On the Motivic Spectra Representing

Algebraic Cobordism and Algebraic K-Theory

David Gepner and Victor Snaith

Received: September 9, 2008

Communicated by Lars Hesselholt

Abstract. We show that the motivic spectrum representing algebraic K-theory is a localization of the suspension spectrum of ℙ^{∞}, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of BGL. In particular, working over ℂ and passing to spaces of ℂ-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic K-theory and periodic algebraic cobordism are E_{∞} motivic spectra.

2000 Mathematics Subject Classification: 55N15; 55N22

1. Introduction

1.1. Background and motivation. Let (X, μ) be an E_{∞} monoid in the category of pointed spaces and let β ∈ π_{n}(Σ^{∞}X) be an element in the stable homotopy of X. Then Σ^{∞}X is an E_{∞} ring spectrum, and we may invert the “multiplication by β” map

μ(β) : Σ^{∞}X ≃ Σ^{∞}S^{0} ∧ Σ^{∞}X ⟶^{Σ^{−n}β∧1} Σ^{−n}Σ^{∞}X ∧ Σ^{∞}X ⟶^{Σ^{−n}Σ^{∞}μ} Σ^{−n}Σ^{∞}X.

to obtain an E_{∞} ring spectrum

Σ^{∞}X[1/β] := colim{Σ^{∞}X ⟶^{β_{∗}} Σ^{−n}Σ^{∞}X ⟶^{Σ^{−n}β_{∗}} Σ^{−2n}Σ^{∞}X ⟶ ⋯ }

with the property that μ(β) : Σ^{∞}X[1/β] → Σ^{−n}Σ^{∞}X[1/β] is an equivalence. In fact, as is well-known, Σ^{∞}X[1/β] is universal among E_{∞} Σ^{∞}X-algebras A in which β becomes a unit. It was originally shown in [27] (see also [28] for a simpler proof) that the ring spectra Σ_{+}^{∞}BU[1/β] and Σ_{+}^{∞}ℂℙ^{∞}[1/β], obtained as above by taking X to be BU_{+} or ℙ_{+}^{∞} and β a generator of π_{2}X (a copy of ℤ in both cases), represent periodic complex cobordism and topological K-theory, respectively.

This motivated an attempt in [27] to define algebraic cobordism by replacing BGL(ℂ) in this construction with Quillen’s algebraic K-theory spaces [26]. The result was an algebraic cobordism theory, defined in the ordinary stable homotopy category, which was far too large. By analogy with topological complex cobordism, algebraic cobordism ought to be the universal oriented algebraic cohomology theory. However, there are at least two algebraic reformulations of the topological theory; as a result, there are at least two distinct notions of algebraic cobordism popular in the literature today. One, due to Levine and Morel [11], [12], constructs a universal “oriented Borel-Moore” cohomology theory Ω by generators and relations in a way reminiscent of the construction of the Lazard ring, and indeed the value of Ω on the point is the Lazard ring. However, Ω is not a generalized motivic cohomology theory in the sense of Morel and Voevodsky [20], so it is not represented by a motivic ring spectrum. The other notion, and the one relevant to this paper, is Voevodsky’s spectrum MGL [34]. It is a bona fide motivic cohomology theory in the sense that it is defined directly on the level of motivic spectra. Although the coefficient ring of MGL is still not known (at least in all cases), the orientability of MGL implies that it is an algebra over the Lazard ring, as it carries a formal group law. Provided one defines an orientation as a compatible family of Thom classes for vector bundles, it is immediate that MGL represents the universal oriented motivic cohomology theory; moreover, as shown in [23], and just as in the classical case, the splitting principle implies that it is enough to specify a Thom class for the universal line bundle. The infinite Grassmannian

BGL_{n} ≃ Grass_{n,∞} := colim_{k} Grass_{n,k}

represents, in the 𝔸^{1}-local homotopy category, the functor which associates to a variety X the set of isomorphism classes of rank n vector bundles on X. In particular, tensor product of line bundles and Whitney sum of stable vector bundles endow ℙ^{∞} ≃ BGL_{1} and BGL ≃ colim_{n} BGL_{n} with the structure of abelian group objects in the 𝔸^{1}-homotopy category. Note that, over ℂ, the spaces ℙ^{∞}(ℂ) and BGL(ℂ) underlying the associated complex-analytic varieties are equivalent to the usual classifying spaces ℂℙ^{∞} and BU. We might therefore hypothesize, by analogy with topology, that there are equivalences of motivic ring spectra

Σ_{+}^{∞}BGL[1/β] ⟶ P MGL  and  Σ_{+}^{∞}ℙ^{∞}[1/β] ⟶ K

where P MGL denotes a periodic version of the algebraic cobordism spectrum MGL. The purpose of this paper is to prove this hypothesis. In fact, it holds over an arbitrary Noetherian base scheme S of finite Krull dimension, provided one interprets K properly: the Thomason-Trobaugh K-theory of schemes [33] is not homotopy invariant, and so it cannot possibly define a motivic cohomology theory. Rather, the motivic analogue of K-theory is Weibel’s homotopy Ktheory [38]; the two agree for any regular scheme.

Motivic Spectra Representing Cobordism and K-Theory 361

1.2. Organization of the paper. We begin with an overview of the theory of oriented motivic ring spectra. The notion of an orientation is a powerful one, allowing us to compute first the oriented cohomology of flag varieties and Grassmannians. We use our calculations to identify the primitive elements in the Hopf algebra R^{0}(ℤ × BGL) with R^{0}(BGL_{1}), a key point in our analysis of the abelian group R^{0}(K) of spectrum maps from K to R. The second section is devoted to algebraic cobordism, in particular the proof that algebraic cobordism is represented by the motivic spectrum Σ_{+}^{∞}BGL[1/β]. We recall the construction of MGL as well as its periodic version P MGL and note the functors they (co)represent as monoids in the homotopy category of motivic spectra. We show that P MGL is equivalent to ⋁_{n} Σ^{∞}MGL_{n}[1/β] and use the isomorphism R^{0}(BGL) ≅ ∏_{n} R^{0}(MGL_{n}) to identify the functors Rings(Σ_{+}^{∞}BGL[1/β], −) and Rings(⋁_{n} Σ^{∞}MGL_{n}[1/β], −). The third section provides the proof that algebraic K-theory is represented by the motivic spectrum Σ_{+}^{∞}ℙ^{∞}[1/β]. First we construct a map; to see that it’s an equivalence, we note that it’s enough to show that the induced map R^{0}(K) → R^{0}(Σ_{+}^{∞}ℙ^{∞}[1/β]) is an isomorphism for any P MGL-algebra R. An element of R^{0}(K) amounts to a homotopy class of an infinite loop map ℤ × BGL ≃ Ω^{∞}K → Ω^{∞}R; since loop maps ℤ × BGL → Ω^{∞}R are necessarily additive, we are reduced to looking at maps ℙ^{∞} → Ω^{∞}R. We use this to show that the spaces map(K, R) and map(Σ_{+}^{∞}ℙ^{∞}[1/β], R) both arise as the homotopy inverse limit of the tower associated to the endomorphism of the space map(Σ_{+}^{∞}ℙ^{∞}, R) induced by the action of the Bott map ℙ^{1} ∧ ℙ^{∞} → ℙ^{∞}, and are therefore homotopy equivalent. We conclude the paper with a couple of corollaries. The first is a quick proof of the motivic Conner-Floyd theorem, namely that the map

MGL^{∗,∗}(X) ⊗_{MGL^{∗,∗}} K^{∗,∗} ⟶ K^{∗,∗}(X),

induced by an MGL-algebra structure on K, is an isomorphism for any compact motivic spectrum X. This was first obtained by Panin-Pimenov-Röndigs [24] and follows from a motivic version of the Landweber exact functor theorem [21]. We include a proof because, using the aforementioned structure theorems, we obtain a simplification of the (somewhat similar) method in [24], but which is considerably more elementary than that of [21]. Second, it follows immediately from our theorems that both K and P MGL are E_{∞} as motivic spectra. An E_{∞} motivic spectrum is a coherently commutative object in an appropriate symmetric monoidal model category of structured motivic spectra, such as P. Hu’s motivic 𝕊-modules [6] or J.F. Jardine’s motivic symmetric spectra [7]; in particular, this is a much stronger than the assertion that algebraic K-theory defines a presheaf of (ordinary) E_{∞} spectra on an appropriate site. This is already known to be the case for algebraic cobordism, where it is clear from the construction of MGL, but does not appear to be known either for periodic algebraic cobordism or algebraic K-theory.

This is important because the category of modules over an E_{∞} motivic spectrum R inherits a symmetric monoidal structure, at least in the higher categorical sense of [16]. As a result, there is a version of derived algebraic geometry which uses E_{∞} motivic spectra as its basic building blocks. In [13], J. Lurie shows that spec Σ_{+}^{∞}ℙ^{∞}[1/β] is the initial derived scheme over which the derived multiplicative group 𝔾_{R} := spec R ∧ Σ_{+}^{∞}ℤ acquires an “orientation”, in the sense that the formal group of 𝔾_{R} may be identified with the formal spectrum ℙ^{∞} ⊗ spec R. Since Σ_{+}^{∞}ℂℙ^{∞}[1/β] represents topological K-theory, this is really a theorem about the relation between K-theory and the derived multiplicative group, and is the starting point for Lurie’s program to similarly relate topological modular forms and derived elliptic curves. Hence the motivic version of the K-theory result may be seen as a small step towards an algebraic version of elliptic cohomology.

1.3. Acknowledgements. We are very grateful to Mike Hopkins and Rick Jardine for helpful lunch-break discussions during a workshop at the Fields Institute in May 2007, as well as to an anonymous referee for their careful reading of the paper and constructive criticism. We thank M. Spitzweck and P.A. Østvær for bringing to our attention their methods, which give an alternate proof of Theorem 4.17 (see [32]), and for suggesting a number of valuable comments. The first author would also like to thank Sarah Whitehouse for illuminating conversations about operations in K-theory and John Greenlees for his interest in this project and its equivariant analogues.

2. Oriented Cohomology Theories

2.1. Motivic spaces. Throughout this paper, we write S for a Noetherian base scheme of finite Krull dimension.

Definition 2.1. A motivic space is a simplicial sheaf on the Nisnevich site of smooth schemes over S.

We often write 0 for the initial motivic space ∅, the simplicial sheaf with constant value the set with zero elements, and 1 for the final motivic space S, the simplicial sheaf with constant value the set with one element. We assume that the reader is familiar with the Morel-Voevodsky 𝔸^{1}-local model structure on the category of motivic spaces used to define the unstable motivic homotopy category [20]. We adhere to this treatment with one exception: we adopt a different convention for indexing the simplicial and algebraic spheres. The simplicial circle is the pair associated to the constant simplicial sheaves

S^{1,0} := (Δ^{1}, ∂Δ^{1});

its smash powers are the simplicial spheres

S^{n,0} := (Δ^{n}, ∂Δ^{n}).

The algebraic circle is the multiplicative group scheme 𝔾 := 𝔾_{m} := 𝔸^{1} − 𝔸^{0}, pointed by the identity section 1 → 𝔾; its smash powers define the algebraic spheres

S^{0,n} := (𝔾, 1)^{∧n}.

Putting the two together, we obtain a bi-indexed family of spheres

S^{p,q} := S^{p,0} ∧ S^{0,q}.

It is straightforward to show that

(𝔸^{n} − 𝔸^{0}, 1) ≃ S^{n−1,n}

and

(𝔸^{n}, 𝔸^{n} − 𝔸^{0}) ≃ (ℙ^{n}, ℙ^{n−1}) ≃ S^{n,n}.

We emphasize that, according to the more usual grading convention, S^{p,q} is written S^{p+q,q}; we find it more intuitive to separate the simplicial and algebraic spheres notationally. Moreover, for this purposes of this paper, the diagonal spheres

S^{n,n} ≃ (𝔸^{n}, 𝔸^{n} − 𝔸^{0}) ≃ (ℙ^{n}, ℙ^{n−1})

are far and away the most important, so they will be abbreviated

S^{n} := S^{n,n}.

This allows us to get by with just a single index most of the time. We extend this convention to suspension and loop functors. That is, Σ(−) denotes the endofunctor on pointed motivic spaces (or spectra) defined by

ΣX := S^{1} ∧ X := S^{1,1} ∧ X.

Similarly, its right adjoint Ω(−) is defined by

ΩX := map_{+}(S^{1}, X) := map_{+}(S^{1,1}, X).

Note that Σ is therefore not the categorical suspension, which is to say that the cofiber of the unique map X → 1 is given by S^{1,0}∧X instead of S^{1,1}∧X = ΣX. While this may be confusing at first, we feel that the notational simplification that results makes it worthwhile in the end.

2.2. Motivic spectra. To form the stable motivic category, we formally add desuspensions with respect to the diagonal spheres S^{n} = S^{n,n} = (𝔸^{n}, 𝔸^{n} −𝔸^{0}).

Definition 2.2. A motivic prespectrum is a sequence of pointed motivic spaces

{X(0), X(1), …},

equipped with maps Σ^{p}X(q) → X(p + q), such that the resulting squares

\xymatrix{Σ^{p}Σ^{q}X(r) \ar[r] \ar[d] & Σ^{p+q}X(r) \ar[d] \\ Σ^{p}X(q + r) \ar[r] & X(p + q + r)}

commute.

Definition 2.3. A motivic prespectrum is a motivic spectrum if, for all natural numbers p, q, the adjoints X(q) → Ω^{p}X(p + q) of the prespectrum structure maps Σ^{p}X(q) → X(p + q) are weak equivalences.

A pointed motivic space X = (X, 1) gives rise to the suspension spectrum Σ^{∞}X, the spectrum associated to the prespectrum with

(Σ^{∞}X)(p) := Σ^{p}X

and structure maps

Σ^{q}Σ^{p}X ⟶ Σ^{p+q}X.

If X isn’t already pointed, we usually write Σ_{+}^{∞}X for Σ^{∞}X_{+}, where X_{+} is the pointed space (X_{+}, 1) ≃ (X, 0). If X happens to be the terminal object 1, we write 𝕊 := Σ_{+}^{∞}1 for the resulting suspension spectrum, the motivic sphere. We will need that the category of motivic spectra is closed symmetric monoidal with respect to the smash product. However, we do not focus on the details of its construction, save to say that either P. Hu’s theory of motivic 𝕊-modules [6] or J.F. Jardine’s motivic symmetric spectra [7] will do. In particular, the category of motivic spectra is tensored and cotensored over itself via the smash product and the motivic function spectrum bifunctors. We may also regard it as being tensored and cotensored over pointed motivic spaces via the suspension spectrum functor. Given a motivic spectrum R and a pointed motivic space X, we write X ∧ R for the motivic spectrum Σ^{∞}X ∧ R and R^{X} for the motivic spectrum of maps from Σ^{∞}X to R. Here Σ^{∞}X is the motivic spectrum associated to the motivic prespectrum whose value in degree n is the pointed motivic space Σ^{n}X. As a functor from pointed motivic spaces to motivic spectra, Σ^{∞} admits a right adjoint Ω^{∞} which associates to a motivic spectrum its underlying motivic “infinite-loop”. There are also a number of symmetric monoidal categories over which the category of motivic spectra is naturally enriched. We write Y^{X} for the motivic function spectrum of maps from the motivic spectrum X to the motivic spectrum Y, map(X, Y) = (Ω^{∞}Y^{X})(S) for the (ordinary) space of maps from X to Y, and [X, Y] = Y^{0}(X) = π_{0} map(X, Y) for the abelian group of homotopy classes of maps from X to Y.

2.3. Motivic ring spectra. In this paper, unless appropriately qualified, a motivic ring spectrum will always mean a (not necessarily commutative) monoid in the homotopy category of motivic spectra. We reiterate that a motivic spectrum is a ℙ^{1}-spectrum; that is, it admits desuspensions by algebraic spheres as well as simplicial spheres.

Definition 2.4. A motivic ring spectrum R is periodic if the graded ring π_{∗}R contains a unit μ ∈ π_{1}R in degree one.

Remark 2.5. Since π_{1}R is by definition π_{0} map_{+}(ℙ^{1}, Ω^{∞}R), and over spec ℂ, ℙ^{1}(ℂ) ≃ ℂℙ^{1}, the topological 2-sphere, this is compatible with the notion of an even periodic ring spectrum so common in ordinary stable homotopy theory.

Proposition 2.6. If R is periodic then R ≃ Σ^{n}R for all n.

Proof. Let μ ∈ π_{1}R be a unit with inverse μ^{−1} ∈ π_{−1}R. Then for any n, the multiplication by μ^{−n} map

R ⟶ Σ^{n}R

is an equivalence, since multiplication by μ^{n} provides an inverse. □

Let P 𝕊 denote the periodic sphere, the motivic spectrum

P 𝕊 := ⋁_{n∈ℤ} Σ^{n}𝕊.

With respect to the multiplication induced by the equivalences Σ^{p}𝕊 ∧ Σ^{q}𝕊 → Σ^{p+q}𝕊, the unit in degree one given by the inclusion Σ^{1}𝕊 → P 𝕊 makes P 𝕊 into a periodic 𝕊-algebra. More generally, given an arbitrary motivic ring spectrum R,

P R := P 𝕊∧R ≃ ⋁_{n∈ℤ} Σ^{n}R

is a periodic ring spectrum equipped with a ring map R → P R.

Proposition 2.7. Let R be a motivic ring spectrum. Then homotopy classes of ring maps P 𝕊 → R naturally biject with units in π_{1}R.

Proof. By definition, ring maps P 𝕊 → R are indexed by families of elements r_{n} ∈ π_{n}R with r_{m}r_{n} = r_{m+n} and r_{0} = 1. Hence r_{n} = r_{1}^{n}, and in particular r_{−1} = r_{1}^{−1}. □

Said differently, the homotopy category of periodic motivic ring spectra is equivalent to the full subcategory of the homotopy category of motivic ring spectra which admit a ring map from P 𝕊. This is not the same as the homotopy category of P 𝕊-algebras, in which only those maps which preserve the distinguished unit are allowed.

Corollary 2.8. Let Q be a motivic ring spectrum and R a periodic motivic ring spectrum. Then the set of homotopy classes of ring maps P Q → R is naturally isomorphic to the set of pairs consisting of a homotopy class of ring map Q → R and a distinguished unit μ ∈ π_{1}R.

2.4. Orientations. Let R be a commutative motivic ring spectrum.

Definition 2.9. The Thom space of an n-plane bundle V → X is the pair (V, V − X), where V − X denotes the complement in V of the zero section X → V.

Given two vector bundles V → X and W → Y, the Thom space (V ×W, V ×W − X × Y) of the product bundle V × W → X × Y is equivalent (even isomorphic) to the smash product (V, V − X) ∧ (W, W − Y) of the Thom spaces. Since the Thom space of the trivial 1-dimensional bundle 𝔸^{1} → 𝔸^{0} is the motivic 1-sphere S^{1} ≃ (𝔸^{1}, 𝔸^{1} − 𝔸^{0}), we see that the Thom space of the trivial ndimensional bundle 𝔸^{n} → 𝔸^{0} is the motivic n-sphere S^{n} ≃ (𝔸^{n}, 𝔸^{n}−𝔸^{0}). Note that the complement of the zero section 𝕃 − ℙ^{∞} of the universal line bundle 𝕃 → ℙ^{∞} ≃ B𝔾 is equivalent to the total space of the universal principal 𝔾bundle E𝔾 → B𝔾, which is contractible. Hence the Thom space of 𝕃 → ℙ^{∞} is equivalent to (ℙ^{∞}, ℙ^{0}), and the Thom space of the restriction of 𝕃 → ℙ^{∞} along the inclusion ℙ^{1} → ℙ^{∞} is equivalent to (ℙ^{1}, ℙ^{0}) ≃ S^{1}.

Definition 2.10. An orientation of R is the assignment, to each m-plane bundle V → X, of a class θ(V/X) ∈ R^{m}(V, V − X), in such a way that

(1) for any f : Y → X, the class θ(f^{∗}V/Y) of the restriction f^{∗}V → Y of V → X is equal to the restriction f^{∗}θ(V/X) of the class θ(V/X) in R^{m}(f^{∗}V, f^{∗}V − Y),

(2) for any n-plane bundle W → Y, the (external) product θ(V/X) × θ(W/Y) of the classes θ(V/X) and θ(W/Y) is equal to the class θ(V × W/X × Y) of the (external) product of V → X and W → Y in R^{m+n}(V × W, V × W − X × Y), and

(3) if 𝕃 → ℙ^{∞} is the universal line bundle and i : ℙ^{1} → ℙ^{∞} denotes the inclusion, then i^{∗}θ(𝕃/ℙ^{∞}) ∈ R^{1}(f^{∗}𝕃, f^{∗}𝕃 − ℙ^{1}) corresponds to 1 ∈ R^{0}(S^{0}) via the isomorphism R^{0}(S^{0}) ≅ R^{1}(S^{1}) ≅ R^{1}(f^{∗}𝕃, f^{∗}𝕃 − ℙ^{1}).

Given an orientation of R, the class θ(V/X) ∈ R^{n}(V, V − X) associated to a n-plane bundle V → X is called the Thom class of V → X. The main utility of Thom classes is that they define R^{∗}(X)-module isomorphisms R^{∗}(X) → R^{∗+n}(V, V − X) (cf. [23]).

Remark 2.11. The naturality condition implies that it is enough to specify Thom classes for the universal vector bundles V_{n} → BGL_{n}. We write MGL_{n} for the Thom space of V_{n} → BGL_{n} and θ_{n} for θ(V_{n}/BGL_{n}) ∈ R^{n}(MGL_{n}).

2.5. Basic calculations in oriented cohomology. In this section we fix an oriented commutative motivic ring spectrum R equipped with a unit μ ∈ π_{1}R. Note that we can use μ to move the Thom classes θ_{n} ∈ R^{n}(MGL_{n}) to degree zero Thom classes ϑ_{n} := μ^{n}θ_{n} ∈ R^{0}(MGL_{n}). The following calculations are well known (cf. [1], [4], [23]). Note that all (co)homology is implicitly the (co)homology of a pair. In particular, if X is unpointed, then R^{0}(X) := R^{0}(X, 0), where 0 → X is the unique map from the inital object 0; if X is pointed, then R^{0}(X) := R^{0}(X, 1), where 1 → X is the designated map from the terminal object 1.

Proposition 2.12. The first Chern class of the tautological line bundle on ℙ^{n} defines a ring isomorphism R^{0}[λ]/(λ^{n+1}) → R^{0}(ℙ^{n}).

Proof. Inductively, one has a morphism of exact sequences

\xymatrix{λ^{n}R^{0}[λ]/(λ^{n+1}) \ar[r] \ar[d] & R^{0}[λ]/(λ^{n+1}) \ar[r] \ar[d] & R^{0}[λ]/(λ^{n}) \ar[d] \\ R^{0}(ℙ^{n}, ℙ^{n−1}) \ar[r] & R^{0}(ℙ^{n}) \ar[r] & R^{0}(ℙ^{n−1})}

in which the left and right, and hence also the middle, vertical maps are isomorphisms. □

Proposition 2.13. The first Chern class of the tautological line bundle on ℙ^{∞} defines a ring isomorphism R^{0}[[λ]] ≅ lim R^{0}[λ]/(λ^{n}) → R^{0}(ℙ_{+}^{∞}).

Proof. The lim^{1} term in the exact sequence

0 ⟶ lim^{1}R^{−1,0}(ℙ^{n}) ⟶ R^{0}(ℙ^{∞}) ⟶ lim R^{0}(ℙ^{n})

vanishes because the maps R^{−1,0}(ℙ^{n}) ⟶ R^{−1,0}(ℙ^{n−1}) are surjective. □

Corollary 2.14. For each n, the natural map

R_{0}(ℙ^{n}) ⟶ hom_{R^{0}}(R^{0}(ℙ^{n}), R^{0})

is an isomorphism.

Proof. The dual of 2.12 shows that R_{0}(ℙ^{n}) is free of rank n + 1 over R_{0}. □

Proposition 2.15 (Atiyah [4]). Let p : Y → X be a map of quasicompact S-schemes and let y_{1}, …, y_{n} be elements of R^{0}(Y). Let M be the free abelian group on the y_{1}, …, y_{n}, and suppose that X has a cover by open subschemes U such that for all open V in U, the natural map

R^{0}(V) ⊗ M ⟶ R^{0}(p^{−1}V)

is an isomorphism. Then, for any open W in X, the map

R^{0}(X, W) ⊗ M ⟶ R^{0}(Y, p^{−1}W)

is an isomorphism.

Proof. Apply Atiyah’s proof [4], mutatis mutandis. □

Proposition 2.16. Let Z be an S-scheme such that, for any homotopy commutative R-algebra A, A^{0}(Z) ≅ R^{0}(Z) ⊗_{R^{0}} A^{0}. Then, for any S-scheme X, R^{0}(Z × X) ≅ R^{0}(Z) ⊗_{R^{0}} R^{0}(X).

Proof. The diagonal of X induces a homotopy commutative R-algebra structure on A = R^{X}, the cotensor of the motivic space X with the motivic spectrum R. Hence

A^{0}(Z) ≅ R^{0}(Z) ⊗_{R^{0}} A^{0} ≅ R^{0}(Z) ⊗_{R^{0}} R^{0}(X).
□

Corollary 2.17. Let p : V → X be a rank n vector bundle over a quasicompact S-scheme X and let L → ℙ(V) be the tautological line bundle. Then the map which sends λ to the first Chern class of L induces an isomorphism

R^{0}(X)[λ]/(λ^{n} − λ^{n−1}c_{1}V + ⋯ + (−1)^{n}c_{n}V) ⟶ R^{0}(ℙ(V))

of R^{0}-algebras.

Proof. If V is trivial then ℙ(V) ≅ ℙ_{X}^{n−1}, and the result follows from Propositions 2.12 and 2.16. In general, the projection ℙ(V) → X is still locally trivial, so we may apply Proposition 2.15, with W empty and {y_{i}} the image in R^{0}(ℙ(V)) of a basis for R^{0}(X)[λ]/(λ^{n} − … + (−1)^{n}c_{n}V) as a free R^{0}(X)- module. □

Proposition 2.18. Let V → X be a rank n vector bundle over a quasicompact S-scheme X, Flag(V) → X the associated flag bundle, and σ_{k}(x_{1}, …, x_{n}), 1 ≤ k ≤ n, the k^{th} elementary symmetric function in the indeterminates λ_{i}. Then the map

R^{0}(X)[λ_{1}, …, λ_{n}]/({c_{k}(V) − σ_{k}(λ_{1}, …, λ_{n})}_{k>0}) ⟶ R^{0}(Flag(V))

which sends the λ_{i} to the first Chern classes of the n tautological line bundles on Flag(V), is an isomorphism of R^{0}-algebras.

Proof. The evident relations among the Chern classes imply that the map is well-defined. Using Proposition 2.15 and a basis for the free R^{0}- module R^{0}(Flag(𝔸^{n−1}), it follows inductively from the fibration Flag(𝔸^{n−1}) → Flag(𝔸^{n}) → ℙ^{n−1} that

R^{0}[λ_{1}, …, λ_{n}]/({σ_{k}(λ_{1}, …, λ_{n})}) ⟶ R^{0}(Flag(𝔸^{n}))

is an isomorphism. Using Proposition 2.16, we deduce the desired result for trivial vector bundles V. For the general case, we apply Proposition 2.15 again, with a basis of the free R^{0}(X)-module R^{0}(X)[λ_{1}, …, λ_{n}]/({c_{k}(V)−σ_{k}}) giving the necessary elements of R^{0}(Flag(V)). □

Proposition 2.19. Let p : V → X be an rank n vector bundle over a quasicompact S-scheme X, let q : Grass_{m}(V) → X be the Grassmannian bundle of m-dimensional subspaces of V, let ξ_{m}(V) → Grass_{m}(V) be the tautological m-plane bundle over Grass_{m}(V), and write q^{∗}(V)/ξ_{m}(V) for the quotient (n − m)-plane bundle. Then the map

R^{0}(X)[σ_{1}, …, σ_{m}, τ_{1}, …, τ_{n−m}] ⟶ R^{0}(Grass_{m}(V))

which sends σ_{i} to c_{i}(ξ_{m}(V)) and τ_{j} to c_{j}(q^{∗}(V)/ξ_{m}(V)) induces an isomorphism

R^{0}(X)[σ_{1}, …, σ_{m}, τ_{1}, …, τ_{n−m}]/({c_{k}(V) − Σ_{i+j=k}σ_{i}τ_{j}}) ⟶ R^{0}(Grass_{m}(V))

of R^{0}-algebras (as usual, c_{k}(V) = 0 for k > n and c_{0}(V) = σ_{0} = τ_{0} = 1).

Proof. The identity q^{∗}c(V) = c(q^{∗}V) = c(ξ_{m}(V))c(q^{∗}V/ξ_{m}(V)) implies that each c_{k}(V) − Σσ_{i}τ_{j} is sent to zero, so the map is well-defined. Just as in the case of flag bundles, use induction together with the fibration Flag(𝔸^{m}) × Flag(𝔸^{n−m}) → Flag(𝔸^{n}) → Grass_{m}(𝔸^{n}) to see that

R^{0}[σ_{1}, …, σ_{m}, τ_{1}, …, τ_{n−m}]/({ Σ_{i+j=k}σ_{i}τ_{j}}) ⟶ R^{0}(Grass_{m}(𝔸^{n}))

is an isomorphism. This implies the result for trivial vector bundles by Proposition 2.16, and we deduce the general result from Proposition 2.15. □

Proposition 2.20. There are isomorphisms

R^{0}(BGL_{n}) ⟶ R^{0}(BGL_{1}^{n})^{Σ_{n}} ≅ R^{0}[[λ_{1}, …, λ_{n}]]^{Σ_{n}} ≅ R^{0}[[σ_{1}, …, σ_{n}]].

Proof. Writing V_{n} → BGL_{n} for the tautological vector bundle, we have an equivalence Flag(V_{n}) ≃ BGL_{1}^{n}. Inductively, we have isomorphisms

R^{0}[[λ_{1}, …, λ_{n}]] ⟶ R^{0}(BGL_{1}^{n})

and the map

R^{0}(BGL_{n}) ⟶ R^{0}(BGL_{1}^{n}) ≅ R^{0}[[λ_{1}, …, λ_{n}]]

factors through the invariant subring R^{0}[[λ_{1}, …, λ_{n}]]^{Σ_{n}} ≅ R^{0}[[σ_{1}, …, σ_{n}]]. By Proposition 2.18, R^{0}(BGL_{1}^{n}) is free of rank n! over R^{0}(BGL_{n}), so it follows that R^{0}(BGL_{n}) ≅ R^{0}(BGL_{1}^{n})^{Σ_{n}}. □

Corollary 2.21. The natural map

R^{0}(BGL_{n}) ⟶ hom_{R_{0}}(Sym_{R_{0}}^{n} R_{0}(ℙ^{∞}), R_{0})

is an isomorphism.

Proof. By Proposition 2.20, we need only check this for n = 1. But

R^{0}(ℙ^{m}) ≅ hom_{R_{0}}(R_{0}(ℙ^{m}), R_{0}),

both being free of rank m + 1 over R^{0}, and

R^{0}(ℙ^{∞}) ≅ lim R^{0}(ℙ^{m}) ≅ hom_{R_{0}}(colim R_{0}(ℙ^{m}), R_{0}) ≅ hom_{R_{0}}(R_{0}(ℙ^{∞}), R_{0})

by Proposition 2.12. □

Corollary 2.22. There are isomorphisms R^{0}(BGL) ≅ lim_{n} R^{0}(BGL_{n}) ≅ R^{0}[[σ_{1}, σ_{2}, …]].

Proof. The lim^{1} term in the short exact sequence

0 ⟶ lim_{n}^{1}R^{1,0}(BGL_{n}) ⟶ R^{0}(BGL) ⟶ lim_{n}R^{0}(BGL_{n})

vanishes since the maps R^{1,0}(BGL_{n}) → R^{1,0}(BGL_{n−1}) are surjective. □

2.6. The oriented cohomology of BGL_{+} ∧ Z. Let R be an oriented periodic commutative motivic ring spectrum and let Z be an arbitrary motivic spectrum. Recall (cf. [5]) that a motivic spectrum is cellular if belongs to the smallest full subcategory of motivic spectra which is closed under homotopy colimits and contains the spheres S^{p,q} for all p, q ∈ ℤ, and that a motivic space is stably cellular if its suspension spectrum is cellular.

Proposition 2.23. Let X := colim_{n} X_{n} be a telescope of finite cellular motivic spectra such that each R^{∗,∗}(X_{n}) is a finite free R^{∗,∗}-module and, for any motivic spectrum Z, the induced maps

R^{0}(X_{n} ∧ Z) ⟶ R^{0}(X_{n−1} ∧ Z)

are surjective. Then the natural map

R^{0}(X) ˆ⊗_{R^{0}}R^{0}(Z) ⟶ R^{0}(X ∧ Z)

is an isomorphism.

Proof. This is an immediate consequence of the motivic Künneth spectral sequence of Dugger-Isaksen [5]. Indeed, for each n, R^{∗,∗}(X_{n}) is a free R^{∗,∗}- module, so the spectral sequence

Tor_{∗}^{R^{∗,∗}}(R^{∗,∗}(X_{n}), R^{∗,∗}(Z)) ⇒ R^{∗,∗}(X_{n} ∧ Z)

collapses to yield the isomorphism

R^{∗,∗}(X_{n}) ˆ⊗_{R^{∗,∗}}R^{∗,∗}(Z) ≅ R^{∗,∗}(X_{n} ∧ Z).

Moreover, by hypothesis, each of the relevant lim^{1} terms vanish, so that

R^{∗,∗}(X) ˆ⊗_{R^{∗,∗}}R^{∗,∗}(Z) ≅ lim_{n} R^{∗,∗}(X_{n}) ⊗_{R^{∗,∗}}R^{∗,∗}(Z)

≅ lim_{n} R^{∗,∗}(X_{n} ∧ Z) ≅ R^{∗,∗}(X ∧ Z).

□

Corollary 2.24. Let Z be a motivic spectrum. Then there are natural isomorphisms

R^{0}(ℙ^{∞}) ˆ⊗_{R^{0}}R^{0}(Z) ⟶ R^{0}(ℙ_{+}^{∞} ∧ Z)

and

R^{0}(BGL) ˆ⊗_{R^{0}}R^{0}(Z) ⟶ R^{0}(BGL_{+} ∧ Z)

Proof. For each m,

BGL_{m} ≃ colim_{n} Grass_{m}(𝔸^{n})

is a colimit of finite stably cellular motivic spaces such that, for each n, R^{∗,∗}(Grass_{m}(𝔸^{n})) is a free R^{∗,∗}-module and

R^{0}(Grass_{m}(𝔸^{n})) ⊗_{R^{0}} R^{0}(Z) ≅ R^{0}(Grass_{m}(𝔸^{n})_{+} ∧ Z) ⟶
R^{0}(Grass_{m}(𝔸^{n−1})_{+} ∧ Z) ≅ R^{0}(Grass_{m}(𝔸^{n−1})) ⊗_{R_{0}} R^{0}(Z)

is (split) surjective. It therefore follows from Proposition 2.23 that, for each m,

R^{0}(Grass_{m}(𝔸^{∞})_{+}) ˆ⊗_{R^{0}}R^{0}(Z) ≅ R^{0}(Grass_{m}(𝔸^{∞})_{+} ∧ Z).

Taking m = 1 yields the result for ℙ^{∞}; for BGL, we must consider the sequence

BGL ≃ colim_{m} BGL_{m} ≃ Grass_{m}(𝔸^{∞}) in which the maps come from a fixed isomorphism 𝔸^{1} ⊕ 𝔸^{∞} ≅ 𝔸^{∞}. Note, however, that it follows from the above, together with the (split) surjection

R^{0}(Grass_{m}(𝔸^{∞}) ⟶ R^{0}(Grass_{m−1}(𝔸^{∞}))

of Proposition 2.20, that, for each m,

R^{0}(Grass_{m}(𝔸^{∞})_{+} ∧ Z) ⟶ R^{0}(Grass_{m−1}(𝔸^{∞})_{+} ∧ Z)

is (split) surjective, so that the lim^{1} term vanishes and

R^{0}(BGL) ˆ⊗_{R^{0}}R^{0}(Z) ≅ lim_{m} R^{0}(Grass_{m}(𝔸^{∞})_{+} ∧ Z) ≅ R^{0}(BGL_{+} ∧ Z).

□

2.7. Primitives in the oriented cohomology of BGL. Let R be an oriented periodic commutative motivic ring spectrum. As is shown in Section 4.3 of [20], the group completion

BGL_{ℤ} ≃ Ω^{1,0}B(BGL_{ℕ})

(usually written ℤ × BGL) of the additive monoid BGL_{ℕ} = ∐_{n∈ℕ} BGL_{n} fits into a fibration sequence

BGL ⟶ BGL_{ℤ} ⟶ ℤ,

where BGL ≃ colim_{n} BGL_{n}. As BGL_{ℕ} is commutative up to homotopy, BGL_{ℤ} is an abelian group object in the motivic homotopy category.

Lemma 2.25. Let Add(BGL_{ℤ}, Ω^{∞}R) denote the abelian group of homotopy classes of additive maps BGL_{ℤ} → Ω^{∞}R. Then the inclusion

Add(BGL_{ℤ}, Ω^{∞}R) ⟶ R^{0}(BGL_{ℤ})

identifies Add(BGL_{ℤ}, Ω^{∞}R) with the abelian group of primitive elements in the Hopf algebra R^{0}(BGL_{ℤ}).

Proof. By definition, there is an equalizer diagram

\xymatrix{Add(BGL_{ℤ}, Ω^{∞}R) \ar[r] & R^{0}(BGL_{ℤ}) \ar[r] \ar[r] & (BGL_{ℤ} × BGL_{ℤ})}

associated to the square

\xymatrix{BGL_{ℤ} × BGL_{ℤ} \ar[r] \ar[d] & BGL_{ℤ} \ar[d] \\ Ω^{∞}R × Ω^{∞}R \ar[r] & Ω^{∞}R}

in which the horizontal maps are the addition maps. Let δ denote the Hopf algebra diagonal

δ : R^{0}(BGL_{ℤ}) ⟶ R^{0}(BGL_{ℤ} × BGL_{ℤ}) ≅ R^{0}(BGL_{ℤ}) ˆ⊗_{R^{0}}R^{0}(BGL_{ℤ}).

Then the equalizer consists of those f ∈ R^{0}(BGL_{ℤ}) such that δ(f) = f ⊗ 1 + 1 ⊗ f. This identifies Add(BGL_{ℤ}, Ω^{∞}R) with the primitive elements in R^{0}(BGL_{ℤ}). □

Lemma 2.26. There are natural isomorphisms

Add(BGL_{ℤ}, Ω^{∞}R) ≅ Add(BGL, Ω^{∞}R) × Add(ℤ, Ω^{∞}R)
≅ Add(BGL, Ω^{∞}R) × R^{0}.

Proof. The product of additive maps is additive, and, in any category with finite products and countable coproducts, ℤ = ∐_{ℤ} 1 is the free abelian group on the terminal object 1. □

Proposition 2.27. The map

Add(BGL_{ℤ}, Ω^{∞}R) ⟶ R^{0}(BGL_{1}),

obtained by restricting an additive map BGL_{ℤ} → Ω^{∞}R along the inclusion BGL_{1} → BGL_{ℤ}, is an isomorphism.

Proof. By Lemma 2.26, it’s enough to show that the inclusion (BGL_{1}, 1) → (BGL, 1) induces an isomorphism

Add(BGL, Ω^{∞}R) ⟶ R^{0}(BGL_{1}, 1).

Thus let M = R_{0}(BGL_{1}, 1), and consider the R_{0}-algebra

A := ⨁_{n≥0} Sym_{R_{0}}^{n} M

together with its augmentation ideal

I := ⨁_{n>0} Sym_{R_{0}}^{n} M.

We have isomorphisms of split short exact sequences

\xymatrix{0 \ar[r] & R^{0}(BGL, 1) \ar[r] \ar[d] & R^{0}(BGL) \ar[r] \ar[d] & R^{0} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & hom_{R_{0}}(I, R_{0}) \ar[r] & hom_{R_{0}}(A, R_{0}) \ar[r] & hom_{R_{0}}(R_{0}, R_{0}) \ar[r] & 0}

and

\xymatrix{0 \ar[r] & R^{0}(BGL^{×2}, BGL^{∨2}) \ar[r] \ar[d] & R^{0}(BGL^{×2}) \ar[r] \ar[d] & … \\ 0 \ar[r] & hom_{R_{0}}(I ⊗_{R_{0}} I, R_{0}) \ar[r] & hom_{R_{0}}(A ⊗_{R_{0}} A, R_{0}) \ar[r] & …}

\xymatrix{… \ar[r] & R^{0}(BGL^{∨2}) \ar[r] \ar[d] & 0 \\ … \ar[r] & hom_{R_{0}}(R_{0} ⊕ I^{⊕2}, R_{0}) \ar[r] & 0}

Documenta Mathematica 14 (2009) 359–396

of R^{0}-modules. According to Lemmas 2.25 and 2.26, we have an exact sequence

0 ⟶ Add(BGL, Ω^{∞}R) ⟶ R^{0}(BGL, 1) ⟶ R^{0}(BGL^{×2}, BGL^{∨2})

in which the map on the right is the cohomology of the map

μ − p_{1} − p_{2} : (BGL^{×2}, BGL^{∨2}) ⟶ (BGL, 1)

(μ is the addition and the p_{i} are the projections); moreover, this map is the R_{0}-module dual of the multiplication I ⊗_{R_{0}} I → I. Hence these short exact sequences assemble into a diagram

\xymatrix{0 \ar[r] & hom_{R_{0}}(I/I^{2}, R_{0}) \ar[r] \ar[d] & Add(BGL, Ω^{∞}R) \ar[r] \ar[d] & … \\ 0 \ar[r] & hom_{R_{0}}(I, R_{0}) \ar[r] \ar[d] & R^{0}(BGL) \ar[r] \ar[d] & … \\ 0 \ar[r] & hom_{R_{0}}(I ⊗_{R_{0}} I, R_{0}) \ar[r] & R^{0}(BGL^{×2}) \ar[r] & …}

\xymatrix{… \ar[r] & 0 \ar[d] \ar[r] & 0 \\ … \ar[r] & R^{0} \ar[d] \ar[r] & 0 \\ … \ar[r] & hom_{R_{0}}(R_{0} ⊕ I^{⊕2}, R_{0}) \ar[r] & 0}

of short exact sequences by the snake lemma. In particular, we see that Add(BGL, Ω^{∞}R) is naturally identified with the dual hom_{R_{0}}(I/I^{2}, R_{0}) of the module of indecomposables I/I^{2}. But I/I^{2} ≅ M = R_{0}(BGL_{1}, 1), the duality map

R^{0}(BGL_{1}, 1) ⟶ hom_{R_{0}}(R_{0}(BGL_{1}, 1), R_{0})

is an R^{0}-module isomorphism, and the restriction R^{0}(BGL, 1) → R^{0}(BGL_{1}, 1) is dual to the inclusion M → I. □

3. Algebraic Cobordism

3.1. The representing spectrum. For each natural number n, let V_{n} → BGL_{n} denote the universal n-plane bundle over BGL_{n}. Then the Thom spaces

MGL_{n} := (V_{n}, V_{n} − BGL_{n})

come equipped with natural maps

MGL_{p} ∧ MGL_{q} ⟶ MGL_{p+q}

defined as the composite of the isomorphism

(V_{p}, V_{p} − BGL_{p}) ∧ (V_{q}, V_{q} − BGL_{q}) ⟶ (V_{p} × V_{q}, V_{p} × V_{q} − BGL_{p} × BGL_{q}) and the map on Thom spaces associated to the inclusion of vector bundles

\xymatrix{V_{p} × V_{q} \ar[d] \ar[r] & V_{p+q} \ar[d] \\ BGL_{p} × BGL_{q} \ar[r] & BGL_{p+q}.}

Restricting this map of vector bundles along the inclusion 1×BGL_{q} → BGL_{p}× BGL_{q} gives a map of Thom spaces

(𝔸^{p}, 𝔸^{p} − 𝔸^{0}) ∧ MGL_{q} → MGL_{p+q},

and these maps comprise the structure maps of the prespectrum MGL. The associated spectrum is defined by

MGL(p) := colim_{q} Ω^{q}MGL_{p+q},

as evidently the adjoints

MGL(q) ≃ colim_{r} Ω^{r}MGL_{q+r} ≃ colim_{r} Ω^{p+r}MGL_{p+q+r} ≃ Ω^{p}MGL(p + q)

of the structure maps Σ^{p}MGL(q) → MGL(p + q) are equivalences. The last equivalence uses the fact that ℙ^{1} is a compact object of the motivic homotopy category.

Definition 3.1 (Voevodsky [34]). Algebraic cobordism is the motivic cohomology theory represented by the motivic spectrum MGL.

3.2. Algebraic cobordism is the universal oriented motivic spectrum. Just as in ordinary stable homotopy theory, the Thom classes θ_{n} ∈ R^{n}(MGL_{n}) coming from an orientation on a commutative motivic ring spectrum R assemble to give a ring map θ : MGL → R. We begin with a brief review of this correspondence.

Proposition 3.2 (Panin, Pimenov, Röndigs [23]). Let R be a commutative monoid in the homotopy category of motivic spectra. Then the set of monoidal maps MGL → R is naturally isomorphic to the set of orientations on R.

Proof. The classical analysis of complex orientations on ring spectra R generalizes immediately. A spectrum map θ : MGL → R is determined by a compatible family of maps θ_{n} : MGL_{n} → R^{n}, which is to say a family of universal Thom classes θ_{n} ∈ R^{n}(MGL_{n}). An arbitrary n-plane bundle V → X, represented by a map X → BGL_{n}, induces a map of Thom spaces V/V − X → MGL_{n}, so θ_{n} restricts to a Thom class in R^{n}(V/V − X). Moreover, these Thom classes are multiplicative and unital precisely when θ : MGL → R is monoidal. Conversely, an orientation on R has, as part of its data, Thom classes θ_{n} ∈ R^{n}(MGL_{n}) for the universal bundles V_{n} → BGL_{n}, and these assemble to form a ring map θ : MGL → R. □

Again, just as in topology, an orientation on R is equivalent to a compatible family of R-theory Chern classes for vector bundles V → X. This follows from the Thom isomorphism R^{∗}(BGL_{n}) ≅ R^{∗}(MGL_{n}).

More difficult is the fact that an orientation on a ring spectrum R is uniquely determined by the first Thom class alone; that is, a class θ_{1} ∈ R^{1}(BGL_{1}) = R whose restriction i^{∗}θ_{1} ∈ R^{1}(S^{1}) along the inclusion S^{1} → MGL_{1} corresponds to 1 ∈ R^{0}(S^{0}) via the suspension isomorphism R^{1}(S^{1}) ≅ R^{0}(S^{0}). This is a result of the splitting principle, which allows one to construct Thom classes (or Chern classes) for general vector bundles by descent from a space over which they split. See Adams [1] and Panin-Pimenov-Röndigs [23] for details.

3.3. A ring spectrum equivalent to P MGL. The wedge

⋁_{n∈ℕ}Σ^{∞}MGL_{n}

forms a ring spectrum with unit 𝕊 ≃ Σ^{∞}MGL_{0} and multiplication

⋁_{p}Σ^{∞}MGL_{p} ∧ ⋁_{q}Σ^{∞}MGL_{q} ⟶ ⋁_{p,q}Σ^{∞}MGL_{p} ∧ MGL_{q} ⟶ ⋁_{n}Σ^{∞}MGL_{n}

induced by the maps MGL_{p} ∧ MGL_{q} → MGL_{p+q}. Evidently, a (homotopy class of a) ring map ⋁_{n} Σ^{∞}MGL_{n} → R is equivalent to a family of degree zero Thom classes

ϑ_{n} ∈ R^{0}(MGL_{n})

with ϑ_{0} = 1 ∈ R^{0}(MGL_{0}) = R^{0} such that ϑ_{p+q} restricts via MGL_{p}∧MGL_{q} → MGL_{p+q} to the product ϑ_{p}ϑ_{q}. This is not the same as an orientation on R, as there is nothing forcing ϑ_{1} ∈ R^{0}(MGL_{1}) to restrict to a unit in R^{0}(S^{1}). Clearly we should impose this condition, which amounts to inverting β : ℙ^{1} → ℙ^{∞}.

Proposition 3.3. A ring map P MGL → R induces a ring map ⋁_{n} Σ^{∞}MGL_{n}[1/β] → R.

Proof. A ring map θ : P MGL → R consists of a ring map MGL → R and a unit μ ∈ π_{1}R. This specifies Thom classes θ_{n} ∈ R^{n}(MGL_{n}), and therefore Thom classes

ϑ_{n} := μ^{n}θ_{n} ∈ R^{0}(MGL_{n})

such that

ϑ_{p}ϑ_{q} = μ^{p+q}θ_{p}θ_{q} = μ^{p+q}i^{∗}θ_{p+q} = i^{∗}ϑ_{p+q} ∈ R^{0}(MGL_{p} ∧ MGL_{q}),

where i is the map MGL_{p} ∧ MGL_{q} → MGL_{p+q}. This gives a ring map ϑ : ⋁_{n} Σ^{∞}MGL_{n} → R, and therefore the desired map, provided β is sent to a unit. But this is clear: as a class in R^{0}(S^{1}),

ϑ(β) = β^{∗}ϑ_{1} = μβ^{∗}θ_{1},

and β^{∗}θ_{1} ∈ R^{1}(S^{1}) is the image of 1 ∈ R^{0}(S^{0}) under the isomorphism R^{0}(S^{0}) ≅ R^{1}(S^{1}). □

Proposition 3.4. The ring map ⋁_{n∈ℕ}Σ^{∞}MGL_{n}[1/β] → P MGL is an equiva-

lence.

Proof. Write

M := ⋁_{n∈ℕ}Σ^{∞}MGL_{n}[1/β] → P MGL,

and consider the natural transformation of set-valued functors

Rings(P MGL, −) ⟶ Rings(M, −).

Given a ring spectrum R, we have seen that the set Rings(M, R) is naturally isomorphic to the set of collections {ϑ_{n}}_{n∈ℕ} with ϑ_{n} ∈ R^{0}(MGL_{n}) such that ϑ_{p+q} restricts to ϑ_{p}ϑ_{q}, ϑ_{1} restricts to a unit in R^{−1}, and ϑ_{0} = 1 ∈ R^{0}(S^{0}). Similarly, the set Rings(P MGL, R) is naturally isomorphic to the product of the set of units in R^{−1} and the set of collections {θ_{n}}_{n∈ℕ} with R^{n}(MGL_{n}) such that θ_{p+q} restricts to θ_{p}θ_{q}, θ_{1} restricts to the image of 1 ∈ R^{0}(S^{0}) in R^{1}(S^{1}), and θ_{0} = 1 ∈ R^{0}(S^{0}). The map Rings(P MGL, R) → Rings(M, R) sends μ ∈ R^{−1} and θ_{n} ∈ R^{n}(MGL_{n}) to ϑ_{n} = μ^{n}θ_{n}. We get a natural map back which sends ϑ_{n} ∈ R^{0}(MGL_{n}) to θ_{n} = μ^{−n}ϑ_{n}, where μ ∈ R^{−1} in the unit corresponding to β^{∗}ϑ_{1} ∈ R^{0}(S^{1}). Clearly the composites are the respective identities, and we conclude that M → P MGL is an equivalence. □

3.4. Σ_{+}^{∞}BGL[1/β] is orientable. Recall from [23] that, just as in the usual stable homotopy category, an orientation on a ring spectrum R is equivalent to a class in R^{1}(MGL_{1}) which restricts, under the inclusion i : S^{1} → MGL_{1} of the bottom cell, to the class in R^{1}(S^{1}) corresponding to the unit 1 ∈ R^{0}(S^{0}) under the suspension isomorphism R^{0}(S^{0}) → R^{1}(S^{1}). Note also that in the case R is periodic with Bott element β ∈ R^{0}(S^{1}), corresponding under the suspension isomorphism to the unit μ ∈ R^{−1}(S^{0}) with inverse μ^{−1} ∈ R^{1}(S^{0}), then the suspension isomorphism R^{0}(S^{0}) → R^{1}(S^{1}) sends 1 to μ^{−1}β. Now there’s a canonical class θ_{1} ∈ Σ_{+}^{∞}BGL[1/β]^{1}(MGL_{1}) such that

μ^{−1}β = i^{∗}θ_{1} ∈ Σ_{+}^{∞}BGL[1/β]^{1}(S^{1}).

Namely, set θ_{1} := μ^{−1}ϑ_{1}, where ϑ_{1} ∈ Σ_{+}^{∞}BGL[1/β]^{0}(MGL_{1}) is the class of the composite

Σ^{∞}MGL_{1} ≃ Σ^{∞}BGL_{1} ⟶ Σ_{+}^{∞}BGL ⟶ Σ_{+}^{∞}BGL[1/β].

Then β = i^{∗}μθ, so μ^{−1}β = i^{∗}θ.

Proposition 3.5. There is a canonical ring map θ : P MGL → Σ_{+}^{∞}BGL[1/β].

Proof. The Thom class θ_{1} ∈ Σ_{+}^{∞}BGL[1/β]^{0}(MGL_{1}) extends, as in [1] or [23], to a ring map MGL → Σ_{+}^{∞}BGL[1/β], and we have a canonical unit μ ∈ R^{−1}(S^{0}), the image of β ∈ R^{0}(S^{1}) under the suspension isomorphism R^{0}(S^{1}) ≅ R^{−1}(S^{0}). □

Corollary 3.6. There is a canonical ring map ϑ : ⋁_{n} Σ^{∞}MGL_{n}[1/β] → Σ_{+}^{∞}BGL[1/β].

Proof. Precompose the map from the previous Proposition 3.5 with the equivalence

⋁_{n} Σ^{∞}MGL_{n}[1/β] → P MGL.

□

3.5. ϑ is an equivalence. We analyze the effect of ϑ : ⋁_{n} Σ^{∞}MGL_{n}[1/β] → Σ_{+}^{∞}BGL[1/β] on cohomology. To this end, fix an oriented periodic commutative motivic ring spectrum R; we aim to show that the induced map

R^{0}(Σ_{+}^{∞}BGL[1/β]) ⟶ R^{0}(⋁_{n} Σ^{∞}MGL_{n}[1/β])

is an isomorphism.

Lemma 3.7. Let R be a commutative ring and let A = colim_{n} A_{n} be a filtered commutative R-algebra with the property that A_{p} ⊗_{R} A_{q} → A ⊗_{R} A → A factors through the inclusion A_{p+q} → A (that is, the multiplication is compatible with the filtration). Suppose that, for each n, the maps A_{n−1} → A_{n} are split injections, so that the isomorphisms A_{n−1} ⊕ A_{n}/A_{n−1} → A_{n} define an Rmodule isomorphism

grA := ⨁_{n} A_{n}/A_{n−1} ⟶^{≃} colim_{n} A_{n} = A

of A with its associated graded. Then the multiplication A_{p}/A_{p−1} ⊗_{R} A_{q}/A_{q−1} → A_{p+q}/A_{p+q−1} makes grA = ⨁_{n} A_{n}/A_{n−1} into a commutative R-algebra in such a way that grA → A is an R-algebra isomorphism. □

Proposition 3.8. There is a commuting square of R^{0}-module maps

\xymatrix{R^{0}(BGL) \ar[r] \ar[d] & ∏_{n} R^{0}(MGL_{n}) \ar[d] \\ R^{0}(BGL × BGL) \ar[r] & ∏_{p,q} R^{0}(MGL_{p} ∧ MGL_{q})} ,

in which the vertical maps are induced by the multiplication on BGL and ⋁_{n} MGL_{n}, respectively, and the horizontal maps are isomorphisms.

Proof. Set A := colim_{n} Sym_{R_{0}}^{n} R_{0}(ℙ^{∞}), where the map Sym_{R_{0}}^{n−1} R_{0}(ℙ^{∞}) → Sym_{R_{0}}^{n} R_{0}(ℙ^{∞}) is induced by the the inclusion R_{0} ≅ R_{0}(ℙ^{0}) → R_{0}(ℙ^{∞}). Applying R_{0} to the cofiber sequence BGL_{n−1} → BGL_{n} → MGL_{n} yields split short exact sequences

\xymatrix{Sym^{n−1} R_{0}(ℙ^{∞}) \ar[r] \ar[d]^{≅} & Sym^{n} R_{0}(ℙ^{∞}) \ar[r] \ar[d]^{≅} & Sym^{n} R_{0}(ℙ^{∞})/ Sym^{n−1} R_{0}(ℙ^{∞}) \ar[d]^{≅} \\ R_{0}(BGL_{n−1}) \ar[r] & R_{0}(BGL_{n}) \ar[r] & R_{0}(MGL_{n})}

Documenta Mathematica 14 (2009) 359–396

with A ≅ colim_{n} R_{0}(BGL_{n}) ≅ R_{0}(BGL) a filtered commutative R-algebra. By the lemma, we have a commutative square

\xymatrix{⨁_{p,q} Sym^{p} R_{0}(ℙ^{∞})/ Sym^{p−1} R_{0}(ℙ^{∞}) ⊗_{R_{0}} Sym^{q} R_{0}(ℙ^{∞})/ Sym^{q−1} R_{0}(ℙ^{∞}) \ar[r] \ar[d] & A ⊗_{R_{0}} A \ar[d] \\ ⨁_{n} Sym^{n} R_{0}(ℙ^{∞})/ Sym^{n−1} R_{0}(ℙ^{∞}) \ar[r] & A}

in which the vertical maps are multiplication and the horizontal maps are R_{0}-algebra isomorphisms. The desired commutative square is obtained by taking R_{0}-module duals. □

Theorem 3.9. The map of oriented periodic motivic ring spectra

ϑ : ⋁_{n} Σ^{∞}MGL_{n}[1/β] → Σ_{+}^{∞}BGL[1/β]

is an equivalence.

Proof. We show that the induced natural transformation

ϑ^{∗} : Rings(Σ_{+}^{∞}BGL[1/β], −) ⟶ Rings(⋁_{n} Σ^{∞}MGL_{n}[1/β], −)

is in fact a natural isomorphism. The result then follows immediately from Yoneda’s Lemma. Fix a ring spectrum R, and observe that, for another ring spectrum A, Rings(A, R) is the equalizer of the pair of maps from R^{0}(A) to R^{0}(A∧A)×R^{0}(𝕊) which assert the commutativity of the diagrams

\xymatrix{A ∧ A \ar[r] \ar[d] & A \ar[d] \\ R ∧ R \ar[r] & R}  and  \xymatrix{𝕊 \ar[dr] \ar[r] & A \ar[d] \\  & R}.

Given a map β : Σ^{1}𝕊 → A, the set Rings(A[1/β], R) is the equalizer of the pair of maps from Rings(A, R)×R^{0}(Σ^{−1}𝕊) to R^{0}(𝕊) which assert that the ring map A → R is such that there’s a spectrum map Σ^{−1}𝕊 → R for which the product

𝕊 ≃ Σ^{1}𝕊 ∧ Σ^{−1}𝕊 ⟶ A ∧ R ⟶ R ∧ R ⟶ R

is equivalent to the unit 𝕊 → R. Putting these together, we may express Rings(A[1/β], R) as the equalizer of natural pair of maps from R^{0}(A)×R^{0}(𝕊^{−1}) to R^{0}(A ∧ A) × R^{0}(𝕊) × R^{0}(𝕊). We therefore get a map of equalizer diagrams

\xymatrix{R^{0}(BGL) × R^{0}(Σ^{−1}𝕊) \ar[r] \ar[r] \ar[d] & R^{0}(BGL × BGL) × R^{0}(𝕊) × R^{0}(𝕊) \ar[d] \\ ∏_{n} R^{0}(MGL_{n}) × R^{0}(Σ^{−1}𝕊) \ar[r] \ar[r] & ∏_{p,q} R^{0}(MGL_{p} ∧ MGL_{q}) × R^{0}(𝕊) × R^{0}(𝕊) ,}

the equalizer of which is ϑ^{∗}. Now if R does not admit the structure of a P MGL-algebra, then clearly there cannot be any ring maps from either of the P MGL-algebras ⋁_{n} Σ^{∞}MGL_{n}[1/β] of Σ_{+}^{∞}BGL[1/β]. Hence we may assume that R is also an oriented periodic ring spectrum, in which case Proposition 3.8 implies that the vertical maps are isomorphisms. □

Corollary 3.10. The map of periodic oriented motivic ring spectra

θ : P MGL → Σ_{+}^{∞}BGL[1/β]

is an equivalence.

Proof. ⋁_{n} Σ^{∞}MGL_{n}[1/β] → P MGL is an equivalence. □

4. Algebraic K-Theory

4.1. The representing spectrum. Let BGL_{ℤ} ≃ ℤ×BGL denote the group completion of the monoid

BGL_{ℕ} := ∐_{n∈ℕ} BGL_{n}.

Given a motivic space X, write K^{0}(X) := π_{0} map_{S}(X, BGL_{ℤ}). If S = spec ℤ and X is a scheme, this agrees with the homotopy algebraic K-theory of X as defined by Weibel [38], and if in addition X is smooth, this also agrees with Thomason-Trobaugh algebraic K-theory of X [33]; see Proposition 4.3.9 of [20] for details. As the name suggests, homotopy algebraic K-theory is a homotopy invariant version of the Thomason-Trobaugh algebraic K-theory, and homotopy invariance is of course a prerequisite for any motivic cohomology theory. It turns out that the motivic space BGL_{ℤ}, pointed by the inclusion

1 ≃ BGL_{0} ⟶ BGL_{ℕ} ⟶ BGL_{ℤ},

is the zero space of the motivic spectrum K representing (homotopy) algebraic K-theory. This is a direct corollary of the following famous fact.

Proposition 4.1 (Motivic Bott Periodicity). The adjoint

(BGL_{ℤ}, BGL_{0}) ⟶ Ω(BGL_{ℤ}, BGL_{0})

of the map Bott map Σ(BGL_{ℤ}, BGL_{0}) → (BGL_{ℤ}, BGL_{0}) classifying the tensor product of (L − 1) and V, where L → ℙ^{1} is the restriction of the universal line bundle and V → BGL_{ℤ} is the universal virtual vector bundle, is an equivalence.

Proof. Quillen’s projective bundle theorem [26] implies that the tensor product of vector bundles induces an isomorphism

K^{0}(ℙ^{1}) ⊗_{K^{0}} K^{0}(X) ⟶ K^{0}(ℙ^{1} × X)

of abelian groups. It follows that there’s an isomorphism of split short exact sequences of K^{0}(X)-modules

\xymatrix{λK^{0}(X)[λ]/(λ^{2}) \ar[r] \ar[d] & K^{0}(X)[λ]/(λ^{2}) \ar[r] \ar[d] & K^{0}(X) \ar[d] \\ K^{0}((ℙ^{1}, ℙ^{0}) ∧ (X, 0)) \ar[r] & K^{0}(ℙ^{1} × X) \ar[r] & K^{0}(ℙ^{0} × X) .}

In particular, K^{0}((ℙ^{1}, ℙ^{0}) ∧ (X, 0)) ≅ K^{0}(X) and similarly K^{0}((ℙ^{1}, ℙ^{0}) ∧ (X, 1)) ≅ K^{0}(X, 1). □

Define a sequence of pointed spaces K(n) by

K(n) := (BGL_{ℤ}, BGL_{0})

for all n ∈ ℕ. By Proposition 4.1, each K(n) comes equipped with an equivalence

K(n) = (BGL_{ℤ}, BGL_{0}) ⟶ Ω(BGL_{ℤ}, BGL_{0}) = ΩK(n + 1),

making K := (K(0), K(1), …) into a motivic spectrum.

4.2. A map Σ_{+}^{∞}ℙ^{∞}[1/β] → K. Let β : ℙ^{1} → ℙ^{∞} be the map classifying the tautological line bundle on ℙ^{1}. We construct a ring map Σ_{+}^{∞}ℙ^{∞} → K which sends β to a unit in K, thus yielding a ring map Σ_{+}^{∞}ℙ^{∞}[1/β] → K. There is a homotopy commutative ring structure on the motivic space BGL_{ℤ} ≃ Ω^{∞}K in which addition is induced by the sum of vector bundles and multiplication is induced by the tensor product of vector bundles. Ring maps Σ_{+}^{∞}ℙ^{∞} → K are adjoint to monoidal maps ℙ^{∞} → GL_{1}K, the multiplicative monoid of units (up to homotopy) in the ring space Ω^{∞}K ≃ BGL_{ℤ}. Since π_{0}BGL_{ℤ} contains a copy of ℤ, the multiplicative units contain the subgroup {±1} → ℤ, giving a map

{±1} × BGL ⟶ GL_{1}K.

But the inclusion BGL_{1} → BGL is monoidal with respect to the multiplicative structure on BGL, so we get a monoidal map

ℙ^{∞} ≃ BGL_{1} ⟶ {+1} × BGL ⟶ GL_{1}K

and therefore a ring map Σ_{+}^{∞}ℙ^{∞} → K.

Proposition 4.2. The class of the composite

Σ^{∞}S^{1} ≃ Σ^{∞}(ℙ^{1}, ℙ^{0}) ⟶ Σ_{+}^{∞}ℙ^{∞} ⟶ K

is equal to that of the K-theory Bott element β, i.e. the class of the reduced tautological line bundle L − 1 on ℙ^{1}.

Proof. The map Σ_{+}^{∞}ℙ^{∞} → K classifies the tautological line bundle on ℙ^{∞}, so the pointed version Σ^{∞}ℙ^{∞} → K corresponds to the reduced tautological line bundle on ℙ^{∞}. This restricts to the reduced tautological line bundle on ℙ^{1}. □

Corollary 4.3. There’s a canonical ring map ψ : Σ_{+}^{∞}ℙ^{∞}[1/β] → K. □

4.3. Comparing R^{0}(K) and R^{0}(L). Let L denote the localized motivic ring spectrum

L := Σ_{+}^{∞}ℙ^{∞}[1/β].

That is, L is the colimit

L = colim_{n} Σ^{∞−n}ℙ_{+}^{∞}

of a telescope of desuspended suspension spectra. We show that ψ : L → K is an equivalence by showing the induced map R^{0}(L) → R^{0}(K) is an isomorphism for a sufficiently large class of motivic spectra R. Throughout this section, we will be considering the motivic space BGL_{ℤ} (pointed by {0}×BGL_{0}) multiplicatively, as a homotopy commutative monoid with respect to the smash product. Accordingly, Σ^{∞}BGL_{ℤ} is a ring spectrum, and the monoidal map

ℙ_{+}^{∞} ≃ BGL_{0} + BGL_{1} ⟶ BGL_{ℕ} ⟶ BGL_{ℤ}

gives Σ^{∞}BGL_{ℤ} the structure of a homotopy commutative Σ^{∞}ℙ_{+}^{∞}-algebra. In particular, the Bott element β ∈ π_{1}Σ^{∞}ℙ_{+}^{∞} determines a Bott element β ∈ π_{1}Σ^{∞}BGL_{ℤ} as well as a Bott element β ∈ π_{1}K. If R is a homotopy commutative ring spectrum equipped with homotopy element α ∈ π_{n}R, we write

μ(α) = Σ^{−n}(μ ∘ (α ∧ R)) : R ⟶ Σ^{−n}R

for the “multiplication by α” map, the n-fold desuspension of the composite

Σ^{n}R ≃ Σ^{n}𝕊 ∧ R ⟶^{α∧R} R ∧ R ⟶^{μ} R.

If α ∈ π_{n}R is a unit, then this map has an inverse μ(α)^{−1} : Σ^{−n}R → R, the n-fold desuspension of the multiplication by α^{−1} map μ(α^{−1}) : R → Σ^{n}R. For our purposes, R will typically admit a periodic orientation, and α will be the image of the Bott element β ∈ π_{1}P MGL ≅ π_{1}Σ_{+}^{∞}BGL[1/β] under some ring map P MGL → R. Finally, we also write

μ(β) : BGL_{ℤ} ⟶ ΩBGL_{ℤ}

for multiplication by β in the homotopy commutative monoid BGL_{ℤ} (regarded multiplicatively). This is Ω^{∞} applied to the multiplication by β map μ(β) : K → Σ^{−1}K on K-theory, and thus it is the equivalence adjoint to the Bott map

ΣBGL_{ℤ} ⟶ BGL_{ℤ}

The following lemma is formal.

Lemma 4.4. Let ε : ΣΩBGL_{ℤ} → BGL_{ℤ} denote the counit of the adjunction (Σ, Ω) applied to BGL_{ℤ}. Then the composite

ΣBGL_{ℤ} ⟶^{Σμ(β)} ΣΩBGL_{ℤ} ⟶^{ε} BGL_{ℤ}

is the Bott map ΣBGL_{ℤ} ⟶ BGL_{ℤ}.

Proof. More generally, if (Σ, Ω) is any adjunction and β^{∗} : ΣX → Y is a map left adjoint to β_{∗} : X → ΩY, then β^{∗} = ε_{Y} ∘ Σβ_{∗}. □

Proposition 4.5. The square

\xymatrix{Σ^{∞+1}ℙ_{+}^{∞} \ar[r]^{Σμ(β)} \ar[d] & Σ^{∞}ℙ_{+}^{∞} \ar[d] \\ Σ^{∞+1}BGL_{ℤ} \ar[r]^{Σμ(β)} & Σ^{∞}BGL_{ℤ} ,}

in which vertical maps come from the inclusion i : ℙ_{+}^{∞} ≃ BGL_{0} + BGL_{1} → BGL_{ℤ} and the horizontal maps are the Bott maps, commutes up to homotopy.

Proof. The inclusion Σ^{∞}ℙ_{+}^{∞} → Σ^{∞}BGL_{ℤ} is a map of homotopy commutative ring spectra, and the Bott element Σ^{∞}ℙ^{1} → Σ^{∞}BGL_{ℤ} factors through the Bott element Σ^{∞}ℙ^{1} → Σ^{∞}ℙ_{+}^{∞}. □

Proposition 4.6. Let R be a homotopy commutative P MGL-algebra. Then the space map(K, R) of maps from K to R is equivalent to the homotopy inverse limit

map(K, R) ≃ holim_{n}{⋯ ⟶^{f} map(Σ^{∞}BGL_{ℤ}, R) ⟶^{f} map(Σ^{∞}BGL_{ℤ}, R)},

where f = μ(α)^{−1} ∘ Σ^{−1} ∘ μ(β) is the endomorphism of map(Σ^{∞}BGL_{ℤ}, R) which sends a map x : Σ^{∞}BGL_{ℤ} → R to the composite

Σ^{∞}BGL_{ℤ} ⟶^{μ(β)} Σ^{∞−1}BGL_{ℤ} ⟶^{Σ^{−1}x} Σ^{−1}R ⟶^{μ(α)^{−1}} R.

Proof. In general, for motivic spectra M and N,

map(M, N) ≃ holim{⋯ ⟶ map(M(1), N(1)) ⟶ map(M(0), N(0))},

where the maps send a map x : M(n) → N(n) to Ωx : M(n − 1) ≃ ΩM(n) → ΩN(n) ≃ N(n − 1). By adjunction, we may rewrite this as

map(M, N) ≃ holim{⋯ ⟶ map(Σ^{∞}Ω^{∞}ΣM, ΣN) ⟶ map(Σ^{∞}Ω^{∞}M, N)}.

Now K and R are periodic via equivalences μ(β) : K → Σ^{−1}K and μ(α) : R → Σ^{−1}R, the diagram

\xymatrix{map(BGL_{ℤ}, Ω^{∞}R) \ar[r]^{Ω} \ar[d]^{≃} & map(ΩBGL_{ℤ}, Ω^{∞+1}R) \ar[r]^{μ(β)^{∗}} \ar[d]^{≃} & map(BGL_{ℤ}, Ω^{∞+1}R) \ar[d]^{≃} \\ map(Σ^{∞}BGL_{ℤ}, R) \ar[r]^{Σ^{∞}ε^{∗}} & map(Σ^{∞+1}ΩBGL_{ℤ}, R) \ar[r]^{Σ^{∞+1}μ(β)^{∗}} & map(Σ^{∞+1}BGL_{ℤ}, R) ,}

in which the vertical arrows are adjunction equivalences, commutes, and according to Lemma 4.4 above, the Σ^{∞} applied to the composite ε ∘ Σμ(β) is Σμ(β) : Σ^{∞+1}BGL_{ℤ} → Σ^{∞}BGL_{ℤ}. Hence

map(K, R) ≃ holim{⋯ ⟶ map(Σ^{∞}BGL_{ℤ}, R) ⟶ map(Σ^{∞}BGL_{ℤ}, R)} is the homotopy inverse limit of the tower determined by the composite

map(Σ^{∞}BGL_{ℤ}, R) ⟶^{Σμ(β)^{∗}} map(Σ^{∞+1}BGL_{ℤ}, R) ⟶^{μ(α)_{∗}^{−1}∘Σ^{−1}} map(Σ^{∞}BGL_{ℤ}, R).

That is, the endomorphism f of map(Σ^{∞}BGL, R) sends x : Σ^{∞}BGL → R to the composite μ(α)^{−1} ∘ Σ^{−1}(x ∘ Σμ(β)) = μ(α)^{−1} ∘ Σ^{−1}(x) ∘ μ(β), which is to say that f = μ(α)^{−1} ∘ Σ^{−1} ∘ μ(β). □

Proposition 4.7. Let R be a homotopy commutative P MGL-algebra. Then the space map(L, R) of maps from L to R is equivalent to the homotopy inverse limit

map(L, R) ≃ holim_{n}{⋯ ⟶^{g} map(Σ^{∞}ℙ_{+}^{∞}, R) ⟶^{g} map(Σ^{∞}ℙ_{+}^{∞}, R)},

where g = μ(α)^{−1} ∘ Σ^{−1} ∘ μ(β) is the endomorphism of map(Σ^{∞}ℙ_{+}^{∞}, R) which sends a map y : Σ^{∞}ℙ_{+}^{∞} → R to the composite

Σ^{∞}ℙ_{+}^{∞} ⟶^{μ(β)} Σ^{∞−1}ℙ_{+}^{∞} ⟶^{Σ^{−1}y} Σ^{−1}R ⟶^{μ(α)^{−1}} R.

Proof. By definition, L = Σ_{+}^{∞}ℙ^{∞}[1/β] = hocolim_{n} Σ^{∞−n}ℙ_{+}^{∞}, where the map

Σ^{∞−n}ℙ_{+}^{∞} → Σ^{∞−n−1}ℙ_{+}^{∞}

is the n-fold desuspension of the multiplication by β map μ(β) : Σ^{∞}ℙ_{+}^{∞} → Σ^{∞−1}ℙ_{+}^{∞}. Hence

map(L, R) ≃

≃ holim_{n}{⋯ ⟶^{Σ^{−1}∘Σμ(β)^{∗}} map(Σ^{∞}ℙ_{+}^{∞}, ΣR) ⟶^{Σ^{−1}∘Σμ(β)^{∗}} map(Σ^{∞}ℙ_{+}^{∞}, R)}.

Again, since R is periodic via the multiplication by α map μ(α) : R → Σ^{−1}R, we may compose with μ(α)^{−1} in order to rewrite this as

map(L, R) ≃ holim_{n}{⋯ ⟶^{g} map(Σ^{∞}ℙ_{+}^{∞}, R) ⟶^{g} map(Σ^{∞}ℙ_{+}^{∞}, R)},

where g is the endomorphism of map(Σ^{∞}ℙ_{+}^{∞}, R) which sends the map y : Σ^{∞}ℙ_{+}^{∞} → R to the map g(y) = μ(α)^{−1} ∘ Σ^{−1}(y) ∘ μ(β). □

Corollary 4.8. Let R be a homotopy commutative P MGL-algebra. Then the square

\xymatrix{map(Σ^{∞}BGL_{ℤ}, R) \ar[r]^{f} \ar[d]^{i^{∗}} & map(Σ^{∞}BGL_{ℤ}, R) \ar[d]^{i^{∗}} \\ map(Σ^{∞}ℙ_{+}^{∞}, R) \ar[r]^{g} & map(Σ^{∞}ℙ_{+}^{∞}, R) ,}

in which the vertical map are induced by the inclusion i : ℙ_{+}^{∞} ≃ BGL_{0} + BGL_{1} → BGL_{ℤ}, commutes up to homotopy. In particular, ψ^{∗} : map(K, R) → map(L, R) is the homotopy inverse limit of the map of towers

\xymatrix{map(K, R) \ar[r]^{≃} \ar[d]^{ψ^{∗}} & holim\{⋯ \ar[r]^{f} & map(Σ^{∞}BGL_{ℤ}, R) \ar[r]^{f} \ar[d]^{i^{∗}} & map(Σ^{∞}BGL_{ℤ}, R)\} \ar[d]^{i^{∗}} \\ map(L, R) \ar[r]^{≃} & holim\{⋯ \ar[r]^{g} & map(Σ^{∞}ℙ_{+}^{∞}, R) \ar[r]^{g} & map(Σ^{∞}ℙ_{+}^{∞}, R)\}}

obtained from iterating this commuting square.

Proof. This is immediate from Lemmas 4.6 and 4.7. □

4.4. A useful splitting. To complete the analysis of ψ^{∗} : map(K, R) → map(L, R), we must split the space of additive maps from BGL_{ℤ} to Ω^{∞}R off of the space of all maps from BGL_{ℤ} to Ω^{∞}R.

Proposition 4.9. Let R be a motivic spectrum equipped with an equivalence μ(α) : R → Σ^{−1}R. Then given a map x : Ω^{∞}K → Ω^{∞}R, the map

Ω^{∞}μ(α)^{−1} ∘ Ω(x) ∘ Ω^{∞}μ(β) : Ω^{∞}K ⟶ Ω^{∞+1}K ⟶ Ω^{∞+1}R ⟶ Ω^{∞}R

is a homomorphism for the additive structures on Ω^{∞}K and Ω^{∞}R.

Proof. If X is a motivic space equipped with an equivalence X → ΩY, then X is a group object in the homotopy category of motivic spaces; if in addition Y ≃ ΩZ, then X is an abelian group object. In particular, the additions on Ω^{∞}K and Ω^{∞}R are induced by the equivalences Ω^{∞}μ(β) : Ω^{∞}K → Ω^{∞+1}K and Ω^{∞}μ(α) : Ω^{∞}R → Ω^{∞+1}R, respectively, and Ω^{∞}μ(α)^{−1} ∘ Ω(x) ∘ Ω^{∞}μ(β) is a map of loop spaces and therefore respects this addition. □

Proposition 4.10. Let R be a homotopy commutative P MGL-algebra. Then there exists a canonical section s : R^{ℙ_{+}^{∞}} → R^{BGL_{ℤ}} of the restriction r = i^{∗} : R^{BGL_{ℤ}} → R^{ℙ_{+}^{∞}} induced by the inclusion i : ℙ_{+}^{∞} ≃ BGL_{0} + BGL_{1} → BGL_{ℤ}.

Proof. Set Y = R^{BGL_{ℤ}} and Z = R^{ℙ_{+}^{∞}}. By Proposition 4.11, the additive maps from BGL_{ℤ} to Ω^{∞}R define a canonical section π_{0}Z → π_{0}Y of the surjection π_{0}Y → π_{0}Z. We must lift this to a map of spectra s : Z → Y. By Proposition 2.24, we have isomorphisms R^{0}(ℙ_{+}^{∞} ⊗ Z) ≅ R^{0}(Z) ⊗_{π_{0}R} π_{0}Z ≅ Z^{0}(Z). Combined with the section π_{0}Y → π_{0}Z, this induces a map

Z^{0}(Z) → R^{0}(Z) ⊗_{π_{0}R} π_{0}Z → R^{0}(Z) ⊗_{π_{0}R} π_{0}Y → Y^{0}(Z).

Take s ∈ Y^{0}(Z) to be the image of 1 ∈ Z^{0}(Z) under this map. □

Corollary 4.11. Let R be a homotopy commutative P MGL-algebra. Then

map(Σ^{∞}BGL_{ℤ}, R) ≃ map(Σ^{∞}ℙ_{+}^{∞}, R) × X

for some space X.

Proof. Take X to be the global points of the motivic space obtained by applying Ω^{∞} to the fiber of r : R^{BGL_{ℤ}} → R^{ℙ_{+}^{∞}}. □

We write

r : map(Σ^{∞}BGL_{ℤ}, R) ⟶ map(Σ^{∞}ℙ_{+}^{∞}, R)

for the restriction and

s : map(Σ^{∞}ℙ_{+}^{∞}, R) ⟶ map(Σ^{∞}BGL_{ℤ}, R)

for the section corresponding to the inclusion of the additive maps from BGL_{ℤ} to Ω^{∞}R into all maps from BGL_{ℤ} to Ω^{∞}R. By Corollary 4.8, we have that r ∘ f = g ∘ r; however, the next proposition shows that in fact f ≃ s ∘ g ∘ r, which is stronger since r ∘ f ≃ r ∘ s ∘ g ∘ r ≃ g ∘ r as s is a section of r.

Proposition 4.12. Let R be a homotopy commutative P MGL-algebra. Then

f ≃ s ∘ g ∘ r : map(Σ^{∞}BGL_{ℤ}, R) ⟶ map(Σ^{∞}BGL_{ℤ}, R).

Proof. Note that f and g are induced from corresponding maps f : R^{BGL_{ℤ}} → R^{BGL_{ℤ}} and g : R^{ℙ_{+}^{∞}} → R^{ℙ_{+}^{∞}}, respectively, and that r and s come similarly from maps r : R^{BGL_{ℤ}} → R^{ℙ_{+}^{∞}} and s : R^{ℙ_{+}^{∞}} → R^{BGL_{ℤ}}. Thus it’s enough to check that

f = s ∘ g ∘ r ∈ π_{0} map(R^{BGL_{ℤ}}, R^{BGL_{ℤ}}).

Since r ∘ f ≃ g ∘ r, we may instead show that f ≃ s ∘ r ∘ f. By definition,

f ∈ π_{0} map(R^{BGL_{ℤ}}, R^{BGL_{ℤ}}) ≅ R^{0}(BGL_{ℤ}) ˆ⊗_{R^{0}}R^{0}(R^{BGL_{ℤ}})

is the image of

1 ∈ π_{0} map(R^{BGL_{ℤ}}, R^{BGL_{ℤ}}) ≅ R^{0}(BGL_{ℤ}) ˆ⊗_{R^{0}}R^{0}(R^{BGL_{ℤ}})

under the map obtained by applying (−)ˆ⊗_{R^{0}}R^{0}(R^{BGL_{ℤ}}) to the map

Ω^{∞}μ(α)_{∗}^{−1}Ω^{∞}μ(β)^{∗}Ω : π_{0} map(BGL_{ℤ}, Ω^{∞}R) ⟶ π_{0} map(BGL_{ℤ}, Ω^{∞}R).

One similarly checks that r and s are obtained by applying (−)ˆ⊗_{R_{0}}R^{0}(R^{BGL_{ℤ}}) to the restriction R^{0}(BGL_{ℤ}) → R^{0}(ℙ^{∞}) and its section R^{0}(ℙ^{∞}) → R^{0}(BGL_{ℤ}), respectively. Now, according to Proposition 4.9, as a map of motivic loop spaces, Ω^{∞}μ(α)_{∗}^{−1}Ω^{∞}μ(β)^{∗}Ω sends x : BGL_{ℤ} → Ω^{∞}R to the additive map

Ω^{∞}μ(α)^{−1} ∘ Ω(x) ∘ Ω^{∞}μ(β) : BGL_{ℤ} ⟶ ΩBGL_{ℤ} ⟶ Ω^{∞+1}R ⟶ Ω^{∞}R,

which is to say that it factors through the inclusion Add(BGL_{ℤ}, Ω^{∞}R) ≅ R^{0}(ℙ_{+}^{∞}) → R^{0}(BGL_{ℤ}). Hence f is in the image of

\xymatrix{R^{0}(ℙ_{+}^{∞})ˆ⊗_{R^{0}}R^{0}(R^{BGL_{ℤ}}) \ar[r]^{s} \ar[d]^{≅} & R^{0}(BGL_{ℤ})ˆ⊗_{R^{0}}R^{0}(R^{BGL_{ℤ}}) \ar[d]^{≅} \\ π_{0} map(R^{BGL_{ℤ}}, R^{ℙ_{+}^{∞}}) \ar[r]^{s} & π_{0} map(R^{BGL_{ℤ}}, R^{BGL_{ℤ}}) ,}

so f = s∘f̃for some f̃: R^{BGL_{ℤ}} → R^{ℙ_{+}^{∞}}. But then s∘r∘f ≃ s∘r∘s∘f̃ ≃ s∘f̃ ≃ f. □

Lemma 4.13. Suppose given a homotopy commutative diagram of (ordinary) spectra

\xymatrix{Y \ar[r]^{f} \ar[d]_{r} & Y \ar[d]^{r} \\ Z \ar[r]^{g} & Z}

such that r : Y → Z admits a section s : Z → Y with

f ≃ s ∘ g ∘ r : Y → Y.

Then the natural map from the homotopy limit of the tower {⋯ → Y → Y }, obtained by iterating f, to the homotopy limit of the tower {⋯ → Z → Z}, obtained by iterating g, is an equivalence.

Proof. Since Z is a retract of Y, we may write Y ≃ Z × X for some spectrum X such that the fiber of f over g is the trivial map X → X. Now consider the diagram

\xymatrix{W \ar[r] \ar[d] & ∏_{n} X \ar[r] \ar[d] & ∏_{n} X \ar[d] \\ holim\{⋯ → Y → Y \} \ar[r] \ar[d] & ∏_{n} Y \ar[r]^{1−f} \ar[d] & ∏_{n} Y \ar[d] \\ holim\{⋯ → Z → Z\} \ar[r] & ∏_{n} Z \ar[r]^{1−g} & ∏_{n} Z}

in which the rows and columns are fiber sequences. Then the fiber of 1 − f over 1 − g is the identity ∏_{n} X → ∏_{n} X, so W is trivial and holim{⋯ → Y → Y } ≃ holim{⋯ → Z → Z}. □

Proposition 4.14. Let R be an orientable commutative motivic ring spectrum and let α ∈ π_{1}R be a unit. Then

ψ^{∗} : map(K, R) ⟶ map(L, R)

is an equivalence.

Proof. Set Y = map(Σ^{∞}BGL_{ℤ}, R) and Z = map(Σ^{∞}ℙ_{+}^{∞}, R), so that Y ≃ Z × X via r : Y → Z and its section s : Z → Y. By Proposition 4.12, f ≃ s ∘ g ∘ r : Y → Y, and since r and s are infinite loop maps we may regard them as maps of (ordinary) connective spectra. The result then follows from Lemma 4.13. □

Corollary 4.15. The map ψ : L → K induces an isomorphism ψ^{∗} : R^{0}(K) → R^{0}(L) for any orientable periodic commutative motivic ring spectrum R.

Proof. This is immediate from Proposition 4.14 above, since the spectrum of motivic spectrum maps from L to R admits precisely the same description as that of the spectrum of motivic spectrum maps from K to R. Indeed,

L ≃ colim_{n}{Σ_{+}^{∞}ℙ^{∞} ⟶^{β_{∗}} Σ^{−1}Σ_{+}^{∞}ℙ^{∞} ⟶^{β_{∗}} ⋯ },

and we see that

R^{L} ≃ holim_{n}{⋯ ⟶^{g} R^{ℙ_{+}^{∞}} ⟶^{g} R^{ℙ_{+}^{∞}}},

where the map g : R^{ℙ_{+}^{∞}} → R^{ℙ_{+}^{∞}} sends λ : Σ_{+}^{∞}ℙ^{∞} → R to α_{∗}^{−1} ∘ Σ^{−1}λ ∘ β_{∗}, the composite

Σ_{+}^{∞}ℙ^{∞} → Σ^{−1}Σ_{+}^{∞}ℙ^{∞} → Σ^{−1}R → R,

just as above. □

By the homotopy category of orientable periodic spectra, we mean the full subcategory of the homotopy category of spectra on the orientable and periodic objects. In other words, R is an orientable periodic spectrum if there exists a homotopy commutative ring structure on R which admits a ring map P MGL → R. Note that, according to this definition, maps between orientable periodic spectra need not preserve potential orientations or even ring structures.

Proposition 4.16. ψ induces an isomorphism ψ^{∗} : [K, −] → [L, −] of functors from the homotopy category of orientable periodic spectra to abelian groups.

Proof. Let R be an orientable periodic spectrum. Then

ψ^{∗} : [K, R] = R^{0}(K) → R^{0}(L) = [L, R]

is an isomorphism by Corollary 4.15, and this isomorphism is natural in spectrum maps R → R′, provided of course that R′ is also orientable and periodic. □

Theorem 4.17. The ring map ψ : L → K is an equivalence.

Proof. Let φ^{∗} : [L, −] → [K, −] be the inverse of the isomorphism ψ^{∗} : [K, −] → [L, −] of Proposition 4.16, and let φ : K → L be the map obtained by applying φ^{∗} to the identity 1 ∈ [L, L]. It follows from the Yoneda lemma that φ^{∗} is precomposition with φ. The equations φ^{∗} ∘ ψ^{∗} = 1_{K}^{∗} and ψ^{∗} ∘ φ^{∗} = 1_{L}^{∗} imply that ψ ∘ φ = 1_{K} and φ ∘ ψ = 1_{L} in the homotopy category of orientable periodic spectra, and therefore that ψ ∘ φ = 1_{K} and φ ∘ ψ = 1_{L} in the homotopy category of spectra. Hence ψ : L → K is an equivalence with inverse φ : K → L. □

5. Applications

5.1. The motivic Conner-Floyd theorem. The classical theorem of Conner and Floyd shows that complex cobordism determines complex K-theory by base change. More precisely, writing P MU for periodic complex cobordism and KU for complex K-theory, then, for any finite spectrum X, the natural map

P MU^{0}(X) ⊗_{P MU^{0}} KU^{0} → KU^{0}(X)

is an isomorphism of KU^{0}-modules.

Remark 5.1. This is the precursor of the more general notion of Landweber exactness. In [8], P. Landweber gives a necessary and sufficient condition on an MU_{∗}-module G so that the functor (−) ⊗_{MU_{∗}} G, from (MU_{∗}, MU_{∗}MU)- comodules to graded abelian groups, is exact. For G = K_{∗}, it follows that the natural map

MU^{∗}(−) ⊗_{MU^{∗}} K^{∗} → K^{∗}(−)

is an isomorphism. See [21] for the motivic analogue of Landweber exactness.

We now turn to the motivic version of the theorem of Conner and Floyd. A motivic spectrum X is said to be compact if [X, −], viewed as a functor from motivic spectra to abelian groups, commutes with filtered colimits.

Proposition 5.2. Let X be a compact motivic spectrum. Then the natural map

P MGL^{0}(X) ⊗_{P MGL^{0}} K^{0} ⟶ K^{0}(X)

is surjective.

Proof. Set B := Σ_{+}^{∞}BGL and A := Σ_{+}^{∞}ℙ^{∞}. Then the determinant map r : B → A admits a section s : A → B, so, for each n, Σ^{−n}A is a retract of Σ^{−n}B and Σ^{−n}B^{0}(X) → Σ^{−n}A^{0}(X) is surjective. Since X is compact, the colimit

B[1/β]^{0}(X) ≅ colim_{n} Σ^{−n}B^{0}(X) ⟶ colim_{n} Σ^{−n}A^{0}(X) ≅ A[1/β]^{0}(X)

is also surjective, and we see that

B[1/β]^{0}(X) ⊗_{B[1/β]^{0}} A[1/β]^{0} ⟶ A[1/β]^{0}(X) ⊗_{B[1/β]^{0}} A[1/β]^{0} ≅ A[1/β]^{0}(X)

is surjective as well. □

Theorem 5.3. Let X be a compact motivic spectrum. Then the natural map

P MGL^{0}(X) ⊗_{P MGL^{0}} K^{0} ⟶ K^{0}(X)

is an isomorphism.

Proof. According to Proposition 5.2, it’s enough to show that the map is injective. For simplicity of notation, set R := P MGL, define a contravariant functor J^{0}(−) from compact motivic spectra to R^{0}-modules by the rule

J^{0}(X) := ker{R^{0}(X) → K^{0}(X)},

and write J^{0} for J^{0}(𝕊). Since the tensor product is right exact, the map

J^{0}(X) ⊗_{R^{0}} K^{0} ⟶ ker{R^{0}(X) ⊗_{R^{0}} K^{0} → K^{0}(X) ⊗_{R^{0}} K^{0}}

is surjective, so in light of the isomorphism

K^{0}(X) ⊗_{R^{0}} K^{0} ≅ K^{0}(X) ⊗_{K^{0}} K^{0} ≅ K^{0}(X)

it’s enough to show that J^{0}(X) ⊗_{R^{0}} K^{0} is zero, or, equivalently, that

J^{0}(X) ⊗_{R^{0}} J^{0} ⟶ J^{0}(X) ⊗_{R^{0}} R^{0} ≅ J^{0}(X)

is surjective. To this end, set

I^{0}(X) := im{J^{0}(X) ⊗_{R^{0}} J^{0} → J^{0}(X) ⊗_{R^{0}} R^{0} ≅ J^{0}(X)};

we must show that I^{0}(X) ≅ J^{0}(X). Now, writing B := Σ_{+}^{∞}BGL and A := Σ_{+}^{∞}ℙ^{∞} as above, and using the compactness of X, we see that any element of

J^{0}(X) ≅ ker{colim_{n}[X, Σ^{−n}B] → colim_{n}[X, Σ^{−n}A]}
≅ colim_{n} ker{[X, Σ^{−n}B] → [X, Σ^{−n}A]}

is represented by a map

x : Σ^{n}X → B ≃ colim_{p} colim_{q} Σ_{+}^{∞} Grass_{p,q},

which, by compactness, factors as f_{x} : Σ^{n}X → Y_{x} followed by y : Y_{x} → B for Y_{x} ≃ Σ_{+}^{∞} Grass_{p,q} the suspension spectrum of a finite Grassmannian. This yields a commuting diagram

\xymatrix{Σ^{n}X \ar[r]^{x} \ar[d]_{f_{x}} & B \ar[d]^{r} \\ Y_{x} \ar[r] \ar[ur]^{y} & A}

in which r ∘ x is trivial and the determinant map r : B → A admits a section s : A → B. Of course, as r ∘ y need not be trivial, set y′ := y − s ∘ r ∘ y, so that

y′ ∘ f ≃ (y − s ∘ r ∘ y) ∘ f ≃ y ∘ f − s ∘ r ∘ y ∘ f ≃ x − s ∘ r ∘ x ≃ x

and r ∘ y′ ≃ 0, which is to say that y′ ∈ J^{0}(X) and f_{x}^{∗}y′ = x. Finally, according to Proposition 2.19, R^{0}(Y_{x}) ⊗_{R^{0}} K^{0} ≅ K^{0}(Y_{x}) for each x ∈ J^{0}(X), so we must have surjections

J^{0}(Y_{x}) ⊗_{R^{0}} J^{0} ⟶ J^{0}(Y_{x}) ⊗_{R^{0}} R^{0} ≅ J^{0}(Y_{x}).

Adding these together, we obtain a morphism of short exact sequences

\xymatrix{0 \ar[r] & ⨁_{x} I^{0}(Y_{x}) \ar[r] \ar[d] & ⨁_{x}J^{0}(Y_{x}) \ar[r] \ar[d]^{⨁_{x} f_{x}^{∗}} & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I^{0}(X) \ar[r] & J^{0}(X) \ar[r] & J^{0}(X) ⊗_{R^{0}} K^{0} \ar[r] & 0}

such that ⨁_{x} f_{x}^{∗} : ⨁_{x} J^{0}(Y_{x}) → J^{0}(X) is surjective. It follows that I^{0}(X) ≅ J^{0}(X). □

5.2. P MGL and K are E_{∞} motivic spectra. As a final application, we show that P MGL and K are E_{∞} motivic spectra. As we shall see, this is an immediate consequence of the fact that P MGL and K are obtained through a localization of the category of E_{∞} R-algebras for some E_{∞} motivic spectrum R. Roughly, given an element β ∈ π_{p,q}R, the functor which sends the Rmodule M to M[1/β] := M ∧_{R} R[1/β] defines a monoidal localization of the category of R-modules, so it extends to a localization of the category of E_{∞} R-algebras. Taking R = Σ_{+}^{∞}BGL and β the Bott element, we see that P MGL is the localization of the initial E_{∞} R-algebra and K is the localization of the determinant E_{∞} R-algebra. In order to make this precise, we fix a suitable symmetric monoidal model category (Mod_{𝕊}, ∧_{𝕊}) of motivic spectra, such as motivic 𝕊-modules [6] or motivic symmetric spectra [7]. For sake of definiteness, we adopt the formalism of the latter; nevertheless, we refer to motivic symmetric spectra as 𝕊-modules, as they are indeed modules over the symmetric motivic sphere 𝕊. Recall that a motivic symmetric sequence is a functor from the groupoid Σ of finite sets and isomorphisms to pointed motivic spaces. It is sometimes convenient to use a skeleton of Σ, so we simply write n for a finite set with n elements and Σ(n) for its automorphism group. Motivic symmetric sequences form a symmetric monoidal category under the smash product defined by

(X ∧ Y)(n) := ⋁_{n=p+q} Σ(n)_{+} ∧_{Σ(p)×Σ(q)} X(p) ∧ Y (q).

The motivic sphere 𝕊 has a natural interpretation as the motivic symmetric sequence in which 𝕊(n) is the pointed Σ(n)-space associated to the pair (𝔸^{n}, 𝔸^{n}−𝔸^{0}), where Σ(n) acts by permutation of coordinates. The Σ(p)×Σ(q)- equivariant maps

(𝔸^{p}, 𝔸^{p} − 𝔸^{0}) ∧ (𝔸^{q}, 𝔸^{q} − 𝔸^{0}) ⟶ (𝔸^{p+q}, 𝔸^{p+q} − 𝔸^{0})

give 𝕊 the structure of a commutative monoid for this smash product. An 𝕊-module is then a motivic symmetric sequence equipped with an action of 𝕊, which is to say a sequence X(p) of pointed Σ(p)-equivariant motivic spaces equipped with Σ(p) × Σ(q)-equivariant maps

(𝔸^{p}, 𝔸^{p} − 𝔸^{0}) ∧ X(q) ⟶ X(p + q); the fact that 𝕊 is a commutative monoid implies that ∧ extends to a smash product ∧_{𝕊} on the category Mod_{𝕊} of 𝕊-modules. There are monoidal functors

{Motivic spaces}→{Motivic symmetric spaces}→{Motivic symmetric spectra}

in which the righthand map is the free 𝕊-module functor, the left hand map sends the motivic space X to the constant motivic symmetric space X_{+}, and the composite is a structured version of the suspension spectrum functor Σ_{+}^{∞}.

Proposition 5.4. The 𝕊-modules Σ_{+}^{∞}BGL and Σ_{+}^{∞}ℙ^{∞} are equivalent to strictly commutative 𝕊-algebras in such a way that the determinant map Σ_{+}^{∞}BGL → Σ_{+}^{∞}ℙ^{∞} is equivalent to a map of strictly commutative 𝕊-algebras.

Proof. For each n, write GL(n) for the group S-scheme of linear automorphisms of 𝔸^{n}. Then Σ(n) acts on GL(n) by conjugation via the embedding Σ(n) → GL(n), so that GL(n) is the value at n of a symmetric sequence GL in group Sschemes such that the determinant map GL → GL_{1} is a morphism of symmetric sequences in group S-schemes, where we regard GL_{1} as a constant symmetric sequence. Taking classifying spaces, we obtain a morphism of commutative monoid symmetric sequences BGL → BGL_{1} in unpointed motivic spaces. Now let 𝕊[BGL] and 𝕊[BGL_{1}] denote the 𝕊-modules defined by

𝕊[BGL](n) := 𝕊(n) ∧ BGL(n)_{+}  and  𝕊[BGL_{1}](n) := 𝕊(n) ∧ BGL_{1+},

respectively, where Σ(n) acts diagonally; note that 𝕊[BGL_{1}] is the free 𝕊module on the motivic symmetric sequence BGL_{1+}, whereas the action of 𝕊 on 𝕊[BGL] is induced by the canonical Σ(p) × Σ(q)-equivariant inclusions

BGL(q) ⟶ BGL(p) × BGL(q) → BGL(p + q)

coming from the fact that BGL(p) has a canonical basepoint which is fixed by the action of Σ(p). The monoidal structure on BGL_{1} extends to a strictly commutative 𝕊-algebra structure on 𝕊[BGL_{1}], and the strictly commutative 𝕊-algebra structure on 𝕊[BGL] comes from Σ(p) × Σ(q)-equivariant pairing

𝕊(p) ∧ BGL(p)_{+} ∧ 𝕊(q) ∧ BGL(q)_{+} ⟶ 𝕊(p + q) ∧ BGL(p + q)_{+};

moreover, it is clear that the determinant map 𝕊[BGL] → 𝕊[BGL_{1}] is monoidal with respect to these multiplicative structures. Hence we are done, provided the underlying ordinary motivic spectra (obtained by forgetting the actions of the symmetric groups) of 𝕊[BGL] and 𝕊[BGL_{1}] are equivalent to the motivic spectra Σ_{+}^{∞}BGL and Σ_{+}^{∞}ℙ^{∞}, respectively. This is immediate for 𝕊[BGL_{1}], whose underlying spectrum is the suspension spectrum Σ_{+}^{∞}BGL_{1}; on the other hand, the underlying spectrum of 𝕊[BGL] is the prespectrum {S^{n} ∧ BGL_{n+}}. But the motivic spectrum associated to Σ_{+}^{∞}BGL is given by

colim_{p} Ω^{p}Σ^{p} colim_{q} BGL_{q} ≃ colim_{p} colim_{q} Ω^{p}Σ^{p}BGL_{q} ≃ colim_{n} Ω^{n}Σ^{n}BGL_{n},

so the two motivic prespectra are stably equivalent. □

Instead of considering localization in the context of symmetric monoidal model categories, it will be enough to consider localization in the context of symmetric monoidal ∞-categories, in the sense of Lurie [16]. Indeed, if (ℳ , ⊗) is a symmetric monoidal model category, then, in the notation of [16], the commutative algebra objects of the associated symmetric monoidal ∞-category N(ℳ , ⊗)^{∘} correspond to coherently homotopy commutative, or E_{∞}, objects of (ℳ , ⊗). Here N denotes the simplicial nerve of a simplicial category; if the simplicial category comes from a symmetric monoidal model category, then its simplicial nerve is symmetric monoidal as an ∞-category. See [17] for facts about ∞-categories and simplicial nerves, and [16] for a treatment of commutative algebra in the ∞-categorical context. Recall (cf. [17]) that a map F : 𝒞 → 𝒟 of ∞-categories is said to be a localization if F admits a fully faithful right adjoint G. In this case, it is common to identify 𝒟 with the full subcategory of 𝒞 consisting of those objects in the essential image of G (the “local objects”), and suppress 𝒟 and G from the notation by writing L for the composite G ∘ F : 𝒞 → 𝒟 → 𝒞. If 𝒞 is the underlying ∞-category of a symmetric monoidal ∞-category (𝒞 , ⊗), then we may ask when a localization L : 𝒞 → 𝒞 extends to a lax symmetric monoidal functor on (𝒞 , ⊗). Given a localization L : 𝒞 → 𝒞, an L-equivalence is a map which becomes an equivalence after applying L.

Definition 5.5 ([16], Example 1.7.5). Let (𝒞 , ⊗) be a symmetric monoidal ∞-category and let L : 𝒞 → 𝒞 be a localization of the underlying ∞-category. Then L is said to be compatible with ⊗ if, for all L-equivalences A → A′ and all objects B of 𝒞, A ⊗ B → A′ ⊗ B is an L-equivalence.

Proposition 5.6 ([16], Proposition 1.7.6). Let (𝒞 , ⊗) be a symmetric monoidal ∞-category, let L : 𝒞 → 𝒞 be a localization of the underlying ∞-category, and suppose that L is compatible with ⊗. Then L extends to a lax symmetric monoidal functor

(L, ⊗) : (𝒞 , ⊗) → (𝒞 , ⊗).

In particular, L preserves algebra and commutative algebra objects of (𝒞 , ⊗).

Let N(Mod_{𝕊}, ∧_{𝕊})^{∘} denote the symmetric monoidal ∞-category which arises as the simplicial nerve of the symmetric monoidal simplicial model category (Mod_{𝕊}, ∧_{𝕊}) of 𝕊-modules. Since commutative algebra objects of N(Mod_{𝕊}, ∧_{𝕊})^{∘} are modeled by algebras over a suitable E_{∞} operad, we refer to commutative algebra objects of N(Mod_{𝕊}, ∧_{𝕊})^{∘} as E_{∞} 𝕊-algebras. Given an E_{∞} 𝕊-algebra R, we write (Mod_{R}, ∧_{R}) for the resulting symmetric monoidal ∞-category of R-modules, and refer to commutative algebra objects of (Mod_{R}, ∧_{R}) as E_{∞} R-algebras.

Proposition 5.7. Let R be an E_{∞} 𝕊-algebra, let f ∈ π_{p,q}R be an arbitrary element, and write L_{f} : Mod_{R} → Mod_{R} for the functor which sends the Rmodule M to the R-module

M[1/f] := M ∧_{R} R[1/f].

Then L_{f} is a localization functor which is compatible with the symmetric monoidal structure ∧_{R} on Mod_{R}; in particular, L_{f} extends to a lax monoidal functor L_{f} : (Mod_{R}, ∧_{R}) → (Mod_{R}, ∧_{R}).

Proof. Say that an R-module M is f-local if the multiplication by f map f_{∗} : M → Σ^{−p,−q}M is an equivalence. Given an f-local R-module M, the induced map

Map(R[1/f], M) ≃ lim{M ← Σ^{p,q}M ← ⋯ } ≃ M

is an equivalence, so

map(N[1/f], M) ≃ map(N, Map(R[1/f], M)) ≃ map(N, M)

is an equivalence for any R-module N. Hence L_{f} is left adjoint to the inclusion of the full subcategory of f-local R-modules, and is therefore a localization. Moreover, it is compatible with ∧_{R}, since if M → M′ is an L_{f}-equivalence then so is M ∧_{R} N → M′ ∧_{R} N for any R-module N, for if M[1/f] → M′[1/f] is an equivalence then

(M ∧_{R} N)[1/f] ≃ M[1/f] ∧_{R} N ⟶ M′[1/f] ∧_{R} N ≃ (M′ ∧_{R} N)[1/f]

is as well. Hence, by Proposition 5.6, L_{f} extends to a lax symmetric monoidal endofunctor (L_{f}, ∧_{R}) of (Mod_{R}, ∧_{R}). □

Corollary 5.8. Let R be an E_{∞} 𝕊-algebra and let f ∈ π_{p,q} be a fixed element. Then R[1/f] is an E_{∞} R-algebra, and therefore also an E_{∞} 𝕊-algebra.

Proof. By Proposition 5.7, L_{f} is a lax symmetric monoidal functor with L_{f}R ≃ R[1/f]. Since lax symmetric monoidal functors preserve commutative algebra objects, we see that R[1/f] is a commutative algebra object in (Mod_{R}, ∧_{R}). Lastly, as the forgetful functor from R-modules to 𝕊-modules is lax symmetric monoidal, it follows that R[1/f] is also an E_{∞} 𝕊-algebra. □

Corollary 5.9. MGL, P MGL and K are E_{∞} 𝕊-algebras.

Proof. The MGL case is already well-known (cf. [6], for example). By Proposition 5.4, Σ_{+}^{∞}BGL and Σ_{+}^{∞}ℙ^{∞} are equivalent to strictly commutative 𝕊-algebras, so they are naturally commutative algebra objects in the symmetric monoidal ∞-category N(Mod_{𝕊}, ∧_{𝕊}). Applying Proposition 5.8, we see that P MGL ≃ Σ_{+}^{∞}BGL[1/β] is an E_{∞} Σ_{+}^{∞}BGL-algebra, and likewise that K ≃ Σ_{+}^{∞}ℙ^{∞}[1/β] is an E_{∞} Σ_{+}^{∞}ℙ^{∞}-algebra. In particular, P MGL and K are E_{∞} 𝕊-algebras. □

Proposition 5.10. K is an E_{∞} P MGL-algebra.

Proof. Note that the Bott element Σ𝕊 → Σ_{+}^{∞}ℙ^{∞} factors as the composite of the Bott element Σ𝕊 → Σ_{+}^{∞}BGL followed by the determinant map Σ_{+}^{∞}BGL → Σ_{+}^{∞}ℙ^{∞}. By Proposition 5.4, the determinant map Σ_{+}^{∞}BGL → Σ_{+}^{∞}ℙ^{∞} is a map of E_{∞} Σ_{+}^{∞}BGL-algebras, so by Proposition 5.8, the localization K ≃ Σ_{+}^{∞}ℙ^{∞}[1/β] is an E_{∞} algebra over P MGL ≃ Σ_{+}^{∞}BGL[1/β]. □

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David Gepner Department of Mathematics University of Illinois at Chicago 851 S. Morgan Street Chicago, IL 60607 USA Victor Snaith Department of Pure Mathematics University of Sheffield Hounsfield Road Sheffield S3 7RH United Kingdom