On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory

I. Panin^{∗}^{†}  K. Pimenov^{†}  O. Röndigs^{‡}

May 31, 2007^{§}

Abstract

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^{1}-spectrum MGL of Voevodsky is considered as a commutative ring P^{1}-spectra. Setting MGL^{i} = ⊕_{2q−p=i} MGL^{p,q} we regard the bigraded theory MGL^{p,q} as just a graded theory. There is a unique ring morphism ϕ: MGL^{0}(k) → ℤ which sends the class [X]_{MGL} of a smooth projective k-variety X to the Euler characteristic χ(X, 𝒪_{X}) of the structure sheaf 𝒪_{X}. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories on the category 𝒮m𝒪p/k

φ: MGL^{∗}(X, U) ⊗_{MGL^{0}(k)} ℤ → K_{−∗}^{T T} (X, U) = K′_{−∗}(X − U),

in the sense of [PS1], where K_{∗}^{T T} is the Thomason-Trobaugh K-theory and K′_{∗} is Quillen’s K′-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented in the sense of [PS1] and φ respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism [CF].

∗Universität Bielefeld, SFB 710, Bielefeld, Germany †Steklov Institute of Mathematics at St. Petersburg, Russia ‡Institut für Mathematik, Universität Osnabrück, Osnabrück, Germany §The authors thank the SFB-701 at the Universität Bielefeld, the RTN-Network HPRN-CT-2002-00287, the RFFI-grant 03-01-00633a, and INTAS-03-51-3251 for their support.

1 A motivic version of a theorem by Conner and Floyd

Our main result relates Voevodsky’s algebraic cobordism theory MGL^{∗,∗} to Quillen’s K′-theory. We refer to [PPR1, Appendix] for the basic terminology, notation, constructions, definitions, results. Let S be a Noetherian separated finite-dimensional scheme S. One may think of S being the spectrum of a field or the integers. A motivic space over S is a functor

A: 𝒮m/S^{op} → sSet

(see [PPR1, Appendix]). The category of motivic spaces over S is denoted M(S). This definition of a motivic space is different from the one considered by Morel and Voevodsky in [MV] – they consider only those simplicial presheaves which are sheaves in the Nisnevich topology on 𝒮m/S. With our definition the Thomason-Trobaugh K-theory functor obtained by using big vector bundles is a motivic space on the nose. It is not a simplicial Nisnevich sheaf. This is why we prefer to work with the above notion of “space”.

We write H_{•}^{cm}(S) for the pointed motivic homotopy category and SH^{cm}(S) for the stable motivic homotopy category over S as constructed in [PPR1, A.3.9, A.5.6]. By [PPR1, A.3.11 resp. A.5.6] there are canonical equivalences to H_{•}(S) of [MV] resp. SH(S) of [V1]. Both H_{•}^{cm}(S) and SH^{cm}(S) are equipped with closed symmetric monoidal structures such that the P^{1}- suspension spectrum functor is a strict symmetric monoidal functor

Σ_{P^{1}}^{∞}: H_{•}^{cm}(S) → SH^{cm}(S).

Here P^{1} is considered as a motivic space pointed by ∞ ∈ P^{1}. The symmetric monoidal structure (∧, 𝕀_{S} = Σ_{P^{1}}^{∞}S_{+}) on the homotopy category SH^{cm}(S) is constructed on the model category level by employing the category MSS(S) of symmetric P^{1}-spectra. It satisfies the properties required by Theorem 5.6 of Voevodsky congress talk [V1]. From now on we will usually omit the superscript (−)^{cm}.

Given a P^{1}-spectrum E one has a cohomology theory on the category of pointed spaces. Namely, for a pointed space (A, a) set E^{p,q}(A, a) = Hom_{H_{•}^{cm}(S)}(Σ_{P^{1}}^{∞}(A, a), Σ^{p,q}(E)) and E^{∗,∗}(A, a) = ⊕_{p,q}E^{p,q}(A, a). A cohomology theory on the category of non-pointed spaces is defined as follows. For a non-pointed space A set E^{p,q}(A) = E^{p,q}(A_{+}, +) and E^{∗,∗}(A) = ⊕_{p,q}E^{p,q}(A).

Each X ∈ 𝒮m/S defines a motivic space constant in the siplicial direction taking an S-smooth U to Mor_{S}(U, X). This motivic space is non-pointed. So we regard S-smooth varieties as motivic spaces (non-pointed) and set

E^{p,q}(X) = E^{p,q}(X_{+}, +).

Given a P^{1}-spectrum E we will reduce the double grading on the cohomology theory E^{∗,∗} to a grading. Namely, set E^{m} = ⊕_{m=p−2q}E^{p,q} and E^{∗} = ⊕_{m}E^{m}. We often will write E^{∗}(k) for E^{∗}(Spec(k)) below in this text.

A P^{1}-ring spectrum is a monoid (E, μ, e) in (SH(S), ∧, 𝕀_{S}). A commutative P^{1}-ring spectrum is a commutative monoid (E, μ, e) in (SH(S), ∧, 1).

The cohomology theory E^{∗} defined by a P^{1}-ring spectrum is a ring cohomology theory. The cohomology theory E^{∗} defined by a commutative P^{1}-ring spectrum is a ring cohomology theory, however it is not necessary graded commutative. The cohomology theory E^{∗} defined by an oriented commutative P^{1}-ring spectrum is a graded commutative ring cohomology theory.

Occasionally a P^{1}-ring spectrum (E, μ, e) might have a model (E′, μ′, e′) which is a symmetric P^{1}-ring spectrum, that is, a symmetric P^{1}-spectrum E′ equipped with a strict multiplication μ′: E′ ∧ E′ → E′ which is strictly associative and strictly unital for the unit e′: Σ_{P^{1}}^{∞}(S_{+}) → E′. This is the case for the algebraic cobordism P^{1}-ring spectrum MGL, as described below. Such a model for the algebraic K-theory P^{1}-ring spectrum BGL is currently not known to us.

For the rest of the paper let k be a field and S = Spec(k). Usually S will be replaced by k in the notation. We work in this text with the algebraic cobordism P^{1}-spectrum MGL and the algebraic K-theory P^{1}-spectrum BGL as described in [PPR1, Defn. 1.2.4] and [PPR2, Sect. 2.1] respectively. The spectrum MGL is a commutative ring P^{1}-spectrum by that construction. The spectrum BGL is equipped with a structure of a commutative P^{1}-ring spectrum as explained in [PPR1, Thm. 2.1.1]. Let K_{∗}^{T T} be Thomason-Trobaugh K-theory functor [TT]. There is a canonical isomorphism

Ad: K_{−∗}^{T T} → BGL^{∗,0}

of ring cohomology theories on the category 𝒮m𝒪p/S in the sense of [PS1]. An invertible Bott element β ∈ BGL^{2,1}(Spec(k)) is constructed in [PPR1, Section 1.3]. For every pointed motivic space A the morphism

BGL^{∗,0}(A) ⊗ BGL^{0}(Spec(k)) → BGL^{∗,∗}(A)  (1)

given by a⊗b ↦ a∪b is a ring isomorphism by [PPR1, Sect. 1.3]. Furthermore BGL^{0}(Spec(k)) = ℤ[β, β^{−1}] is the ring of Laurent polynomials on the Bott element β. To say the same in a different way,

BGL^{∗,0}(A)[β, β^{−1}] ≅ BGL^{∗,∗}(A).  (2)

The special case A = X/(X ∖ Z) where X is a smooth k-variety and Z ⊂ X is a closed subset implies the following result [PPR1, Cor. 1.3.6].

Corollary 1.0.1. Let X be a smooth k-scheme, Z a closed subset of X and U = X ∖ Z its open complement. Then there are isomorphisms

K_{−∗,Z}^{T T}(X)[β, β^{−1}] ≅ BGL^{∗,∗}(X/U) = BGL^{∗}(X/U)  (3)
K_{−∗,Z}^{T T}(X) ≅ BGL^{∗,∗}(X/U)/(β + 1)BGL^{∗,∗}(X/U)  (4)

of ring cohomology theories on 𝒮m𝒪p/k in the sense of [PS1].

We refer to [PPR2] for a construction of the commutative ring P^{1}-spectrum MGL. For the purposes of the present preprint we will need to know only two properties of that spectrum. Those properties are: Quillen universality and MGL-cellularity (see Subsection 2.1 below).

1.1 Oriented commutative ring spectra

Following Adams and Morel we define an orientation of a commutative P^{1}- ring spectrum. However we prefer to use Thom classes instead of Chern classes. Consider the pointed motivic space P^{∞} = colim_{n≥0} P^{n} having base point g_{1} : S = P^{0} ↪ P^{∞}.

The tautological “vector bundle” 𝒯(1) = 𝒪_{P^{∞}}(−1) is also known as the Hopf bundle. It has zero section z: P^{∞} ↪ 𝒯(1). The fiber over the point g_{1} ∈ P^{∞} is 𝔸^{1}. For a vector bundle V over a smooth S-scheme X with zero section z: X ↪ V consider a Nisnevich sheaf associated with the presheaf Y ↦ V (Y)/(V ∖ z(X))(Y) on the Nisnevich site 𝒮m/S. The Thom space Th(V) of V is defined as that Nisnevich sheaf regarded as a presheaf. In particular Th(V) is a pointed motivic space in the sense of [PPR1, Defn. A.1.1]. Its Nisnevich sheafification coincides with Voevodsky’s Thom space [V1, p. 422], since Th(V) is already a Nisnevich sheaf. The Thom space of the Hopf bundle is then defined as the colimit Th(𝒯(1)) = colim_{n≥0} Th(𝒪_{P^{n}}(−1)). Abbreviate T = Th(A_{S}^{1}) = A_{S}^{1}/(A_{S}^{1} ∖ {0}).

Let E be a commutative ring P^{1}-spectrum. The unit gives rise to an element 1 ∈ E^{0,0}(Spec(k)_{+}). Applying the P^{1}-suspension isomorphism to that element we get an element Σ_{P^{1}}(1) ∈ E^{2,1}(P^{1}/{∞})). The canonical covering of P^{1} defines motivic weak equivalences

\xymatrix{P^{1}/\{∞\} \ar[r]^{∼} & P^{1}/A^{1} & A^{1}/A^{1} ∖ \{0\} = T, \ar[l]_{∼}}

which in turn define pull-back isomorphisms E(P^{1}/{∞}) ← E(A^{1}/A^{1} ∖ {0}) → E(T) . Denote Σ_{T} (1) the image of Σ_{P^{1}}(1) in E^{2,1}(T).

Definition 1.1.1. Let E be a commutative ring P^{1}-spectrum. A Thom orientation of E is an element th ∈ E^{2,1}(Th(𝒯(1)) such that its restriction to the Thom space of the fibre over the distinguished point coincides with the element Σ_{T} (1) ∈ E^{2,1}(T). A Chern orientation of E is an element c ∈ E^{2,1}(P^{∞}) such that c|_{P^{1}} = −Σ_{P^{1}}(1). An orientation of E is either a Thom orientation or a Chern orientation. Two Thom orientations of E coincide if respecting Thom elements coincides. Two Chern orientations of E coincide if respecting Chern elements coincides. One says that a Thom orientation th of E coincides with a Chern orientation c of E provided that c = z^{∗}(th) or equivalently the element th coincides with the one th(𝒪(−1)) given by (6) below.

Remark 1.1.2. The element th should be regarded as a Thom class of the tautological line bundle 𝒯(1) = 𝒪(−1) over P^{∞}. The element c should be regarded as a Chern class of the tautological line bundle 𝒯(1) = 𝒪(−1) over P^{∞}.

Example 1.1.3. The following orientations given right below are relevant for our work. Here MGL denotes the P^{1}-ring spectrum representing algebraic cobordism obtained in [PPR2, Defn 2.1.1] and BGL denotes the P^{1}-ring spectrum representing algebraic K-theory constructed in [PPR1, Theorem 2.2.1].

• Let u_{1} : Σ_{P^{1}}^{∞}(Th(𝒯(1)))(−1) → MGL be the canonical map of P^{1}- spectra. Set th^{MGL} = u_{1} ∈ MGL^{2,1}(Th(𝒯(1))). Since th^{MGL}|_{Th(1)} = Σ_{P^{1}}(1) in MGL^{2,1}(Th(1)), the class th^{MGL} is an orientation of MGL.

• Set c = (−β) ∪ ([𝒪] − [𝒪(1)]) ∈ BGL^{2,1}(P^{∞}). The relation (11) from [PPR1] shows that the class c is an orientation of BGL.

2 Oriented cohomology theories

Let (E, c) be an oriented commutative P^{1}-ring spectrum. In this Section we compute the E-cohomology of infinite Grassmannians and their products. The results are the expected ones 2.0.6.

The oriented P^{1}-ring spectrum (E, c) defines an oriented cohomology theory on 𝒮m𝒪p in the sense of [PS1, Defn. 3.1] as follows. The restriction of the functor E^{∗,∗} to the category 𝒮m/S is a ring cohomology theory. By [PS1, Th. 3.35] it remains to construct a Chern structure on E^{∗,∗}|_{𝒮m𝒪p} in the sense of [PS1, Defn.3.2]. Let H(k) be the homotopy category of spaces over k. The functor isomorphism Hom_{H(k)}(−, P^{∞}) → Pic(−) on the category 𝒮m/S provided by [MV, Thm. 4.3.8] sends the class of the identity map P^{∞} → P^{∞} to the class of the tautological line bundle 𝒪(−1) over P^{∞}. For a line bundle L over X ∈ 𝒮m/S let [L] be the class of L in the group Pic(X). Let f_{L}: X → P^{∞} be a morphism in H(k) corresponding to the class [L] under the functor isomorphism above. For a line bundle L over X ∈ 𝒮m/S set c(L) = f_{L}^{∗}(c) ∈ E^{2,1}(X). Clearly, c(𝒪(−1)) = c. The assignment L/X ↦ c(L) is a Chern structure on E^{∗,∗}|_{𝒮m𝒪p} since c|_{P^{1}} = −Σ_{P^{1}}(1) ∈ E^{2,1}(P^{1}, ∞). With that Chern structure E^{∗,∗}|_{𝒮m𝒪p} is an oriented ring cohomology theory in the sense of [PS1]. In particular, (BGL, c^{K}) defines an oriented ring cohomology theory on 𝒮m𝒪p.

Given this Chern structure, one obtains a theory of Thom classes V/X ↦ th(V) ∈ E^{2rank(V),rank(V)}(Th_{X}(V)) on the cohomology theory E^{∗,∗}|_{𝒮m𝒪p/S} in the sense of [PS1, Defn. 3.32] as follows. There is a unique theory of Chern classes V ↦ c_{i}(V) ∈ E^{2i,i}(X) such that for every line bundle L on X one has c_{1}(L) = c(L). For a rank r vector bundle V over X consider the vector bundle W := 1 ⊕ V and the associated projective vector bundle P(W) of lines in W. Set

t̄h(V) = c_{r}(p^{∗}(V) ⊗ 𝒪_{P(W)}(1)) ∈ E^{2r,r}(P(W)).  (5)

It follows from [PS1, Cor. 3.18] that the support extension map

E^{2r,r}(P(W)/(P(W) ∖ P(1))) → E^{2r,r}(P(W))

is injective and t̄h(E) ∈ E^{2r,r}(P(W)/(P(W) ∖ P(1))). Set

th(E) = j^{∗}(t̄h(E)) ∈ E^{2r,r}(Th_{X}(V)),  (6)

where j: Th_{X}(V) → P(W)/(P(W) ∖ P(1)) is the canonical motivic weak equivalence of pointed motivic spaces induced by the open embedding V ↪ P(W). The assignment V/X to th(V) is a theory of Thom classes on E^{∗,∗}|_{𝒮m𝒪p} (see the proof of [PS1, Thm. 3.35]). So the Thom classes are natural, multiplicative and satisfy the following Thom isomorphism property.

Theorem 2.0.4. For a rank r vector bundle p: V → X on X ∈ 𝒮m/S with zero section z: X ↪ V, the map

∪th(V): E^{∗,∗}(X) → E^{∗+2r,∗+r}(V/(V ∖ z(X)))

is an isomorphism of the two-sided E^{∗,∗}(X)-modules, where − ∪ th(V) is written for the composition map (∪th(V)) ∘ p^{∗}.

Proof. See [PS1, Defn. 3.32.(4)].

Analogous to [V1, p. 422] one obtains for vector bundles V → X and W → Y in 𝒮m/S a canonical map of pointed motivic spaces Th(V) ∧ Th(W) → Th(V ×_{S} W) which is a motivic weak equivalence as defined in [PPR1, Defn. 3.1.6]. In fact, the canonical map becomes an isomorphism after Nisnevich (even Zariski) sheafification. Taking Y = S and W = 1 the trivial line bundle yields a motivic weak equivalence Th(V)∧T → T h(V ⊕1). The canonical covering of P^{1} defines motivic weak equivalences

\xymatrix{T = A^{1}/A^{1} ∖ \{0\} \ar[r]^{∼} & P^{1}/A^{1} & P^{1} \ar[l]_{∼}}

and the arrow T = A^{1}/A^{1} ∖ {0} → P^{1}/P^{1} ∖ {0} is an isomorphism. Hence one may switch between T and P^{1} as desired.

Corollary 2.0.5. For W = V ⊕ 1 consider the composite motivic weak equivalence ϵ: Th(V) ∧ P^{1} → Th(V) ∧ P^{1}/A^{1} ← Th(V) ∧ T → Th(W) in H_{•}(S). Then the diagram

E^{∗+2r,∗+r}(T h(V)) ⟶^{Σ_{P^{1}}} E^{∗+2r+2,∗+r+1}(T h(V) ∧ P^{1})
id↑  ^{ϵ^{∗}}↑
E^{∗+2r,∗+r}(T h(V)) ⟶^{Σ_{T}} E^{∗+2r+2,∗+r+1}(T h(W))
∪th(V)↑  ^{∪th(W)}↑
E^{∗,∗}(X)  ⟶^{id}  E^{∗,∗}(X).

commutes.

Theorem 2.0.6. Let c_{i} = c_{i}(𝒯(n)) ∈ E^{2i,i}(Gr(n)) be the i-th Chern class of the tautological bundle 𝒯(n). Then

E^{∗,∗}(Gr(n)) = E^{∗,∗}(k)[[c_{1}, c_{2}, …, c_{n}]]

is the formal power series on the c_{i}’s. The inclusion i: Gr(n) ↪ Gr(n + 1) satisfies i^{∗}(c_{m}) = c_{m} for m < n + 1 and i^{∗}(c_{n+1}) = 0.

2.1 A general result

The main result of this Section is Theorem 2.1.4. The complex cobordism spectrum, equipped with its natural orientation, is a universal oriented ring cohomology theory by Quillen’s universality theorem [Qu1]. A motivic version of this universality theorem is proved in [PPR2] (see [Ve] for the original statement). We consider MGL with the commutative monoid structure described in [PPR2, Defn 2.1.1] and with the orientation th^{MGL} described in 1.1.3.

By a cofibration we mean below in the text a cofibration with respect to the closed model structure on the category M(S) (see [PPR1, Appendix]).

Recall that for a P^{1}-spectrum E and a cofibration Y → X the group E^{p,q}(X, Y) is defined as the cohomology E^{p,q}(X/Y, Y/Y) of the pointed space (X/Y, +) (if Y is the empty set, then one should take the group E^{p,q}(X_{+}, +) for E^{p,q}(X, Y)).

Definition 2.1.1 (Universality Property). Let (U, u) be an oriented commutative ring P^{1}-spectrum over a field k. We say that (U, u) satisfies Quillen universality property, if for each commutative ring P^{1}-spectrum E over k the assignment φ ↦ φ(u) ∈ U^{2,1}(Th(𝒯(1))) identifies the set of monoid morphisms

φ: U → E  (7)

in the motivic stable homotopy category SH^{cm}(S) with the set of orientations of E.

Let (U, u) be an oriented commutative ring P^{1}-spectrum over k. Let (E, th) oriented commutative ring P^{1}-spectrum over k. Let

φ: U → E  (8)

be a monoid morphism in SH^{cm}(k) such that φ(u) = th. For every space X over k and a cofibration Y → X and a unique morphism f: X/Y → Spec(k) one has a commutative diagram of U^{0}(k)-module homomorphisms.

\xymatrix{U^{∗}(X, Y) \ar[r]^{φ_{X,Y}} & E^{∗}(X, Y) \\ U^{0}(k) \ar[u]^{f^{∗}} \ar[r]^{φ_{S}^{0}} & E^{0}(k) \ar[u]_{f^{∗}}}

It is known that for each oriented commutative ring P^{1}-spectrum (F, v) and each space A the ring F^{0}(A) is contained in the center of F^{∗}(A). The last commutative diagram induces two homomorphisms

φ̄_{X,Y} : U^{∗}(X, Y) ⊗_{U^{0}(k)} E^{0}(k) → E^{∗}(X, Y)  (9)

φ̄_{X,Y}^{0} : U^{0}(X, Y) ⊗_{U^{0}(k)} E^{0}(k) → E^{0}(X, Y)  (10)

which are natural in a cofibration Y → X.

Since this moment choose (BGL, th^{K}) for (E, th) (see Example 1.1.3). Set Ū^{∗}(X, Y) = U^{∗}(X, Y)⊗_{U^{0}(k)} BGL^{0}(k), Ū^{0}(X, Y) = U^{0}(X, Y)⊗_{U^{0}(k)} BGL^{0}(k).

Definition 2.1.2 (Weakly MGL-Cellular). A Quillen universal oriented commutative ring P^{1}-spectrum (U, u) is called weakly MGL-cellular if there exists an integer N such that the map φ̄_{U_{n},∗}^{0} is an isomorphism for n ≥ N .

Remark 2.1.3. By the Universality Theorem [Ve] or [PPR2] the P^{1}-spectrum MGL is Quillen universal. That is why we choose to write MGL-cellular in the definition above. The following theorem motivates two last Definitions.

Theorem 2.1.4. Let (U, u) be an oriented commutative ring P^{1}-spectrum over a field k satisfying the Quillen universality property. Suppose (U, u) is weakly MGL-cellular. Then for each cofibration Y → Xof small spaces over the field k the homomorphism φ̄_{X,Y} is an isomorphism.

Proof. The proof consists of several steps. Our first aim is to prove that homomorphisms φ̄_{X,Y}^{0} are isomorphisms. We beging with constructing a section of the natural transformation

φ^{0,0} : U^{0,0} → BGL^{0,0}

of functors on the category of cofibrations of small spaces. To do this we begin with recalling that for every oriented commutative P^{1}-ring spectrum (E, th) the ring cohomology theory E^{∗,∗}|_{𝒮m𝒪p} is an oriented cohomology theory on the category 𝒮m𝒪p (see Section 2). Let 𝔽_{E,th} be the induced commutative formal group law over the ring E^{0}(k). Let Ω be the complex cobordism ring and let l_{E,th}: Ω → E^{0}(k) be the unique ring homomorphism, which takes the universal formal group 𝔽_{Ω} to 𝔽_{E,th}. Set

[P^{n}]_{E} = l_{E,th}([ℂℙ^{n}]),  (11)

where [ℂℙ^{n}] is the class of the complex projective space ℂℙ^{n} in Ω. Although the class [P^{n}]_{E} depends on the orientation class th, we use the notation [P^{n}]_{E} instead. If (E′, th′) is another oriented commutative P^{1}-ring spectrum and ψ: E → E′ is a monoid homomorphism in the category SH^{cm}(S) which preserves orientation classes, then it sends the formal group law 𝔽_{E,th} to 𝔽_{E′,th′}. In particular ψ([P^{n}]_{E}) = [P^{n}]_{E′}. Applying this observation to the monoid homomorphism φ one obtains

φ([P^{1}]_{U}) = [P^{1}]_{BGL}.

To compute [P^{1}]_{BGL} recall that the coefficient at XY in the formal group law 𝔽_{Ω} coincides with the class −[ℂℙ^{1}] in Ω. The formal group law 𝔽_{BGL} coincides with X + Y + β^{−1}XY, since c^{BGL}(L) = ([1] − [L^{∨}])(−β). Thus one gets

[P^{1}]_{BGL} = −β^{−1}

We are ready to construct a section. Consider the map

s: Σ_{P^{1}}^{∞}(ℤ × Gr) → U  (12)

in the stable homotopy category category SH^{cm}(S) given by the element

c_{1}^{U}(∞ − τ_{∞}^{∨}) ∪ [P^{1}]_{U} ∈ U^{0,0}(ℤ × Gr).

Claim 2.1.5. One has φ(c_{1}^{U}(∞ − τ_{∞}^{∨}) ∪ [P^{1}]_{U}) = τ_{∞} − ∞ ∈ BGL^{0,0}(ℤ × Gr).

In fact,

φ(c_{1}^{U}(∞ − τ_{∞}^{∨}) ∪ [P^{1}]_{U}) = c_{1}^{BGL}(∞ − τ_{∞}^{∨}) ∪ [P^{1}]_{BGL} = (∞ − τ_{∞}) ∪ β ∪ (−β^{−1})

= τ_{∞} − ∞.

Claim 2.1.5 shows that the composite map

φ ∘ s: Σ_{P^{1}}^{∞}(ℤ × Gr) → BGL

coincides with the adjoint of the motivic weak equivalence i: ℤ × Gr → 𝒦 = 𝒦_{0} from [PPR1, Lemma 1.2.2]. Thus for every cofibration Y → X of small motivic spaces the map

s_{X,Y} : BGL^{0,0}(X/Y) = [X/Y, 𝒦_{0}] = [X/Y, ℤ×Gr] → [Σ_{P^{1}}^{∞}(X/Y), U] = U^{0,0}(X/Y)

is a section of the map φ_{X,Y}^{0,0} : U^{0,0}(X, Y) → BGL^{0,0}(X, Y). Moreover, the section s_{X,Y} is natural in the cofibration Y → X.

Next we extend the section s to a section s̄^{0} : BGL^{0} → \overline{U}^{0} of the natural transformation φ̄^{0} : \overline{U}^{0} → BGL^{0} of functors on the category of cofibrations. To achieve this, recall that

BGL^{0} = BGL^{0,0}[β, β^{−1}]

for the Bott element β ∈ BGL^{2,1}(k) (see (2)). Thus for every cofibration Y → X every element α ∈ BGL^{0}(X, Y) can be presented in a unique way in the form a ∪ β^{i} with a ∈ BGL^{0,0}(X, Y). Define

s̄_{X,Y}^{0} : BGL^{0} → \overline{U}^{0}  (13)

by s̄_{X,Y}^{0} (a∪β^{i}) = s_{X,Y} (a)⊗β^{i} ∈ \overline{U}^{0}(A), where a ∈ BGL^{0,0}(X, Y). It is immediate that s_{A}^{0} is natural in cofibration Y → X. The following computation proves the claim which is right below the row of computation

φ̄_{A}^{0}(s̄^{0}(a ∪ β^{i})) = φ̄_{A}^{0}(s(a) ⊗ β^{i}) = φ(s(a)) ∪ β^{i} = a ∪ β^{i}.

Claim 2.1.6. The map s̄_{X,Y}^{0} is a section of φ̄_{X,Y}^{0} .

Now observe the following. If for a cofibration Y → X the map φ̄_{X,Y}^{0} is an isomorphism, then s̄_{X,Y}^{0} is an isomorphism inverse to φ̄_{X,Y}^{0} . In particular, one has s̄_{X,Y}^{0} ∘ φ̄_{X,Y}^{0} = id.

The homomorphism φ̄_{X,Y}^{0} is an isomorphism for cofibrations of the form ∗ → U_{n} with n ≥ N, since U is weakly MGL-cellular. Taking ∗ → U_{n} as a cofibration Y → X and the class [u_{n}] ∈ U^{2n,n}(U_{n}, ∗) of the canonical morphism u_{n}: Σ_{P^{1}}^{∞}U_{n}(−n) → U we get the following relation:

(s̄_{U_{n},∗}^{0} ∘ φ_{U_{n},∗}^{0})([u_{n}]) = [u_{n}] ⊗ 1 ∈ \overline{U}^{0}(U_{n}, ∗).  (14)

Now we are ready to check that φ̄_{A}^{0} is an isomorphism for all cofibrations Y → X of small motivic spaces. Recall that for a cofibrations Y → X of small motivic spaces there is a canonical isomorphism of the form

U^{2i,i}(X, Y) = colim_{n}[Σ^{2n,n}(X/Y, Y/Y), U_{i+n}]_{H_{•}(S)}  (15)

where Σ^{2n,n} = Σ_{P^{1}}^{n} (if Y is empty then one should replace the pair (X/Y, Y/Y) by the one (X_{+}, +)). This isomorphism implies that for every element a ∈ U^{2i,i}(X, Y) there exists an integer n ≥ 0 such that Σ^{2n,n}(a) = f^{∗}([u_{n}]) for an appropriate map f: Σ^{2n,n}(X/Y) → U_{i+n} in the homotopy category H_{•}^{cm}(S). Here Σ^{2n,n}(a) is the n-fold Σ_{P^{1}}-suspension of a.

The surjectivity of φ̄_{X,Y}^{0} is clear, since s̄_{X,Y}^{0} is its section. It remains to check the injectivity of φ̄_{X,Y}^{0} . Take a homogeneous element α ∈ \overline{U}^{2i,i}(X, Y) ⊆ \overline{U}^{0}(X, Y) such that φ̄_{X,Y}^{0} (α) = 0. It has the form α = a ⊗ β^{m} for a homogeneous element a ∈ U^{0,0}(X, Y). Since the element β is invertible in BGL^{∗,∗}(k), one concludes φ_{X,Y}^{0} (a) = 0.

Choose an integer n ≥ 0 such that Σ^{2n,n}(a) = f^{∗}([u_{n}]) and write A for X/Y to short the notation. The map φ of P^{1}-spectra respects the suspension isomorphisms. Thus φ_{Σ^{2n,n}A}(Σ^{2n,n}(a)) = Σ^{2n,n}(φ_{A}(a)) = 0 and (s̄_{Σ^{2n,n}A}^{0} ∘ φ_{Σ^{2n,n}A})(Σ^{2n,n}(a)) = 0 too. The chain of relations in \overline{U}^{0}(Σ^{2n,n}A) given by

0 = (s̄_{Σ^{2n,n}A}^{0} ∘ φ_{Σ^{2n,n}A})(Σ^{2n,n}(a)) = (s̄_{Σ^{2n,n}A}^{0} ∘ φ_{Σ^{2n,n}A})(f^{∗}([u_{n}]))
= f^{∗}((s̄_{U_{n+i}}^{0} ∘ φ_{U_{n+i}})([u_{n}])) = f^{∗}([u_{n}] ⊗ 1) = f^{∗}([u_{n}]) ⊗ 1
= Σ^{2n,n}(a) ⊗ 1

implies that Σ^{2n,n}(a ⊗ 1) = Σ^{2n,n}(a) ⊗ 1 = 0. Because the n-fold suspension map

Σ^{2n,n}: \overline{U}^{0}(X/Y, Y/Y) → \overline{U}^{0}(Σ^{2n,n}(X/Y, Y/Y))

is an isomorphism, a⊗1 = 0 in \overline{U}^{0}(X/Y) = \overline{U}^{0}(X, Y). This proves the injectivity and hence the bijectivity of φ̄_{X,Y}^{0} for cofibrations of all small motivic spaces.

To prove that φ̄_{X,Y} is an isomorphism for cofibrations of all small motivic spaces we will use the fact that φ̄_{X,Y} respects the P^{1}-suspension isomorphisms. Set A = X/Y.

For every integer i ∈ ℤ choose an integer n ≥ 0 with n ≥ i. Then for a pointed motivic space A one may form the suspension 𝔾_{m}^{∧n} ∧ S_{s}^{n−i} ∧ A = S^{n,n} ∧ S^{n−i,0} ∧ A in the category of pointed motivic spaces, which supplies the commutative diagram

\xymatrix{BGL^{i}(A) \ar[r]^{Σ^{2n,n}}_{≅} & BGL^{i}(S^{2n,n} ∧ A) & BGL^{0}(S^{n,n} ∧ S^{n−i,0} ∧ A) \ar[l]_{Σ^{i,0}}^{≅} \\ U^{i}(A) \ar[r]^{Σ^{2n,n}}_{≅} \ar[u]^{φ̄_{A}^{i}} & U^{i}(S^{2n,n} ∧ A) \ar[u]^{φ̄_{S^{2n,n}∧A}^{i}} & U^{0}(S^{n,n} ∧ S^{n−i,0} ∧ A) \ar[l]_{Σ^{i,0}}^{≅} \ar[u]_{φ̄_{S^{n,n}∧S^{n−i,0}∧A}^{0}}^{≅}}

with the suspension isomorphisms Σ^{2n,n} = Σ_{P^{1}}^{n} and Σ^{i,0}. The map φ̄_{B}^{0} is an isomorphism for B a small pointed motivic space, hence so is φ̄_{A}^{i}. We proved that the map φ̄_{X,Y} is an isomorphism. Theorem 2.1.4 is proven.

2.2 The MGL-cellularity of the MGL

Theorem 2.2.1. The oriented commutative ring P^{1}-spectrum (MGL, th^{MGL}) from Example 1.1.3 is weakly MGL-cellular.

Proof. We must check that the homomorphism φ̄_{X,Y}^{0} is an isomorphism for (X, Y) being (T h(𝒯_{n}), ∗) = (MGL_{n}, ∗). We check that inspecting step by step motivic spaces Spec(k), P^{∞}, Gr(n) and the pair (T h(𝒯_{n}), ∗) = (MGL_{n}, ∗).

The map φ̄_{k}^{0} is an isomorphism, since it is the identity map. By the case n = 1 of Theorem 2.0.6 one has \overline{MGL}^{∗}(P^{∞}) = \overline{MGL}^{∗}(k)[[c^{MGL}]], whence

\overline{MGL}^{0}(P^{∞}) = \overline{MGL}^{0}(k)[[c^{MGL}]]

(the formal power series on the first Chern class c^{MGL} of the tautological line bundle 𝒪(−1)). The same holds for BGL. Namely

BGL^{0}(P^{∞}) = BGL^{0}(k)[[c^{BGL}]].

By its definition the morphism φ takes the orientation class th^{MGL} to the orientation class th^{K} and so it preserves the first Chern class. Whence the map φ̄_{P^{∞}}^{0} coincides with a map of formal power series induced by the isomorphism φ̄_{k}^{0} of the coefficients rings. Hence φ̄_{P^{∞}}^{0} is an isomorphism as well.

Consider now X = Gr(n). By Theorem 2.0.6 its MGL-cohomology ring is the ring of formal power series on the Chern classes of the tautological bundle 𝒯_{n} over the coefficient ring MGL^{∗,∗}(k). The same holds for the BGLcohomology ring. As observed above, the map φ preserves the first Chern class, thus it takes Chern classes to the Chern classes. Whence φ̄_{Gr(n)}^{0} is an isomorphism as well.

Now consider (X, Y) = (Th(𝒯_{n}), ∗). The morphism φ respects Thom classes (see (5) and (6)). The vertical arrows in the commutative diagram

\xymatrix{\overline{MGL}^{0}((Th(𝒯_{n}), ∗)) \ar[r]^{φ̄_{Th(𝒯n),∗)}^{0}} & BGL^{0}(Th(𝒯_{n})) \\ \overline{MGL}^{0}(Gr(n)) \ar[r]^{φ̄_{Gr(n)}^{0}} \ar[u]^{_{\overline{thom}_{MGL}}} & BGL^{0}(G(n)) \ar[u]_{_{thom_{BGL}}}}

are isomorphisms induced by the the Thom isomorphism 2.0.4. The map φ̄_{Gr(n)}^{0} is an isomorphism by the preceding case, whence φ̄_{T h(𝒯_{n}),∗)}^{0} is an isomorphism too.

2.3 Main Result

Let k be a field and S = Spec(k). By Theorem [PPR2, Theorem 2.2.1] and Example 1.1.3 there exists a unique monoid morphism

φ: MGL → BGL  (16)

in SH^{cm}(S) such that φ(th^{MGL}) = th^{K}. For every cofibration Y → X of motivic spaces over k a unique morphism f: X/Y → S induces the homomorphism

φ̄_{X,Y} : \overline{MGL}^{∗}(X, Y): = MGL^{∗}(X, Y) ⊗_{MGL^{0}(k)} BGL^{0}(k) → BGL^{∗}(X, Y)(17)

which is natural in cofibration Y → X. Recall that a space A is called small if the covariant functor Σ_{P^{1}}^{∞}A represents on SH^{cm}(S) commutes with arbitrary coproducts.

Theorem 2.3.1. The homomorphism φ̄_{X,Y} is an isomorphism for all cofibrations Y → X of small motivic spaces.

In fact, the (MGL, th^{MGL}) is Quillen universal by [Ve] or by Theorem 2.2.1 from [PPR2] and weakly MGL-cellular by Theorem 2.2.1 above. Theorem 2.1.4 completes the proof.

Remark 2.3.2. There is an unpublished result due to Morel and Hopkins, which states that there is a canonical isomorphism of the form

MGL^{∗,∗}(X) ⊗_{𝕃} ℤ[β, β^{−1}] → BGL^{∗,∗}(X)

where 𝕃 denotes the Lazard ring carrying the universal formal group law. If the canonical homomorphism 𝕃 → MGL^{0}(k) is an isomorphism, Theorem 2.3.1 implies their result.

Let X be a smooth k-scheme and Z ⊆ X a closed subset, with open complement U ⊆ X. Consider the motivic space X/U and take the quotients of both sides of the isomorphism (17) modulo the principal ideal generated by the element 1⊗(β+1). Corollary 1.0.1 then implies the following isomorphism

¯φ̄_{X/U}: \overline{\overline{MGL}}^{∗}(X, Y) = MGL^{∗}(X/U) ⊗_{MGL^{0}(k)} ℤ → K_{−∗,Z}^{T T}(X)  (18)

where K_{∗,Z}^{T T}(X) are the Thomason-Trobaugh K-groups with supports. This family of isomorphisms shows that the functor

(X, X ∖ Z) ↦ MGL^{∗}(X/(X ∖ Z)) ⊗_{MGL^{0}(k)} ℤ =: \overline{\overline{MGL}}^{∗}(X/(X ∖ Z))

is a ring cohomology theory in the sense of [PS1]. This implies the first part of our main result.

Theorem 2.3.3 (Main Theorem). Let X ∈ 𝒮m_{k} and Z ⊆ X be a closed subset.

• The family of isomorphisms

¯φ̄_{X/(X−Z)} : \overline{\overline{MGL}}^{∗}(X/(X ∖ Z)) → K_{−∗,Z}^{T T}(X)  (19)

form an isomorphism ¯ φ̄ of ring cohomology theories on 𝒮m𝒪p/k.

• The ¯ φ̄ respects orientations provided that MGL^{∗} and K_{−∗}^{T T} are considered as oriented cohomology theories in the sense of [PS1] with orientations given by the Thom class th^{MGL} ⊗ 1 from 1.1.3 and the Chern structure L/X ↦ [𝒪] − [L^{−1}]. In particular, the composition

\xymatrix{MGL^{0}(k) \ar[r] & MGL^{0}(k) ⊗ ℤ \ar[r] & ℤ \\ a \ar@{|->}[r] & a ⊗ 1 b ⊗ c \ar@{|->}[r] & φ(b) · c}

sends the class [X] ∈ MGL^{0}(X) of a smooth projective k-variety X to the Euler characteristic χ(X, 𝒪_{X}) of the structure sheaf 𝒪_{X}.

Proof. The first part is already proven. To prove the second one consider the orientations th^{MGL} and th^{K} from 1.1.3. Note that by the very definition of φ it sends th^{MGL} to th^{K}. Thus it respects the Chern structures on MGL^{∗} and BGL^{∗} described in Section 2.

The quotient map BGL^{∗} → K_{−∗}^{T T} takes the Bott element β to (−1). Thus it takes the Chern structure on BGL^{∗} to the Chern structure on K_{−∗}^{T T} given by L/X ↦ [𝒪] − [L^{−1}] ∈ K_{0}(X). This shows that ¯ φ̄: \overline{\overline{MGL}}^{∗} → K_{−∗}^{T T} respects the orientations described in the Theorem 2.3.3.

Let f ↦ f_{MGL} resp. f ↦ f_{K} be the integrations on MGL^{∗} resp. K_{−∗}^{T T} given by these Chern structures via Theorem [PS3, Thm. 4.1.4]. By Theorem [PS2, Thm. 1.1.10] the composition MGL^{∗} → BGL^{∗} → K_{−∗}^{T T} respects the integrations on MGL^{∗} and K_{−∗}^{T T} since it preserves the Chern structures. In particular, given a smooth projective S-scheme f: X → Spec(k), the diagram

\xymatrix{\overline{\overline{MGL}}^{0}(X) \ar[r]^{¯φ̄} \ar[d]_{f_{MGL}} & K_{0}^{T T} (X) \ar[d]^{f_{K}} \\ \overline{\overline{MGL}}^{0}(k) \ar[r]^{¯φ̄} & K_{0}^{T T} (k)}

commutes where f_{MGL} and f_{K} are the push-forward maps for MGL^{∗} and K_{−∗}^{T T} respectively. The integration f ↦ f_{K} on K_{−∗}^{T T} respecting the Chern structure L ↦ [𝒪] − [L^{−1}] coincides with the one given by the higher direct images by Theorem [PS2, Thm. 1.1.11]. The last one sends the class [V] ∈ K_{0}(X) of a vector bundle V over a smooth projective variety X to the Euler characteristic χ(X, 𝒱) of the sheaf 𝒱 of sections of V.

Recall that for an oriented cohomology theory A with a Chern structure L ↦ c(L) and for a smooth projective variety f: X → Spec(k) its class [X]_{A} ∈ A^{even}(Spec(k)) is defined as f_{A}(1), where f_{A}: A(X) → A(Spec(k)) is the push-forward respecting the Chern structure (see [PS3, Thm. 4.1.4]). The f_{A} depends on the Chern structure. However we write just f_{A} for the push-forward operator. Taking the element 1 ∈ MGL^{0,0}(X) and using the commutativity of the very last diagram we see that

¯φ̄([X]_{MGL} ⊗ 1) = χ(X, 𝒪_{X}).

Whence the Theorem.

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