On Voevodsky’s algebraic K-theory spectrum BGL

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

April 17, 2007^{§}

Abstract Under a certain normalization assumption we prove that the P^{1}-spectrum BGL of Voevodsky which represents algebraic K-theory is unique over Spec(ℤ). Following an idea of Voevodsky, we equip the P^{1}-spectrum BGL with the structure of a commutative ring P^{1}-spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec(ℤ). For an arbitrary Noetherian base scheme S we pull this structure back to get a distinguished monoidal structure on BGL.

1 Preliminaries

We refer to the Appendix A for the basic terminology, notation, constructions, definitions, results concerning homotopy theory. For a scheme S we write M(S), M_{•}(S), H_{•}^{cm}(S) and SH(S) for the category of motivic spaces, the category of pointed motivic spaces, the pointed motivic homotopy category and the stable motivic homotopy category over S. These categories are equipped with symmetric monoidal structures. In particular, a symmetric monoidal structure (∧, 𝕀) is constructed on the motivic stable homotopy category and its basic properties are proved. This structure is used intensively over the present text.

Let S be a regular scheme, and let K^{0}(S) denote the Grothendieck group of vector bundles over S. Morel and Voevodsky proved in [MV, Thm. 4.3.13] that the Thomason-Trobaugh K-theory [TT] is represented in the pointed motivic homotopy category H_{•}(S) by the space ℤ × Gr pointed by (0, x_{0}). Here Gr is the union of the finite Grassmann varieties ⋃_{n=0}^{∞} Gr(n, 2n), considered as motivic spaces. There is an element τ_{∞} − ∞ ∈ K^{0}(ℤ × Gr) which corresponds to the identity morphism id: ℤ × Gr → ℤ × Gr. It follows that there exists a unique morphism

μ_{⊗}: (ℤ × Gr) ∧ (ℤ × Gr) → ℤ × Gr

∗Steklov Institute of Mathematics at St. Petersburg, Russia

†Universität Bielefeld, SFB 701, Bielefeld, Germany

‡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.

in H_{•}^{cm}(S) such that the composition (ℤ×Gr)×(ℤ×Gr) → (ℤ×Gr)∧(ℤ×Gr) ⟶^{μ_{⊗}} ℤ×Gr represents the element (τ_{∞} − ∞) ⊗ (τ_{∞} − ∞) in K^{0}((ℤ × Gr) × (ℤ × Gr)) (see Cor. B.1.2). Let e_{⊗} : S^{0} → ℤ × Gr be the map which corresponds to the point (1, x_{0}) ∈ ℤ × Gr. The triple

(ℤ × Gr, μ_{⊗}, e_{⊗})  (1)

is a commutative monoid in H_{•}^{cm}(S).

Using this fact Voevodsky constructed in [V1] a P^{1}-spectrum

BGL = (𝒦_{0}, 𝒦_{1}, 𝒦_{2}, …)

with structure maps e_{i}: 𝒦_{i} ∧ P^{1} → 𝒦_{i+1} such that

1. there is a motivic weak equivalence w: ℤ × Gr → 𝒦_{0}, and for all i one has 𝒦_{i} = 𝒦_{0} and e_{i} = e_{0},

2. the morphism

\xymatrix{ℤ × Gr × P^{1} \ar[r]^{can} & (ℤ × Gr) ∧ P^{1} \ar[r]^{w∧P^{1}} & K_{i} ∧ P^{1} \ar[r]^{e_{i}} & 𝒦_{i+1} \ar[r]^{w^{−1}} & ℤ × Gr}

in H_{•}(S) represents the element (τ_{∞} − ∞) ⊗ ([𝒪(−1)] − [𝒪]) ∈ K^{0}(ℤ × Gr × P^{1}),

3. and the adjoint 𝒦_{i} → Ω_{P^{1}}(𝒦_{i+1}) of e_{i} is a motivic weak equivalence.

With this spectrum in hand given a smooth X over S we may identify K^{0}(X) with BGL^{2i,i}(X) as follows

K^{0}(X) = Hom_{H_{•}(S)}(X_{+}, ℤ × Gr) = Hom_{H_{•}(S)}(X_{+}, 𝒦_{i}) = BGL^{2i,i}(X)  (2)

Our first aim is to recall Voevodsky construction to show that this spectrum is essentially unique. This has also been obtained in [R]. Our second and more important aim is to give a commutative monoidal structure to the P^{1}-spectrum BGL which respects the naive multiplicative structure on the functor X ↦ BGL^{2∗,∗}(X). To be more precise, we construct a product

μ_{BGL} : BGL ∧ BGL → BGL  (3)

in the stable motivic homotopy category SH(S) such that for any X ∈ 𝒮m/S the diagram

\xymatrix{K^{0}(X) × K^{0}(X) \ar[r]^{⊗} \ar[d]_{≅} & K^{0}(X) \ar[d]^{≅} \\ BGL^{2i,i}(X) × BGL^{2j,j}(X) \ar[r]^{μ_{BGL}} & BGL^{2(i+j),i+j}(X)}

commutes. We show that there is a unique product μ_{BGL} ∈ Hom_{SH(ℤ)}(BGL ∧ BGL, BGL) satisfying this property. This induces a product μ_{BGL} ∈ Hom_{SH(S)}(BGL ∧ BGL, BGL) for an arbitrary regular scheme S by pull-back along the structural morphism S → Spec(ℤ). As well, we show that the product is associative, commutative and unital. The resulting multiplicative structure on the bigraded theory BGL^{∗,∗} coincides with the Waldhausen multiplicative structure on the Thomason-Trobaugh K-theory, as we prove in Proposition 1.2.2.

1.1 Voevodsky’s K-theory spectrum BGL

The basic definitions, constructions and model structures used in the text are given in the Appendix A. Let S be a Noetherian finite-dimensional scheme. A motivic space over S is a simplicial presheaf on the site 𝒮m/S of smooth quasi-projective S-schemes. A pointed motivic space over S is a pointed simplicial presheaf on the site 𝒮m/S. Let S^{n} = S^{n,0} denote the n-fold smash product of the constant simplicial presheaf Δ^{1}/∂Δ^{1} with itself. We write M_{•}(S) for the category of pointed motivic spaces over S. A closed motivic model structure M_{•}^{cm}(S) is constructed in A.3.9. The adjective “closed” refers to the fact that closed embeddings in 𝒮m/S are forced to become cofibrations. The resulting homotopy category H_{•}^{cm}(S) is called the motivic homotopy category A.3.9 of S. By A.3.11 it is equivalent to the Morel-Voevodsky A^{1}-homotopy category [MV], and we may drop the superscript in H_{•}^{cm}(S) for convenience. The closed motivic model structure has the properties that

1. for any closed S-point x_{0} : S ↪ X in a smooth S-scheme, the pointed motivic space (X, x_{0}) is cofibrant in M_{•}^{cm}(S),

2. a morphism f: S → S′ of base schemes induces a left Quillen functor f^{∗}: M_{•}^{cm}(S′) → M_{•}^{cm}(S), and

3. taking complex points is a left Quillen functor R_{ℂ}: M_{•}^{cm}(ℂ) → Top_{•}.

Conditions 2 and 3 do not hold for the Morel-Voevodsky model structure, condition 1 fails for the so-called projective model structure [DRØ]. For a morphism f: A → B of pointed motivic spaces we will write [f] for the class of f in H_{•}(S).

We will consider P^{1} as a pointed space over S pointed by ∞: S ↪ P^{1}. A P^{1}-spectrum E over S consists of a sequence E_{0}, E_{1}, … of pointed motivic spaces over S, together with structure maps σ_{n}: E_{n} ∧ P^{1} → E_{n+1}. Let SH(S) denote the homotopy category of P^{1}- spectra, as described in A.5. By Theorem A.5.6 it is canonically equivalent to the motivic stable homotopy category constructed in [V1] and [J]. As we will see below there exists an essentially unique P^{1}-spectrum BGL over S = Spec(ℤ) satisfying properties 1 to 2 from Section 1. In the following, we will construct BGL in a slightly different way than Voevodsky did originally in [V1]. In order to achieve this, we begin with a description of the known monoidal structure on the Thomason-Trobaugh K-theory [TT].

1.2 A construction of BGL

Let S be a regular scheme. For every S-scheme X consider the category Big(X) of big vector bundles over X (see for instance [FS] for the definition and basic properties). The assignments X ↦ Big(X) and (f: Y → X) ↦ f^{∗}: Big(X) → Big(Y) form a functor from schemes to the category of small categories. The reason is that there is an equality (f ∘ g)^{∗} = g^{∗} ∘ f^{∗}, not just a unique natural isomorphism. In what follows we will always consider the Thomason-Trobaugh K-theory space of X as the space obtained by applying Waldhausen’s 𝒮_{•}-construction [W] applied to the category Big(X) rather than to the category Vect(X) of usual vector bundles on X. This has the advantage that the assignment taking an S-scheme X to the Thomason-Trobaugh K-theory space of X becomes a functor on the category of S-schemes, and in particular a pointed motivic space over S.

Let 𝕂^{TT} be the pointed motivic space defined in A.3.4. It has the properties that it is fibrant in M_{•}^{cm}(S) and that 𝕂^{TT}(X) is naturally weakly equivalent to the Waldhausen K-theory space associated to the category of big vector bundles on X. For X ∈ 𝒮m/S and S regular, 𝕂^{TT}(X) is thus a Kan simplicial set having the Thomason-Trobaugh K-theory groups K_{∗}^{TT}(X) as its homotopy groups. Note that since X is regular, these K-theory groups coincide with Quillen’s higher K-theory groups. It follows immediately from the adjunction isomorphism

Hom_{H_{•}(S)}(S^{p,0} ∧ X_{+}, 𝕂^{TT}) ≅ Hom_{H_{•}}(S^{p}, 𝕂^{TT}(X)) = K_{p}^{TT}(X)  (4)

that 𝕂^{TT} regarded as an object in the motivic homotopy category H_{•}(S) (see A.3.9) represents the Thomason-Trobaugh K-theory on 𝒮m/S. Using a result of Morel and Voevodsky we describe now a commutative monoidal structure on 𝕂^{TT} as follows.

Let (Gr, x_{0}) and (ℤ×Gr, (0, x_{0})) be the pointed motivic spaces described in Section 1.

Lemma 1.2.1. The canonical map

Hom_{H_{•}^{cm}(S)}(Gr, 𝕂^{TT}) → \underleftarrow{lim} Hom_{H_{•}(S)}(Gr(n, 2n), 𝕂^{TT}) = \underleftarrow{lim}K_{0}^{TT}(Gr(n, 2n))

is an isomorphism. A similar statement holds for the pointed motivic spaces ℤ × Gr and (ℤ × Gr) × (ℤ × Gr).

Proof. The pointed motivic spaces (Gr(n, 2n), x_{0}) are closed cofibrant and the inclusions (Gr(n, 2n), x_{0}) ⟶^{i_{n}} (Gr(n + 1, 2n + 2), x_{0}) are closed cofibrations. Thus the colimit Gr is a model for the homotopy colimit. Since we already know that 𝕂^{TT} is weakly equivalent to the zeroth term in a fibrant P^{1}-spectrum BGL, there is an adjunction isomorphism

Hom_{H_{•}^{cm}(S)}(A, 𝕂^{TT}) ≅ Hom_{SH(S)}(Σ_{P^{1}}^{∞}A, BGL).

It follows from Lemma A.5.10 that there is an exact sequence of the form

0 → \underleftarrow{lim}^{1}[S^{1,0} ∧ Gr(n, 2n), 𝕂^{TT}] → [Gr, 𝕂^{TT}] → \underleftarrow{lim}[Gr(n, 2n), 𝕂^{TT}] → 0

It remains to check that the \underleftarrow{lim}^{1}-group vanishes. To prove this, note that by (4) there is an identification

\underleftarrow{lim}^{1}[S^{1,0} ∧ Gr(n, 2n), 𝕂^{TT}] = \underleftarrow{lim}^{1}K_{1}^{TT}(Gr(n, 2n)).

Lemma B.1.4 implies that \underleftarrow{lim}^{1}K_{1}^{TT}(Gr(n, 2n)) = \underleftarrow{lim}^{1}K_{1}^{TT}(S) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(Gr(n, 2n)), and the connecting homomorphisms in this tower are given by the tensor products id⊗ i_{n}^{∗}, where i_{n}^{∗}: K_{0}^{TT}(Gr(n + 1, 2n + 2)) → K_{0}^{TT}(Gr(n, 2n)). It is known that i_{n}^{∗} is surjective for all n ≥ 0. Thus the \underleftarrow{lim}^{1}-group vanishes. The other cases are handled similarly.

For an integer n let [−n, n] = {a ∈ ℤ: − n ≤ a ≤ n}. ¿From now on we will identify K_{0}^{TT}(ℤ × Gr) with the group \underleftarrow{lim}K_{0}^{TT}([−n, n] × Gr(n, 2n)). Following Lemma 1.2.1 we will identify Hom_{H_{•}^{cm}(S)}(ℤ × Gr, 𝕂^{TT}) with K_{0}^{TT}(ℤ × Gr).

Let τ_{n} be the tautological vector bundle over Gr(n, 2n), and let m be the trivial vector bundle of rank m. Let τ_{∞} − ∞ ∈ K_{0}^{TT}(ℤ × Gr) be the unique element such that for each integer m ≥ 0 its restriction to the subspace

{m} × Gr(n, 2n) ⊆ ℤ × Gr

coincides with the element [τ_{n}] − n + m ∈ K_{0}^{TT}(Gr(n, 2n)). Let f: ℤ × Gr → 𝕂^{TT} be the unique morphism in H_{•}^{cm}(S) representing τ_{∞} − ∞. By [MV, Thm. 4.3.13] the morphism f: ℤ × Gr → 𝕂^{TT} is an isomorphism in H_{•}^{cm}(S).

The space ℤ × Gr is closed cofibrant and the space 𝕂^{TT} is closed motivic fibrant. Thus by Quillen’s Theorem A.2.1 there is a motivic weak equivalence ℤ × Gr ⟶^{w} 𝕂^{TT} in M_{•}^{cm}(S) such that [w] ∈ Hom_{H_{•}(S)}(ℤ × Gr, 𝕂^{TT}) = K_{0}^{TT}(ℤ × Gr) coincides with the element τ_{∞} − ∞ ∈ K_{0}^{TT}(ℤ × Gr). Note that w is unique up to a simplicial homotopy.

Set K_{0}^{TT}((ℤ×Gr)×(ℤ×Gr)) := \underleftarrow{lim}K_{0}^{TT}(([−n, n]×Gr(n, 2n))×([−n, n]×Gr(n, 2n))). Following 1.2.1, identify Hom_{H_{•}(S)}((ℤ×Gr)×(ℤ×Gr), 𝕂^{TT}) with K_{0}^{TT}((ℤ×Gr)×(ℤ×Gr)). The space ℤ×Gr pointed by (0, x_{0}) is a pointed S-cellular scheme (see B.1 for the definition of a pointed S-cellular scheme). The element τ_{∞} − ∞ vanishes at the S-point (0, x_{0}). Corollary B.1.2 and Lemma 1.2.1 show that there exists a unique morphism

μ^{TT} ∈ Hom_{H_{•}(S)}(𝕂^{TT} ∧ 𝕂^{TT}, 𝕂^{TT})  (5)

such that the composite morphism in H_{•}(S)

(ℤ × Gr) × (ℤ × Gr) → (ℤ × Gr) ∧ (ℤ × Gr) ⟶^{w∧w} 𝕂^{TT} ∧ 𝕂^{TT} ⟶^{μ^{TT}} 𝕂^{TT}  (6)

represents the class (τ_{∞} − ∞) ⊗ (τ_{∞} − ∞) in K_{0}^{TT}((ℤ × Gr) × (ℤ × Gr)). In fact, w ∧ w is a weak equivalence in M_{•}^{cm}(S) by Theorem A.3.9. Now define e^{TT} as the composite morphism S^{0} ⟶^{e_{⊗}} ℤ × Gr ⟶^{w} 𝕂^{TT}. in M_{•}(S). Clearly the triple

(𝕂^{TT}, μ^{TT}, [e^{TT}])  (7)

is a commutative monoidal structure on 𝕂^{TT} in H_{•}(S).

Proposition 1.2.2. The commutative monoidal structure (7) on 𝕂^{TT} regarded as an object in H_{•}(S) coincides with the structure defined by Waldhausen in [W].

A sketch of the proof will be given in Appendix B. Now we continue our exposition.

Lemma 1.2.3. There is a trivial cofibration i: ℤ × Gr → 𝒦 in M_{•}^{cm}(S) and a trivial fibration p: 𝒦 → 𝕂^{TT} such that w = p ∘ i. In particular, the motivic space 𝒦 is cofibrant and fibrant in M_{•}^{cm}(S).

Proof. This follows from factoring w as a cofibration followed by a trivial fibration in the model category M_{•}^{cm}(S).

Choose and fix i and p as required by the Lemma. The morphisms [w], [i] and [p] in H_{•}^{cm}(S) are isomorphisms. The structure (7) of a commutative monoid on 𝕂^{TT} in H_{•}(S) induces via the isomorphism [w] (resp. [p]) the structure of a commutative monoid (ℤ × Gr, μ^{Gr}, e^{Gr}) (resp. (𝒦, μ̄, ē)) on the motivic space ℤ × Gr (resp. 𝒦) in H_{•}^{cm}(S) such that [i] and [p] are monoid isomorphisms. The space 𝒦 is closed motivic fibrant and closed cofibrant, thus the morphisms μ̄ and ē in H_{•}^{cm}(S) can be represented by certain morphisms in M_{•}(S) by Theorem A.2.1. We will do this as follows.

Let μ: (ℤ × Gr) ∧ (ℤ × Gr) → 𝒦 be a morphism in M_{•}(S) representing the morphism [i∘μ^{Gr}] in H_{•}^{cm}(S). There exists such a μ since the space (ℤ×Gr)∧(ℤ×Gr) is cofibrant and the space 𝒦 is fibrant in M_{•}^{cm}(S). There exists a unique (up to a homotopy) morphism μ^{𝒦} : 𝒦 ∧ 𝒦 → 𝒦 in M_{•}(S) such that [μ^{𝒦} ∘ (i ∧ i)] = [μ] in H_{•}^{cm}(S). Here one uses that the smash product of cofibrant objects in M_{•}^{cm}(S), and in particular 𝒦 ∧ 𝒦, is cofibrant by A.3.9. Define e^{𝒦} in M_{•}(S) as the composition S^{0} ⟶^{e_{⊗}} ℤ × Gr ⟶^{i} 𝒦. The maps μ^{𝒦} and e^{𝒦} represent the classes μ̄ and ē respectively. So

(𝒦, [μ^{𝒦} ], [e^{𝒦} ])  (8)

is a commutative monoid in H_{•}^{cm}(S) which coincides with (𝒦, μ̄, ē).

Remark 1.2.4. The functors Hom_{H_{•}^{cm}(S)}(−, 𝒦) and Hom_{H_{•}^{cm}(S)}(−, 𝕂^{TT}) are isomorphic via the isomorphism sending g to [p]∘g. Under this identification the class [i] of i represents the element τ_{∞} − ∞ ∈ K_{0}^{TT}(ℤ × Gr) since w = p ∘ i and [w] ∈ [ℤ × Gr, 𝕂^{TT}] represents τ_{∞} − ∞.

The composition of the inclusion P^{1} = Gr(1, 2) ↪ Gr and the map i is denoted b: P^{1} → 𝒦. This map represents the class [𝒪(−1)]−[𝒪] in the kernel of the homomorphism ∞^{∗}: K^{0}(P^{1}) → K^{0}(k) induced by ∞: Spec(k) → P^{1}. Consider the map of pointed motivic spaces given by the composition

ϵ : 𝒦 ∧ P^{1} ⟶^{id∧b} 𝒦 ∧ 𝒦 ⟶^{μ^{𝒦}} 𝒦.  (9)

Definition 1.2.5. Define BGL as the P^{1}-spectrum of the form (𝒦_{0}, 𝒦_{1}, 𝒦_{2}, …) with 𝒦_{i} = 𝒦 for all i and with the structure maps e_{i}: 𝒦_{i} ∧ P^{1} → 𝒦_{i+1} equal to the map ϵ: 𝒦 ∧ P^{1} → 𝒦.

It will be proved in Section 1.5 that in the case S = Spec(ℤ) this spectrum is essentially unique. In the next section, we will construct a monoidal structure on BGL regarded as an object in the stable homotopy category SH(S). In the case of S = Spec(ℤ) such a monoidal structure is unique. Pulling it back via the structural morphism S′ → Spec(ℤ) we get a monoidal structure on BGL in SH(S′) for an arbitrary Noetherian finite-dimensional base scheme S′.

To complete this section we prove that BGL is an Ω_{P^{1}}-spectrum which represents the Thomason-Trobaugh K-theory on 𝒮m/S. For X ∈ 𝒮m/S we abbreviate BGL(X_{+}) as BGL(X), which forces us to write BGL(X, x_{0}) for a pointed S-scheme (X, x_{0}).

Lemma 1.2.6. The spectrum BGL is an Ω_{P^{1}}-spectrum.

Proof. For Y ∈ 𝒮m/S and a closed subscheme Z ↪ Y write K_{Z}^{n}(Y) for the n-th Thomason-Trobaugh K-group of perfect complexes on Y with support on Z. It may be obtained as the n-th homotopy group of the homotopy fiber of the map 𝕂^{TT}(Y) → 𝕂^{TT}(X ∖ Z). For each smooth X over S the map

K^{n}(X) = [S^{n,0} ∧ X_{+}, 𝒦_{i}] → [S^{n,0} ∧ X_{+} ∧ P^{1}, 𝒦_{i} ∧ P^{1}]
→ [S^{n,0} ∧ X_{+} ∧ P^{1}, 𝒦_{i+1}] ≅ K_{X×{∞}}^{n}(X × P^{1})

induced by the structure map e_{i} coincides with the multiplication by the class [𝒪(−1)]−[𝒪] in K_{{∞}}^{0}(P^{1}). This multiplication is known to be an isomorphism for the Thomason-Trobaugh K-groups, by the projective bundle theorem [TT, Thm. 4.1] for X×P^{1}. Whence the Lemma.

Corollary 1.2.7. For each pointed motivic space A over S the adjunction map

Hom_{H_{•}(S)}(A, 𝒦_{0}) → Hom_{SH(S)}(Σ_{P^{1}}^{∞}A, BGL)

is an isomorphism. In particular, for every smooth scheme X over S and each closed subscheme Z in X one has K_{Z,p}^{TT}(X) = BGL^{−p,0}(X/(X ∖ Z)). The family of these isomorphisms form an isomorphism Ad : K_{∗}^{TT} → BGL^{−∗,0} of cohomology theories on the category 𝒮m𝒪p/S in the sense of [PS1]. Moreover the adjunction map [A, 𝒦_{i}] → [Σ_{P^{1}}^{∞}(A)(−i), BGL] is an isomorphism. In particular, for every smooth scheme X over S and each closed subscheme Z in X one has K_{Z,p}^{TT}(X) = BGL^{2i−p,i}(X/(X − Z)).

The family of pairings 𝒦_{i} ∧ 𝒦_{j} ⟶^{m_{ij}} 𝒦_{i+j} with m_{ij} = μ^{𝒦} defines a family of pairings

∪ : BGL^{p,i}(A) ⊗ BGL^{q,j}(B) → BGL^{p+i,q+j}(A ∧ B)  (10)

for pointed motivic spaces A and B. We will refer to the latter as the naive product structure on the functor BGL^{∗,∗} on the category M_{∗}(S). With this naive product in hand one has the following

Corollary 1.2.8. The isomorphism Ad: K_{∗}^{TT} → BGL^{−∗,0} of cohomology theories on 𝒮m𝒪p/S is an isomorphism of ring cohomology theories in the sense of [PS1].

1.3 The Bott element

The aim of this Section is to construct an element β ∈ BGL^{2,1}(Spec(k)), to show that it is invertible and to check that for any pointed motivic space A one has

BGL^{∗,0}(A)[β, β^{−1}] = BGL^{∗,∗}(A)

(the Laurent polynomials over BGL^{∗,0}(A)). We will use the naive product structure on BGL described just above Corollary 1.2.8.

Definition 1.3.1. Set β: = [S^{0} ⟶^{e_{𝒦}} 𝒦 = 𝒦_{1}] ∈ BGL^{2,1}(Spec(k)), where e_{𝒦} is the unit of the monoid 𝒦 (see (8)).

Lemma 1.3.2. Let b: P^{1} ↪ 𝒦 be the map described just above (9). It represents the element [𝒪(−1)] − [𝒪] in BGL^{0,0}(P^{1}, ∞) = Ker(∞^{∗}: K^{0}(P^{1}) → K^{0}(k)). There is a relation

β ∪ ([𝒪(−1)] − [𝒪]) = Σ_{P^{1}}(1) ∈ BGL^{2,1}(P^{1}, ∞),  (11)

where Σ_{P^{1}} is the suspension isomorphism and 1 ∈ BGL^{0,0}(Spec(k)) is the unit. There is another relation

β ∪ ([𝒪(1)] − [𝒪]) = −Σ_{P^{1}}(1) ∈ BGL^{2,1}(P^{1}, ∞).  (12)

Proof. The element Σ_{P^{1}}(1) is represented by the map

S^{0} ∧ P^{1} ⟶^{e_{𝒦} ∧id} 𝒦_{0} ∧ P^{1} ⟶^{id∧b} 𝒦_{0} ∧ 𝒦_{1} ⟶^{m_{01}} 𝒦_{1},

where m_{01} is from Theorem 2.2.1. The element β ∪ ([𝒪(−1)] − [𝒪]) is represented by the map

S^{0} ∧ P^{1} ⟶^{e_{𝒦} ∧b} 𝒦_{1} ∧ 𝒦_{0} ⟶^{m_{01}} 𝒦_{1}.

Since 𝒦_{0} = 𝒦_{1} = 𝒦 one has (id ∧ b) ∘ (e_{𝒦} ∧ id) = e_{𝒦} ∧ b. This implies the relation (11). Relation (12) follows from the first one since [𝒪(1)]−[𝒪] = −[𝒪(−1)]+[𝒪] in K^{0}(P^{1}).

Lemma 1.3.3. Let u ∈ BGL^{−2,−1}(Spec(k)) be the unique element such that Σ_{P^{1}}(u) = [𝒪(1)] − [𝒪] in BGL^{0,0}(P^{1}, ∞). Then β ∪ u = 1.

Proof. Consider the commutative diagram

\xymatrix{BGL^{2,1}(Spec(k)) ⊗ BGL^{0,0}(P^{1}, ∞) \ar[r]^{∪} & BGL^{2,1}(P^{1}, ∞) \\ BGL^{2,1}(Spec(k)) ⊗ BGL^{−2,−1}(Spec(k)) \ar[r]^{∪} \ar[u]^{id⊗Σ_{P^{1}}} & BGL^{0,0}(Spec(k)). \ar[u]_{Σ_{P^{1}}}}

Now the Lemma follows from the relation (12).

Definition 1.3.4. For P^{1}-spectra E and F set E^{alg}(F) = ⊕_{−∞}^{+∞}E^{2i,i}(F).

Proposition 1.3.5. For every pointed motivic space A the map

BGL^{∗,0}(A) ⊗ BGL^{alg}(Spec(k)) → BGL^{∗,∗}(A)  (13)

given by a ⊗ b ↦ a ∪ b is a ring isomorphism and BGL^{alg}(Spec(k)) = ℤ[β, β^{−1}] is the Laurent polynomial ring. One can rewrite this ring isomorphism as

BGL^{∗,0}(A)[β, β^{−1}] = BGL^{∗,∗}(A)  (14)

Proof. In fact, BGL^{∗,0}(A) ⟶^{∪β} BGL^{∗+2,1}(A) is an isomorphism since β is invertible. Since BGL^{0,0}(Spec(k)) = K^{0}(Spec(k)) = ℤ the map (13) is a ring isomorphism.

Using the isomorphism Ad : K_{∗}^{TT} → BGL^{−∗,0} of ring cohomology theories from Corollary 1.2.8 we get the following statement.

Corollary 1.3.6. For every X ∈ 𝒮m/S and every closed subscheme Z ↪ X one has

K_{Z,−∗}^{TT}(X)[β, β^{−1}] = BGL_{Z}^{∗,∗}(X).  (15)

The family of these isomorphisms form an isomorphism of the ring cohomology theories on 𝒮m𝒪p in the sense of [PS1]. As well, there is an isomorphism

K_{Z,−∗}^{TT}(X) = BGL_{Z}^{∗,∗}(X)/(β + 1)BGL_{Z}^{∗,∗}(X).  (16)

The family of these isomorphisms form an isomorphism of the ring cohomology theories on 𝒮m𝒪p in the same sense.

Remark 1.3.7. The identification (18) of BGL^{−1,0}(X) with BGL^{2i−1,i}(X) coincides with

the Bott periodicity isomorphism BGL^{−1,0}(X) ⟶^{∪β^{i}} BGL^{2i−1,i}(X).

1.4 BGL^{∗,∗} as an oriented ring cohomology theory

The restriction of the functor BGL^{∗,∗} to the category 𝒮m𝒪p is a ring cohomology theory. In this section we promote it to an oriented ring cohomology theory in the sense of [PS1]. For that it suffices to construct a Chern structure on BGL^{∗,∗}|_{𝒮m𝒪p} in the sense of [PS1]. Let P^{∞} be the motivic space pointed by ∞ ∈ P^{1} ↪ P^{∞}. A functor isomorphism Hom_{H_{•}(S)}(−, P^{∞}) ≅ Pic(−) on the category 𝒮m/S is constructed in [MV, Prop. 4.3.8]. That functor isomorphism takes the homotopy class of the canonical map P_{+}^{∞} → (P^{∞}, ∞) to the class of the tautological line bundle 𝒪(−1) over P^{∞}. Consider the element

c = (−β) ∪ ([𝒪] − [𝒪(1)]) ∈ BGL^{2,1}(P^{∞}, ∞).

For a line bundle L over X ∈ 𝒮m/S set c(L) = f_{L}^{∗}(c) ∈ BGL^{2,1}(X), where f_{L} ∈ Hom_{H_{•}(S)}(X_{+}, P^{∞}) is the morphism corresponding to the class [L] of L in the group Pic(X). Clearly, c(𝒪(−1)) = c. The assignment L/X ↦ c(L) is a Chern structure on BGL^{∗,∗}|_{𝒮m𝒪p} since c|_{P^{1}} = −Σ_{P^{1}}(1) ∈ BGL^{2,1}(P^{1}) by (12). With that Chern structure, BGL^{∗,∗}|_{𝒮m𝒪p} is an oriented ring cohomology theory in the sense of [PS1].

If the Thomason-Trobaugh K-theory is equipped with the Chern structure given by L/X ↦ [𝒪] − [L^{∨}] ∈ K_{0}^{TT}(X), then the isomorphism (16) of the ring cohomology theories respects the orientations.

1.5 Uniqueness of BGL

We prove in this Section that over S = Spec(ℤ) the spectrum BGL is essentially unique regarded as an object in the stable homotopy category SH(S). This has also been obtained in [R]. By [MV, Thm. 4.3.13] and Corollary B.1.2 there exists a unique morphism

ϵ: (ℤ × Gr) ∧ P^{1} → ℤ × Gr

in H_{•}(S) such that the composite morphism (ℤ × Gr) × P^{1} → (ℤ × Gr) ∧ P^{1} ⟶^{ϵ} ℤ × Gr represents the element (τ_{∞} − ∞) ⊗ ([𝒪(−1)] − [𝒪]) ∈ K^{0}((ℤ × Gr) × P^{1}). Let

ℤ × Gr ⟶^{i} 𝒦

be a motivic weak equivalence and let e: 𝒦 ∧ P^{1} → 𝒦 be a morphism in M_{•}(S) such that the diagram

\xymatrix{(ℤ × Gr) ∧ P^{1} \ar[r]^{ϵ} \ar[d]_{i∧id} & ℤ × Gr \ar[d]^{i} \\ 𝒦 ∧ P^{1} \ar[r]^{e} & 𝒦}

commutes in H_{•}^{cm}(S). Let i′: ℤ × Gr → 𝒦′ be another motivic weak equivalence and let e′: 𝒦′ ∧ P^{1} → 𝒦′ be a morphism in M_{•}(S) such that e′ ∘ (i′ ∧ id) = i′ ∘ ϵ in H_{•}^{cm}(S).

Definition 1.5.1. Define BGL as the P^{1}-spectrum of the form (𝒦_{0}, 𝒦_{1}, 𝒦_{2}, …) with 𝒦_{i} = 𝒦 for all i, and with the structure maps e_{i}: 𝒦_{i} ∧ P^{1} → 𝒦_{i+1} equal to the map ϵ: 𝒦 ∧ P^{1} → 𝒦. Define BGL′ in the same way using 𝒦′ and e′.

Proposition 1.5.2. There exists a unique morphism θ: BGL → BGL′ in SH(S) such that for every integer i ≥ 0 the diagram

\xymatrix{Σ_{P^{1}}^{∞}𝒦_{i}(−i) \ar[r]^{u_{i}} \ar[d]_{Σ_{P^{1}}^{∞}ϕ_{i}(−i)} & BGL \ar[d]^{θ} \\ Σ_{P^{1}}^{∞}𝒦′_{i}(−i) \ar[r]^{u′_{i}} & _{BGL_{′}}}

commutes, where ϕ_{i} = i′ ∘ i^{−1} ∈ [𝒦_{i}, 𝒦′_{i}]_{H_{•}(S)} and u_{i}, u′_{i} are the canonical morphisms. Similarly, there exists a unique morphism θ′: BGL′ → BGL in SH(S) such that for every integer i ≥ 0 the diagram

\xymatrix{Σ_{P^{1}}^{∞}𝒦′_{i}(−i) \ar[r]^{u′_{i}} \ar[d]_{Σ_{P^{1}}^{∞}ϕ′_{i}(−i)} & BGL′ \ar[d]^{θ′} \\ Σ_{P^{1}}^{∞}𝒦_{i}(−i) \ar[r]^{u_{i}} & BGL}

commutes, where θ_{i} = i ∘ (i′)^{−1} ∈ [𝒦′_{i}, 𝒦_{i}]_{H_{•}(S)}.

Proof. Consider the exact sequence

0 → \underleftarrow{lim}^{1}BGL^{2i−1,i}(𝒦′_{i}) → BGL^{0,0}(BGL′) → \underleftarrow{lim}BGL^{2i,i}(𝒦′_{i}) → 0

from Lemma A.5.10. The family of elements (u_{i} ∘ Σ_{P^{1}}^{∞}θ′_{i}(−i)) is an element of the group \underleftarrow{lim}BGL^{2i,i}(𝒦′_{i}). Thus there exists the required morphism θ′. To prove its uniqueness, observe that the \underleftarrow{lim}^{1}-group vanishes by Proposition 1.7.1. Whence BGL^{0,0}(BGL′) = \underleftarrow{lim}BGL^{2i,i}(𝒦′_{i}) and θ′ is indeed unique. By symmetry there also exists a unique morphism θ with the required property.

Proposition 1.5.3. The morphism θ: BGL → BGL′ is the inverse of θ′: BGL′ → BGL in SH(S), and in particular an isomorphism.

Proof. The composite morphism θ′ ∘ θ: BGL → BGL has the property that for every integer i ≥ 0 the diagram

\xymatrix{Σ_{P^{1}}^{∞}𝒦_{i}(−i) \ar[r]^{u_{i}} \ar[d]_{id} & BGL \ar[d]^{θ′∘θ} \\ Σ_{P^{1}}^{∞}𝒦_{i}(−i) \ar[r]^{u_{i}} & _{BGL_{′}}}

commutes. However, the identity morphism id: BGL → BGL has the same property. Thus θ′ ∘ θ = id. by the uniqueness in Proposition 1.5.2, and similarly θ ∘ θ′ = id.

Remark 1.5.4. The isomorphisms θ and θ′ are monoid isomorphisms provided that BGL and BGL′ are equipped with the monoidal structures given by Theorem 2.2.1.

1.6 Preliminary computations I

In this section we prepare for the next section, in which we show that certain \underleftarrow{lim}^{1}-groups vanish. Let BGL be the P^{1}-spectrum defined in 1.2.5. We will identify in this section the functors BGL^{0,0} and BGL^{2i,i} on the category H_{•}(S) as follows:

BGL^{0,0}(A) ≅ Hom_{H_{•}(S)}(A, 𝒦_{0}) = Hom_{H_{•}(S)}(A, 𝒦_{0}) ≅ BGL^{2i,i}(A).  (17)

Similarly,

BGL^{−1,0}(A) ≅ Hom_{H_{•}(S)}(S^{1,0} ∧ A, 𝒦_{0}) = Hom_{H_{•}(S)}(S^{1,0} ∧ A, 𝒦_{i}) ≅ BGL^{2i−1,i}(A). (18)

These identifications respect the naive product structure (10) on the functor BGL^{∗,∗}. In particular, the following diagram commutes for every pointed motivic space A over S.

\xymatrix{BGL^{−1,0}(S) ⊗ BGL^{0,0}(A) \ar[r] \ar[d]_{≅} & BGL^{−1,0}(A) \ar[d]^{≅} \\ BGL^{−1,0}(S) ⊗ BGL^{2i,i}(A) \ar[r] & BGL^{2i−1,i}(A)}  (19)

Lemma 1.6.1. Let S be a regular scheme. For every integer i the map

BGL^{−1,0}(S) ⊗_{BGL^{0,0}(S)} BGL^{2i,i}(𝒦) → BGL^{2i−1,i}(𝒦)

induced by the naive product structure is an isomorphism. The same holds if 𝒦 ∧ P^{1} replaces 𝒦.

Proof. The commutativity of the diagram (19) shows that it suffices to consider the case i = 0. Furthermore we may replace the pointed motivic space 𝒦 with ℤ×Gr since the map i: ℤ × Gr → 𝒦 = 𝒦 is a weak equivalence. The functor isomorphism K_{∗}^{TT} → BGL^{−∗,0} is a ring cohomology isomorphism by Corollary 1.2.8. Thus it remains to check that the map

K_{1}^{TT}(S) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(Gr) → K_{1}^{TT}(Gr)

is an isomorphism. This is well-known. See Lemma B.1.4) for a proof. The assertion concerning 𝒦 ∧ P^{1} is proved similarly using Lemma B.1.3 instead.

To state the next lemma, consider the scheme morphism f: S = Spec(ℤ) → Spec(ℂ), the pull-back functor f^{∗}: SH(ℤ) → SH(ℂ) described in Proposition A.7.4, and the topological realization functor R_{ℂ}: SH(ℂ) → SH_{ℂP^{1}} described in Section A.7. Set r = R_{ℂ} ∘ f^{∗}: SH(S) → SH_{ℂP^{1}}. The functor r will be called for short the realization functor below in this Section.

Lemma 1.6.2. Let 𝔹U be the periodic complex K-theory ℂP^{1}-spectrum with terms ℤ×BU. There is a termwise zigzag weak equivalence 𝔹U ⟵ E ⟶ rBGL of ℂP^{1}-spectra.

Proof. This follows from A.7.3 and the fact that Grassmann varieties pull back.

Lemma 1.6.3. Let S = Spec(ℤ) and let r: SH(ℤ) → SH_{ℂP^{1}} be the topological realization functor. Then for every integer i the realization homomorphism BGL^{2i,i}(𝒦) → (rBGL)^{2i}(r𝒦) is an isomorphism.

Proof. Clearly it suffices to prove the case i = 0. We may replace the pointed motivic space 𝒦_{i} with ℤ × Gr as in the proof of Lemma 1.6.1. It remains to check that the topological realization homomorphism BGL^{0,0}(Gr) → (rBGL)^{0}(Gr) is an isomorphism.

Since Gr(n, 2n) is a smooth cellular S-scheme, Lemma 1.6.4 below implies that the map BGL^{0,0}(Gr(n, 2n)) → (rBGL)^{0}(Gr(n, 2n)) is an isomorphism for every n. To conclude the statement for Gr = ∪Gr(n, 2n), use the short exact sequence from Lemma A.5.10. In the resulting diagram

\xymatrix{0 \ar[r] & \underleftarrow{lim}^{1} BGL^{−1,0}(Gr(n, 2n)) \ar[r] \ar[d] & BGL^{0,0}(Gr) \ar[r] \ar[d] & \underleftarrow{lim} BGL^{0,0}(Gr(n, 2n)) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \underleftarrow{lim}^{1} rBGL^{−1,0}(rGr(n, 2n)) \ar[r] & rBGL^{0,0}(rGr) \ar[r] & \underleftarrow{lim} rBGL^{0,0}(rGr(n, 2n)) \ar[r] & 0}

the map on the right hand side is then an isomorphism. Furthermore one concludes from [Sw, Thm. 16.32] that \underleftarrow{lim}^{1} rBGL^{−1,0}(rGr(n, 2n)) = \underleftarrow{lim}^{1} K_{top}^{1}(rGr(n, 2n)) = 0. On the other hand \underleftarrow{lim}^{1} BGL^{−1,0}(Gr(n, 2n)) = 0, as we already mentioned in the proof of 1.2.1. The result follows.

Lemma 1.6.4. Let X ∈ 𝒮m/S, where S = Spec(ℤ), and let X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} = X be a filtration by closed subsets such that for every integer i ≥ 0 the S-scheme X_{i} − X_{i−1} is isomorphic to a disjoint union of several copies of the affine space A_{S}^{i}. The map BGL^{0,0}(X) → (rBGL)^{0}(rX) is an isomorphism.

Proof. Consider the class ℛ of P^{1}-spectra E such that BGL^{0,0}(E) → rBGL^{0}(rE) is an isomorphism. It contains S^{0,0} because in this case we obtain the isomorphism BGL^{0,0}(S^{0,0}) ≅ K^{0}(ℤ) ≅ ℤ ≅ K_{top}^{0}(S^{0} which identifies the class of an algebraic resp. complex topological vector bundle over Spec(ℤ) resp. • with its rank. The Bott periodicity isomorphisms for BGL and rBGL which are compatible by A.7.3 imply that S^{2m,m} ∈ ℛ for all m ∈ ℤ. Finally, if E → F → G → S^{1,0} ∧ E is a distinguished triangle in SH(S) such that E and G are in ℛ, then so is F.

For i ≥ 0 write U^{i}: = X ∖ X_{i}, so that U^{i} is an open subset of U^{i−1}. In particular we have U^{n} = ∅ and U^{−1} = X. The closed subscheme X_{i} ∖ X_{i−1} = X_{i} ∩ U^{i−1} ↪ U^{i−1} is isomorphic to a disjoint union m_{i} copies of affine spaces A^{i}, and is in particular smooth over S. Furthermore the normal bundle is trivial. The homotopy purity theorem [MV, 3.2.29] supplies a distinguished triangle

Σ_{P^{1}}^{∞}U_{+}^{i} → Σ_{P^{1}}^{∞}U_{+}^{i−1} → Σ_{P^{1}}^{∞}U^{i−1}/U^{i} ≅ ∨_{j=1}^{m_{i}}S^{2(n−i),(n−i)}

of P^{1}-spectra. Since ℛ contains Σ_{P^{1}}^{∞}U^{n} = • we obtain inductively that ℛ contains Σ_{P^{1}}^{∞}U^{−1} = Σ_{P^{1}}^{∞}X_{+}.

Lemma 1.6.5. Let 𝔹^{0}U be the sub-spectrum of 𝔹U with the n-th term equal to the connected component BU of the topological space ℤ × BU containing the basepoint •. The inclusion 𝔹^{0}U ⟶ 𝔹U is a weak equivalence of ℂP^{1}-spectra.

Proof. One has to check that the inclusion induces an isomorphism on stable homotopy groups. This follows because the structure map (ℤ × BU) ∧ ℂP^{1} → ℤ × BU factors over {0} × BU.

Lemma 1.6.6. There exists a sub-spectrum 𝔹^{f}U of the ℂP^{1}-spectrum 𝔹U with the n-th term Gr(b(n), 2b(n)) such that the inclusion 𝔹^{f}U ⟶ 𝔹U is a stable equivalence.

Proof. The sequence b(n) will be constructed such that b(n) ≥ 2n + 1. Set b(0) = 1. We may assume that the structure map e_{0} : BU ∧ ℂP^{1} → BU is cellular. Since rGr(b(0), 2b(0))∧ℂP^{1} is a finite cell complex, it lands in a Grassmannian rGr(b(1), 2b(1)) for some integer b(1) ≥ 2 · 1 + 1. Continuing this process produces the required sequence of b(n)’s. The inclusions induce an isomorphism colim_{n≥0} Gr(b(n), 2b(n)) ≅ Gr.

To observe that the inclusion j: 𝔹^{f}U ⟶ 𝔹U is then a stable equivalence, recall that the number of 2k-cells in Gr(n, m) is given by the number of partitions of k into at most n each of which is ≥ m − n. In particular, the 2k-skeleton of BU coincides with the 2k-skeleton of rGr(k, 2k). To prove the surjectivity of π_{i}(j) choose an element α ∈ π_{i}𝔹U. It is represented by a cellular map a: S^{i+2m} → BU for some m with i + 2m ≥ 0. We may choose m such that m ≥ i. Thus a lands in rGr(b(m), 2b(m)) and gives rise to an element in π_{i}𝔹^{f}U mapping to α. To prove the injectivity of π_{i}(j), choose an element α ∈ π_{i}𝔹^{f}U such that π_{i}(j)(α) = 0. We may represent α by some map a: π_{i+2m}(j)Gr(b(m), 2b(m)) for some m with i + 2m > 0 and m ≥ i. The composition

S^{i+2m} → aGr(b(m), 2b(m)) ↪ BU

is nullhomotopic since π_{i+2m}BU ≅ π_{i}𝔹U via the homomorphism induced by the structure map. The nullhomotopy may be chosen to be cellular and thus lands in Gr(b(m), 2b(m)). This completes the proof.

1.7 Vanishing of certain groups I

Consider the stable equivalence hocolim_{i≥0} Σ_{P^{1}}^{∞}𝒦_{i}(−i) ≅ BGL (see (21)) and the respecting short exact sequence

0 → \underleftarrow{lim}^{1}BGL^{2i−1,i}(𝒦_{i}) → BGL^{0,0}(BGL) → \underleftarrow{lim} BGL^{2i,i}(𝒦_{i}) → 0

We prove in this section the following result

Proposition 1.7.1. Let S = Spec(ℤ), then \underleftarrow{lim}^{1}BGL^{2i−1,i}(𝒦_{i}) = 0.

Proof. The connecting homomorphism in the tower of groups for the \underleftarrow{lim}^{1}-term is the composite map

BGL^{2i−1,i}(𝒦_{i}) ⟵^{Σ_{P^{1}}^{−1}} BGL^{2i+1,i+1}(𝒦_{i} ∧ P^{1}) ⟵^{e_{i}^{∗}} BGL^{2i+1,i+1}(𝒦_{i+1})

where Σ_{P^{1}}^{−1} is the inverse to the P^{1}-suspension isomorphism and e_{i}^{∗} is the pull-back induced by the structure map e_{i}. Set A = BGL^{−1,0}(S) and consider the diagram

\xymatrix{BGL^{2i−1,i}(𝒦_{i}) & BGL^{2i+1,i+1}(𝒦_{i} ∧ P^{1}) \ar[l]_{Σ_{P^{1}}^{−1}} & BGL^{2i+1,i+1}(𝒦_{i+1}) \ar[l]_{e_{i}^{∗}} \\ A ⊗ BGL^{2i,i}(𝒦_{i}) \ar[u] & A ⊗ BGL^{2(i+1),i+1}(𝒦_{i} ∧ P^{1}) \ar[l]_{id⊗Σ_{P^{1}}^{−1}} \ar[u] & A ⊗ BGL^{2(i+1),i+1}(𝒦_{i+1}) \ar[u] \ar[l]_{id⊗e_{i}^{∗}}}

where the vertical arrows are induced by the naive product structure on the functor BGL^{∗,∗}. Clearly it commutes. Since S is regular, the vertical arrows are isomorphisms by Lemma 1.6.1. It follows that \underleftarrow{lim}^{1}BGL^{2i−1,i}(𝒦_{i}) = \underleftarrow{lim}^{1}(A⊗BGL^{2i,i}(𝒦_{i})) where in the last tower of groups the connecting maps are id ⊗ (Σ_{P^{1}}^{−1} ∘ e_{i}^{∗}). It remains to prove the following assertion.

Claim. \underleftarrow{lim}^{1}(A ⊗ BGL^{2i,i}(𝒦_{i})) = 0 Since S = Spec(ℤ) one gets A = BGL^{−1,0}(S) = K_{1}^{TT}(ℤ) = ℤ/2ℤ. Thus A⊗BGL^{2i,i}(𝒦_{i}) = BGL^{2i,i}(𝒦_{i})/mBGL^{2i,i}(𝒦_{i}) with m = 2 and the connecting maps in the tower are just the mod-m reduction of the maps Σ_{P^{1}}^{−1} ∘ e_{i}^{∗}. Now a chain of isomorphisms completes the proof of the Claim.

\underleftarrow{lim}^{1}BGL^{2i,i}(𝒦_{i})/m ≅ \underleftarrow{lim}^{1}(rBGL)^{2i}(r𝒦_{i})/m ≅ \underleftarrow{lim}^{1}𝔹U^{2i}(r𝒦_{i})/m
≅ \underleftarrow{lim}^{1}𝔹U^{2i}(ℤ × BU)/m ≅ \underleftarrow{lim}^{1}𝔹U^{2i}(ℤ × BU; ℤ/m)
≅ K_{top}^{1}(𝔹U; ℤ/m) ≅ K_{top}^{1}(𝔹^{0}U; ℤ/m)
≅ \underleftarrow{lim}^{1}K_{top}^{2i}(Gr(b(i), 2b(i)); ℤ/m) = 0

The first isomorphism follows from Lemma 1.6.3. The second isomorphism is induced by the levelwise weak equivalence 𝔹U ≃ rBGL mentioned in Lemma 1.6.2. The third isomorphism is induced by the image of the weak equivalence 𝒦_{i} ≃ ℤ × Gr under topological realization. The forth and fifth isomorphism hold since 𝔹U^{2i+1}(ℤ × BU) = 0. The sixth isomorphism is induced by the stable equivalence 𝔹^{0}U ≃ 𝔹U from Lemma 1.6.5, the seventh one is induced by the stable equivalence 𝔹^{f}U ≃ 𝔹^{0}U from Lemma 1.6.6. The last one holds since all groups in the tower are finite.

2 Smash-product, pull-backs, topological realization

In this section we construct a smash-product ∧ of P^{1}-spectra, check its basic properties, consider its behavior with respect to pull-back and realization functors. We follow here an idea of Voevodsky [V1, Comments to Thm. 5.6] and use results of Jardine [J].

2.1 The smash product

Definition 2.1.1. Let V : SH(S) → SH^{Σ}(S) and U: SH^{Σ}(S) → SH(S) be the equivalence described in Theorem A.6.4. For a pair of P^{1}-spectra E and F set

E ∧ F := U(V E ∧ V F)

as in Remark A.6.5.

Proposition 2.1.2. Let S be a Noetherian finite-dimensional base scheme. The smashproduct of P^{1}-spectra over S induces a closed symmetric monoidal structure (∧, 𝕀) on the motivic stable homotopy category SH(S) having the properties required by Theorem 5.6 of Voevodsky congress talk [V1]:

1. There is a canonical isomorphism E ∧ Σ_{P^{1}}^{∞}A ≅ (E_{i} ∧ A, e_{i} ∧ id) for every pointed motivic space A and every P^{1}-spectrum.

2. There is a canonical isomorphism (⊕E_{α}) ∧ F ≅ ⊕(E_{α} ∧ F) for P^{1}-spectra E_{i}, F.

3. Smashing with a P^{1}-spectrum preserves distinguished triangles. To be more precise, if E ⟶^{f} F → cone(f) ⟶^{ϵ} E[1] is a distinguished triangle and G is a P^{1}-spectrum, the sequence E ∧ G ⟶^{f} F ∧ G → cone(f) ∧ G ⟶^{ϵ} E ∧ G[1] is a distinguished triangle, where the last morphism is the composition of ϵ∧id_{G} with the canonical isomorphism E[1] ∧ G → (E ∧ F)[1].

Proof. Follows from Remark A.6.5 and Theorem A.6.4.

Lemma 2.1.3. Let E = hocolim_{i≥0}E_{i} is a sequential homotopy colimit of P^{1}-spectra. For every P^{1}-spectrum F there is an exact sequence of abelian groups

0 → \underleftarrow{lim}^{1}F^{p−1,q}(E_{i}) → F^{p,q}(E) → \underleftarrow{lim} F^{p,q}(E_{i}) → 0.  (20)

Proof. This is Lemma A.5.10.

By Lemma A.5.9, any P^{1}-spectrum E can be expressed as the homotopy colimit

hocolim Σ_{P^{1}}^{∞}E_{i}(−i) ≅ E.  (21)

Corollary 2.1.4. For two P^{1}-spectra E and F there is a canonical short exact sequence

0 → \underleftarrow{lim}^{1}F^{p+2i−1,q+i}(E_{i}) → F^{p,q}(E) → \underleftarrow{lim} F^{p+2i,q+i}(E_{i}) → 0.  (22)

Corollary 2.1.5. For a pair of spectra E and F and each spectrum G one has a canonical exact sequence of the form

0 → \underleftarrow{lim}^{1}G^{p+4i−1,q+2i}(E_{i} ∧ F_{i}) → G^{p,q}(E ∧ F) → \underleftarrow{lim} G^{p+4i,q+2i}(E_{i} ∧ E_{i}) → 0. (23)

Proof. For a pair of spectra E and F one has a canonical isomorphism of the form

hocolim (Σ_{P^{1}}^{∞}(E_{i} ∧ F_{i})(−2i)) ≅ E ∧ F  (24)

as deduced in Lemma A.6.8. The result follows from Corollary 2.1.4.

2.2 A monoidal structure on BGL

For a P^{1}-spectrum E and an integer i ≥ 0 u_{i}: Σ_{P^{1}}^{∞}E_{i}(−i) → E denotes the canonical map. The aim of this section is to prove the following

Theorem 2.2.1. Consider the family of pairings 𝒦_{i} ∧ 𝒦_{j} ⟶^{m_{ij}} 𝒦_{i+j} with m_{ij} = μ^{𝒦} . For S = Spec(ℤ) there is a unique morphism μ_{BGL} : BGL∧BGL ⟶ BGL in the motivic stable homotopy category SH(S) such that for every i the diagram

\xymatrix{Σ_{P^{1}}^{∞}𝒦_{i}(−i) ∧ Σ_{P^{1}}^{∞}𝒦_{i}(−i) \ar[r]^{Σ^{∞}(m_{ii})} \ar[d]_{u_{i}∧u_{i}} & Σ_{P^{1}}^{∞}𝒦_{2i}(−2i) \ar[d]^{u_{2i}} \\ BGL ∧ BGL \ar[r]^{μ_{BGL}} & BGL}

commutes. Let e_{BGL} : 𝕀 → BGL in SH(S) be adjoint to the unit e_{𝒦} : S^{0,0} → 𝒦. Then

(BGL, μ_{BGL}, e_{BGL})

is a commutative monoid in SH(S).

Proof. The morphism μ_{BGL} we are looking for is an element of the group BGL^{0,0}(BGL ∧ BGL). This group fits in the exact sequence

0 → \underleftarrow{lim}^{1}BGL^{4i−1,2i}(𝒦_{i} ∧ 𝒦_{i}) → BGL^{0,0}(BGL ∧ BGL) → \underleftarrow{lim} BGL^{4i,2i}(𝒦_{i} ∧ 𝒦_{i}) → 0

by Corollary 2.1.5. The family of elements \{u_{2i} ∘Σ^{∞}(m_{ii})\} is an element of the \underleftarrow{lim} group. The \underleftarrow{lim}^{1} group vanishes by Proposition 2.4.1 below, whence there exist a unique element μ_{BGL} whose image in the \underleftarrow{lim} group coincides with the element \{u_{2i} ∘ Σ^{∞}(m_{ii})\}. That morphism μ_{BGL} is the required one.

In fact, the identities u_{2i} ∘ Σ^{∞}(m_{ii}) = μ_{BGL} ∘ (u_{i} ∧ u_{i}) hold by the very construction of μ_{BGL}. The operation μ_{BGL} is associative because the group \underleftarrow{lim}^{1}BGL^{8i−1,4i}(𝒦_{i} ∧ 𝒦_{i} ∧ 𝒦_{i}) vanishes by Proposition 2.5.1). That μ_{BGL} is commutative follows from the vanishing of the group \underleftarrow{lim}^{1}BGL^{4i−1,2i}(𝒦_{i} ∧ 𝒦_{i}) (see Proposition 2.4.1). The fact that e_{BGL} is a twosided unit for the multiplication μ_{BGL} follows by Proposition 1.7.1, which shows that the group \underleftarrow{lim}^{1}BGL^{2i−1,i}(𝒦_{i}) vanishes.

Definition 2.2.2. Let S be a regular scheme, with structural morphism f: S → Spec(ℤ). Let f^{∗}: SH(ℤ) → SH(S) be the strict symmetric monoidal pull-back functor from A.7.4. Set

μ_{BGL}^{S}: = f^{∗}(BGL) ∧ f^{∗}(BGL) ⟶^{can} f^{∗}(BGL ∧ BGL) ⟶^{f^{∗}(μ_{BGL})} f^{∗}(BGL)
e_{BGL}^{S}: = S^{0} ⟶^{can} f^{∗}(S^{0}) ⟶^{f^{∗}(e_{BGL})} f^{∗}(BGL)

and BGL^{S} = f^{∗}(BGL). Then (BGL^{S}, μ_{BGL}^{S}, e_{BGL}^{S}) is a commutative monoid in SH(S).

We will sometimes refer to a monoid in SH(S) as a P^{1}-ring spectrum.

Corollary 2.2.3. The multiplicative structure on the functor BGL^{∗,∗} induced by the pairing μ_{BGL}^{S} and the unit e_{BGL}^{S} coincides with the naive product structure (10).

Proof. Follows from Theorem 2.2.1.

Corollary 2.2.4. The functor isomorphism [X, 𝒦_{0}] → [Σ_{P^{1}}^{∞}(X), BGL] respects the multiplicative structures on both sides. The isomorphism Ad: K_{∗}^{TT} → BGL^{−∗,0} of cohomology theories on 𝒮m𝒪p/S is an isomorphism of ring cohomology theories in the sense of [PS1].

Proof. Follows from Theorem 2.2.1.

2.3 Preliminary computations II

We will identify in this section the functors BGL^{0,0} and BGL^{2i,i}, BGL^{−1,0} and BGL^{2i−1,i} on the motivic unstable category H_{•}(S) as in Section 1.6.

Lemma 2.3.1. Let S be a regular base scheme. For every integer i the map

BGL^{−1,0}(S) ⊗_{BGL^{0,0}(S)} BGL^{2i,i}(𝒦_{i} ∧ 𝒦_{i}) → BGL^{2i−1,i}(𝒦_{i} ∧ 𝒦_{i})

induced by the naive product structure is an isomorphism. The same holds if we replace 𝒦_{i} ∧ 𝒦_{i} by 𝒦_{i} ∧ 𝒦_{i} ∧ P^{1}.

Proof. Since diagram (19) commutes, it suffices to consider the case i = 0. Furthermore we may replace the pointed motivic space 𝒦_{i} with ℤ×Gr since the map i: ℤ×Gr → 𝒦_{i} = 𝒦 is a motivic weak equivalence. The functor isomorphism 𝕂_{∗}^{TT} → BGL^{−∗,0} is an isomorphism of ring cohomology theories. Thus it remains to check that the map

K_{1}^{TT}(S) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(Gr ∧ Gr) → K_{1}^{TT}(Gr ∧ Gr)

is an isomorphism. This holds by Lemma B.1.3. The case of 𝒦_{i} ∧ 𝒦_{i} ∧ P^{1} is proved by the same arguments.

Lemma 2.3.2. Let S = Spec(ℤ) and let r: SH(S) → SH_{ℂP^{1}} be the topological realization functor. Then for every integer i the homomorphism BGL^{2i,i}(𝒦 ∧ 𝒦) → (rBGL)^{2i}(r(𝒦 ∧ 𝒦)) ≅ (rBGL)^{2i}(r𝒦 ∧ r𝒦) is an isomorphism.

Proof. Since the Bott periodicity isomorphisms in the algebraic and complex topological setting correspond by A.7.3, it suffices to consider the case i = 0. We may replace the pointed motivic space 𝒦 with ℤ × Gr as in the proof of Lemma 1.6.1. It remains to check that the realization homomorphism

BGL^{0,0}(Gr ∧ Gr) → (rBGL)^{0}(r(Gr ∧ Gr)) = (rBGL)^{0}(r(Gr) ∧ r(Gr)))

is an isomorphism. By Example A.5.8 Σ_{P^{1}}^{∞}Gr ∧ Gr is a retract of Σ_{P^{1}}^{∞}Gr × Gr in SH(S), whence it suffices to consider the topological realization homomorphism for Gr × Gr. Since the latter is an increasing union of the cellular S-schemes Gr(n, 2n) × Gr(m, 2m), the result follows with the help of Lemma 1.6.4 as in the proof of Lemma 1.6.3.

2.4 Vanishing of certain groups II

Consider the stable equivalence hocolim Σ_{P^{1}}^{∞}(𝒦∧𝒦)(−2i) ≅ BGL∧BGL displayed in (24) and the corresponding short exact sequence

0 → \underleftarrow{lim}^{1}BGL^{4i−1,2i}(𝒦 ∧ 𝒦) → BGL^{0,0}(BGL ∧ BGL) → \underleftarrow{lim} BGL^{4i,2i}(𝒦 ∧ 𝒦) → 0

from Corollary 2.1.5. We prove in this section the following result

Proposition 2.4.1. If S = Spec(ℤ) then \underleftarrow{lim}^{1}BGL^{4i−1,2i}(𝒦 ∧ 𝒦) = 0.

Proof. For a pointed motivic space A we abbreviate A ∧ A as A^{∧2}. The connecting homomorphism in the tower of groups forming the \underleftarrow{lim}^{1}-term is the composite map

BGL^{4i−1,2i}(𝒦∧𝒦) ⟵^{(Σ∘Σ)^{−1}∘tw} BGL^{4(i+1)−1,2(i+1)}((𝒦∧P^{1})^{∧2}) ⟵^{(ϵ∧ϵ)^{∗}} BGL^{4(i+1)−1,2(i+1)}(𝒦^{∧2})

where Σ is the P^{1}-suspension isomorphism, tw is induced by interchanging the two pointed motivic spaces in the middle of the four-fold smash product, and ϵ: 𝒦 ∧ P^{1} → 𝒦 is the structure map of BGL. Set A = BGL^{−1,0}(S) and write B for BGL. Consider the diagram

\xymatrix{B^{4i−1,2i}(𝒦^{∧2}) & B^{4i+3,2(i+1)}((𝒦 ∧ P^{1})^{∧2}) \ar[l]_{(Σ∘Σ)^{−1}∘tw} & B^{4i+3,2i+2}(𝒦^{∧2}) \ar[l]_{(ϵ∧ϵ)^{∗}} \\ A ⊗ B^{4i,2i}(𝒦^{∧2}) \ar[u] & A ⊗ B^{4i+4,2i+2}((𝒦 ∧ P^{1})^{∧2}) \ar[l]_{id⊗(Σ∘Σ)^{−1}∘tw} \ar[u] & A ⊗ B^{4i+4,2i+2}(𝒦^{∧2}) \ar[l]_{id⊗(ϵ∧ϵ)^{∗}} \ar[u]}

where the vertical arrows are induced by the naive product structure on the functor BGL^{∗,∗}. Clearly it commutes. The vertical arrows are isomorphisms by Lemma 1.6.1. It follows that \underleftarrow{lim}^{1}BGL^{4i−1,2i}(𝒦^{∧2}) = \underleftarrow{lim}^{1}(A ⊗ BGL^{4i,2i}(𝒦^{∧2})) where in the last tower of groups the connecting maps are id ⊗ ((Σ ∘ Σ)_{P^{1}}^{−1} ∘ tw) ∘ (ϵ ∧ ϵ)^{∗}). It remains to prove the following claim.

Claim. \underleftarrow{lim}^{1}(A ⊗ BGL^{4i,2i}(𝒦^{∧2})) = 0. Since A = BGL^{−1,0}(S) = K_{1}^{TT}(S) = ℤ/2ℤ, there is an isomorphism A ⊗ BGL^{4i,2i}(𝒦^{∧2}) = BGL^{4i,2i}(𝒦^{∧2})/mBGL^{4i,2i}(𝒦^{∧2}) with m = 2. The connecting map in the tower are just the mod-m reduction of the maps (Σ ∘ Σ)^{−1} ∘ tw ∘ (ϵ ∧ ϵ)^{∗}. Now a chain of isomorphisms completes the proof of the Claim.

\underleftarrow{lim}^{1}BGL^{4i,2i}(𝒦^{∧2})/m ≅ \underleftarrow{lim}^{1}(rBGL)^{4i}(r𝒦 ∧ r𝒦)/m ≅ \underleftarrow{lim}^{1}𝔹U^{4i}(r𝒦 ∧ r𝒦)/m
≅ \underleftarrow{lim}^{1}𝔹U^{4i}((ℤ × BU) ∧ (ℤ × BU))/m
≅ \underleftarrow{lim}^{1}𝔹U^{4i}((ℤ × BU) ∧ (ℤ × BU); ℤ/m)
≅ K_{top}^{1}(𝔹U ∧ 𝔹U; ℤ/m) ≅ K_{top}^{1}(𝔹^{0}U ∧ 𝔹^{0}U; ℤ/m)
≅ \underleftarrow{lim}^{1}K_{top}^{4i}(Gr(b(i), 2b(i)) ∧ Gr(b(i), 2b(i)); ℤ/m) = 0

The first isomorphism follows from Lemma 2.3.2. The second isomorphism is induced by the levelwise weak equivalence 𝔹U ≃ rBGL mentioned in Lemma 1.6.2. The third isomorphism is induced by the image of the weak equivalence 𝒦_{i} ≃ ℤ × Gr under topological realization. The forth and fifth isomorphism hold since 𝔹U^{4i+1}(ℤ × BU) = 0. The sixth isomorphism is induced by the stable equivalence 𝔹^{0}U ≃ 𝔹U from Lemma 1.6.5, the seventh one is induced by the stable equivalence 𝔹^{f}U ≃ 𝔹^{0}U from Lemma 1.6.6. The last one holds since all groups in the tower are finite.

2.5 Vanishing of certain groups III

Consider the stable equivalence

hocolim Σ_{P^{1}}^{∞}(𝒦 ∧ 𝒦 ∧ 𝒦)(−3i) ≅ BGL ∧ BGL ∧ BGL

from (24) and the induced short exact sequence

0 → \underleftarrow{lim}^{1}BGL^{8i−1,4i}(𝒦^{∧3}) → BGL^{0,0}(BGL ∧ BGL ∧ BGL) → \underleftarrow{lim} BGL^{8i,4i}(𝒦^{∧3}) → 0.

Proposition 2.5.1. If S = Spec(ℤ) then \underleftarrow{lim}^{1}BGL^{8i−1,4i}(𝒦 ∧ 𝒦 ∧ 𝒦) = 0.

Proof. This is proved in the same way as Proposition 2.4.1.

2.6 BGL as an oriented commutative P^{1}-ring spectrum

Following Adams and Morel, we define an orientation of a commutative P^{1}-ring spectrum. However we prefer to use a Thom class rather than a Chern class. Let P^{∞} = ⋃ P^{n} be the motivic space pointed by ∞ ∈ P^{1} ↪ P^{∞}. 𝒪(−1) be the tautological line bundle over P^{∞}.

It is also known as the Hopf bundle. If V → X is a vector bundle over X ∈ 𝒮m/S, with zero section z: X ↪ V, let Th_{X}(V) = V/(V ∖z(X)) be the Thom space of V, considered as a pointed motivic space over S. For example Th_{X}(A_{X}^{n}) ≃ S^{2n,n}. Define Th_{P^{∞}}(𝒪(−1)) as the obvious colimit of the Thom spaces Th_{P^{n}}(𝒪(−1)).

Definition 2.6.1. Let E be a commutative P^{1}-ring spectrum. An orientation of E is an element th ∈ E^{2,1}(Th_{P^{∞}}(𝒪(−1))) = E_{P^{∞}}^{2,1}(𝒪(−1)) such that its restriction to the Thom space of the fibre over the distinguished point coincides with the element Σ_{P^{1}}(1) ∈ E^{2,1}(Th(1)) = E^{2,1}(P^{1}, ∞).

Remark 2.6.2. Let th be an orientation of E. Set c := z^{∗}(th) ∈ E^{2,1}(P^{∞}). It is proved in [PY, Prop. 6.5.1] that c|_{P^{1}} = −Σ_{P^{1}}(1). The class th(𝒪(−1)) ∈ E_{P^{∞}}^{2,1}(𝒪(−1)) given by (26) coincides with the element th (see [PS1, Thm.3.5]). Thus another possible definition of an orientation of E is the following.

Definition 2.6.3. Let E be a commutative ring P^{1}-spectrum. An orientation of E is an element c ∈ E^{2,1}(P^{∞}) such that c|_{P^{1}} = −Σ_{P^{1}}(1) (of course the element c should be regarded as the first Chern class of the Hopf bundle 𝒪(−1) on P^{∞}).

Remark 2.6.4. Let c be an orientation of E. Consider th(𝒪(−1)) ∈ E_{P^{∞}}^{2,1}(𝒪(−1)) given by (26) and set th = th(𝒪(−1)). It is straightforward to check that th|_{Th(1)} = Σ_{P^{1}}(1). Thus th is an orientation of E. Clearly c = z^{∗}(th) ∈ E^{2,1}(P^{∞}), whence the two definitions of orientations of E are equivalent.

Example 2.6.5. Set c^{K} = (−β) ∪ ([𝒪] − [𝒪(1)]) ∈ BGL^{2,1}(P^{∞}). The relation (12) shows that c^{K} is an orientation of BGL. Consider th(𝒪(−1)) ∈ BGL_{P^{∞}}^{2,1}(𝒪(−1)) given by (26) and set th^{K} = th(𝒪(−1)). The class th^{K} is the same orientation of BGL.

The orientation of BGL described in Example 2.6.5 has the following property. The map (16)

BGL^{∗,∗} → K_{∗}^{TT}

which takes β to −1 is an oriented morphism of oriented cohomology theories, provided that K_{∗}^{TT} is oriented via the Chern structure L/X ↦ [𝒪] − [L^{−1}] ∈ K_{0}(X).

2.7 BGL^{∗,∗} as an oriented ring cohomology theory

An 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 𝒮m𝒪p 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]. The functor isomorphism Hom_{H_{•}(S)}(−, P^{∞}) → Pic(−) on the category 𝒮m/S provided by [MV, Thm. 4.3.8] takes the class of the canonical map P_{+}^{∞} → P^{∞} to the class of the tautological line bundle 𝒪(−1) over P^{∞}. Now for a line bundle L over X ∈ 𝒮m/S set c(L) = f_{L}^{∗}(c) ∈ E^{2,1}(X), where the morphism f_{L}: X_{+} → P^{∞} in H_{•}(S) corresponds to the class [L] of L in the group Pic(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 E^{∗,∗}|_{𝒮m𝒪p} 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). Now 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)).  (25)

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)),  (26)

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 2.7.1. For a rank r vector bundle p: V → X on X ∈ 𝒮m/S the map

− ∪ th(V): E^{∗,∗}(X) → E^{∗+2r,∗+r}(Th_{X}(V))

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)].

A Motivic homotopy theory

The aim of this section is to present details on the model structures we use to perform homotopical calculations. Our reference on model structures is [Ho]. For the convenience of the reader who is not familiar with model structures, we recall the basic features and purposes of the theory below, after discussing categorical prerequisites.

A.1 Categories of motivic spaces

Let S be a Noetherian separated scheme of finite Krull dimension (base scheme for short). The category of smooth quasi-projective S-schemes is denoted 𝒮m/S. A smooth morphism is always of finite type. In particular, 𝒮m/S is equivalent to a small category.

The category of compactly generated topological spaces is denoted Top, the category of simplicial sets is denotes sSet. The set of n-simplices in K is K_{n}.

Definition A.1.1. A motivic space over S is a functor A: 𝒮m/S^{op} → sSet. The category of motivic spaces over S is denoted M(S).

For X ∈ 𝒮m/S the motivic space sending Y ∈ 𝒮m/S to the discrete simplicial set Hom_{𝒮m/S}(Y, X) is denoted X as well. More generally, any scheme X over S defines a motivic space X over S. Any simplicial set K defines a constant motivic space K. A pointed motivic space is a pair (A, a_{0}), where a_{0} : S → A. Usually the basepoint will be omitted from the notation. The resulting category is denoted M_{•}(S).

Definition A.1.2. A morphism f: S → S′ of base schemes defines the functor

f_{∗}: M_{•}(S) → M_{•}(S′)

sending A to (Y → S′) ↦ A(S ×_{S′} Y). Left Kan extension produces a left adjoint f^{∗}: M_{•}(S′) → M_{•}(S) of f_{∗}.

If A is a motivic space, let A_{+} denote the pointed motivic space (A ∐ S, i), where i: S → A ∐ S is the canonical inclusion. The category M_{•}(S) is closed symmetric monoidal, with smash product A ∧ B defined by the sectionwise smash product

(A ∧ B)(X): = A(X) ∧ B(X)  (27)

and with internal hom \underline{Hom}_{M_{•}(S)}(A, B) defined by

\underline{Hom}_{M_{•}(S)}(A, B)(x: X → S)_{n}: = Hom_{M_{•}(S)}(A ∧ Δ_{+}^{n}, x_{∗}x^{∗}B).  (28)

In particular, M_{•}(S) is also enriched over the category of pointed simplicial sets, with enrichment sSet_{•}(A, B): = \underline{Hom}_{M_{•}(S)}(A, B)(S). The mapping cylinder of a map f: A → B is the pushout of the diagram

\xymatrix{A ∧ ∂Δ_{+}^{1} \ar[d]^{ρ} \ar[r]^{≅} & A ∐ A \ar[r]^{id_{A} ∐ f} & A ∐ B \ar[d] \\ A ∧ Δ_{+}^{1} \ar[rr] &  & Cyl(f)} .  (29)

The composition of the canonical maps A ↪ Cyl(f) → B is f.

The pushout product of two maps f: A → C and g: B → D of motivic spaces over S is the map f ⊔ g: A ∧ D ∪_{A∧B} C ∧ B → C ∧ D induced by the commutative diagram

\xymatrix{A ∧ B \ar[r] \ar[d] & A ∧ D \ar[d] \\ C ∧ B \ar[r] & C ∧ D.}  (30)

The functor f^{∗}: M_{•}(S′) → M_{•}(S) is strict symmetric monoidal, since so is the pullback functor sending X ∈ 𝒮m_{S′} to S ×_{S′} X ∈ 𝒮m/S. This ends the categorical considerations.

A.2 Model categories

The basic purpose of a model structure is to give a framework for the construction of a homotopy category. Suppose w𝒞 is a class of morphisms in a bicomplete 𝒞 one wants to be invertible. Call them weak equivalences. One can define the homotopy “category” of the pair (𝒞, w𝒞) to be the universal “functor” Γ: 𝒞 → Ho(𝒞, w𝒞) such that every weak equivalence is mapped to an isomorphism. In general, this homotopy “category” may not be a category: it has hom-classes, but not necessarily hom-sets. If one requires the existence of two auxiliary classes of morphisms f𝒞 (the fibrations) and c𝒞 (the cofibrations), together with certain compatibility axioms, one does get a homotopy category Ho(𝒞, w𝒞) and an explicit description of the hom-sets in it.

Theorem A.2.1 (Quillen). Let (w𝒞, f𝒞, c𝒞) be a model structure on a bicomplete category 𝒞. Then the homotopy category Γ: 𝒞 → Ho(𝒞, w𝒞) exists and is the identity on objects. The set of morphisms in Ho(𝒞, w𝒞) from ΓA to ΓB is the set of morphisms in 𝒞 from A to B modulo a homotopy equivalence relation, provided that ∅ → A is a cofibration and B → ∗ is a fibration.

Here ∅ is the initial object and ∗ is the terminal object in 𝒞. An object A resp. B as in Theorem A.2.1 is called cofibrant resp. fibrant. Every object ΓA in the homotopy category is isomorphic to an object ΓC, where C is both fibrant and cofibrant. A (co)fibration which is also a weak equivalence is usually called a trivial or acyclic (co)fibration.

To describe the standard way to construct model structures on a bicomplete category, one needs a definition.

Definition A.2.2. Let f: A → B and g: C → D be morphisms in 𝒞. If every commutative diagram

\xymatrix{A \ar[r] \ar[d]_{f} & C \ar[d]^{g} \\ B \ar[r] & D}

admits a morphism h: B → C such that the resulting diagram

\xymatrix{A \ar[r] \ar[d]_{f} & C \ar[d]^{g} \\ B \ar[r] \ar@{-->}[ur]^{h} & D}

commutes (a lift for short), then f has the left lifting property with respect to g, and g has the right lifting property with respect to f.

Here is the standard way of constructing a model structure on a given bicomplete category. Choose the class of weak equivalences such that it contains all identities, is closed under retracts and satisfies the two-out-of-three axiom. Pick a set I (the generating cofibrations) and define a cofibration to be a morphism which is a retract of a transfinite composition of cobase changes of morphisms in I. Pick a set J (the generating acyclic cofibrations) of weak equivalences which are also cofibrations and define the fibrations to be those morphisms which have the right lifting property with respect to every morphism in J. Some technical conditions have to be fulfilled in order to conclude that this indeed is a model structure, which is then called cofibrantly generated. See [Ho, Theorem 2.1.19].

Example A.2.3. In Top, let the weak equivalences be the weak homotopy equivalences, and set

I = {∂D^{n} ↪ D^{n}}_{n≥0} J = {D^{n} × {0} ↪ D^{n} × I}_{n≥0}.

Then the fibrations are precisely the Serre fibrations, and the cofibrations are retracts of generalized cell complexes (“generalized” refers to the fact that cells do not have to be attached in order of dimension). In sSet, let the weak equivalences be those maps which map to (weak) homotopy equivalences under geometric realization. Set

I = {∂Δ^{n} ↪ Δ^{n}}_{n≥0} J = {Λ_{j}^{n} ↪ Δ^{n}}_{n≥1,0≤j≤n}

where Λ_{j}^{n} is the sub-simplicial set of ∂Δ^{n} obtained by removing the j-th face. Then the fibrations are precisely the Kan fibrations, and the cofibrations are the inclusions.

Example A.2.4. For the purpose of this paper, model structures on presheaf categories Fun(𝒞^{op}, sSet) with values in simplicial sets are relevant. There is a canonical one, due to Quillen, which is usually referred to as the projective model structure. It has as weak equivalences those morphisms f: A → B such that f(c): A(c) → B(c) is a weak equivalence for every c ∈ Ob𝒞 (the objectwise or sectionwise weak equivalences). Set

I = {Hom_{𝒮}(−, c) × (∂Δ^{n} ↪ Δ^{n})}_{n≥0} J = {Hom_{𝒮}(−, c) × (Λ_{j}^{n} ↪ Δ^{n})}_{n≥1,0≤j≤n}

so that by adjointness, the fibrations are precisely the sectionwise Kan fibrations. There is another one with the same weak equivalences, due to Heller [He], such that the cofibrations are precisely the injective morphisms (whence the name injective model structure).The description of J involves the cardinality of the set of morphisms in 𝒞 and is not explicit. Neither is the characterization of the fibrations.

The morphisms of model categories are called Quillen functors. A Quillen functor of model categories ℳ → 𝒩 is an adjoint pair (F, G): ℳ → 𝒩 such that F preserves cofibrations and G preserves fibrations. This condition ensures that (F, G) induces an adjoint pair on homotopy categories (ℒF, ℛG), where ℒF is the total left derived functor of F. A Quillen functor is a Quillen equivalence if the total left derived is an equivalence. For example, geometric realization is a strict symmetric monoidal left Quillen equivalence | − |: sSet → Top, and similarly in the pointed setting.

If a model category has a closed symmetric monoidal structure as well, one has the following statement.

Theorem A.2.5 (Quillen). Let 𝒞 be a bicomplete category with a model structure. Suppose that (𝒞, ⊗, 𝕀) is closed symmetric monoidal. Suppose further that these structures are compatible in the following sense:

• The pushout product of two cofibrations is a cofibration, and

• the pushout product of an acyclic cofibration with a cofibration is an acyclic cofibration.

Then A ⊗ − is a left Quillen functor for all cofibrant objects A ∈ 𝒞. In particular, there is an induced (total derived) closed symmetric monoidal structure on Ho(𝒞, w𝒞).

One abbreviates the hypotheses of Theorem A.2.5 by saying that 𝒞 is a symmetric monoidal model category. This ends our introduction to model category theory.

A.3 Model structures for motivic spaces

To equip M_{•}(S) with a model structure suitable for the various requirements (compatibility with base change, taking complex points, finiteness conditions, having the correct motivic homotopy category), we construct a preliminary model structure first. Start with the following construction, which is a special case of the considerations in [I]. Choose any X ∈ Sm_{S} and a finite set

{i^{j}: Z^{j} ↬ X}_{j=1}^{m}

of closed embeddings in 𝒮m/S. Regarding i_{j} as a monomorphism of motivic spaces, one may form the categorical union (not the categorical coproduct!) ⋆_{j=1}^{m}i^{j}: ∪_{j=1}^{m} Z^{j} ↪ X. That is, ∪_{j=1}^{m}Z^{j} is the coequalizer in the category of motivic spaces of the diagram

∐_{j=1}^{m} Z_{j} ⇉ ∐_{j,j′} Z_{j} ×_{X} Z_{j′}  (31)

Call the resulting monomorphism ⋆_{j=1}^{m}i^{j}: ∪_{j=1}^{m}Z^{j} ↪ X acceptable. The closed embedding ∅ ↬ X is acceptable as well. Consider the set Ace of acceptable monomorphisms. Let I_{S}^{c} be the set of maps

{i_{+} ⊔ (∂Δ^{n} ↪ Δ^{n})_{+}}_{i∈Ace,n≥0}  (32)

and let J_{S}^{c} be the set of maps

{i_{+} ⊔ (Λ_{j}^{n} ↪ Δ^{n})_{+}}_{i∈Ace,n≥1,0≤j≤n}  (33)

Definition A.3.1. A map f: A → B in M_{•}(S) is a schemewise weak equivalence if f: A(X) → B(X) is a weak equivalence of simplicial sets for all X ∈ 𝒮m/S. It is a closed schemewise fibration if f: A → B has the right lifting property with respect to J_{S}^{c}. It is a closed cofibration if it has the left lifting property with respect to all acyclic closed schemewise fibrations (closed schemewise fibrations which are also schemewise weak equivalences).

Theorem A.3.2. The classes defined in A.3.1 are a closed symmetric monoidal model structure on M_{•}(S), denoted M_{•}^{cs}(S). A morphism f: S → T of base schemes induces a strict symmetric monoidal left Quillen functor f^{∗}: M_{•}(T) → M_{•}(S).

Proof. The existence of the model structure follows from [I]. The pushout product axiom follows, because the pushout product of two acceptable monomorphism is again acceptable. To conclude the last statement, it suffices to check that f^{∗} maps any map in I_{T}^{c} resp. J_{T}^{c} to a closed cofibration resp. schemewise weak equivalence. In fact, if i = ⋆_{j=1}^{m}i^{j}: ∪_{j=1}^{m} Z^{j} ↪ X is an acceptable monomorphism in 𝒮m_{T}, then f^{∗}(i) is the acceptable monomorphism obtained from the closed embeddings

{T ×_{S} Z_{j} ↬ T ×_{S} X.}

Because f^{∗} is strict symmetric monoidal and a left adjoint, it preserves the pushout product. Hence f^{∗} even maps the set I_{T}^{c} to the set I_{S}^{c}, and likewise for J_{T}^{c}. The result follows.

The resulting homotopy category is equivalent – via the identity functor – to the usual homotopy category of the diagram category M_{•}(S) (obtained via the projective model structure), since the weak equivalences are just the objectwise ones. The model structure M_{•}^{cs}(S) has the advantage that for any S-point x_{0} : S ↬ X in a smooth Sscheme, the pair (X, x_{0}) is closed cofibrant. Not all pointed motivic spaces are closed cofibrant. Let (−)^{cs} → Id_{M_{•}(S)} denote a cofibrant replacement functor, for example the one obtained from applying the small object argument to I_{S}^{c}. That is, the map A^{cs} → A is a natural closed schemewise fibration and a schemewise weak equivalence, and A^{cs} is closed cofibrant. Dually, let Id_{M_{•}(S)} → (−)^{cf} denote the fibrant replacement functor obtained by applying the small object argument to J_{S}^{c}. The closed schemewise fibrations may be characterized explicitly.

Lemma A.3.3. A map f: A → B is a closed schemewise fibration if and only if the following two conditions hold.

1. f(X): A(X) → B(X) is a Kan fibration for every X ∈ 𝒮m/S, and

2. for every finite set {Z_{j} ↬ X}_{j=1}^{m} of closed embeddings in 𝒮m/S, the induced map

A(X) → B(X) ×_{sSet_{•}(∪_{j=1}^{m}Z_{j},B)} sSet_{•}(∪_{j=1}^{m}Z_{j}, A)

is a Kan fibration.

Proof. Follows by adjointness from the definition.

To obtain a motivic model structure, one localizes M_{•}^{cs}(S) as follows. Recall that an elementary distinguished square (or simply Nisnevich square) is a pullback diagram

\begin{matrix}V & ⟶ & Y \\ ↓ & & ↓^{p} \\ U & ⟶^{j} & X \end{matrix}

in 𝒮m/S, where j is an open embedding and p is an étale morphism inducing an isomorphism Y − V ≅ X − U of reduced closed subschemes. Say that a pointed motivic space C is closed motivic fibrant if it is closed schemewise fibrant, the map

C(X ×_{S} 𝔸_{S}^{1} ⟶^{pr} X)

is a weak equivalence of simplicial sets for every X ∈ 𝒮m/S, the square

C(V) ⟵ C(Y)
↓  ↓
C(U) ⟵ C(X)

is a homotopy pullback square of simplicial sets for every Nisnevich square in 𝒮m/S and C(∅) is contractible.

Example A.3.4. Let X ↦ K^{TT}(X) be the pointed motivic space sending X ∈ 𝒮m/S to the first term of the Waldhausen K-theory spectrum [W] associated to the category of big vector bundles over X [FS], with isomorphisms as weak equivalences. That is,

K^{TT}(X) = Ω Sing |w𝒮_{•}(Vect^{big}(X), iso)|

Suppose now that S is regular. Then so is X, and K^{TT}(X) has the same homotopy type as the first term of the Thomason-Trobaugh K-theory spectrum of X [TT, Cor. 3.9]. It follows that the projection induces a weak equivalene K^{TT}(X) → K^{TT}(X ×_{S} 𝒜_{S}^{1}) [TT, Prop. 6.8]. By [TT, Thm. 10.8] the square

\xymatrix{K^{TT}(X) \ar[r] \ar[d]_{K^{TT}(p)} & K^{TT}(U) \ar[d] \\ K^{TT}(Y) \ar[r] & K^{TT}(p^{−1}(U))}

associated to a Nisnevich square is a homotopy pullback square. Hence K^{TT} is fibrant in the projective motivic model structure on M_{•}(S). However, if i: Z ↪ X is a closed embedding in 𝒮m/S, the map K^{TT}(i): K^{TT}(X) → K^{TT}(Z) is not necessarily a Kan fibration. In particular, K^{TT} is not closed schemewise fibrant. Choose a closed cofibration which is also a schemewise weak equivalence K^{TT} → 𝕂^{TT} such that 𝕂^{TT} is closed schemewise fibrant. It follows immediately that 𝕂^{TT} is closed motivic fibrant, and that ^{TT}(X) has the homotopy type of the first term of the Thomason-Trobaugh K-theory spectrum of X.

Definition A.3.5. A map f: A → B is a motivic weak equivalence if the map

sSet_{•}(f^{cs}, C): sSet_{•}(B^{cs}, C) → sSet_{•}(A^{cs}, C)

is a weak equivalence of simplicial sets for every closed motivic fibrant C. It is a closed motivic fibration if it has the right lifting property with respect to all acyclic closed cofibrations (closed cofibrations which are also motivic equivalences).

Example A.3.6. Suppose that f: A → B is a map in M_{•}(S) inducing weak equivalences x^{∗}f: x^{∗}A → x^{∗}B of simplicial sets on all Nisnevich stalks x^{∗}: M_{•}(S) → sSet. Then f is a motivic weak equivalence. If f: A → B is an A^{1}-homotopy equivalence (for example, the projection of a vector bundle), then it is a motivic weak equivalence.

Example A.3.7. The canonical covering of P^{1} shows that it is motivic weakly equivalent as a pointed motivic space to the suspension S^{1} ∧ (𝒜^{1} − {0}, 1), where S^{1} = Δ^{1}/∂Δ^{1}. Set S^{1,0} : = S^{1} and S^{1,1} : = (𝒜^{1} − {0}, 1), and define

S^{p,q}: = (S^{1,0})^{∧p−q} ∧ (S^{1,1})^{∧q} for p ≥ q ≥ 0

To generalize the example of P^{1}, one can show that if P^{n−1} ↪ P^{n} is a linear embedding, then P^{n}/P^{n−1} is motivic weakly equivalent to S^{2n,n}.

To prove that the classes from Definition A.3.5 are part of a model structure, it is helpful to characterize the closed motivic fibrant objects via a lifting property. Let J_{S}^{cm} be the union of the set J_{S}^{c} from (33) and the set J_{S}^{m} of pushout products of maps (∂Δ^{n} ↪ Δ^{n})_{+} with maps of the form

\xymatrix{X_{+} \ar[r]^{^{zero^{+}}} & (𝒜_{S}^{1} ×_{S} X)_{+} \\ U_{+} ∪_{V_{+}} Cylh_{+} \ar[r] & Cyl(U_{+} ∪_{V_{+}} Cyl(h_{+}) → X_{+}) \\ ∗ \ar[r] & ∅_{+}}  (34)

where h is the open embedding appearing on top of a Nisnevich square

\begin{matrix}V & ⟶^{h} & Y \\ ↓ & & ^{p}↓ \\ U & ⟶ & X \end{matrix}

in 𝒮m/S.

Lemma A.3.8. A pointed motivic space C is closed motivic fibrant if and only if the map C → ∗ has the right lifting property with respect to the set J_{S}^{cm}.

Proof. This follows from adjointness, the Yoneda lemma and the construction of J_{S}^{cm}.

Theorem A.3.9. The classes of motivic weak equivalences, closed motivic fibrations and closed cofibrations constitute a symmetric monoidal model structure on M_{•}(S), denoted M_{•}^{cm}(S). The resulting homotopy category is denoted H_{•}^{cm}(S) and called the pointed motivic unstable homotopy category of S. A morphism f: S → T of base schemes induces a strict symmetric monoidal left Quillen functor f^{∗}: M_{•}^{cm}(T) → M_{•}^{cm}(S).

Proof. The existence of the model structure follows by standard Bousfield localization techniques. Here are some details. The problem is that J_{S}^{cm} might be to small in order to characterize all closed motivic fibrations. Let κ be a regular cardinal strictly bigger than the cardinality of the set of morphisms in 𝒮m/S. A motivic space A is κ-bounded if the union

∐_{n≥0,X∈𝒮m/S} A(X)_{n}

has cardinality ≤ κ. Let J_{S}^{κ} be a set of isomorphism classes of acyclic monomorphisms whose target is κ-bounded. One may show that given an acyclic monomorphism j: A ↪ B and a κ-bounded subobject C ⊆ B, there exists a κ-bounded subobject C′ ⊆ B containing C such that j^{−1}(C′) ↪ C′ is an acyclic monomorphism. Via Zorn’s lemma, one then gets that a map f ∈ M_{•}(S) has the right lifting property with respect to all acyclic monomorphisms if (and only if) it has the right lifting property with respect to the set J_{S}^{κ}. Such a map is in particular a closed motivic fibration. Any given map f: A → B can now be factored (via the small object argument) as an acyclic monomorphism j: A ↪ C followed by a closed motivic fibration. Factoring j as a closed cofibration followed by an acyclic closed schemewise fibration in the model structure of Theorem A.3.2 implies the existence of the model structure.

To prove that the model structure is symmetric monoidal, it suffices – by the corresponding statement A.3.2 for M_{•}^{cs}(S) – to check that the pushout product of a generating closed cofibration and an acyclic closed cofibration is again a motivic equivalence. However, from the fact that the injective motivic model structure is symmetric monoidal, one knows that motivic equivalences are closed under smashing with arbitrary motivic spaces [DRØ]. The first sentence is now proven.

Concerning the third sentence, Theorem A.3.2 already implies that f^{∗} preserves closed cofibrations. To prove that f^{∗} is a left Quillen functor, it suffices (by Dugger’s lemma [D, Cor. A2]) to check that it maps the set J_{T}^{cm} to motivic weak equivalences in M_{•}(S). One may calculate that f^{∗}(J_{T}^{cm}) = J_{S}^{cm}, whence the statement.

The closed motivic model structure is cofibrantly generated. As remarked in the proof of Theorem A.3.9, the set J_{S}^{cm} is perhaps not big enough to yield a full set of generating trivial cofibrations. By localization theory and Lemma A.3.8, a map with closed motivic fibrant codomain is a closed motivic fibration if and only if it has the right lifting property with respect to J_{S}^{cm}. Note that the domains and codomains of the maps in J_{S}^{cm} are closed cofibrant and finitely presentable. The following Lemma is an easy consequence.

Lemma A.3.10. Motivic equivalences and closed motivic fibrations with closed motivic fibrant codomain are closed under filtered colimits.

Proof. This follows, since the domains of the maps in I_{S}^{c} and J_{S}^{cm} are finitely presentable. See [DRØ] for the corresponding statement for the projective motivic model structure.

Let ℳ_{•}(S) be the category of simplicial objects in the category of pointed Nisnevich sheaves on 𝒮m/S. The functor mapping a (pointed) presheaf to its associated (pointed)

Nisnevich sheaf determines by degreewise application a functor a_{Nis}: M_{•}(S) → ℳ_{•}(S). Let ℳ_{•}(S) ⟶^{i} M_{•}(S) be the inclusion functor, the right adjoint of a_{Nis}.

Theorem A.3.11. The pair (a_{Nis}, i) is a Quillen equivalence to the Morel-Voevodsky model structure. The functor a_{Nis} is strict symmetric monoidal. In particular, the total left derived functor of a_{Nis} is a strict symmetric monoidal equivalence

H_{•}^{cm}(S) → H_{•}(S)

to the unstable pointed A^{1}-homotopy category from [MV].

Proof. Recall that the cofibrations in the Morel-Voevodsky model structure are precisely the monomorphisms. Since every closed cofibration is a monomorphism and Nisnevich sheafification preserves these, a_{Nis} preserves cofibrations. The unit A → i(a_{Nis}(A)) of the adjunction is an isomorphism on all Nisnevich stalks, hence a motivic weak equivalence by Example A.3.6 for every motivic space A. In particular, a_{Nis} maps schemewise weak equivalences as well as the maps in J_{S}^{m} described in (34) to weak equivalences. Let Id_{M_{•}(S)} → (−)^{fib} be the fibrant replacement functor in M_{•}^{cm}(S) obtained from the small object argument applied to J_{S}^{cm}. Hence if f is a motivic weak equivalence, then f^{fib} is a schemewise weak equivalence. One concludes that a_{Nis} preserves all weak equivalences, thus is a left Quillen functor. Since the unit A → i(a_{Nis}(A)) is a motivic weak equivalence for every A, the functor a_{Nis} is a Quillen equivalence.

Note that a map f in M_{•}(S) is a motivic weak equivalence if and only if a_{Nis}(f) is a weak equivalence in the Morel-Voevodsky model structure on simplicial sheaves. Conversely, a map of simplicial sheaves is a weak equivalence if and only if it is a motivic weak equivalence when considered as a map of motivic spaces.

Remark A.3.12. Starting with the injective model structure on simplicial presheaves mentioned in Example A.2.4, there is a model structure M_{•}^{im}(S) on the category of pointed motivic spaces with motivic weak equivalences as weak equivalences and monomorphisms as cofibrations. It has the advantage that every object is cofibrant, but the disadvantage that it does not behave well under base change [MV, ??] or geometric realization (to be defined below). The identity functor is a left Quillen equivalence Id: M_{•}^{cm}(S) → M_{•}^{im}(S), since the homotopy categories coincide.

Further, let Spc_{•}(S) be the category of pointed Nisnevich sheaves on 𝒮m/S. Recall the cosimplicial smooth scheme over S whose value at n is

Δ_{S}^{n} = Spec(𝒪_{S}[X_{0}, …, X_{n}]/(∑_{i=0}^{n} X_{i} = 1))  (35)

The functor Spc_{•}(S) → ℳ_{•}(S) sending A to the simplicial object Sing_{S}(A)_{n} = A(−×Δ_{S}^{n}) has a left adjoint | − |_{S}: ℳ_{•}(S). It maps B to the coend

|B|_{S} = ∫_{n∈Δ} B_{n} × Δ_{S}^{n}

in the category of pointed Nisnevich sheaves. The following statement is proved in [MV].

Theorem A.3.13 (Morel-Voevodsy). There is a model structure on the category Spc_{•}(S) such that the pair (| − |_{S}, Sing_{S}) is a Quillen equivalence to the Morel-Voevodsky model structure. The functor | − |_{S} is strict symmetric monoidal. In particular, the total left derived functor of | − |_{S} is a strict symmetric monoidal equivalence from Voevodsky’s pointed homotopy category to the unstable pointed A^{1}-homotopy category

A.4 Topological realization

In the case where the base scheme is the complex numbers, there is a topological realization functor R_{ℂ}: M_{•}^{cm}(ℂ) → Top_{•} which is a strict symmetric monoidal left Quillen functor. It is defined as follows. If X ∈ 𝒮m_{ℂ}, the set X(ℂ) of complex points is a topological space when equipped with the analytic topology. Call this topological space X^{an}. It is a smooth manifold, and in particular a compactly generated topological space. One may view X ↦ X^{an} as a functor 𝒮m_{ℂ} → Top. Note that if i: Z ↬ X is a closed embedding in 𝒮m_{ℂ}, then the resulting map i^{an} is the closed embedding of a smooth submanifold, and in particular a cofibration of compactly generated topological spaces. Every motivic space A is a canonical colimit

colim_{X×Δ^{n}→A} X × Δ^{n} ⟶^{≅} A

and one defines

R_{ℂ}(A): = colim_{X×Δ^{n}→A} X^{an} × |Δ^{n}| ∈ Top.

Observe that if A is pointed, then so is R_{ℂ}(A).

Theorem A.4.1. The functor R_{ℂ}: M_{•}^{cm}(ℂ) → Top_{•} is a strict symmetric monoidal left Quillen functor.

Proof. The right adjoint of R_{ℂ} maps the compactly generated topological space Z to the motivic space Sing_{ℂ}(Z) which sends X ∈ 𝒮m_{ℂ} to the simplicial set sSet_{Top}(X^{an}, Z). To conclude that R_{ℂ} is strict symmetric monoidal, it suffices to observe that there is a canonical homeomorphism (X ×Y)^{an} ≅ X^{an}×Y^{an}, and that geometric realization is strict symmetric monoidal.

Suppose now that i: ∪_{j=1}^{m} Z_{j} ↪ X is an acceptable monomorphism. One computes R_{ℂ}(∪_{j=1}^{m}Z_{j}) as the coequalizer

∐_{j=1}^{m} Z_{j}^{an} ⇉ ∐_{j,j′}(Z_{j} ×_{X} Z_{j′})^{an}.

in Top. Every map (Z_{j} ×_{X} Z_{j′})^{an} → Z_{j}^{an} is a closed embedding of smooth submanifolds of complex projective space. In particular, one may equip Z_{j}^{an} with a cell complex structure such that (Z_{j} ×_{X} Z_{j′})^{an} is a subcomplex for every j′. Then R_{ℂ}(∪_{j=1}^{m}Z_{j}) is the union of these subcomplexes, and in particular again a subcomplex. It follows that R_{ℂ}(i) is a cofibration of topological spaces. Since R_{ℂ} is compatible with pushout products, it maps the generating closed cofibrations to cofibrations of topological spaces.

To conclude that R_{ℂ} preserves trivial cofibrations as well, it suffices by Dugger’s lemma [D, Cor. A2] to check that R_{ℂ} maps every map in J_{ℂ}^{m} to a weak homotopy equivalence. In fact, since the domains and codomains of the maps ∂Δ^{m} ↪ Δ^{m} are cofibrant, it suffices to check the latter for the maps in diagram (34). In the first case, one obtains the map X^{an} ↪ (𝒜_{ℂ}^{1}×X)^{an} ≅ ℝ^{2} ×X^{an}, in the second case one obtains up to simplicial homotopy equivalence the canonical map U^{an} ∪_{p^{−1}(V)^{an}} Y^{an} → X^{an} for a Nisnevich square

\xymatrix{V \ar[r] \ar[d] & Y \ar[d]^{p} \\ U \ar[r] & X}

This is in fact a homeomorphism of topological spaces. The result follows.

Suppose now that R ↪ ℂ is a subring of the complex numbers. Let f: Spec(ℂ) → Spec(R) denote the resulting morphism of base schemes. The realization with respect to R (or better f) is defined as the composition

R_{R} = R_{ℂ} ∘ f^{∗}: M_{•}(R) → M_{•}(ℂ) → Top_{•}.  (36)

It is a strict symmetric monoidal Quillen functor. The most relevant case is R = ℤ.

Example A.4.2. The topological realization of the finite Grassmannian Gr(m, n) (over any base with a complex point) is the complex Grassmannian with the usual topology. Since R_{ℂ} commutes with filtered colimits, R_{ℂ}(Gr) is the infinite complex Grassmannian, which in turn is the classifying space BU for the infinite unitary group. Because R_{ℂ} is a left Quillen functor, the topological realization of any closed cofibrant motivic space weakly equivalent to Gr (such as 𝒦) is homotopy equivalent to BU.

A.5 Spectra

Definition A.5.1. Let P_{S}^{1} denote the pointed projective line over S. The category MS(S) of P^{1}-spectra over S has the following objects. A P^{1}-spectrum E consists of a sequence (E_{0}, E_{1}, E_{2}, …) of pointed motivic spaces over S, together with structure maps σ_{n}^{E}: E_{n} ∧ P^{1} → E_{n+1} for every n ≥ 0. A map of P^{1}-spectra is a sequence of maps of pointed motivic spaces which is compatible with the structure maps.

Example A.5.2. Any pointed motivic space B over S gives rise to a P_{S}^{1}-suspension spectrum

Σ_{P^{1}}^{∞}B = (B, B ∧ P^{1}, B ∧ P^{1} ∧ P^{1}, …)

having identities as structure maps. More generally, let Fr_{n}B denote the P^{1}-spectrum having values

(Fr_{n}B)_{n+m} = \begin{cases}B ∧ P^{1}^{∧m} & m ≥ 0 \\ ∗ & m < 0 \end{cases}

and identities as structure maps, except for σ_{n−1}^{Fr_{n}B}. The functor B ↦ Fr_{n}B is left adjoint to the functor sending the P^{1}-spectrum E to E_{n}.

Remark A.5.3. In Definition A.5.1, one may replace P^{1} by any pointed motivic space A, giving the category MS_{A}(S) of A-spectra over S. Essentially the only relevant example for us is when A is weakly equivalent to the pointed projective line P^{1}. The Thom space T = 𝒜^{1}/𝒜^{1} − {0} of the trivial line bundle over S admits motivic weak equivalences

\xymatrix{P^{1} \ar[r]^{∼} & P^{1}/𝒜^{1} & T \ar[l]^{∼}}  (37)

The motivic space P^{1} itself is not always the ideal suspension coordinate. For example, the algebraic cobordism spectrum MGL naturally comes as a T -spectrum. In order to switch between T -spectra and P^{1}-spectra, consider the following general construction. A map ϕ: A → B induces a functor ϕ^{∗}: MS_{B}(S) → MS_{A}(S) sending the B-spectrum (E_{0}, E_{1}, …, σ_{n}^{E}) to the A-spectrum

(E_{0}, E_{1}, …) with structure maps σ_{n}^{ϕ^{∗}E} = σ_{n}^{E} ∘ (E_{n} ∧ ϕ)

Its left adjoint ϕ_{♯} maps the A-spectrum (F_{0}, F_{1}, …, σ_{n}^{F}) to the B-spectrum

(F_{0}, B ∧ F_{0} ∪_{A∧F_{0}} F_{1}, B ∧ (B ∧ F_{0} ∪_{A∧F_{0}} F_{1}) ∪_{A∧F_{1}} F_{2}), …)  (38)

having the canonical maps as structure maps. Note that for the purpose of constructing a model structure on A-spectra over S, the pointed motivic space A has to be cofibrant in the model structure under consideration.

The next goal is to construct a model structure on MS(S) having the motivic stable homotopy category as its homotopy category.

Definition A.5.4. Let Ω_{P^{1}} = \underline{Hom}_{M_{•}(S)}(P_{S}^{1}, −) denote the right adjoint of P^{1} ∧−. For a P^{1}-spectrum E with structure maps σ_{n}^{E}: E_{n} ∧ P^{1} → E_{n+1}, let ω_{n}^{E}: E_{n} → Ω_{P^{1}}E_{n+1} denote the adjoint structure map. A P^{1}-spectrum E is closed stably fibrant if

• E_{n} is closed motivic fibrant for every n ≥ 0, and

• ω_{n}^{E}: E_{n} → Ω_{P^{1}}E_{n+1} is a motivic weak equivalence for every n ≥ 0.

Any P^{1}-spectrum E admits a closed stably fibrant replacement. First replace E by a levelwise P^{1}-spectrum E^{ℓ} as follows. Let E_{0}^{ℓ} = E_{0}^{fib} for a fibrant replacement in M_{•}^{cm}(S). Given E_{n} → E_{n}^{ℓ}, set

E_{n+1}^{ℓ}: = (E_{n}^{ℓ} ∧ P^{1} ∪_{E_{n}∧P^{1}} E_{n+1})^{fib}

which yields a levelwise motivic weak equivalence E → E^{ℓ} of P^{1}-spectra. To continue, observe that the adjoint structure maps of any P^{1}-spectrum F may be viewed as a natural transformation

q: F → Q(F)

where Q(F) is the P^{1}-spectrum with terms Ω_{P^{1}}F_{1}, Ω_{P^{1}}F_{2}, … and structure maps

Ω_{P^{1}}(ω_{n+1}^{F}): Ω_{P^{1}}F_{n+1} → Ω_{P^{1}}^{2}F_{n+2}.

Define Q^{∞}(E) as the colimit of the sequence \xymatrix{E^{ℓ} \ar[r]^{q} & Q(E^{ℓ}) \ar[r]^{Q(q)} & Q^{2}(E^{ℓ}) \ar[r] & ⋯} Definition A.5.5. A map f: E → F of P^{1}-spectra is a stable equivalence if the map Q^{∞}(f)_{n} is a weak equivalence for every n ≥ 0. It is a closed stable fibration if f_{n}: E_{n} → F_{n} is a closed motivic fibration and the induced map E_{n} → F_{n} ×_{Q^{∞}(E)_{n}} Q^{∞}(F)_{n} is a motivic weak equivalence for every n ≥ 0. It is a closed cofibration if f_{n}: E_{n} → F_{n} and F_{n} ∧ P^{1} ∪_{E_{n}∧P^{1}} E_{n+1} → F_{n+1} are closed cofibrations for every n ≥ 0.

Theorem A.5.6. The classes from Definition A.5.5 are a model structure on the category of P^{1}-spectra, denoted MS^{cm}(S). The identity functor on P^{1}-spectra from Jardine’s stable model structure to MS^{cm}(S) is a left Quillen equivalence. In particular, the homotopy category SH^{cm}(S): = Ho(MS^{cm}(S)) is equivalent to the motivic stable homotopy category SH(S).

Proof. Recall that P^{1} is closed cofibrant. The existence of the model structure follows as in [J, Thm. 2.9]. Moreover, the stable equivalences coincide with the ones in [J], because so do the stabilization constructions and the unstable weak equivalences. Since every closed cofibration of motivic spaces is in particular a monomorphism, Id_{MS(S)} is a left Quillen equivalence. Note that the closed cofibrations are generated by the set

{Fr_{m}(X × ∂Δ^{n} ↪ X × Δ^{n})_{+})}_{m,n≥0,X∈𝒮m/S}  (39)

One may also describe a set of generating acyclic cofibrations.

Remark A.5.7. Note that one may form the smash product of a motivic space A and a P^{1}-spectrum E by setting (A ∧ E)_{n}: = A ∧ E_{n} and σ_{n}^{A∧E}: = A ∧ σ_{n}^{E}. If A is closed cofibrant, A∧− is a left Quillen functor by Theorem A.3.9. From Theorem A.5.6 one may deduce that P^{1} ∧ −: MS(S) → MS(S) is a left Quillen equivalence. Since ¶ ≃ S^{2,1} = S^{1} ∧ (𝒜^{1} − {0}, 1), also S^{1} ∧ − is a left Quillen equivalence. In particular, SH^{cm}(S) is triangulated, with the total left derived of S^{1} ∧ − as the shift functor. The triangles are those which are isomorphic to the image of

E ↪ F → F/E ← E ∧ Δ^{1} ∪_{E} F → S^{1} ∧ E

where E ↪ F is an inclusion of P^{1}-spectra. As well, one has sphere spectra S^{p,q} ∈ SH^{cm}(S) for all integers p, q ∈ ℤ.

Example A.5.8. Since SH(S) is an additive category, the canonical map E ∨F → E ×F is a stable equivalence. In the special case of P^{1}-suspension spectra, the canonical map factors as

Σ_{P^{1}}^{∞}A ∨ Σ_{P^{1}}^{∞}B ≅ Σ_{P^{1}}^{∞}(A ∨ B) → Σ_{P^{1}}^{∞}(A × B) → Σ_{P^{1}}^{∞}A × Σ_{P^{1}}^{∞}B

which shows that Σ_{P^{1}}^{∞}(A×B) contains Σ_{P^{1}}^{∞}(A∧B) as a retract in SH(S). Thus it is even a direct summand. The latter can be deduced as follows. The (reduced) join (A, a_{0})∗(B, b_{0}) is defined as the pushout in the diagram

\xymatrix{A × B × ∂Δ^{1} ∪ \{a_{0}\} × \{b_{0}\} × Δ^{1} \ar[r] \ar[d] & A ∨ B \ar[d] \\ A × B × Δ^{1} \ar[r] & A ∗ B.}

of pointed motivic spaces over S. Attaching A ∧ (Δ^{1}, 0) and B ∧ (Δ^{1}, 0) to A ∗ B via A ∨ B produces a pointed motivic space C which is equipped with a sectionwise weak equivalence C → (A × B) ∧ S^{1,0}. Collapsing {a_{0}} ∗ B and A ∗ {b_{0}} inside C yields a sectionwise weak equivalence C → (A ∧ B ∧ S^{1,0}) ∨ (A ∧ S^{1,0}) ∨ (B ∧ S^{1,0}). Since S^{1,0} is invertible in SH(S), one gets a splitting Σ_{P^{1}}^{∞}(A × B) ≃ (Σ_{P^{1}}^{∞}(A ∧ B)) ∨ (Σ_{P^{1}}^{∞}A) ∨ (Σ_{P^{1}}^{∞}B) in SH(S).

For a P^{1}-spectrum E let Tr_{n}E denote the P^{1}-spectrum with

(Tr_{n}E)_{m} = \begin{cases}E_{m} & m ≤ n \\ E_{n} ∧ (P^{1})^{∧m−n} = (Fr_{n}E_{n})_{m} & m ≥ n \end{cases}

and with the obvious structure maps. The structure maps of E determine maps Tr_{n}E → Tr_{n+1}E such that E = colim_{n} Tr_{n}E. Since the canonical map Fr_{n}E_{n} → Tr_{n}E adjoint to the identity id: E_{n} → E_{n} is an identity in all levels ≥ n, it is in particular a stable equivalence. The identity id: E_{n} ∧ (P^{1})^{∧n} → (Fr_{0}E_{n})_{n} leads by adjointness to the map

\xymatrix{Fr_{n}(E_{n}) ∧ (P^{1})^{∧n} \ar[r]^{≅} & Fr_{n}(E_{n} ∧ (P^{1})^{∧n}) \ar[r] & Fr_{0}E_{n}}  (40)

and hence to a map Fr_{n}E_{n} → Ω_{P^{1}}^{n}(Fr_{0}E_{n}). Since the map (40) is an isomorphism in all levels ≥ n, it is a stable equivalence. Because Ω_{P^{1}} is a Quillen equivalence, the map Fr_{n}E_{n} → Ω_{P^{1}}^{n}((Fr_{0}E_{n})^{fib}) is a stable equivalence as well if E_{n} is closed cofibrant. In fact, the condition on E_{n} can be removed since Fr_{n} preserves all weak equivalences. This leads to the following statement.

Lemma A.5.9. Any P^{1}-spectrum E is the colimit of a natural sequence

\xymatrix{Tr_{0}E \ar[r] & Tr_{1}E \ar[r] & Tr_{2}E \ar[r] & ⋯}  (41)

of P^{1}-spectra in which the n-th term is naturally stably equivalent to Ω_{P^{1}}^{n}((Σ_{P^{1}}^{∞}E_{n})^{f}).

One may use the description in Lemma A.5.9 for computations as follows. Say that a P^{1}-spectrum F is finite if it is stably equivalent to a P^{1}-spectrum F′ such that ∗ → E′ is obtained by attaching finitely many cells from the set (39).

Lemma A.5.10. Let D(0) → D(1) → D(2) → ⋯ be a sequence of P^{1}-spectra, with colimit D(∞).

1. Suppose that F is a finite P^{1}-spectrum. The canonical map

colim_{i≥0} Hom_{SH(S)}(F, D(i)) → Hom_{SH(S)}(F, D(∞))

is an isomorphism.

2. For any P^{1}-spectrum E there is a canonical short exact sequence

0 → lim_{i≥0}^{1}[S^{1,0} ∧ D(i), E] → [D(∞), E] → lim_{i≥0}[D(i), E] → 0  (42)

of abelian groups, where [−, −] denotes Hom_{SH(S)}(−, −).

Proof. Observe first that stable equivalences and closed stable fibrations are detected by the functor Q^{∞} which is defined in A.5.5 as a sequential colimit. Lemma A.3.10 implies that stable equivalences and closed stable fibrations with closed stably fibrant codomain are closed under filtered colimits. Thus by Theorem A.2.1 one may compute

Hom_{SH(S)}(F, colim_{i≥0} D(i)) ≅ Hom_{MS(S)}(F, colim_{i≥0} D(i)^{fib})/ ≃

for any cofibrant P^{1}-spectrum F, where ≃ denotes the equivalence relation “simplicial homotopy”. This implies statement 1 because Hom_{MS(S)}(F, −) commutes with filtered colimits if ∗ → F is obtained by attaching finitely many cells.

To prove the second statement, let C be the coequalizer of the diagram

\xymatrix{⋁_{i≥0} D(i) \ar[r]^{f} \ar[r]_{g} & ⋁_{i≥0} Δ_{+}^{1} ∧ D(i)}

where f resp. g is defined on the i-th summand D(i) as D(i) = 1_{+} ∧ D(i) ↪ Δ_{+}^{1} ∧ D(i) resp. D(i) → D(i + 1) = 0_{+} ∧ D(i + 1) ↪ Δ_{+}^{1} ∧ D(i + 1). The canonical map C → colim_{i≥0} D(i) induced by the Δ_{+}^{1} ∧ D(i) → D(i) → colim_{i≥0} D(i) is a weak equivalence. In the stable homotopy category, which is additive, one may take the difference of f and g, and thus describe colim_{i≥0} D(i) via the distinguished triangle

\xymatrix{⋁_{i≥0} D(i) \ar[r]^{f−g} & ⋁_{i≥0} D(i) \ar[r] & colim_{i≥0} D(i) \ar[r] & ⋁_{i≥0} S^{1,0} ∧ D(i).}  (43)

Applying [−, E]: = Hom_{SH(S)}(−, E) to the triangle (43) produces a long exact sequence

\xymatrix{⋯ & ∏_{i≥0}[D(i), E] \ar[l]_{f^{∗}−g^{∗}} & [D(∞), E] \ar[l] & ∏_{i≥0}[S^{1,0} ∧ D(i), E] \ar[l] & ⋯ \ar[l]}

which may be split into the short exact sequence

\xymatrix{0 & lim_{i≥0}[D(i), E] \ar[l] & [D(∞), E] \ar[l] & lim_{i≥0}^{1}[S^{1,0}∧ D(i), E] \ar[l] & 0 \ar[l]}

A.6 Symmetric spectra

There seems to be no reasonable (i.e. symmetric monoidal) smash product for P^{1}-spectra inducing a decent symmetric monoidal smash product on SH(S). This will be solved as in [HSS] and [J].

Definition A.6.1. A symmetric P^{1}-spectrum E over S consists of a sequence (E_{0}, E_{1}, …) of pointed motivic spaces over S, together with group actions (Σ_{n})_{+} ∧ E_{n} → E_{n} and structure maps σ_{n}^{E}: E_{n} ∧ P^{1} → E_{n+1} for all n ≥ 0. Iterations of these structure maps are required to be as equivariant as they can, using the permutation action of Σ_{n} on (P^{1})^{∧n}. A map of motivic symmetric P^{1}-spectra is a sequence of maps of pointed motivic spaces which is compatible with all the structure (group actions and structure maps). Call the resulting category MSS(S).

Example A.6.2. Analogous to Example A.5.2, the n-th shifted suspension spectrum Fr_{n}^{Σ}A of a pointed motivic space A has as values

(Fr_{n}^{Σ}B)_{m+n} = \begin{cases}Σ_{m+n}^{+} ∧_{Σm×\{1\}} A ∧ (P^{1})^{∧m} & m ≥ 0 \\ ∗ & m < 0 \end{cases}

where the m-th fold smash product (P^{1})^{∧m} carries the natural permutation action.

Every symmetric P^{1}-spectrum determines a P^{1}-spectrum by forgetting the symmetric group actions. Call the resulting functor u: MSS(S) → MS(S). It has a left adjoint v, which is characterized uniquely up to unique isomorphism by the fact that

v(Fr_{n}A) = Fr_{n}^{Σ}A.  (44)

The smash product E ∧ F of two symmetric P^{1}-spectra E and F is constructed as follows. Set (E ∧ F)_{n} as the coequalizer of the diagram

\xymatrix{_{∐r+1+s=n}Σ_{n}^{+} ∧_{Σ_{r}×Σ_{1}×Σ_{s}} E_{r} ∧ P^{1} ∧ F_{s} \ar[r]_{σ_{r}^{E}∧(F_{s}∘twist)} \ar[r]^{σ_{r}^{E}∧F_{s}} & ∐_{r+s=n} Σ_{n}^{+} ∧_{Σ_{r}×Σ_{s}} E_{r} ∧ F_{s}} (45)

where the coequalizer is taken in the category of pointed Σ_{n}-motivic spaces. The structure map σ_{n}^{E∧F} is induced by the structure maps σ_{0}^{F} , …, σ_{n}^{F} of F. One may provide natural coherence isomorphisms for associativity, commutativity and unitality, where the unit is 𝕀_{S} = (S_{+}, P^{1}, P^{1} ∧P^{1}, …, (P^{1})^{∧n}, …) with the obvious permutation action and identities as structure maps.

We proceed with the homotopy theory of symmetric P^{1}-spectra, as in [J].

Definition A.6.3. A map ϕ: E → F of motivic symmetric P^{1}-spectra is a levelwise acyclic fibration if ϕ_{n}: E_{n} → F_{n} is an acyclic closed motivic fibration of pointed motivic spaces over S for all n ≥ 0. A map ϕ: E → F of motivic symmetric P^{1}-spectra is a closed cofibration if it has the left lifting property with respect to all levelwise acyclic fibrations. There exists a cofibrant replacement functor (−)^{cof} → Id_{MSS(S)}. A levelwise fibrant motivic symmetric P^{1}-spectrum E is closed stably fibrant if the adjoint E_{n} → M_{X}(¶, E_{n+1}) of the structure map is a weak equivalence for every n ≥ 0. A map ϕ: E → F is a stable equivalence if the map

sSet_{MSS_{X}}(ϕ^{cof}, G): sSet_{MSS_{X}}(F^{cof}, G) → sSet_{MSS_{X}}(E^{cof}, G)

is a weak equivalence of pointed motivic spaces for all closed stably fibrant motivic symmetric P^{1}-spectra G. The closed stable fibrations are then defined by the right lifting property.

Theorem A.6.4 (Jardine). The classes of stable equivalences, closed cofibrations and closed stable fibrations from Definition A.6.3 constitute a symmetric monoidal model structure on MSS(S). The forgetful functor u: MSS^{cm}(S) → MS^{cm}(S) is a right Quillen equivalence.

Proof. The proof of the first statement follows as in [J, Thm. 4.15, Prop. 4.19]. Note that the stable equivalences in MSS^{cm}(S) and in Jardine’s model structure MSS^{Jar}(S) coincide, since the unstable weak equivalences do so by Remark A.3.12. In particular, the identity Id: MSS^{cm}(S) → MSS^{Jar}(S) is a left Quillen equivalence. The closed cofibrations in MSS^{cm}(S) are generated by the inclusions

{Fr_{m}^{Σ}(X × ∂Δ^{n} ↪ X × Δ^{n})_{+})}_{m,n≥0,X∈𝒮m/S}  (46)

By formula (44), the left adjoint v of u sends the generating cofibrations to the generating cofibrations. It follows that v is a left Quillen functor. Since v: MS^{Jar}(S) → MSS^{Jar}(S) is a Quillen equivalence by [J, Thm. 4.31], so is the functor v: MS^{cm}(S) → MSS^{cm}(S).

Remark A.6.5. In particular, the category Ho(MS^{cm}(S)) inherits a closed symmetric monoidal product ∧ by setting

E ∧ F: = ℛu(ℒv(E) ∧ ℒv(F))

In other words, if E and F are closed cofibrant P^{1}-spectra, their smash product in SH(S) is given by the P^{1}-spectrum u((v(E) ∧ v(F))^{fib}). The unit is ℛu(ℒv(𝕀)) ≅ 𝕀 the sphere P^{1}-spectrum.

Notation A.6.6. For P^{1}-spectra E and F over S define the E-cohomology and the Ehomology of F as

E^{p,q}(F) = Hom_{SH(S)}(F, S^{p,q} ∧ E)  (47)
E_{p,q}(F) = Hom_{SH(S)}(S^{p,q}, F ∧ E).  (48)

In the special case F = Σ_{P^{1}}^{∞}A, where A is a pointed motivic space over S one writes E^{p,q}(A) and E_{p,q}(A) instead. Sometimes we abbreviate S^{2n,n} ∧ E by E(n).

Remark A.6.7. Since (v, u) is a Quillen adjoint pair of stable model categories the total derived pair respects in particular the triangulated structures. Note that since u preserves all colimits, both ℒv and ℛu preserve arbitrary coproducts.

Lemma A.6.8. Let E and F be P^{1}-spectra. Then E ∧F ∈ SH(S) may be obtained as the sequential colimit of a sequence whose n-th term is stably equivalent to Ω_{P^{1}}^{2n}(Σ_{P^{1}}^{∞}E_{n}∧F_{n})^{fib}.

Proof. We may assume that both E and F are closed cofibrant. By Lemma A.5.9 E and F can be expressed as sequential colimits of their truncations. Since v preserves colimits, v(E) is the sequential colimit of the diagram

v(Tr_{0}E) = v(Fr_{0}E_{0}) = Fr_{0}^{Σ}E_{0} → v(Tr_{1}E) → ⋯

and similarly for F. The stable equivalence Fr_{n}E_{n} → Tr_{n}E of cofibrant P^{1}-spectra induces a stable equivalence v(Fr_{n}E_{n}) = Fr_{n}^{Σ}E_{n} → v(Tr_{n}E). Since smashing with a symmetric P^{1}-spectrum preserves colimits, one has

v(Tr_{m}E) ∧ colim_{n} v(Tr_{n}F) ≅ colim_{n}(v(Tr_{m}E) ∧ v(Tr_{n}F))

for every n. It follows that v(E) ∧ v(F) is the filtered colimit of the diagram sending (m, n) to v(Tr_{m}E) ∧ v(Tr_{n}F). Since the diagonal is a final subcategory in ℕ × ℕ, there is a canonical isomorphism colim_{n} v(Tr_{n}E) ∧ v(Tr_{n}F) ≅ v(E) ∧ v(F). Theorem A.6.4 says that MSS^{cm}(S) is symmetric monoidal, thus the canonical map

Fr_{2n}^{Σ}(E_{n} ∧ F_{n}) ≅ Fr_{n}^{Σ}E_{n} ∧_{n}^{Σ} E_{n} → v(Tr_{n}E) ∧ v(Tr_{n}F)  (49)

is a stable equivalence. There is a canonical map Fr_{2n}^{Σ}(E_{n} ∧F_{n}) → Ω_{P^{1}}^{2n}Fr_{0}^{Σ}(E_{n} ∧F_{n}) which is adjoint to the unit E_{n} ∧ F_{n} → Ω_{P^{1}}^{2n}(P^{1})^{∧2n} ∧ (E_{n} ∧ F_{n}). As in the case of P^{1}-spectra, the map

Fr_{2n}^{Σ}(E_{n} ∧ F_{n}) → Ω_{P^{1}}^{2n}Fr_{0}^{Σ}(E_{n} ∧ F_{n}) → Ω_{P^{1}}^{2n}(Fr_{0}^{Σ}(E_{n} ∧ F_{n}))^{fib}

is a stable equivalence. It follows that v(E) ∧ v(F) is the colimit of a sequence whose n-th term is stably equivalent to Ω_{P^{1}}^{2n}(Fr_{0}^{Σ}(E_{n} ∧ F_{n}))^{fib}. Hence it also follows that a fibrant replacement of v(E) ∧ v(F) may be obtained as the colimit of a sequence of closed stably fibrant symmetric P^{1}-spectra whose n-th term is stably equivalent to Ω_{P^{1}}^{2n}(Fr_{0}^{Σ}(E_{n}∧ F_{n}))^{fib}. Since the forgetful functor u preserves colimits, stable equivalences of closed stably fibrant symmetric P^{1}-spectra and Ω_{P^{1}}, the P^{1}-spectrum u(v(E) ∧ v(F)) is the colimit of a sequence of P^{1}-spectra whose n-th term is stably equivalent to Ω_{P^{1}}^{2n}u((Fr_{0}^{Σ}(E_{n} ∧F_{n}))^{fib}). The map

Fr_{0}(E_{n} ∧ F_{n}) → u((vFr_{0}(E_{n} ∧ F_{n}))^{fib})

is a stable equivalence because (v, u) is a Quillen equivalence A.6.4, whence the result.

As in the case of non-symmetric spectra, one may change the suspension coordinate A.5.3. If A is a pointed motivic space over S, let MSS_{A}(S) denote the category of symmetric A-spectra over S.

Lemma A.6.9. A map A → B in M_{•}(S) induces a strict symmetric monoidal functor MSS_{A}(S) → MSS_{B}(S) having a right adjoint. If the map is a motivic weak equivalence of closed cofibrant pointed motivic spaces, this pair is a Quillen equivalence.

Proof. This is quite formal. For a proof consider [Ho2, Thm. 9.4].

Because the change of suspension coordinate functors are lax symmetric monoidal, they preserve (commutative) monoid objects, that is, (commutative) symmetric ring spectra.

A.7 Stable topological realization

Let Sp = Sp(Top, ℂP^{1}) be the category of ℂP^{1}-spectra (in Top). An object in Sp is thus a sequence of pointed compactly generated topological spaces E_{0}, E_{1}, … with structure maps E_{n} ∧ ℂP^{1} → E_{n+1}. The model structure on Sp is obtained as follows: Cofibrations are generated by

{Fr_{m}^{Top}(|∂Δ^{n} ↪ Δ^{n}|_{+})}_{m,n≥0}

so that every ℂP^{1}-spectrum E has a cofibrant replacement E^{cof} → E mapping to E via a levelwise acyclic Serre fibration. For any ℂP^{1}-spectrum E and any n ∈ ℤ let

π_{n}E: = colim_{2m+n ≥ 0, m ≥ 0} π_{n+2m}E_{m} → π_{n+2m+2}E_{m} ∧ ℂP^{1} → π_{n+2(m+1)}E_{1+m} → ⋯

be the n-th stable homotopy group of E. Note that homotopy groups of non-degenerately based compactly generated topological spaces commute with filtered colimits. If E is cofibrant, E_{n} is in particular non-degenerately based for all n ≥ 0. A map f: E → F of ℂP^{1}-spectra is a stable equivalence if the induced map π_{n}f: π_{n}E^{cof} → π_{n}F^{cof} is an isomorphism for all n ∈ ℤ. It is a stable fibration if it has the right lifting property with respect to all stable acyclic cofibrations.

Similarly, one may form the category Sp^{Σ} = Sp^{Σ}(Top, ℂP^{1}) of symmetric ℂP^{1}-spectra in Top. Cofibrations are generated by

{Fr_{m}^{Top,Σ}(|∂Δ^{n} ↪ Δ^{n}|_{+})}_{m,n≥0}

and a symmetric ℂP^{1}-spectrum is stably fibrant if its underlying ℂP^{1}-spectrum is stably fibrant. A map f: E → F of symmetric ℂP^{1}-spectra is a stable equivalence if the induced map sSet_{Sp}^{Σ}(f^{cof}, G) of simplicial sets of maps is an isomorphism for all stably fibrant symmetric ℂP^{1}-spectra G. It is a stable fibration if it has the right lifting property with respect to all stable acyclic cofibrations.

Theorem A.7.1. Stable equivalences, stable fibrations and cofibrations form (symmetric monoidal) model structures on the categories of (symmetric) ℂP^{1}-spectra in Top. The functor forgetting the symmetric group actions is a right Quillen equivalence. There is a zig-zag of strict symmetric monoidal left Quillen functors connecting Sp^{Σ}(Top, ℂP^{1}) and Sp^{Σ}(Top, S^{1}). In particular, the homotopy category of (symmetric) ℂP^{1}-spectra is equivalent as a closed symmetric monoidal and triangulated category to the stable homotopy category.

Proof. The statement about the model structures follows as in [HSS] if one replaces S^{1} by ℂP^{1} and simplicial sets by compactly generated topological spaces. The same holds for the statement about the functor forgetting the symmetric group actions. To construct the zig-zag, consider the corresponding stable model structure on the category of symmetric S^{1}-spectra in the category Sp^{Σ}(Top, ℂP^{1}), which is isomorphic as a symmetric monoidal model category to the category of symmetric ℂP^{1}-spectra in the category Sp^{Σ}(Top, S^{1}) of topological symmetric S^{1}-spectra. The suspension spectrum functors give a zig-zag

Sp^{Σ}(Top, ℂP^{1}) → Sp^{Σ}(Sp^{Σ}(Top, ℂP^{1}), S^{1}) ≅ Sp^{Σ}(Sp^{Σ}(Top, S^{1})ℂP^{1}) ← Sp^{Σ}(Top, S^{1})

(50)

of strict symmetric monoidal left Quillen functors. Since ℂP^{1} ∧ − is a left Quillen equivalence on the left hand side in the zig-zag (50) and S^{1} ∧ S^{1} ≅ ℂP^{1}, S^{1} ∧ − is a left Quillen equivalence on the left hand side as well. By [Ho2, Theorem 9.1], the arrow pointing to the right in the zig-zag (50) is a Quillen equivalence. A similar argument works for the arrow on the right hand side, which completes the proof.

Given a P^{1}-spectrum E over ℂ, one gets a ℂP^{1}-spectrum R_{ℂ}(E) = (R_{ℂ}E_{0}, R_{ℂ}E_{1}, …) with structure maps R_{ℂ}(E_{n}) ∧ P^{1} ≅ R_{ℂ}(E_{n} ∧ P^{1}) → R_{ℂ}(E_{n+1}). The right adjoint for the resulting functor R_{ℂ}: MS(ℂ) → Sp is also obtained by a levelwise application of Sing_{ℂ}. The same works for symmetric P^{1}-spectra over ℂ.

Theorem A.7.2. The functors R_{ℂ}: MS(ℂ) → Sp and R_{ℂ}: MSS(ℂ) → Sp^{Σ} are left Quillen functors, the latter being strict symmetric monoidal.

Proof. Since the diagrams

\xymatrix{MSS(ℂ) \ar[r]^{u} \ar[d]_{_{Sing_{ℂ}}} & MS(ℂ) \ar[r]^{E↦E_{n}} \ar[d]_{_{Sing_{ℂ}}} & M_{•}(ℂ) \ar[d]^{_{Sing_{ℂ}}} \\ _{Sp_{Σ}} \ar[r]^{u} & Sp \ar[r]^{E↦E_{n}} & _{Top_{•}}}

commute, R_{ℂ} preserves the generating cofibrations by A.4.1. Then Dugger’s Lemma [D, Cor. A.2] implies that R_{ℂ} is a left Quillen functor, because Sing_{ℂ} preserves weak equivalences and fibrations between fibrant objects. The fact that R_{ℂ}: MSS(ℂ) → Sp^{Σ} is strict symmetric monoidal follows from the definition of the smash product (45).

Example A.7.3. Let BGL be the P^{1}-spectrum over ℂ constructed in 1.2. Its n-th term is a closed cofibrant pointed motivic space 𝒦 weakly equivalent to ℤ × Gr. Thus the n-th term of R_{ℂ}(BGL) is weakly equivalent to BU. To show that the ℂP^{1}-spectrum R_{ℂ}(BGL) is the one representing complex K-theory, it suffices to check that the structure map 𝒦 ∧ P^{1} → 𝒦 realizes to the structure map BU ∧ ℂP^{1} → BU of complex K-theory. Consider the diagram

\xymatrix{Hom_{H_{•}(ℂ)}(ℤ × Gr, ℤ × Gr) \ar[r]^{≅} \ar[d] & K_{0}^{alg}(ℤ × Gr) \ar[d] \\ Hom_{Ho(Top_{•})}(R_{ℂ}(ℤ × Gr), R_{ℂ}(ℤ × Gr)) \ar[r]^{≅} & K_{0}^{top}(R_{ℂ}(ℤ × Gr))}

where the vertical map on the left hand side is induced by R_{ℂ} and the vertical map on the right hand side is induced by the passage from algebraic to topological complex vector bundles. The latter is a ring homomorphism, and in particular preserves the unit, which is the image of the identity map under the horizontal isomorphisms. By naturality, it follows that the diagram

\xymatrix{Hom_{H_{•}(ℂ)}(A, ℤ × Gr) \ar[r]^{≅} \ar[d] & K_{0}^{alg}(ℤ × Gr) \ar[d] \\ Hom_{Ho(Top_{•})}(R_{ℂ}(A), R_{ℂ}(ℤ × Gr)) \ar[r]^{≅} & K_{0}^{top}(R_{ℂ}(ℤ × Gr))}  (51)

commutes for every pointed motivic space A over ℂ. In particular, the structure map of BGL which corresponds to (τ_{∞} − ∞) ⊗ ([𝒪(−1)] − [𝒪]) maps to the structure map of the complex K-theory spectrum, since it corresponds to the same bundle, viewed as a topological bundle.

Proposition A.7.4. A morphism f: S → S′ of base schemes induces a strict symmetric monoidal left Quillen functor

f^{∗}: MSS(S′) → MSS(S)

such that (f^{∗}(E))_{n} = f^{∗}(E_{n}).

Proof. The structure maps of f^{∗}(E) are defined via the canonical map

\xymatrix{f^{∗}(E)_{n} ∧ P_{S′}^{1} ≅ f^{∗}(E_{n}) ∧ f^{∗}(P_{S}^{1}) ≅ f^{∗}(E_{n} ∧ P_{S}^{1}) \ar[r]^{f^{∗}(σ_{n}^{E})} & f^{∗}(E_{n+1}) = f^{∗}(E)_{n+1}.}

It follows that f^{∗} has the functor f_{∗} as right adjoint, where (f_{∗}(E))_{n} = f_{∗}(E_{n}) and

\xymatrix{σ_{n}^{f_{∗}E} = f_{∗}E_{n} ∧ P_{S′}^{1} \ar[r] & f_{∗}E_{n} ∧ f_{∗}f^{∗}P_{S′}^{1} ≅ f_{∗}E_{n} ∧ f_{∗}P_{S}^{1} \ar[r] & f_{∗}(E_{n} ∧ P_{S}^{1}) \ar[r]^{f_{∗}σ_{n}^{E}} & f_{∗}E_{n+1}.}

Theorem A.3.9 and Dugger’s lemma imply that f^{∗} preserves cofibrations and f_{∗} preserves fibrations. Since f^{∗}: M_{•}(S′) → M_{•}(S) is strict symmetric monoidal and preserves all colimits, then so is f^{∗}: MSS(S′) → MSS(S) by the definition of the smash product (45).

In particular, any complex point f: Spec(ℂ) → S of a base scheme S induces a strict symmetric monoidal left Quillen functor

MSS(S) → MSS(ℂ) → Sp^{Σ}(Top, ℂP^{1})  (52)

to the category of topological ℂP^{1}-spectra.

B Products in K-theory

One of the aim of the Appendix is to prove Proposition 1.2.2. For that we need Proposition B.1.1 and Corollary B.1.2.

B.1 Cellular schemes

Suppose that S is a regular base scheme. Recall that an S-cellular scheme is an S-scheme X equipped with a filtration X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} = X by closed subsets such that for every integer i ≥ 0 the S-scheme X_{i}∖X_{i−1} is a disjoint union of several copies of the affine space A_{S}^{i}. We do not assume that X is connected. A pointed S-cellular scheme is an S-cellular scheme equipped with a closed S-point x: S ↪ X such that s(X) is contained in one of the open cells (a cell which is an open subscheme of X). The examples we are interested in are Grassmannians, projective lines and their products. We need the following proposition which will be proved below. Its proof uses only cup-products of the form K_{i} ⊗ K_{0} → K_{i}. The latter are easily defined without knowing the Waldhausen products since the tensor product with a vector bundle is an exact functor.

Proposition B.1.1. Let (X, x) and (Y, y) be pointed smooth S-schemes. Then the sequence

0 → K_{0}^{TT}(X ∧ Y) → K_{0}^{TT}(X × Y) → K_{0}^{TT}(X ∨ Y) → 0

is short exact.

Proof. This follows from the isomorphism K_{0}^{TT} ≅ BGL^{0,0} and Example A.5.8.

Corollary B.1.2. Under the assumptions of Proposition B.1.1 let a ∈ K_{0}^{TT}(X) and b ∈ K_{0}^{TT}(Y) be such that x^{∗}(a) = 0 = y^{∗}(b) in K_{0}^{TT}(S). Then the element a⊗b ∈ K_{0}^{TT}(X ×Y) belongs to the subgroup K_{0}^{TT}(X ∧ Y). Furthermore the map

K_{0}^{TT}(X/x(S)) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(Y/y(S)) → K_{0}^{TT}(X ∧ Y)

is an isomorphism.

Proof. In fact, clearly the pull-back map K_{0}^{TT}(X ∨ Y) → K_{0}^{TT}(X) ⊕ K_{0}^{TT}(Y) is injective. Since a ⊗ b vanishes on x(S) × Y and on X × x(S) it follows that a ⊗ b vanishes on X ∨ Y. Whence a ⊗ b ∈ K_{0}^{TT}(X ∧ Y). We skip a proof of the last assertion of the Corollary since we do not need it.

With the Corollary B.1.2 in hand we are ready to prove Proposition 1.2.2. For that we use below the notation of Section 1.2. Consider the diagram

𝕂^{TT}(X) ∧ 𝕂^{TT}(X) = Ω_{s}^{1}(W_{1}(X)) ∧ Ω_{s}^{1}(W_{1}(X)) ⟶^{m} Ω_{s}^{2}(W_{2}(X)) ⟵^{ad} 𝕂^{TT}(X)  (53)

with the Waldhausen morphisms m and the adjunction weak equivalence ad. The diagram defines a morphism

𝕂^{TT} ∧ 𝕂^{TT} ⟶^{μ^{W}} 𝕂^{TT}  (54)

in H_{•}^{cm}(S) which is the Waldhausen multiplication on 𝕂^{TT}.

We claim that the morphism μ^{TT} given by (5) coincides with the Waldhausen multiplication μ^{W} on 𝕂^{TT}. In fact, the Waldhausen multiplication μ^{W} has the property that for any smooth X over S the pairing K_{0}^{TT}(X) × K_{0}^{TT}(X) → K_{0}^{TT}(X) coincides with the one given by the tensor product of vector bundles. Taking X = ℤ × Gr and using Corollary B.1.2 we see that the composite morphism

(ℤ × Gr) ∧ (ℤ × Gr) ⟶^{w∧w} 𝕂^{TT} ∧ 𝕂^{TT} ⟶^{μ^{W}} 𝕂^{TT}

represents the class (τ_{∞}−∞)⊗(τ_{∞}−∞) ∈ K_{0}^{TT}((ℤ×Gr)∧(ℤ×Gr)). However there exists a unique class in [𝕂^{TT} ∧ 𝕂^{TT}, 𝕂^{TT}]_{H_{•}(S)} with this property since w ∧ w is an isomorphism in H_{•}^{cm}(S). Whence μ^{W} = μ^{TT}. Similarly the known unit S^{0} → 𝕂^{TT} on the Thomason-Trobaugh K-theory coincides in H_{•}^{cm}(S) with e^{TT} defined above. Whence the monoidal structure (7) on the space 𝕂^{TT} coincides with the Waldhausen one. Proposition 1.2.2 is proved.

The following lemma is useful also.

Lemma B.1.3. Let (X, x) and (Y, y) be pointed smooth S-cellular schemes. Then the map

K_{i}^{TT}(S) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(X ∧ Y) → K_{i}^{TT}(X ∧ Y)

is an isomorphism and K_{0}^{TT}(X ∧ Y) is a projective K_{0}^{TT}(S)-module.

Proof. Consider the commutative diagram

\xymatrix{K_{i}(X ∧ Y) \ar[r]^{α} & K_{i}(X × Y) \ar[r]^{β} & K_{i}(X ∨ Y) \\ K_{i}(S) ⊗ K_{0}(X ∧ Y) \ar[r]^{γ} \ar[u]^{ϵ} & K_{i}(S) ⊗ K_{0}(X × Y) \ar[r]^{δ} \ar[u]_{ρ} & K_{i}(S) ⊗ K_{0}(X ∨ Y) \ar[u]_{θ}}

in which K_{i} is written for K_{i}^{TT} and the tensor product is taken over K_{0}(S). The sequence

0 → K_{i}(X ∧ Y) ⟶^{α} K_{i}(X × Y) ⟶^{β} K_{i}(X ∨ Y) → 0

is short exact as was checked in the proof of Proposition B.1.1. In particular it is short exact for i = 0. Now the sequence

0 → K_{i}(S) ⊗ K_{0}(X ∧ Y) ⟶^{γ} K_{i}(S) ⊗ K_{0}(X × Y) ⟶^{δ} K_{i}(S) ⊗ K_{0}(X ∨ Y) → 0

is short exact since K_{0}(X ∨ Y) is a projective K_{0}(S)-module.

The arrows ρ and θ are isomorphisms by Lemmas B.1.4 and B.1.6 below respectively. Whence ϵ is an isomorphism as well. Finally K_{0}(X ∧ Y) is a projective K_{0}(S)-module since the sequence 0 → K_{0}(X ∧ Y) → K_{0}(X × Y) → K_{0}(X ∨ Y) → 0 is short exact and K_{0}(X × Y) and K_{0}(X ∨ Y) are projective K_{0}(S)-modules.

Lemma B.1.4. Let X be a smooth S-cellular scheme. Then the map

K_{r}^{TT}(S) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(X) → K_{r}^{TT}(X)

is an isomorphism and K_{0}^{TT}(X) is a free K_{0}^{TT}(S)-module of rank equal to the number of cells.

Proof. The Lemma easily follows from a slightly different claim.

Claim B.1.5. Under the assumption of the Lemma the map of Quillen’s K-groups

K_{r}(S) ⊗_{K_{0}(S)} K′_{0}(X_{j}) → K′_{r}(X_{j})

is an isomorphism and K′_{0}(X_{j}) is a free K_{0}(S)-module of the expected rank.

We prove the Claim induction by j. If j = 0, then X = X_{0} is a disjoint union of several copies of the base scheme S. So the Claim is obvious in this case. Assume the Claim holds for all j ≤ d and prove it for j = d + 1. For that consider the subset X_{d+1} − X_{d}. It is a disjoint union of of several copies U_{i} of the affine space A_{S}^{d+1}. Each of the copy is an open subset of X_{d+1}. Thus X_{d}^{1} := X_{d} ∪ U_{1} is a closed subset of X_{d+1}. Consider the long localization sequence of Quillen’s K′-groups [Qu]

⋯ → K′_{r}(X_{d}) ⟶^{i_{∗}} K′_{r}(X_{d}^{1}) ⟶^{j^{∗}} K′_{r}(U_{1}) → …

where i_{∗} is the push-forward map and j^{∗} is the pull-back map. Let p : X_{1} → S be the structure map. Then j^{∗} ∘ p^{∗} is an isomorphism since S is regular. Thus j^{∗} is surjective. So i_{∗} is injective and the sequence splits in short exact sequences (split ones)

0 → K′_{r}(X_{d}) ⟶^{i_{∗}} K′_{r}(X_{d}^{1}) ⟶^{j^{∗}} K′_{r}(U_{1}) → 0.

Taking r = 0 and using the inductive assumption we see that K′_{0}(X_{d}^{1}) is a free K_{0}(S)- module of the expected rank. Taking once again r = 0 and tensoring this sequence with K_{r}(S) over K_{0}(S) we get an exact sequence of the form

0 → K_{r}(S) ⊗_{K_{0}(S)} K′_{0}(X_{d}) ⟶ K_{r}(S) ⊗_{K_{0}(S)} K′_{0}(X_{d}^{1}) ⟶ K_{r}(S) ⊗_{K_{0}(S)} K′_{0}(U_{1}) → 0.

The pairings of the form K_{r}(S)⊗_{K_{0}(S)}K′_{0}(V) → K′_{r}(V) define a morphism of the last short exact sequence to the previous one. Using the inductive assumption we see that the pairing K_{r}(S) ⊗_{K_{0}(S)} K′_{0}(X_{d}^{1}) → K′_{r}(X_{d}^{1}) is an isomorphism. Repeating this procedure several times we get that the pairing K_{r}(S) ⊗_{K_{0}(S)} K′_{0}(X_{d+1}) → K′_{r}(X_{d+1}) is an isomorphism. The induction runs, whence the Claim and the Lemma.

Lemma B.1.6. Let (X, x) and (Y, y) be pointed smooth S-cellular schemes. Then the map

K_{i}^{TT}(S) ⊗_{K_{0}^{TT}(S)} K_{0}^{TT}(X ∨ Y) → K_{i}^{TT}(X ∨ Y)

is an isomorphism and K_{0}^{TT}(X ∨ Y) is a projective K_{0}^{TT}(S)-module.

Proof. Consider the commutative diagram

\xymatrix{K_{i}(X ∨ Y) \ar[r]^{α} & K_{i}(X) ⊕ K_{i}(Y) \ar[r]^{β} & K_{i}(S) \\ K_{i}(S) ⊗ K_{0}(X ∨ Y) \ar[r]^{γ} \ar[u]^{ϵ} & K_{i}(S) ⊗ (K_{0}(X) ⊕ K_{0}(Y)) \ar[r]^{δ} \ar[u]_{ρ} & K_{i}(S) ⊗ K_{0}(S) \ar[u]_{θ}}

in which K_{i} is written for K_{i}^{TT} and the tensor product is taken over K_{0}(S). Its rows are exact, the arrows α and γ are injective, the arrows β and δ are surjective. The arrows ρ and θ are isomorphisms by Lemma B.1.4. Whence ϵ is an isomorphism too.

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