivalence of categories 𝒟_{HR} ⟶ 𝒟_{R}. The functor C_{∗} satisfies H_{∗}(C_{∗}(M)) ≅ π_{∗}(M) and carries the functors ∧_{HR} and F_{HR} to the functors ⊗_{R} and Hom_{R}. The inverse equivalence Φ satisfies π_{∗}(Φ(X)) ≅ H_{∗}(X) and carries the functors ⊗_{R} and Hom_{R} to the functors ∧_{HR} and F_{HR}.

Proof. By construction, we have an adjunction

𝒟_{R}(C_{∗}(M), X) ≅ 𝒟_{HR}(M, Φ(X)),

and one checks that its unit and counit are isomorphisms. The statements relating ∧_{HR} and F_{HR} to ⊗_{R} and Hom_{R} are all consequences of the fact that if M and N are CW HR-modules, then M ∧_{HR} N is a CW HR-module such that

C_{∗}(M ∧_{HR} N) ≅ C_{∗}(M) ⊗_{R} C_{∗}(N). □

9. Universal coefficient and Künneth spectral sequences

Returning to our general commutative S-algebra R, we find spectral sequences for the calculation of our Tor and Ext groups that are analogous to the Eilenberg-Moore (or hyperhomology) spectral sequences in differential homological algebra. Compare [9, 13, 15]. They may be viewed as giving universal coefficient and Künneth spectral sequences for homology and cohomology theories on R-modules, and they specialize to give such spectral sequences for homology and cohomology theories on spectra.

Theorem 9.1. For R-modules M and N, there are natural spectral sequences of differential R_{∗}-modules

E_{p,q}^{2} = Tor_{p,q}^{R_{∗}}(M_{∗}, N_{∗}) ⟹ Tor_{p+q}^{R}(M, N)

and

E_{2}^{p,q} = Ext_{R^{∗}}^{p,q}(M^{∗}, N^{∗}) ⟹ Ext_{R}^{p+q}(M, N).

Moreover, the pairing F_{R}(M, N) ∧_{R} F_{R}(L, M) → F_{R}(L, N) induces a pairing of spectral sequences that coincides with the algebraic Yoneda pairing

Ext_{R^{∗}}^{∗,∗}(M^{∗}, N^{∗}) ⊗_{R^{∗}} Ext_{R^{∗}}^{∗,∗}(L^{∗}, M^{∗}) ⟶ Ext_{R^{∗}}^{∗,∗}(L^{∗}, N^{∗})

on the E_{2}-level and that converges to the induced pairing of Ext groups.

The Tor spectral sequence is of standard homological type, with

d_{p,q}^{r} : E_{p,q}^{r} ⟶ E_{p−r,q+r−1}^{r}.

It lies in the right half-plane, and it converges strongly. The Ext spectral sequence is of standard cohomological type, with

d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1}.

It lies in the right half plane. In the language of Boardman [5] (see also [12, App B]), it is conditionally convergent. It therefore converges strongly if, for each fixed (p, q), there are only finitely many r such that d_{r} is non-zero on E_{r}^{p,q}.

Setting M = 𝔽_{R}X in the two spectral sequences of Theorem 8.1, we obtain a universal coefficient spectral sequence. We have written the stars to indicate the way the grading is usually thought of in cohomology.

Theorem 9.2 (Universal coefficient). For an R-module N and any spectrum X, there are spectral sequences of the form

Tor_{∗,∗}^{R_{∗}}(R_{∗}(X), N_{∗}) ⟹ N_{∗}(X)

and

Ext_{R^{∗}}^{∗,∗}(R_{−∗}(X), N^{∗}) ⟹ N^{∗}(X).

Of course, replacing R and N by Eilenberg-Mac Lane spectra HR and HN for a ring R and R-module N, we obtain the classical universal coefficient theorems. Replacing N by 𝔽_{R}Y and by F_{R}(𝔽_{R}Y, R) in the two universal coefficient spectral sequences, we arrive at Künneth spectral sequences.

Theorem 9.3 (Künneth). For any spectra X and Y, there are spectral sequences of the form

Tor_{∗,∗}^{R_{∗}}(R_{∗}(X), R_{∗}(Y)) ⟹ R_{∗}(X ∧ Y)

and

Ext_{R^{∗}}^{∗,∗}(R_{−∗}(X), R^{∗}(Y)) ⟹ R^{∗}(X ∧ Y).

Adams [1] first observed that one can derive Künneth spectral sequences from universal coefficient spectral sequences, and he observed that, by duality, the four spectral sequences of Theorems 9.2 and 9.3 imply two more universal coefficient and two more Künneth spectral sequences. He derived spectral sequences of this sort under the hypothesis that his given ring spectrum E is the colimit of finite subspectra E_{α} such that H^{∗}(E_{α}; E^{∗}) is E^{∗}-projective and the Atiyah-Hirzebruch spectral sequence converging from H^{∗}(E_{α}; E^{∗}) to E^{∗}(E_{α}) satisfies E_{2} = E_{∞}. Of course, this is an ad hoc calculational hypothesis that requires case-by-case verification. It covers some cases that are not covered by the results above, and conversely. The cited paper of Adams, and his later book [2], are prime sources for the first flowering of stable homotopy theory. While some of their foundational parts may be obsolete, their applications and calculational parts certainly are not.

The following generalized Künneth theorem admits a number of variants; see [11].

Theorem 9.4. Let E and R be commutative S-algebras and M and N be Rmodules. Then there is a spectral sequence of differential E_{∗}(R)-modules of the form

Tor_{p,q}^{E_{∗}(R)}(E_{∗}(M), E_{∗}(N)) ⟹ E_{p+q}(M ∧_{R} N).

10. Algebraic constructions in the derived category of R-modules

If we replace the pair (S, R) by a pair (R, A) in Definition 6.1, we arrive at the notion of an algebra A over a commutative S-algebra R. For example, the S-algebras F_{R}(M, M) of Proposition 7.8 are actually R-algebras. Again, if A is an algebra over a discrete commutative ring R, then HA is an HR-algebra. Proceeding in this line, we can, for instance, construct R-modules whose homotopy groups realize the Hochschild homology of A with coefficients in (A, A)-bimodules.

However, we now proceed in a more homotopical direction, thinking of the derived category of R-modules as an analog of the stable homotopy category. From this point of view, we have the notion of an R-ring spectrum, which is just like the classical notion of a ring spectrum in the stable homotopy category.

Definition 10.1. An R-ring spectrum A is an R-module A with unit η : R ⟶ A and product ϕ : A ∧_{R} A ⟶ A in 𝒟_{R} such that the following left and right unit diagram commutes in 𝒟_{R}.

\xymatrix{R ∧_{R} A \ar[r]^{η∧id} \ar[dr]_{λ} & A ∧_{R} A \ar[d]^{ϕ} & A ∧_{R} R \ar[l]_{id ∧η} \ar[dl]^{λτ} \\  & A}

A is associative or commutative if the appropriate diagram commutes in 𝒟_{R}.

Lemma 10.2. If A and B are R-ring spectrum, then so is A ∧_{R} B. If A and B are associative or commutative, then so is A ∧_{R} B.

By neglect of structure, an R-ring spectrum A is a ring spectrum in the sense of classical stable homotopy theory; its unit is the composite of the unit of R and the unit of A and its product is the composite of the product of A and the canonical map

A ∧ A ≃ A ∧_{S} A ⟶ A ∧_{R} A.

Similarly, for an R-algebra A, we have the evident homotopical notion of an Amodule spectrum. These structures play a role in the study of 𝒟_{R} analogous to the role played by ring spectra and their module spectra in classical stable homotopy theory. When R = S, S-ring spectra and their module spectra are equivalent to classical ring spectra and their module spectra.

We show in this section how to construct quotients M/IM and localizations M[Y^{−1}] of modules over a commutative S-algebra R and indicate in the next section when these constructions inherit a structure of R-ring spectrum from an R-ring spectrum structure on M. When specialized to MU, these results give highly structured versions of spectra that in the past were constructed by means of the Baas-Sullivan theory of manifolds with singularities or the Landweber exact functor theorem. At least at odd primes, the results give an entirely satisfactory, and very simple, treatment of MU-ring structures on the resulting MU-modules.

We are interested in homotopy groups, and we make use of the isomorphisms

(10)  M_{n} = h𝒮 (S^{n}, M) ≅ hℳ_{S}(S_{S}^{n}, M) ≅ hℳ_{R}(S_{R}^{n}, M)

to represent elements as maps of R-modules. For x ∈ R_{n}, the composite map of R-modules

(11)  \xymatrix{S_{R}^{n} ∧_{R} M \ar[r]^{x∧id} & R ∧_{R} M \ar[r]^{λ} & M}

is a module theoretic version of the map x· : Σ^{n}M ⟶ M, and we agree to write Σ^{n}M for S_{R}^{n} ∧_{R} M in this section. By Proposition 7.4, S_{R}^{n} ∧_{R} M is isomorphic as an R-module to S_{S}^{n} ∧_{S} M and, by Theorem 4.7, S_{S}^{n} ∧_{S} M is weakly equivalent as a spectrum to S^{n} ∧ M. Therefore, the R-module Σ^{n}M is a model for the spectrum level suspension of M.

Definition 10.3. Define M/xM to be the cofibre of the map (9.4) and let ρ : M ⟶ M/xM be the canonical map. Inductively, for a finite sequence {x_{1}, …, x_{n}} of elements of R_{∗}, define

M/(x_{1}, …, x_{n})M = N/x_{n}N, where N = M/(x_{1}, …, x_{n−1})M.

For a (countably) infinite sequence X = {x_{i}}, define M/XM to be the telescope of the M/(x_{1}, …, x_{n})M, where the telescope is taken with respect to the successive canonical maps ρ.

Clearly we have a long exact sequence

(12)  ⋯ ⟶ π_{q−n}(M)⟶^{x·}π_{q}(M)⟶^{ρ_{∗}}π_{q}(M/xM) ⟶ π_{q−n−1}(M) ⟶ · · · .

If x is not a zero divisor for π_{∗}(M), then ρ_{∗} induces an isomorphism of R_{∗}-modules

(13)  π_{∗}(M)/x · π_{∗}(M) ≅ π_{∗}(M/xM).

If {x_{1}, …, x_{n}} is a regular sequence for π_{∗}(M), in the sense that x_{i} is not a zero divisor for π_{∗}(M)/(x_{1}, …, x_{i−1})π_{∗}(M) for 1 ≤ i ≤ n, then

(14)  π_{∗}(M)/(x_{1}, …, x_{n})π_{∗}(M) ≅ π_{∗}(M/(x_{1}, …, x_{n})M),

and similarly for a possibly infinite regular sequence X = {x_{i}}. The following result implies that M/XM is independent of the ordering of the elements of the set X. We write R/X instead of R/XR.

Lemma 10.4. For a set X of elements of R_{∗}, there is a natural weak equivalence

(R/X) ∧_{R} M ⟶ M/XM.

In particular, for a finite set X = {x_{1}, …, x_{n}},

R/(x_{1}, …, x_{n}) ≃ (R/x_{1}) ∧_{R} ⋯ ∧_{R} (R/x_{n}).

If I denotes the ideal generated by X, then it is reasonable to define

(15)  M/IM = M/XM.

However, this notation must be used with caution since, if we fail to restrict attention to regular sequences X, the homotopy type of M/XM will depend on the set X and not just on the ideal it generates. For example, quite different modules are obtained if we repeat a generator x_{i} of I in our construction.

We next construct localizations of R-modules at countable multiplicatively closed subsets Y of R_{∗}. Let {y_{i}} be any cofinal sequence of Y, with y_{i} ∈ R_{n_{i}}, so that every y ∈ Y divides some y_{i}. We may represent y_{i} by an R-map S_{R}^{0} ⟶ S_{R}^{−n_{i}}, which we also denote by y_{i}. Let q_{0} = 0 and, inductively, q_{i} = q_{i−1} + n_{i}. The R-map

S_{R}^{0} ∧_{R} M ⟶^{y_{i}∧id}S_{R}^{−n_{i}} ∧_{R} M

represents y_{i}·. Smashing over R with S_{R}^{−q_{i−1}}, we obtain a sequence of R-maps

(16)  S_{R}^{−q_{i−1}} ∧_{R} M ⟶ S_{R}^{−q_{i}} ∧_{R} M.

Definition 10.5. Define the localization of M at Y, denoted M[Y^{−1}], to be the telescope of the sequence of maps (10.11). Since M ≅ S_{R}^{0} ∧_{R} M in 𝒟_{R}, we may regard the inclusion of the initial stage S_{R}^{0} ∧_{R} M of the telescope as a natural map λ : M ⟶ M[Y^{−1}].

Since homotopy groups commute with localization, we see immediately that λ induces an isomorphism of R_{∗}-modules

(17)  π_{∗}(M[Y^{−1}]) ≅ π_{∗}(M)[Y^{−1}].

As in Lemma 10.9, the localization of M is the smash product of M with the localization of R.

Lemma 10.6. For a multiplicatively closed set Y of elements of R_{∗}, there is a natural equivalence

R[Y^{−1}] ∧_{R} M ⟶ M[Y^{−1}].

Moreover, R[Y^{−1}] is independent of the ordering of the elements of Y. For sets X and Y, R[(X ∪ Y)^{−1}] is equivalent to the composite localization R[X^{−1}][Y^{−1}].

11. R-ring structures on localizations and on quotients by ideals

The behavior of localizations with respect to R-ring structures is immediate.

Proposition 11.1. Let Y be a multiplicatively closed set of elements of R_{∗}. If A is an R-ring spectrum, then A[Y^{−1}] is an R-ring spectrum such that λ : A ⟶ A[Y^{−1}] is a map of R-ring spectra. If A is associative or commutative, then so is A[Y^{−1}].

Proof. By Lemmas 10.2 and 10.14, it suffices to observe that R[Y^{−1}] is an associative and commutative R-ring spectrum with unit λ and product the equivalence

R[Y^{−1}] ∧_{R} R[Y^{−1}] ≃ R[Y^{−1}][Y^{−1}] ≃ R[Y^{−1}]. □

This doesn’t work for quotients since (R/X)/X is not equivalent to R/X. However, we can analyze the problem by analyzing the deviation, and, by Lemma 10.9, we may as well work one element at a time. We have a necessary condition for R/x to be an R-ring spectrum that is familiar from classical stable homotopy theory.

Lemma 11.2. Let A be an R-ring spectrum. If A/xA admits an R-ring spectrum structure such that ρ : A ⟶ A/xA is a map of R-ring spectra, then x : A/xA ⟶ A/xA is null homotopic as a map of R-modules.

Thus, for example, the Moore spectrum S/2 is not an S-ring spectrum since the map 2 : S/2 ⟶ S/2 is not null homotopic. To give a criterion for when R/x does have an R-ring spectrum structure, we first note an easy formal lemma.

Lemma 11.3. Let ρ : R ⟶ M be any map of R-modules. Then

(ρ ∧ id) ∘ ρ ≃ (id ∧ρ) ∘ ρ : R ⟶ M ∧_{R} M.

Theorem 11.4. Let x ∈ R_{m} and assume that π_{m+1}(R/x) = 0 and π_{2m+1}(R/x) = 0. Then R/x admits a structure of R-ring spectrum with unit ρ : R ⟶ R/x. Therefore A/XA admits a structure of R-ring spectrum such that ρ : A ⟶ A/XA is a map of R-ring spectra for every R-ring spectrum A and every sequence X of elements of R_{∗} such that π_{m+1}(R/x) = 0 and π_{2m+1}(R/x) = 0 if x ∈ X has degree m.

Proof. Consider the following diagram in the derived category 𝒟_{R}: (18)

\xymatrix{ &  &  & Σ^{2m+1}R \ar[d]^{x} \\  &  &  & Σ^{m+1}R \ar[d]^{ρ} \ar@{-->}[dl]_{ν} \ar[r]^{x} & ΣR \ar[d]^{ρ} \\ Σ^{m}(R/x) \ar[r]^{x} & R/x \ar[r]^{ρ∧id} & (R/x) ∧_{R} (R/x) \ar[r]^{π} \ar@{-->}[l]^{ϕ} & Σ^{m+1}(R/x) \ar[r]^{x} \ar[d]^{π′} \ar@{-->}[l]^{σ} & Σ(R/x) \\  &  &  & Σ^{2m+2}R.}

The map x is that specified by (10.4). The bottom row is the cofibre sequence that results from the equivalence

(R/x) ∧_{R} (R/x) ≃ (R/x)/x

of Lemma 10.9, and the column is also a cofibre sequence. The composite x ∘ ρ is null homotopic since ρ ∘ x is null homotopic and the square commutes. Therefore there is a map ν such that π ∘ ν = ρ, and ν is unique since π_{m+1}(R/x) = 0. Since π ∘ ν ∘ x = ρ ∘ x = 0, ν ∘ x factors through a map Σ^{2m+1}R ⟶ R/x. Since π_{2m+1}(R/x) = 0, such maps are null homotopic. Thus ν ∘ x is null homotopic. Therefore there is a map σ such that σ ∘ ρ = ν. Now π ∘ σ ∘ ρ = π ∘ ν = ρ, hence (π∘σ−id)ρ = 0. Therefore π∘σ−id factors through a map Σ^{2m+2}R ⟶ Σ^{m+1}(R/x). Again, such maps are null homotopic. Therefore π∘σ = id. Thus the bottom cofibre sequence splits (proving in passing that x : Σ^{m}(R/x) ⟶ R/x is null homotopic, as it must be). A choice ϕ of a splitting gives a product on R/x. The unit condition ϕ ∘ (ρ ∧ id) = id is automatic. To see that ϕ ∘ (id ∧ρ) = id, we observe that, by the lemma,

(ϕ ∘ (id ∧ρ) − id) ∘ ρ = ϕ ∘ (id ∧ρ − ρ ∧ id) ∘ ρ = 0.

Therefore ϕ ∘ (id ∧ρ) − id factors through a map Σ^{m+1}R ⟶ R/x. Again, such maps are null homotopic, hence ϕ ∘ (id ∧ρ) = id. This completes the proof that R/x is an R-ring spectrum with unit ρ. The rest follows from Lemmas 10.9 and 10.2. □

The product on R/x can be described a little more concretely. The wedge sum

(19)  (ρ ∧ id) ∨ σ : (R/x) ∨ Σ^{m+1}(R/x) ⟶ (R/x) ∧_{R} (R/x)

is an equivalence. The product ϕ restricts to the identity on the first wedge summand and to the trivial map on the second wedge summand. Thus the product is determined by the choice of σ, and two choices of σ differ by a composite

(20)  \xymatrix{Σ^{m+1}(R/x) \ar[r]^{π′} & Σ^{2m+2}R \ar[r] & (R/x) ∧_{R} (R/x).}

By the splitting (10.6) and the assumption that π_{m+1}(R/x) = 0, we can view the second map as an element of π_{2m+2}(R/x). If x is not a zero divisor, then π′_{∗} = 0 on homotopy groups and any two products have the same effect on homotopy groups.

For an R-ring spectrum A and an element x as in the theorem, we give A/xA ≃ (R/x) ∧_{R} A the product induced by one of our constructed products on R/x and the given product on A. We refer to any such product as a “canonical” product on A/xA. Observe that, by first using the product on A, the product on A/xA can be factored through

ϕ ∧_{R} id : (R/x) ∧_{R} (R/x) ∧_{R} A ⟶ (R/x) ∧_{R} A.

This allows us to smash any diagram giving information about the product on R/x with A and so obtain information about the product on A/xA. Obviously any diagram so constructed is a diagram of right A-modules via the product action of A on itself. This smashing with A can kill obstructions. Clearly, a map of A-modules Σ^{q}A ⟶ M is determined by its restriction S^{q} ⟶ M along the unit of A regarded as a map of spectra (or S-modules), which is just an element of π_{q}(M). This leads to the following result.

Theorem 11.5. Let x ∈ R_{m} and assume that π_{m+1}(R/x) = 0 and π_{2m+1}(R/x) = 0. Let A be an R-ring spectrum and assume that π_{2m+2}(A/xA) = 0. Then there is a unique canonical product on A/xA. If A is commutative, then A/xA is commutative. If A is associative and π_{3m+3}(A/xA) = 0, then A/xA is associative.

Proof. The second arrow of (11.7) becomes zero after smashing with A since it is then given by an element of π_{2m+2}(A/xA) = 0. This proves the uniqueness statement. The commutativity statement follows since if ϕ is a canonical product on A/xA, then so is ϕτ. The associativity statement requires consideration of the restriction of the iterated product to the wedge summands of A/xA ∧_{R} A/xA ∧_{R} A/xA. The details are similar to, but simpler than, those in the proof of Theorem 11.4. □

Iterating and observing that passage to telescopes can kill obstructions, we arrive at the following fundamental conclusion.

Theorem 11.6. Assume that R_{i} = 0 if i is odd. Let X be a sequence of non zero divisors in R_{∗} such that π_{∗}(R/X) is concentrated in degrees congruent to zero mod 4. Then R/X has a unique canonical structure of R-ring spectrum, and it is commutative and associative.

12. The specialization to MU-modules and MU-ring spectra

The classical Thom spectra arise in nature as E_{∞} ring spectra. In fact, it was inspection of their prespectrum level definition in terms of Grassmannians that first led to the theory of E_{∞} ring spectra [19]. Applying the functor S ∧_{ℒ} (?), we obtain models for Thom spectra which are commutative S-algebras. Of course, the homotopy groups of MU are concentrated in even degrees, and every non-zero element is a non zero divisor. Thus the results above have the following immediate corollary.

Theorem 12.1. Let X be a regular sequence in MU_{∗}, let I be the ideal generated by X, and let Y be any sequence in MU_{∗}. Then there is an MU-ring spectrum (MU/X)[Y^{−1}] and a natural map of MU-ring spectra (the unit map)

η : MU ⟶ (MU/X)[Y^{−1}]

such that

η_{∗} : MU_{∗} ⟶ π_{∗}((MU/X)[Y^{−1}])

realizes the natural homomorphism of MU_{∗}-algebras

MU_{∗} ⟶ (MU_{∗}/I)[Y^{−1}].

If MU_{∗}/I is concentrated in degrees congruent to zero mod 4, then there is a unique canonical product on (MU/X)[Y^{−1}], and this product is commutative and associative.

In comparison with earlier constructions of this sort based on the Baas-Sullivan theory of manifolds with singularities or on Landweber’s exact functor theorem (where it applies), we have obtained a simpler proof of a substantially stronger result. We emphasize that an MU-ring spectrum is a much richer structure than just a ring spectrum and that commutativity and associativity in the MU-ring spectrum sense are much more stringent conditions than mere commutativity and associativity of the underlying ring spectrum.

We illustrate by explaining how BP appears in this context. Fix a prime p and write (?)_{p} for localization at p. Let BP be the Brown-Peterson spectrum at p. We are thinking of Quillen’s idempotent construction [24], and we have the splitting maps i : BP ⟶ MU_{p} and e : MU_{p} ⟶ BP. These are maps of commutative and associative ring spectra such that e ∘ i = id. Let I be the kernel of the composite

MU_{∗} ⟶ MU_{p∗} ⟶ BP_{∗}.

Then I is generated by a regular sequence X, and our MU/X is a canonical integral version of BP. For the moment, let BP′ = (MU/X)_{p}. Let ξ : BP ⟶ BP′ be the composite

\xymatrix{BP \ar[r]^{i} & MU_{p} \ar[r]^{η_{p}} & BP′.}

It is immediate that ξ is an equivalence. In effect, since we have arranged that η_{p} has the same effect on homotopy groups as e, ξ induces the identity map of (MU_{∗}/I)_{p} on homotopy groups. By the splitting of MU_{p} and the fact that selfmaps of MU_{p} are determined by their effect on homotopy groups [2, II.9.3], maps MU_{p} ⟶ BP are determined by their effect on homotopy groups. This implies that ξ ∘ e = η_{p} : MU_{p} ⟶ BP′. The product on BP is the composite

\xymatrix{BP ∧ BP \ar[r]^{i∧i} & MU_{p} ∧ MU_{p} \ar[r]^{ϕ} & MU_{p} \ar[r]^{e} & BP.}

Since η_{p} is a map of MU-ring spectra and thus of ring spectra, a trivial diagram chase now shows that the equivalence ξ : BP ⟶ BP′ is a map of ring spectra.

We conclude that our BP′ is a model for BP that is an MU-ring spectrum, commutative and associative if p > 2. The situation for p = 2 is interesting. We conclude from the equivalence that BP′ is commutative and associative as a ring spectrum, although we do not know that it is commutative or associative as an MU-ring spectrum.

Recall that π_{∗}(BP) = ℤ_{(p)}[v_{i}|deg(v_{i}) = 2(p^{i} − 1)], where the generators v_{i} come from π_{∗}(MU) (provided that we use the Hazewinkel generators). We list a few of the spectra derived from BP, with their coefficient rings. Let 𝔽_{p} denote the field with p elements.

\begin{matrix}BP ⟨n⟩ & ℤ_{(p)}[v_{1}, …, v_{n}] & E(n) & ℤ_{(p)}[v_{1}, …, v_{n}, v_{n}^{−1}] \\ P (n) & 𝔽_{p}[v_{n}, v_{n+1}, …] & B(n) & 𝔽_{p}[v_{n}^{−1}, v_{n}, v_{n+1}, …] \\ k(n) & 𝔽_{p}[v_{n}] & K(n) & 𝔽_{p}[v_{n}, v_{n}^{−1}] \end{matrix}

By the method just illustrated, we can construct canonical integral versions of the BP ⟨n⟩ and E(n). All of these spectra fit into the context of Theorem 11.1. If p > 2, they all have unique canonical commutative and associative MU-ring spectrum structures. Further study is needed when p = 2. In any case, this theory makes it unnecessary to appeal to Baas-Sullivan theory or to Landweber’s exact functor theorem for the construction and analysis of spectra such as these.

With more sophisticated techniques, the second author [14] has proven that BP can be constructed as a commutative S-algebra, and in fact admits uncountably many distinct such structures. There is much other ongoing work on the construction and application of new commutative S-algebras, by Hopkins, Miller, McClure, and others, and we have recently proven that the periodic K-theory spectra KO and KU can be constructed as commutative S-algebras. The enriched multiplicative structures on rings and modules that we have discussed are rapidly becoming a standard tool in the study of periodicity phenomena in stable homotopy theory.

References

[1] J.F. Adams. Lectures on generalised cohomology. Springer Lecture Notes in Mathematics Vol. 99, 1969, 1-138.

[2] J.F. Adams. Stable homotopy and generalized homology. University of Chicago. 1974, 1994.

[3] J.F. Adams. A variant of E.H.Brown’s representability theorem. Topology, 10 (1971), 185-198.

[4] J.M. Boardman. Stable homotopy theory. Thesis, Warwick 1964; mimeographed notes from Warwick and Johns Hopkins Universities, 1965-1970.

[5] J.M. Boardman. Conditionally convergent spectral sequences. Preprint. 1981.

[6] E.H. Brown, Jr. Abstract homotopy theory. Trans. Amer. Math. Soc. 119 (1965), 79–85.

[7] R.R. Bruner, J.P. May, J.E. McClure, and M. Steinberger. H_{∞} ring spectra and their applications. Springer Lecture Notes in Mathematics Vol. 1176. 1986.

[8] W.G. Dwyer and J. Spalinski. Homotopy theories and model categories. This volume.

[9] S. Eilenberg and J.C. Moore. Homology and fibrations, I. Comm. Math. Helv. 40 (1966), 199-236.

[10] A.D. Elmendorf, J.P.C. Greenlees, I. Kriz, and J.P. May. Commutative algebra in stable homotopy theory and a completion theorem. Mathematical Research Letters 1 (1994), 225-239.

[11] A.D. Elmendorf, I. Kriz, M. A. Mandell, and J.P. May. Rings, modules, and algebras in stable homotopy theory. In preparation.

[12] J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Memoirs Amer. Math. Soc. To appear.

[13] V.K.A.M. Gugenheim and J.P. May. On the theory and applications of differential torsion products. Memoirs Amer. Math. Soc. 142, 1974.

[14] I. Kriz. Towers of E_{∞} ring spectra with an application to BP. Preprint. 1993.

[15] I. Kriz, and J.P. May. Operads, algebras, modules, and motives. Astérisque. To appear.

[16] L.G. Lewis, Jr., J.P. May, and M. Steinberger (with contributions by J.E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics Vol. 1213. 1986.

[17] S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag. 1971.

[18] J.P. May. The Geometry of Iterated Loop Spaces. Springer Lecture Notes in Mathematics Vol. 271. 1972.

[19] J.P. May (with contributions by N. Ray, F. Quinn and J. Tornehave). E_{∞} ring spaces and E_{∞} ring spectra. Springer Lecture Notes in Mathematics Vol. 577. 1977.

[20] J.P. May. Multiplicative infinite loop space theory. J. Pure and Applied Algebra, 26 (1982), 1-69.

[21] J.P. May. Derived categories in algebra and topology. In the Proceedings of the Eleventh International Conference on Topology, Rendiconti dell Istituto Matematico dell Università di Trieste. To appear.

[22] J.P. May. Equivariant homotopy and cohomology theory. CBMS Lectures. In preparation.

[23] D.G. Quillen. Homotopical algebra. Springer Lecture Notes in Mathematics Vol. 43. 1967.

[24] D.G. Quillen. On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc. 75 (1969), 1293-1298.

[25] A. Robinson. Derived tensor products in stable homotopy theory. Topology 22 (1983), 1-18.

[26] A. Robinson. Spectra of derived module homomorphisms. Math. Proc. Camb. Phil. Soc. 101 (1987), 249-257.

[27] A. Robinson. The extraordinary derived category. Math. Z. 196(1987), 231-238.

[28] J.L. Verdier. Catégories dérivées. Springer Lecture Notes in Mathematics Vol. 569, 1977, 262-311.

[29] G.W. Whitehead. Elements of homotopy theory. Springer-Verlag. 1978.

Purdue University Calumet, Hammond, IN 46323 USA

   E-mail address: aelmendo@@math.purdue.edu

The University of Michigan, Ann Arbor, MI 48109-1003 USA

   E-mail address: ikriz@@math.lsa.umich.edu

The University of Chicago, Chicago, IL 60637 USA

   E-mail address: mandell@@math.uchicago.edu

The University of Chicago, Chicago, IL 60637 USA

   E-mail address: may@@math.uchicago.edu