Slices and Transfers

Marc Levine^{1}

Received: September 28, 2009

Abstract. We study Voevodsky’s slice tower for S^{1}-spectra, and raise a number of questions regarding its properties. We show that the 0th slice does not in general admit transfers, although it does for a ℙ^{1}-loop-spectrum. We define a new tower for each of the higher slices, and show that the layers in these towers have the structure of Eilenberg-Maclane spectra on effective motives.

2010 Mathematics Subject Classification: Primary 14F42, 19E15; Secondary 19E08, 14C25, 55P42 Keywords and Phrases: Algebraic cycles, Morel-Voevodsky stable homotopy category, slice filtration

Contents

Introduction 394 1. Infinite ℙ^{1}-loop spectra 397 2. An example 399 3. Co-transfer 401 4. Co-group structure on ℙ^{1} 407 5. Slice localizations and co-transfer 411 6. Higher loops 425 7. Supports and co-transfers 426 8. Slices of loop spectra 431 9. Transfers on the generalized cycle complex 434 10. The Friedlander-Suslin tower 440 References 442

1Research supported by the NSF grant DMS-0801220 and the Alexander von Humboldt Foundation

Documenta Mathematica · Extra Volume Suslin (2010) 393–443

Introduction

Voevodsky [22] has defined an analog of the classical Postnikov tower in the setting of motivic stable homotopy theory by replacing the simplicial suspension Σ_{s} := − ∧ S^{1} with ℙ^{1}-suspension Σ_{ℙ^{1}} := − ∧ ℙ^{1}; we call this construction the motivic Postnikov tower. Let SH(k) denote the motivic stable homotopy category of ℙ^{1}-spectra. One of the main results on motivic Postnikov tower in this setting is

Theorem 1. Let k be a field of characteristic zero. For E ∈ SH(k), the slices s_{n}E have the natural structure of an Hℤ-module, and hence determine objects in the category of motives DM(k).

The statement is a bit imprecise, as the following expansion will make clear: Röndigs-Østvær [19, 20] have shown that the homotopy category of strict Hℤmodules is equivalent to the category of motives, DM(k). Additionally, Voevodsky [22] and the author [11] have shown that the 0th slice of the sphere spectrum 𝕊 in SH(k) is isomorphic to Hℤ. Each E ∈ SH(k) has a canonical structure of a module over the sphere spectrum 𝕊, and thus the slices s_{n}E acquire an Hℤ-module structure, in SH_{S^{1}}(k). This has been refined to the model category level by Pelaez [17], showing that the slices of a ℙ^{1}-spectrum E have a natural structure of a strict Hℤ-module, hence are motives. Let Spt_{S^{1}}(k) denote the category of S^{1}-spectra, with its homotopy category (for the 𝔸^{1} model structure) SH_{S^{1}}(k). The analog for motives is the category complexes of presheaves with transfer and its 𝔸^{1}-homotopy category DM^{eff}(k), the category of effective motives over k. We consider the motivic Postnikov tower inSH_{S^{1}}(k), and ask the questions:

(1) Is there a ring object in Spt_{S^{1}}(k), Hℤ^{eff}, such that the homotopy category of Hℤ^{eff} modules is equivalent to the category of effective motives DM^{eff}(k)?

(2) What properties (if any) need an S^{1}-spectrum E have so that the slices s_{n}E have a natural structure of Eilenberg-Maclane spectra of a homotopy invariant complex of presheaves with transfer?

Naturally, if Hℤ^{eff} exists as in (1), we are asking the slices in (2) to be (strict) Hℤ^{eff} modules. Of course, a natural candidate for Hℤ^{eff} would be the 0-S^{1}- spectrum of Hℤ, Ω_{ℙ^{1}}^{∞}Hℤ, but as far as I know, this property has not yet been investigated. As we shall see, the 0-S^{1}-spectrum of a ℙ^{1}-spectrum does have the property that its (S^{1}) slices are motives, while one can give examples of S^{1}-spectra for which the 0th slice does not have this property. This suggests a relation of the question of the structure of the slices of an S^{1}-spectrum with a motivic version of the recognition problem:

(3) How can one tell if a given S^{1}-spectrum is an n-fold ℙ^{1}-loop spectrum? In this paper, we prove two main results about the “motivic” structure on the slices of S^{1}-spectra:

Theorem 2. Suppose char k = 0. Let E be an S^{1}-spectrum. Then for each n ≥ 1, there is a tower

… → ρ_{≥p+1}s_{n}E → ρ_{≥p}s_{n}E → … → s_{n}E

in SH_{S^{1}}(k) with the following properties:

(1) the tower is natural in E.

(2) Let s_{p,n}E be the cofiber of ρ_{≥p+1}s_{n}E → ρ_{≥p}s_{n}E. Then there is a homotopy invariant complex of presheaves with transfers π̂_{p}((s_{n}E)^{(n)})^{∗} ∈ DM_{−}^{eff}(k) and a natural isomorphism in SH_{S^{1}}(k),

EM_{𝔸^{1}}(π̂_{p}((s_{n}E)^{(n)})^{∗}) ≅ s_{p,n}E,

where EM_{𝔸^{1}} : DM_{−}^{eff}(k) → SH_{S^{1}}(k) is the Eilenberg-Maclane spectrum functor.

This result is proven in section 9. One can say a bit more about the tower appearing in theorem 2. For instance, holim_{p} fib(ρ_{≥p}s_{n}E → s_{n}E) is weakly equivalent to zero, so the spectral sequence associated to this tower is weakly convergent. If s_{n}E is globally N-connected (i.e., there is an N such that s_{n}E(X) is N-connected for all X ∈ Sm/k) then the spectral sequence is strongly convergent. The “π̂_{p}” appears in the notation due to the construction of π̂_{p}((s_{n}E)^{(n)})^{∗}(X) arising from a “Bloch cycle complex” of codimension n cycles on X × Δ^{∗} with coefficients in π_{p}(Ω^{n}s_{n}E). In other words, the higher slices of an arbitrary S^{1}-spectrum have some sort of transfers “up to filtration”. The situation for the 0th slice appears to be more complicated, but for a ℙ^{1}-loop spectrum we have at least the following result:

Theorem 3. Suppose char k = 0. Take E ∈ SH_{ℙ^{1}}(k). Then for all m, the homotopy sheaf π_{m}(s_{0}Ω_{ℙ^{1}}E) has a natural structure of a homotopy invariant sheaf with transfers.

We actually prove a more precise result (corollary 8.5) which states that the 0th slice s_{0}Ω_{ℙ^{1}}E is itself a presheaf with transfers, with values in the stable homotopy category SH, i.e., s_{0}Ω_{ℙ^{1}}E has “transfers up to homotopy”. This raises the question:

(4) Is there an operad acting on s_{0}Ω_{ℙ^{1}}^{n}E which shows that s_{0}Ω_{ℙ^{1}}^{n}E admits transfers up to homotopy and higher homotopies up to some level?

Part of the motivation for this paper came out of discussions with Hélène Esnault concerning the (admittedly vague) question: Given a smooth projective variety X over some field k, that admits a 0-cycle of degree 1, are there “motivic” properties of X that lead to the existence of a k-point, or conversely, that give obstructions to the existence of a k-point? The fact that the existence of 0-cycles of degree 1 has something to do with the transfer maps from 0-cycles on X_{L} to 0-cycles on X, as L runs over finite field extensions of k, while the lack of a transfer map in general appears to be closely related to the subtlety of the existence of k-points led to our inquiry into the “motivic” nature of the spaces Ω_{ℙ^{1}}^{n}Σ_{ℙ^{1}}^{n}X_{+}, or rather, their associated S^{1}-spectra.

Notation and conventions. Throughout this paper the base-field k will be a field of characteristic zero. Sm/k is the category of smooth finite type k-schemes. We let Spc_{•} denote the category of pointed space, i.e., pointed simplicial sets, and H_{•} the homotopy category of Spc_{•}, the unstable homotopy category. Similarly, we let Spt denote the category of spectra and SH its homotopy category, the stable homotopy category. We let Spc_{•}(k) denote the category of pointed spaces over k, that is, the category of Spc_{•}-valued presheaves on Sm/k, and Spt_{S^{1}}(k) the category of S^{1}-spectra over k, that is, the category of Spt-valued presheaves on Sm/k. We let Spt_{ℙ^{1}}(k) denote the category of ℙ^{1}-spectra over k, which we take to mean the category of Σ_{ℙ^{1}}-spectrum objects over Spt_{S^{1}}(k). Concretely, an object is a sequence (E_{0}, E_{1}, …), E_{n} ∈ Spt_{S^{1}}(k), together with bonding maps ϵ_{n} : Σ_{ℙ^{1}}E_{n} → E_{n+1}. Regarding the categories Spt_{S^{1}}(k), SH_{S^{1}}(k) and SH(k), we will use the notation spelled out in [11]. In addition to this source, we refer the reader to [8, 14, 15, 20, 22]. Relying on these sources for details, we remind the reader that H_{•}(k) is the homotopy category of the category of Spc_{•}(k), for the socalled 𝔸^{1}-model structure. Similarly SH_{S^{1}}(k) and Spt_{ℙ^{1}}(k) have model structures, which we call the 𝔸^{1}-model structures, and SH_{S^{1}}(k), SH(k) are the respective homotopy categories. For details on the category DM^{eff}(k), we refer the reader to [3, 5]. We will be passing from the unstable motivic (pointed) homotopy category over k, H_{•}(k), to the motivic homotopy category of S^{1}-spectra over k, SH_{S^{1}}(k), via the infinite (simplicial) suspension functor

Σ_{s}^{∞} : H_{•}(k) → SH_{S^{1}}(k)

For a smooth k-scheme X ∈ Sm/k and a subscheme Y of X (sometimes closed, sometimes open), we let (X, Y) denote the homotopy push-out in the diagram

\xymatrix{Y \ar[r] \ar[d] & X \\ Spec k}

and as usual write X_{+} for (X ⨿ Spec k, Spec k). We often denote Spec k by ∗. For an object S of H_{•}(k), we often use S to denote Σ_{s}^{∞}S ∈ SH_{S^{1}}(k) when the context makes the meaning clear; we also use this convention when passing to various localizations of SH_{S^{1}}(k). We let [n] denote the set {0, …, n} with the standard total order, and let Ord denote the category with objects [n], n = 0, 1, … and morphisms the order-preserving maps of sets. Let Δ^{n} denote the algebraic n-simplex Spec k[t_{0}, …, t_{n}]/ ∑_{i} t_{i} − 1, with vertices v_{0}^{n}, …, v_{n}^{n}, where v_{i}^{n} is defined by t_{j} = 0 for j ≠ i. As is well-known, sending g : [n] → [m] to the affine-linear extension Δ(g) : Δ^{n} → Δ^{m} of the map on the set of vertices, v_{j}^{n} ↦ v_{g(j)}^{m} defines the cosimplicial k-scheme n ↦ Δ^{n}. We recall that, for a category C, the category of pro-objects in C, pro-C, has as objects functors f : I → C, i ∈ I, where I is a small left-filtering category, a morphism (f : I → C) → (g : J → C) is a pair (ρ : I → J, θ : f → g ∘ ρ), with the evident composition, and we invert morphisms of the form

(ρ : I → J, id : f := g ∘ ρ → g ∘ ρ)

if ρ : I → J has image a left co-final subcategory of J. In this paper we use categories of pro-objects to allow us to use various localizations of smooth finite type k-schemes. This is a convenience rather than a necessity, as all maps and relations lift to the level of finite type k-schemes. I am very grateful to the referee for making a number of perceptive and useful comments, which led to the correction of some errors and an improvement of the exposition. Dedication. This paper is warmly dedicated to Andrei Suslin, who has given me more inspiration than I can hope to tell.

1. Infinite ℙ^{1}-loop spectra

We first consider the case of the 0-S^{1}-spectrum of a ℙ^{1}-spectrum. We recall some constructions and results from [20]. We let

Ω_{ℙ^{1}}^{∞} : SH(k) → SH_{S^{1}}(k)
Ω_{ℙ^{1},mot}^{∞} : DM(k) → DM^{eff}(k)

be the (derived) 0-spectrum (resp. 0-complex) functor, let

EM_{𝔸^{1}} : DM(k) → SH(k)
EM_{𝔸^{1}}^{eff} : DM^{eff}(k) → SH_{S^{1}}(k)

the respective Eilenberg-Maclane spectrum functors. The functors Ω_{ℙ^{1}}^{∞}, Ω_{ℙ^{1},mot}^{∞} are right adjoints to the respective infinite suspension functors

Σ_{ℙ^{1}}^{∞} : SH_{S^{1}}(k) → SH(k)
Σ_{ℙ^{1},mot}^{∞} : DM^{eff}(k) → DM(k)

and the functors EM_{𝔸^{1}}, EM_{𝔸^{1}}^{eff} are similarly right adjoints to the “linearization” functors

ℤ^{tr} : SH(k) → DM(k)
ℤ^{tr} : SH_{S^{1}}(k) → DM^{eff}(k)

induced by the functor ℤ^{tr} from simplicial presheaves on Sm/k to presheaves with transfer on Sm/k sending the representable presheaf Hom_{Sm/k}(−, X) to the free presheaf with transfers ℤ_{X}^{tr} := Hom_{SmCor(k)}(−X), and taking the Kan extension. The discussion in [20, §2.2.1] show that both these adjoint pairs arise from Quillen adjunctions on suitable model categories (followed by a chain of Quillen equivalences), where on the model categories, the functors EM_{𝔸^{1}}, EM_{𝔸^{1}}^{eff} are just forgetful functors and the functors Ω^{∞} just take a sequence E_{0}, E_{1}, … to E_{0}. Thus one has

(1.1)  EM_{𝔸^{1}}^{eff} ∘ Ω_{ℙ^{1},mot}^{∞} ≅ Ω_{ℙ^{1}}^{∞} ∘ EM_{𝔸^{1}}

as one has an identity of the two functors on the model categories.

Theorem 1.1. Fix an integer n ≥ 0. Then there is a functor

Mot^{eff}(s_{n}) : SH(k) → DM^{eff}(k)

and a natural isomorphism

φ_{n} : EM_{𝔸^{1}}^{eff} ∘ Mot^{eff}(s_{n}) → s_{n}^{eff} ∘ Ω_{ℙ^{1}}^{∞}

of functors from SH(k) to SH_{S^{1}}(k).

In other words, for E ∈ SH(k), there is a canonical lifting of the slice s_{n}^{eff}(Ω_{ℙ^{1}}^{∞}E) to a motive Mot^{eff}(s_{n})(E).

Proof. By Pelaez [18, theorem 3.3], there is a functor

Mot(s_{n}) : SH(k) → DM(k)

and a natural isomorphism

Φ_{n} : EM_{𝔸^{1}} ∘ Mot(s_{n}) → s_{n}

i.e., the slice s_{n}E lifts canonically to a motive Mot(s_{n})(E). Now apply the 0-complex functor to define

Mot^{eff}(s_{n}) := Ω_{ℙ^{1},mot}^{∞} ∘ Mot(s_{n}).

We have canonical isomorphisms

EM_{𝔸^{1}}^{eff} ∘ Ω_{ℙ^{1},mot}^{∞} ∘ Mot(s_{n}) ≅ Ω_{ℙ^{1}}^{∞} ∘ EM_{𝔸^{1}} ∘ Mot(s_{n})
≅ Ω_{ℙ^{1}}^{∞} ∘ s_{n}
≅ s_{n}^{eff} ∘ Ω_{ℙ^{1}}^{∞}.

Indeed, the first isomorphism is (1.1) and the second is Pelaez’s isomorphism Φ_{n}. For the third, we have given in [11] an explicit model for s_{n} in terms of the functors s_{m}^{eff} as follows: given a ℙ^{1}-spectrum E, represented as a sequence of S^{1}-spectra E_{0}, E_{1}, … together with bonding maps Σ_{ℙ^{1}}E_{n} → E_{n+1}, suppose that E is fibrant. In particular, the adjoints E_{n} → Ω_{ℙ^{1}}E_{n+1} of the bonding maps are weak equivalences and E_{0} = Ω_{ℙ^{1}}^{∞}E. It follows from [11, theorem 9.0.3] that s_{n}E is represented by the sequence (s_{n}^{eff}E_{0}, s_{n+1}^{eff}E_{1}, …, s_{n+m}^{eff}E_{m}, …), with certain bonding maps (defined in [11, §8.3]). In addition, by [11, theorem 4.1.1] this new sequence is termwise weakly equivalent to its fibrant model. This defines the natural isomorphism Ω_{ℙ^{1}}^{∞}s_{n}E ≅ s_{n}^{eff}E_{0} ≅ s_{n}^{eff}Ω_{ℙ^{1}}E. □

In other words, the slices of an infinite ℙ^{1}-loop spectrum are effective motives.

2. An example

We now show that the 0th slice of an S^{1}-spectrum is not always a motive. In fact, we will give an example of an Eilenberg-Maclane spectrum whose 0th slice does not admit transfers. For this, note the following:

Lemma 2.1. Let p : Y → X be a finite Galois cover in Sm/k, with Galois group G. Let F be a presheaf with transfers on Sm/k. Then the composition

p^{∗} ∘ p_{∗} : F(Y) → F(Y)

is given by

p^{∗} ∘ p_{∗}(x) = ∑_{g∈G} g^{∗}(x)

Proof. Letting Γ_{p} ⊂ Y × X be the graph of p, and Γ_{g} ⊂ Y × Y the graph of g : Y → Y for g ∈ G, one computes that

Γ_{p}^{t} ∘ Γ_{p} = ∑_{g∈G} Γ_{g},

whence the result. □

Now let C be a smooth projective curve over k, having no k-rational points. We assume that C has genus g > 0, so every map 𝔸_{F}^{1} → C_{F} over a field F ⊃ k is constant (C is 𝔸^{1}-rigid). Let ℤ_{C} be the representable presheaf:

ℤ_{C}(Y) := ℤ[Hom_{Sm/k}(Y, C)].

ℤ_{C} is automatically a Nisnevich sheaf; since C is 𝔸^{1}-rigid, ℤ_{C} is also homotopy invariant. Furthermore ℤ_{C} is a birational sheaf, that is, for each dense open immersion U → Y in Sm/k, the restriction map ℤ_{C}(Y) → ℤ_{C}(U) is an isomorphism. To see this, it suffices to show that Hom_{Sm/k}(Y, C) → Hom_{Sm/k}(U, C) is an isomorphism, and for this, take a morphism f : U → C. Then the projection to Y of the closure Γ̄ of the graph of f in Y × C is proper and birational. But since Y is regular, each fiber of Γ̄ → Y is rationally connected, hence maps to a point of C, and thus Γ̄ → Y is birational and 1-1. By Zariski’s main theorem, Γ̄ → Y is an isomorphism, hence f extends to f̄: Y → C, as claimed. Next, ℤ_{C} satisfies Nisnevich excision. This is just a general property of birational sheaves. In fact, let

\xymatrix{V \ar[r]^{j_{V}} \ar[d]_{f_{|V}} & Y \ar[d]^{f} \\ U \ar[r]_{j_{U}} & X}

be an elementary Nisnevich square, i.e., the square is cartesian, f is étale, j_{U} and j_{V} are open immersions, and f induces an isomorphism Y \ V → X \ U.

We may assume that U and V are dense in X and Y. Let F be a birational sheaf on Sm/k, and apply F to this diagram. This gives us the square

\xymatrix{F(X) \ar[r]^{j_{U}^{∗}} \ar[d]_{f^{∗}} & F(U) \ar[d]^{f_{|V}^{∗}} \\ F(Y) \ar[r]_{j_{V}^{∗}} & F(V)}

As the horizontal arrows are isomorphisms, we have the exact sequence

0 → F(X) → F(U) ⊕ F(Y) → F(V) → 0.

Thus, F transforms elementary Nisnevich squares to distinguished triangles in D(Ab); by definition, F therefore satisfies Nisnevich excision. Let EM_{s}(ℤ_{C}) denote presheaf of Eilenberg-Maclane spectra on Sm/k associated to ℤ_{C}, that is, for U ∈ Sm/k, EM_{s}(ℤ_{C})(U) ∈ Spt is the Eilenberg-Maclane spectrum associated to the abelian group ℤ_{C}(U). Since ℤ_{C} is homotopy invariant and satisfies Nisnevich excision, EM_{s}(ℤ_{C}) is weakly equivalent as a presheaf on Sm/k to its fibrant model in SH_{S^{1}}(k) (EM_{s}(ℤ_{C}) is quasifibrant)^{2}. In addition, the canonical map

EM_{s}(ℤ_{C}) → s_{0}(EM_{s}(ℤ_{C}))

is an isomorphism in SH_{S^{1}}(k). Indeed, since EM_{s}(ℤ_{C}) is quasi-fibrant, a quasi-fibrant model for s_{0}(EM_{s}(ℤ_{C})) may be computed by the method of [11, §5] as follows: Take Y ∈ Sm/k and let F = k(Y). Let Δ_{F,0}^{n} be the semi-local algebraic n-simplex over F, that is,

Δ_{F,0}^{n} = Spec(O_{Δ_{F}^{n} ,v}); v = {v_{0}, …, v_{n}}.

The assignment n ↦ Δ_{F,0}^{n} forms a cosimplicial subscheme of n ↦ Δ_{F}^{n} and for a quasi-fibrant S^{1}-spectrum E, there is a natural isomorphism in SH

s_{0}(E)(Y) ≅ E(Δ_{F,0}^{∗}),

where E(Δ_{F,0}^{∗}) denotes the total spectrum of the simplicial spectrum n ↦ E(Δ_{F,0}^{n}). If now E happens to be a birational S^{1}-spectrum, meaning that j^{∗} : E(Y) → E(U) is a weak equivalence for each dense open immersion j : U → Y in Sm/k, then the restriction map

j^{∗} : E(Δ_{Y}^{∗}) → E(Δ_{F,0}^{∗}) ≅ s_{0}(E)(Y)

is a weak equivalence. Thus, as E is quasi-fibrant and hence homotopy invariant, we have the sequence of isomorphisms in SH

E(Y) → E(Δ_{Y}^{∗}) → E(Δ_{F,0}^{∗}) ≅ s_{0}(E)(Y),

and hence E → s_{0}(E) is an isomorphism in SH_{S^{1}}(k). Taking E = EM_{s}(ℤ_{C}) verifies our claim.

2The referee has pointed out that, using the standard model (ℤ_{C}, Bℤ_{C}, …, B^{n}ℤ_{C}, …)

for EM_{s}(ℤ_{C}), EM_{s}(ℤ_{C}) is actually fibrant in the projective model structure.

Documenta Mathematica · Extra Volume Suslin (2010) 393–443

Finally, ℤ_{C} does not admit transfers. Indeed, suppose ℤ_{C} has transfers. Let k → L be a Galois extension such that C(L) ≠ ∅; let G be the Galois group. Since ℤ_{C}(k) = {0} (as we have assumed that C(k) = ∅), the push-forward map

p_{∗} : ℤ_{C}(L) → ℤ_{C}(k)

is the zero map, hence p^{∗} ∘ p_{∗} = 0. But for each L-point x of C, lemma 2.1 tells us that

p^{∗} ∘ p_{∗}(x) = ∑_{g∈G} x^{g} ≠ 0,

a contradiction. Thus the homotopy sheaf

π_{0}(s_{0}EM_{s}(ℤ_{C})) = π_{0}(EM_{s}(ℤ_{C})) = ℤ_{C}

does not admit transfers, giving us the example we were seeking. Even if we ask for transfers in a weaker sense, namely, that there is a functorial separated filtration F^{∗}ℤ_{C} admitting transfers on the associated graded gr_{F}^{∗} ℤ_{C}, a slight extension of the above argument shows that this is not possible as long as the filtration on ℤ_{C}(L) is finite. Indeed, p^{∗}p_{∗} would send F^{n}ℤ_{C}(L) to F^{n+1}ℤ_{C}(L), so (p^{∗}p_{∗})^{N} = 0 for some N ≥ 1, and hence N · ∑_{g∈G} x^{g} = 0, a contradiction.

3. Co-transfer

In this section, k will be an arbitrary perfect field. We recall how one uses the deformation to the normal bundle to define the “co-transfer”

(ℙ_{F}^{1} , 1_{F}) → (ℙ_{F}^{1} (x), 1_{F})

for a closed point x ∈ 𝔸_{F}^{1} ⊂ ℙ_{F}^{1} , with chosen generator f ∈ m_{x}/m_{x}^{2}. For later use, we work in a somewhat more general setting: Let S be a smooth finite type k-scheme and x a regular closed subscheme of ℙ_{S}^{1} \{1} ⊂ ℙ_{S}^{1}, such that the projection x → S is finite. Let m_{x} ⊂ O_{ℙ_{S}^{1}} be the ideal sheaf of x. We assume that the invertible sheaf m_{x}/m_{x}^{2} on x is isomorphic to the trivial invertible sheaf O_{x}, and we choose a generator f ∈ Γ(x, m_{x}/m_{x}^{2}) over O_{x}. We will eventually replace S with a semi-local affine scheme, S = Spec R, for R a smooth semi-local k-algebra, essentially of finite type over k, for instance, R = F a finitely generated separable field extension of k. Although this will take us out of the category H(k), this will not be a problem: when we work with a smooth scheme Y which is essentially of finite type over k, we will consider Y as a pro-object in H(k), and we will be interested in functors on H(k) of the form Hom_{H(k)}(Y, −), which will then be a well-defined filtered colimit of co-representable functors. Let (X_{0} : X_{1}) be the standard homogeneous coordinates on ℙ^{1}. We let s := X_{1}/X_{0} be the standard parameter on ℙ^{1}, and as usual, write 0 = (1 : 0), ∞ = (0 : 1), 1 = (1 : 1). We often write 0, 1, ∞ for the subschemes 0_{X}, 1_{X}, ∞_{X} of ℙ_{X}^{1}.

Let μ : W_{x} → ℙ_{S}^{1} × 𝔸^{1} be the blow-up of ℙ_{S}^{1} × 𝔸^{1} along (x, 0) with exceptional divisor E. Let s_{x}, C_{0} be the proper transforms s_{x} = μ^{−1}[x × 𝔸^{1}], C_{0} = μ^{−1}[ℙ^{1} × 0]. Let t be the standard parameter on 𝔸^{1} and let f̃ be a local lifting of f to a section of m_{x}; the rational function f̃/t restricts to a well-defined rational parameter on E, independent of the choice of lifting, and thus defines a globally defined isomorphism

f/t : E → ℙ_{x}^{1}.

We identify E with ℙ_{x}^{1} by sending s_{x} ∩ E to 0, C_{0} ∩ E to 1 and the section on E defined by f/t = 1 to ∞. Denote this isomorphism by

φ_{f} : ℙ_{x}^{1} → E;

we write (0, 1, ∞) for (s_{x} ∩E, C_{0} ∩E, f/t = 1), when the context makes it clear we are referring to subschemes of E. We let W_{x}^{(s_{x})}, E^{(0)}, (ℙ_{F}^{1} )^{(0)} be following homotopy push-outs

W_{x}^{(s_{x})} := (W_{x}, W_{x} \ s_{x}),
E^{(0)} := (E, E \ 0),
(ℙ_{S}^{1})^{(0)} := (ℙ_{S}^{1}, ℙ_{S}^{1} \ 0).

Since (𝔸_{x}^{1}, 0) ≅ ∗ in H_{•}(k), the respective identity maps induce isomorphisms

(E, 1) → E^{(0)},
(ℙ_{S}^{1}, 1) → (ℙ_{S}^{1})^{(0)}.

Composing with the isomorphism φ_{f} : (ℙ_{x}^{1}, 1) → (E, 1), the inclusion E → W_{x} induces the map

i_{0,f} : (ℙ_{x}^{1}, 1) → W_{x}^{(s_{x})}.

The proof of the homotopy purity theorem of Morel-Voevodsky [15, theorem 2.23] yields as a special case that i_{0,f} is an isomorphism in H_{•}(k). This enables us to define the “co-transfer map” as follows:

Definition 3.1. Let x ⊂ ℙ_{S}^{1} \ 1_{S} be a closed subscheme, smooth over k and finite over S, and suppose that m_{x}/m_{x}^{2} is a free O_{x}-module with generator f. The map

co-tr_{x,f} : (ℙ_{S}^{1}, 1) → (ℙ_{x}^{1}, 1)

in H_{•}(k) is defined to be the composition

(ℙ_{S}^{1}, 1) ⟶^{i_{1}} W_{x}^{(s_{x})} ⟶^{i_{0,f}^{−1}} (ℙ_{x}^{1}, 1).

Let X ∈ H_{•}(k) be a ℙ^{1}-loop space, i.e., X ≅ Ω_{ℙ}^{1}Y := Maps_{•}(ℙ^{1}, Y) for some Y ∈ H_{•}(k). For x ⊂ ℙ_{S}^{1} and f as above, one has the transfer map

X (x) → X (S)

in H_{•} defined by pre-composing with the co-transfer map

co-tr_{x,f} : (ℙ_{S}^{1}, 1) → (ℙ_{x}^{1}, 1).

We will find modification of this construction useful in the sequel, namely, in the proof of lemma 5.9 and lemma 5.11. Let s_{1} := 1_{S} × 𝔸^{1} ⊂ ℙ_{S}^{1} × 𝔸^{1}; as W_{x} → ℙ_{S}^{1} × 𝔸^{1} is an isomorphism over a neighborhood of s_{1}, we view s_{1} as a closed subscheme of W_{x}. We write W for W_{x}, etc., when the context makes the meaning clear.

Lemma 3.2. Let x ⊂ ℙ_{S}^{1} \ 1_{S} be a closed subscheme, smooth over k. Suppose that x → S is finite and étale. Then the identity on W induces an isomorphism

(W, C_{0} ∪ s_{1}) → W^{(s_{x})}

in H_{•}(k).

Proof. As s_{1} ≅ 𝔸_{S}^{1}, with C_{0} ∩ s_{1} = 0_{S}, the inclusion C_{0} → C_{0} ∪ s_{1} is an isomorphism in H(k). Thus, we need to show that (W, C_{0}) → W^{(s_{x})} is an isomorphism in H_{•}(k). As W^{(s_{x})} = (W, W \ s_{x}), we need to show that C_{0} → W \ s_{x} is an isomorphism in H(k). To aid in the proof, we will prove a more general result, namely, let U ⊂ ℙ_{S}^{1} be a open subscheme containing x. We consider W as a scheme over ℙ_{S}^{1} via the composition

W ⟶^{μ} ℙ_{S}^{1} × 𝔸_{S}^{1} ⟶^{p_{1}} ℙ_{S}^{1}

and for a subscheme Z of W, let Z_{U} denote the pull-back Z ×_{ℙ_{S}^{1}} U. Then we will show that

C_{0U} → W_{U} \ s_{x}

is an isomorphism in H(k). We first reduce to the case in which x → S is an isomorphism (in Sm/k). For this, we have the étale map q : ℙ_{x}^{1} → ℙ_{S}^{1} and the canonical x-point of ℙ_{x}^{1}, which we write as x̃. Let U(x) ⊂ ℙ_{x}^{1} be a Zariski open neighborhood of x̃ such that q^{−1}(x) ∩ U(x) = {x̃}. This gives us the elementary Nisnevich square

U(x) :=  \xymatrix{U(x) \ x̃ \ar[r] \ar[d] & U(x) \ar[d] \\ ℙ_{S}^{1} \ x \ar[r] & ℙ_{S}^{1};}

for each ℙ_{S}^{1}-scheme Z → ℙ_{S}^{1} we thus have the elementary Nisnevich square U(x) ×_{ℙ_{S}^{1}} Z, giving a Nisnevich cover of Z. Let V ⊂ ℙ_{S}^{1} be an open subscheme with x ∩ V = ∅. Then W_{V} → V ×_{S} 𝔸_{S}^{1} is an isomorphism and W_{V} ∩ s_{x} = ∅. Similarly C_{0V} → V is an isomorphism, and thus C_{0V} → W_{V} is an isomorphism in H(k). Replacing S with x, and considering the map of elementary Nisnevich squares

U(x) ×_{ℙ_{S}^{1}} C_{0U} → U(x) ×_{ℙ_{S}^{1}} (W_{U} \ s_{x})

induced by C_{0} → W \ s_{x}, we achieve the desired reduction. A similar Mayer-Vietoris argument allows us to replace S with a Zariski open cover of S, so, changing notation, we may assume that x is the point 0 := (1 : 0) of ℙ_{S}^{1}.

Using the open cover of ℙ_{S}^{1} by the affine open subsets U_{0} := ℙ_{S}^{1} \1, U_{1} := ℙ_{S}^{1} \0 and arguing as above, we may assume that U is a subset of U_{0}, which we identify with 𝔸_{S}^{1} by sending (0, ∞) to (0, 1). We may also assume that 0_{S} ⊂ U. Using coordinates (t_{1}, t_{2}) for 𝔸^{2}, (t_{1}, t_{2}, t_{3}) for 𝔸^{3}, the scheme W_{U_{0}} \ s_{0} is isomorphic to the closed subscheme of 𝔸_{S}^{3} defined by t_{2} = t_{1}t_{3}, with μ being the projection (t_{1}, t_{2}, t_{3}) ↦ (t_{1}, t_{2}). C_{0U_{0}} is the subscheme of W_{U_{0}} \ s_{0} defined by t_{3} = 0. The projection p_{13} : W_{U_{0}} \s_{0} → 𝔸_{S}^{2} is thus an isomorphism, sending C_{0} to 𝔸_{S}^{1} × 0. Let y = U \ U_{0}, so y is a closed subset disjoint from 0_{S}. Then

p_{13}(μ^{−1}(y × 𝔸^{1})) = y × 𝔸^{1} ⊂ 𝔸_{S}^{2},

hence p_{13} : W_{U} \ s_{0} → 𝔸_{S}^{2} identifies W_{U} \ s_{0} with U × 𝔸^{1} and identifies C_{0U} with U × 0. Thus C_{0U} → W_{U} \ s_{0} is an isomorphism in H(k), completing the proof. □

Lemma 3.3. With hypotheses as in lemma 3.2, the inclusion E → W and isomorphism φ_{f} : ℙ^{1} → E induces an isomorphism

ĩ_{0,f} : (ℙ_{x}^{1}, 1) → (W_{x}, C_{0} ∪ s_{1})

in H_{•}(k).

Proof. We have the commutative diagram

\xymatrix{(ℙ_{x}^{1}, 1) \ar[r]^{ĩ_{0,f}} \ar[dr]_{i_{0,f}} & (W, C_{0} ∪ s_{1}) \ar[d] \\  & W^{(s_{x})}.}

The diagonal arrow is an isomorphism in H_{•}(k) by Morel-Voevodsky; the vertical arrow is an isomorphism by lemma 3.2. □

Definition 3.4. Let x ⊂ ℙ_{S}^{1} \ 1_{S} be a closed subscheme, smooth over k and finite and étale over S. Suppose that m_{x}/m_{x}^{2} is a free O_{x}-module with generator f. The map

c̃o-tr_{x,f} : (ℙ_{S}^{1}, 1) → (ℙ_{x}^{1}, 1)

in H_{•}(k) is defined to be the composition

(ℙ_{S}^{1}, 1) ⟶^{i_{1}} (W, C_{0} ∪ s_{1}) ⟶^{ĩ_{0,f}^{−1}} (ℙ_{x}^{1}, 1).

Remark 3.5. Given S, x, f as in definition 3.4, we have

c̃o-tr_{x,f} = co-tr_{x,f}.

This follows directly from the commutative diagram

[figure: (ℙ_{S}^{1}, 1); i_{1}; (W, C_{0} ∪ s_{1}); id; (ℙ_{x}^{1}, 1); ĩ_{0,f}; (ℙ_{S}^{1}, 1); i_{1}; W^{(s_{x})}; (ℙ_{x}^{1}, 1).; i_{0,f}]

We examine some properties of co-tr_{x,f}. For any ordering (a, b, c) of {0, 1, ∞}, we let τ_{b,c}^{a} denote the automorphism of ℙ^{1} that fixes a and exchanges b and c. For u ∈ k^{×}, we let μ(u) be the automorphism of ℙ^{1} that fixes 0 and ∞ and sends 1 to u. We first prove the following elementary result

Lemma 3.6. The automorphism ρ := τ_{1,∞}^{0} ∘ μ(−1) ∘ τ_{1,∞}^{0} ∘ τ_{0,∞}^{1} of (ℙ^{1}, 1) is the identity in H_{•}(k).

Proof. τ_{1,∞}^{0}ρτ_{1,∞}^{0} is the automorphism of (ℙ^{1}, ∞) given by the matrix

(\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}) ∈ GL_{2}(k).

Noting that elementary matrices of the form

(\begin{matrix}1 & 0 \\ λ & 1 \end{matrix})

all fix ∞ and thus define automorphisms of (ℙ^{1}, ∞) that are 𝔸^{1}-homotopic to the identity, we see that τ_{1,∞}^{0}ρτ_{1,∞}^{0} = id on (ℙ^{1}, ∞) in H_{•}(k), and thus ρ = id on (ℙ^{1}, 1) in H_{•}(k). □

Lemma 3.7. 1. The map co-tr_{0,−s} : (ℙ^{1}, 1) → (ℙ^{1}, 1) is the identity.

2. The map co-tr_{∞,−s^{−1}} : (ℙ^{1}, 1) → (ℙ^{1}, 1) is the map in H_{•}(k) induced by the automorphism τ_{0,∞}^{1}.

3. The map co-tr_{∞,s^{−1}} : (ℙ^{1}, 1) → (ℙ^{1}, 1) is the identity.

Proof. Since τ_{0,∞}^{1∗}(−s) = −s^{−1}, (2) follows from (1) by applying τ_{0,∞}^{1}. Next, we show that (2) implies (3). It follows directly from the definition of co-tr_{∗,∗} that

co-tr_{∞,s^{−1}} = τ_{1,∞}^{0} ∘ μ(−1) ∘ τ_{1,∞}^{0} ∘ co-tr_{∞,−s^{−1}}.

Thus, assuming (2), we have

co-tr_{∞,s^{−1}} = τ_{1,∞}^{0} ∘ μ(−1) ∘ τ_{1,∞}^{0} ∘ τ_{0,∞}^{1}

in H_{•}(k); (3) then follows from lemma 3.6. We now prove (1). Identify 𝔸^{1} with ℙ^{1} \ {1} sending 0 to 0 and 1 to ∞. The blow-up W := W_{0} is thus identified with an open subscheme of the blow-up μ̄ : W̄ → ℙ^{1} × ℙ^{1} of ℙ^{1} × ℙ^{1} at (0, 0).

The curve C_{0} on W̄ has self-intersection -1, and can thus be blown down via a morphism

ρ̄: W̄ → W̄′.

Letting q : W̄ → ℙ^{1} be the composition

W̄ ⟶^{μ̄} ℙ^{1} × ℙ^{1} ⟶^{p_{2}} ℙ^{1},

the fact that q(C_{0}) = 0 implies that q descends to a morphism

q̄′ : W̄′ → ℙ^{1}.

As the complement W̄_{∞} := W̄ \ W is disjoint from C_{0} and ρ̄ is proper, we have the open subscheme W′ := W̄′ \ ρ̄(W̄_{∞}) of W̄′ and the proper birational morphism

ρ : W → W′,

with ρ(C_{0}) ≅ Spec k and with the restriction W \ C_{0} → W′ \ ρ(C_{0}) an isomorphism. In addition, q̄′ restricts to the proper morphism

q′ : W′ → ℙ^{1} \ 1.

In addition, q′ is a smooth and projective morphism with geometric fibers isomorphic to ℙ^{1}. Finally, we have

q^{′−1}(0) = ρ(E).

Let Δ ⊂ ℙ^{1} × ℙ^{1} \ {1} be the restriction of the diagonal in ℙ^{1} × ℙ^{1}, giving us the proper transform μ^{−1}[Δ] on W and the image Δ′ = ρ(μ^{−1}[Δ]) on W′. Similarly, let s′_{0} = ρ(s_{0}), s′_{1} = ρ(s_{1}); note that ρ(C_{0}) ⊂ s′_{1}. It is easy to check that s′_{0}, Δ′ and s′_{1} give disjoint sections of q′ : W′ → ℙ^{1} \ 1, hence there is a unique isomorphism (over ℙ^{1} \ 1) of W′ with ℙ^{1} × ℙ^{1} \ 1 sending (s′_{0}, s′_{1}, Δ′) to (0, 1, ∞) × ℙ^{1} \ 1. We have in addition the commutative diagram

(3.1)  \xymatrix{(ℙ^{1}, 1) \ar[r]^{i_{0,−s}} \ar[dr]_{i′_{0}} & (W, W \ s_{0}) \ar[d]^{ρ} & (ℙ^{1}, 1) \ar[l]_{i_{1}} \ar[dl]^{i′_{1}} \\  & (W′, W′ \ s′_{0})}

where i′_{0} is the canonical identification of ℙ^{1} with the fiber of W′ over 0, sending (0, 1, ∞) into (s′_{0}, s′_{1}, Δ′), and i′_{1} is defined similarly. We claim that the isomorphism ρ : E → q^{′−1}(0) is a pointed isomorphism

ρ : (E, 0, 1, ∞) → (q^{′−1}(0), q^{′−1}(0) ∩ s′_{0}, q^{′−1}(0) ∩ s′_{1}, q^{′−1}(0) ∩ Δ′).

Indeed, by definition 0 = E ∩ s_{0} and 1 = E ∩ C_{0}. Since ρ(s_{0}) = s′_{0} and ρ(C_{0}) ⊂ s′_{1}, we need only show that ρ(∞) ⊂ Δ′. To distinguish the two factors of ℙ^{1}, we write

x_{1} = p_{1}^{∗}(s), x_{2} = p_{2}^{∗}(s)

where s is the standard parameter on ℙ^{1}. Using this notation, ∞ is the subscheme of E defined by the equation −x_{1}/t = 1, where t is the standard parameter on 𝔸_{S}^{1} = ℙ_{S}^{1} \ 1_{S}. As our identification of ℙ^{1} \ 1_{S} with 𝔸^{1} sends 0 ∈ 𝔸^{1} to 0 ∈ ℙ^{1}, 1 ∈ 𝔸^{1} to ∞ in ℙ^{1}, the standard parameter t goes over to the rational function x_{2}/(x_{2} − 1) on ℙ^{1}. As the image of x_{2}/(x_{2} − 1) in m_{0}/m_{0}^{2} is the same as the image of −x_{2} in m_{0}/m_{0}^{2}, ∞ is defined by x_{1}/x_{2} = 1 on E, which is clearly the subscheme defined by E ∩ Δ. Via the isomorphism ρ, this goes over to q^{′−1}(0) ∩ Δ′, as desired. It follows from the proof of [15, theorem 2.2.3] that all the morphisms in the diagram (3.1) are isomorphisms in H_{•}(k); as i_{0}^{′−1} ∘ i′_{1} is clearly the identity, the lemma is proved. □

The proof of the next result is easy and is left to the reader.

Lemma 3.8. Let S′ → S be a morphism of smooth finite type k-schemes. Let x be a closed subscheme of ℙ_{S}^{1} \ {1}, finite over S. Let x′ = x ×_{S} S′ ⊂ ℙ_{S′}^{1}. We suppose we have a generator f for m_{x}/m_{x}^{2}, and let f′ be the extension to m_{x′}/m_{x′}^{2}. If either S′ → S is smooth, or S → S′ is flat and x → S is étale, then the diagram

\xymatrix{(ℙ_{S′}^{1}, 1) \ar[r]^{^{co-tr^{x′,f′}}} \ar[d] & (ℙ_{x′}^{1}, 1) \ar[d] \\ (ℙ_{S}^{1}, 1) \ar[r]_{_{co-tr_{x,f}}} & (ℙ_{x}^{1}, 1)}

is defined and commutes.

4. Co-group structure on ℙ^{1}

In this section, k will be an arbitrary perfect field. Let 𝔾_{m} = 𝔸^{1}\{0}, which we consider as a pointed scheme with base-point 1. We recall the Mayer-Vietoris square for the standard cover of ℙ^{1}:

\xymatrix{𝔾_{m} \ar[r]^{t_{∞}} \ar[d]_{t_{0}} & 𝔸^{1} \ar[d]^{j_{∞}} \\ 𝔸^{1} \ar[r]_{j_{0}} & ℙ^{1}.}

Here j_{0}, j_{∞}, t_{0}, t_{∞} are defined by j_{0}(t) = (1 : t), j_{∞}(t) = (t : 1), t_{0}(t) = t and t_{∞}(t) = t^{−1}. This gives us the isomorphism in H_{•}(k) of ℙ^{1} with the homotopy push-out in the diagram

(4.1)  \xymatrix{𝔾_{m} \ar[r]^{t_{∞}} \ar[d]_{t_{0}} & 𝔸^{1} \\ 𝔸^{1};}

the contractibility of 𝔸^{1} gives us the canonical isomorphism

(4.2)  α : S^{1} ∧ 𝔾_{m} ⟶^{∼} (ℙ^{1}, 1).

This, together with the standard co-group structure on S^{1}, σ : S^{1} → S^{1} ∨ S^{1}, makes (ℙ^{1}, 1) a co-group object in H_{•}(k); let

σ_{ℙ^{1}} := σ ∧ id_{𝔾_{m}} : (ℙ^{1}, 1) → (ℙ^{1}, 1) ∨ (ℙ^{1}, 1)

be the co-multiplication. In this section, we discuss a more algebraic description of this structure. Let f := (f_{0}, f_{∞}) be a pair of generators for m_{0}/m_{0}^{2}, m_{∞}/m_{∞}^{2}. giving us the collapse map

co-tr_{{0,∞},f} : (ℙ^{1}, 1) → (ℙ^{1}, 1) ∨ (ℙ^{1}, 1).

Lemma 4.1. Let s be the standard parameter X_{1}/X_{0} on ℙ^{1}. For f = (−s, s^{−1}), we have σ_{ℙ^{1}} = co-tr_{{0,∞},f} in H_{•}(k).

Proof. We first unwind the definition of σ_{ℙ^{1}} in some detail. As above, we identify ℙ^{1} with the push-out in the diagram (4.1) and thus (ℙ^{1}, 1) is isomorphic to the push-out in the diagram

\xymatrix{(𝔾_{m}, 1) ∨ (𝔾_{m}, 1) \ar[r]^{(id,id)} \ar[d]_{(t_{0}∨t_{∞})} & (𝔾_{m}, 1) \\ (𝔸^{1}, 1) ∨ (𝔸^{1}, 1).}

Let I denote a simplicial model of the interval admitting a “mid-point” 1/2, for example, we can take I = Δ^{1}_{1}∨_{0} Δ^{1}. The isomorphism α : S^{1} ∧ 𝔾_{m} → (ℙ^{1}, 1) in H_{•}(k) arises via a sequence of comparison maps between push-outs in the following diagrams (we point ℙ^{1}, 𝔸^{1} and 𝔾_{m} with 1):

\xymatrix{𝔾_{m} ∨ 𝔾_{m} \ar[r]^{(id,id)} \ar[dd]_{(t_{0}∨t_{∞})} & 𝔾_{m} &  & 0_{+} ∧ 𝔾_{m} ∨ 1_{+} ∧ 𝔾_{m} \ar[r]^{(ι_{0},ι_{1})} \ar[dd]_{t_{0}∨t_{∞}} & I_{+} ∧ 𝔾_{m} \\  &  & ← \\ 𝔸^{1} ∨ 𝔸^{1} &  &  & 0_{+} ∧ 𝔸^{1} ∨ 1_{+} ∧ 𝔸^{1}}

↓ (4.3)

\xymatrix{0_{+} ∧ 𝔾_{m} ∨ 1_{+} ∧ 𝔾_{m} \ar[r]^{(ι_{0},ι_{1})} \ar[d] & I_{+} ∧ 𝔾_{m} \\ ∗}

the first map is induced by the evident projections and the second by contracting 𝔸^{1} to ∗. Thus, the open immersion 𝔾_{m} → ℙ^{1}, t ↦ (1 : t), goes over to the map

{1/2}_{+} ∧ 𝔾_{m} → I_{+} ∧ 𝔾_{m} → S^{1} ∧ 𝔾_{m},

the second map given by the bottom diagram in (4.3). This gives us the isomorphism

ρ : (ℙ^{1}, 𝔾_{m}) → S^{1} ∧ 𝔾_{m} ∨ S^{1} ∧ 𝔾_{m}

in H_{•}(k), yielding the commutative diagram

\xymatrix{(ℙ^{1}, 1) \ar[r]^{α}_{∼} \ar[d]_{π} & S^{1} ∧ 𝔾_{m} \ar[d]^{σ∧id} \\ (ℙ^{1}, 𝔾_{m}) \ar[r]_{ρ}^{∼} & S^{1} ∧ 𝔾_{m} ∨ S^{1} ∧ 𝔾_{m},}

where π is the canonical quotient map and α is the isomorphism (4.2). If we consider the middle diagram in (4.3), we find a similarly defined isomorphism (in H_{•}(k))

ϵ : (ℙ^{1}, 𝔾_{m}) → (𝔸^{1}, 𝔾_{m})_{t_{0}} ∨ (𝔸^{1}, 𝔾_{m})_{t_{∞}},

where the subscripts t_{0}, t_{∞} refer to the morphism 𝔾_{m} → 𝔸^{1} used. The map from the middle diagram to the last diagram in (4.3) furnishes the commutative diagram of isomorphisms in H_{•}(k):

\xymatrix{(𝔸^{1}, 𝔾_{m})_{t_{0}} ∨ (𝔸^{1}, 𝔾_{m})_{t_{∞}} \ar[r]^{β} \ar[dr]_{ϑ} & S^{1} ∧ 𝔾_{m} ∨ S^{1} ∧ 𝔾_{m} \ar[d]^{α∨α} \\  & (ℙ^{1}, 1) ∨ (ℙ^{1}, 1).}

Putting this all together gives us the commutative diagram in H_{•}(k):

(4.4)  \xymatrix{(ℙ^{1}, 1) \ar[r]_{α}^{∼} \ar[d]_{γ} & S^{1} ∧ 𝔾_{m} \ar[d]^{σ∧id} \\ (ℙ^{1}, 𝔾_{m}) \ar[r]_{ρ}^{∼} \ar[d]^{∼}_{ϵ} & S^{1} ∧ 𝔾_{m} ∨ S^{1} ∧ 𝔾_{m} \ar[d]^{α∨α} \\ (𝔸^{1}, 𝔾_{m})_{t_{0}} ∨ (𝔸^{1}, 𝔾_{m})_{t_{∞}} \ar[ur]^{β}_{∼} \ar[r]_{ϑ}^{∼} & (ℙ^{1}, 1) ∨ (ℙ^{1}, 1).}

Letting δ := ϑ ∘ ϵ, we thus need to show that the map δ ∘ γ : (ℙ^{1}, 1) → (ℙ^{1}, 1) ∨ (ℙ^{1}, 1) is given by co-tr_{{0,1},f}. Write (𝔸^{1}, 𝔾_{m}) := (𝔸^{1}, 𝔾_{m})_{t_{0}}. Letting η be the inverse on 𝔾_{m}, η(t) = t^{−1}, we identify (𝔸^{1}, 𝔾_{m}) with (𝔸^{1}, 𝔾_{m})_{t_{∞}} via the isomorphism (id, η). The maps j_{0} : 𝔸^{1} → ℙ^{1}, j_{∞} : 𝔸^{1} → ℙ^{1} induce the isomorphisms in H_{•}(k)

j̄_{0} : (𝔸^{1}, 𝔾_{m}) → (ℙ^{1}, j_{∞}(𝔸^{1}))
j̄_{∞} : (𝔸^{1}, 𝔾_{m}) → (ℙ^{1}, j_{0}(𝔸^{1}))

giving together the isomorphism τ : (ℙ^{1}, 1) ∨ (ℙ^{1}, 1) → (𝔸^{1}, 𝔾_{m}) ∨ (𝔸^{1}, 𝔾_{m}), defined as the composition:

(ℙ^{1}, 1)∨(ℙ^{1}, 1) ⟶^{id∨id} (ℙ^{1}, j_{∞}(𝔸^{1}))∨(ℙ^{1}, j_{0}(𝔸^{1})) ⟶^{j̄_{0}^{−1}∨j̄_{∞}^{−1}} (𝔸^{1}, 𝔾_{m})∨(𝔸^{1}, 𝔾_{m}).

By comparing with the push-out diagrams in (4.3), we see that τ is the inverse to ϑ. As τ_{0,∞}^{1} exchanges j_{0} and j_{∞}, this gives the identity

(4.5)  ϑ = ϑ_{0} ∨ τ_{0,∞}^{1} ∘ ϑ_{0},

where ϑ_{0} is the composition

\xymatrix{(𝔸^{1}, 𝔾_{m}) \ar[r]^{j_{0}} & (ℙ^{1}, j_{∞}(𝔸^{1})) & (ℙ^{1}, 1). \ar[l]^{∼}_{id}}

Let W → ℙ^{1} × 𝔸^{1} be the blow-up at ({0, ∞}, 0) with exceptional divisor E. Let g be the trivialization g := (−s, −s^{−1}) of m_{0}/m_{0}^{2} × m_{∞}/m_{∞}^{2}. We have the composition of isomorphisms in H_{•}(k)

(4.6) (ℙ^{1}, 𝔾_{m}) ⟶^{i_{1}} (W, W \ s_{{0,∞}}) ⟵^{i_{0}} (E, C_{0} ∩ E) ⟵^{φ_{g}} (ℙ^{1}, 1) ∨ (ℙ^{1}, 1).

The open cover (j_{0}, j_{∞}) : 𝔸^{1} ⨿ 𝔸^{1} → ℙ^{1} of ℙ^{1} gives rise to an open cover of W: Let μ′ : W′ → 𝔸^{1} × 𝔸^{1} be the blow-up at (0, 0), then we have the lifting of (j_{0}, j_{∞}) to the open cover

(j′_{0}, j′_{∞}) : W′ ⨿ W′ → W.

The cover (j_{0}, j_{∞}) induces the excision isomorphism in H_{•}(k)

(ĵ_{0}, ĵ_{∞}) : (𝔸^{1}, 𝔾_{m}) ∨ (𝔸^{1}, 𝔾_{m}) → (ℙ^{1}, 𝔾_{m});

it is easy to see that (ĵ_{0}, ĵ_{∞}) is inverse to the isomorphism ϵ in diagram (4.4). Similarly, letting s′ ⊂ W′ be the proper transform of 0 × 𝔸^{1} to W′, the cover (j′_{0}, j′_{∞}) induces the excision isomorphism in H_{•}(k)

(j̃′_{0}, j̃′_{∞}) : (W′, W′ \ s′) ∨ (W′, W′ \ s′) → (W, W \ s_{{0,∞}}).

This extends to a commutative diagram of isomorphisms in H_{•}(k)

(4.7)  [figure: (𝔸^{1}, 𝔾_{m}) ∨ (𝔸^{1}, 𝔾_{m}); i_{1}∨i_{1}; (ĵ_{0},ĵ_{∞}); (ℙ^{1}, 𝔾_{m}); i_{1}; (W′, W′ \ s′) ∨ (W′, W′ \ s′); (j̃′_{0},j̃′_{∞}); (W, W \ s_{{0,∞}}); (E′, E′ ∩ C′_{0}) ∨ (E′, E′ ∩ C′_{0}); i_{0}∨i_{0}; (j̃′_{E0},j̃′_{E∞}); (E, E ∩ C_{0}); i_{0}; (ℙ^{1}, 1) ∨ (ℙ^{1}, 1); φ_{−s}∨φ_{−s}; (ℙ^{1}, 1) ∨ (ℙ^{1}, 1).; φ_{g}]

Indeed, the commutativity is obvious, except on the bottom square. On the first summand (ℙ^{1}, 1), the commutativity is also obvious, since both φ_{−s} and φ_{g} are defined on this factor using the generator −s for m_{0}/m_{0}^{2}, and on the second factor, the map j̃_{∞} sends −s to −s^{−1}, which gives the desired commutativity. Examining the push-out diagram (4.3), we see that the map

(ĵ_{0}, ĵ_{∞}) : (𝔸^{1}, 𝔾_{m}) ∨ (𝔸^{1}, 𝔾_{m}) → (ℙ^{1}, 𝔾_{m})

is inverse to the map ϵ in diagram (4.4). Let W_{0} → ℙ^{1} × 𝔸^{1} be the blow-up along (0, 0), E^{0} the exceptional divisor, C_{0}^{0} the proper transform of ℙ^{1} × 0. The inclusion j_{0} induces the excision isomorphism in H_{•}(k)

j : (𝔸^{1}, 𝔾_{m}) → (ℙ^{1}, j_{∞}(𝔸^{1}))

and gives us the commutative diagram

[figure: (ℙ^{1}, 1) ∨ (ℙ^{1}, 1); (𝔸^{1}, 𝔾_{m}) ∨ (𝔸^{1}, 𝔾_{m}); ϑ_{0}∨ϑ_{0}; i_{1}∨i_{1}; (j∨j); (ℙ^{1}, j_{∞}(𝔸^{1})) ∨ (ℙ^{1}, j_{∞}(𝔸^{1})); i_{1}∨i_{1}; (W′, W′ \ s′) ∨ (W′, W′ \ s′); (j̃∨j̃); (W_{0}, W_{0} \ s_{0}) ∨ (W_{0}, W_{0} \ s_{0}); (E′, E′ ∩ C′_{0}) ∨ (E′, E′ ∩ C′_{0}); i_{0}∨i_{0}; (j̃_{E′} ∨j̃_{E′}); (E^{0}, E^{0} ∩ C_{0}^{0}) ∨ (E^{0}, E^{0} ∩ C_{0}^{0}); i_{0}∨i_{0}; (ℙ^{1}, 1) ∨ (ℙ^{1}, 1); φ_{−s}∨φ_{−s}; (ℙ^{1}, 1) ∨ (ℙ^{1}, 1).; φ_{−s}∨φ_{−s}]

By lemma 3.7 the composition along the right-hand side of this diagram is the identity on (ℙ^{1}, 1)∨(ℙ^{1}, 1), and thus the composition along the left-hand side is ϑ_{0} ∨ϑ_{0} : (𝔸^{1}, 𝔾_{m})∨(𝔸^{1}, 𝔾_{m}) → (ℙ^{1}, 1)∨(ℙ^{1}, 1). Referring to diagram (4.4), as ϵ = (ĵ_{0}, ĵ_{∞})^{−1}, it follows from (4.5) that the composition along the right-hand side of (4.7) is the map (id ∨ τ_{0,∞}^{1}) ∘ δ. As the right-hand side of (4.7) is the deformation diagram used to define co-tr_{{0,∞},g}, we see that

co-tr_{{0,∞},g} = (id ∨ τ_{0,∞}^{1}) ∘ σ_{ℙ^{1}}.

Noting that f and g differ only by the trivialization at ∞, changing s^{−1} to −s^{−1}, we thus have

co-tr_{{0,∞},f} = (id ∨ τ_{1,∞}^{0} ∘ μ(−1) ∘ τ_{1,∞}^{0}) ∘ co-tr_{{0,∞},g}.

By lemma 3.6, we have

co-tr_{{0,∞},f} = (id ∨ τ_{0,∞}^{1}) ∘ co-tr_{{0,∞},g} = σ_{ℙ^{1}}.

□

5. Slice localizations and co-transfer

In general, the co-transfer maps do not have the properties necessary to give a loop-spectrum Ω_{ℙ^{1}}E an action by correspondences. However, if we pass to a certain localization of SH_{S^{1}}(k) defined by the slice filtration, the co-transfer maps both extend to arbitrary correspondences and respect the composition of correspondences. This will lead to the action of correspondences on s_{0}Ω_{ℙ^{1}}E we wish to construct. In this section, k will be an arbitrary perfect field. We have the localizing subcategory Σ_{ℙ^{1}}^{n}SH_{S^{1}}(k), generated (as a localizing subcategory) by objects of the form Σ_{ℙ^{1}}^{n}E, for E ∈ SH_{S^{1}}(k). We let SH_{S^{1}}(k)/f_{n} denote the localization of SH_{S^{1}}(k) with respect to Σ_{ℙ^{1}}^{n}SH_{S^{1}}(k):

SH_{S^{1}}(k)/f_{n} = SH_{S^{1}}(k)/Σ_{ℙ^{1}}^{n}SH_{S^{1}}(k).

Remark 5.1. Pelaez [17, corollary 3.2.40] has shown that there is a model structure on Spt_{S^{1}}(k) with homotopy category equivalent to SH_{S^{1}}(k)/f_{n}; in particular, this localization of SH_{S^{1}}(k) does exist.

Remark 5.2. In the proofs of some of the next few results we will use the following fact, which relies on our ground field k being perfect: Let V ⊂ U be a Zariski open subset of some U ∈ Sm/k. Then we can filter U by open subschemes

V = U^{N+1} ⊂ U^{N} ⊂ … ⊂ U^{0} = U

such that U^{i+1} = U^{i} \ C_{i}, with C_{i} ⊂ U^{i} smooth and having trivial normal bundle in U^{i} for i = 0, …, N. Indeed, let C = U \ V, with reduced scheme structure. As k is perfect, there is a dense open subscheme C_{sm} of C which is smooth over k, and there is a non-empty open subscheme C_{1} ⊂ C_{sm} such that the restriction of I_{C}/I_{C}^{2} to C_{1} is a free sheaf of rank equal to the codimension of C_{1} in U. We let U^{1} = U \ C_{1}, and then proceed by noetherian induction.

Lemma 5.3. Let V → U be a dense open immersion in Sm/k, n ≥ 1 an integer. Then the induced map

Σ_{ℙ^{1}}^{n}V_{+} → Σ_{ℙ^{1}}^{n}U_{+}

is an isomorphism in SH_{S^{1}}(k)/f_{n+1}.

Proof. Filter U by open subschemes

V = U^{N+1} ⊂ U^{N} ⊂ … ⊂ U^{0} = U

as in remark 5.2. Write U^{i+1} = U^{i} \ C_{i}, with C_{i} having trivial normal bundle in U_{i}, of rank say r_{i}, for i = 0, …, N. By the Morel-Voevodsky purity theorem [15, theorem 2.23], the cofiber of U^{i+1} → U^{i} is isomorphic in H_{•}(k) to Σ_{ℙ^{1}}^{r_{i}}C_{i+}, and thus the cofiber of Σ_{ℙ^{1}}^{n}U_{+}^{i+1} → Σ_{ℙ^{1}}^{n}U_{+}^{i} is isomorphic to Σ_{ℙ^{1}}^{r_{i}+n}C_{i+}. Since V is dense in U, we have r_{i} ≥ 1 for all i, proving the lemma. □

Take W ∈ Sm/k. By excision and homotopy invariance, we have a canonical isomorphism

ψ_{W,r} : 𝔸_{W}^{r} /𝔸_{W}^{r} \ 0_{W} → Σ_{ℙ^{1}}^{r}W_{+}

in H_{•}(k). The action of the group-scheme GL_{r}/k on 𝔸^{r} gives an action of the group of sections GL_{r}(W) on 𝔸_{W}^{r} /𝔸_{W}^{r} \ 0_{W}, giving us for each g ∈ GL_{r}(W) the isomorphism

ψ_{W,r}^{g} :=: ψ_{W,r} ∘ g : 𝔸_{W}^{r} /𝔸_{W}^{r} \ 0_{W} → Σ_{ℙ^{1}}^{r}W_{+}.

Lemma 5.4. For each g ∈ GL_{r}(W), we have ψ_{W,r}^{g} = ψ_{W,r} in SH_{S^{1}}/f_{r+1}.

Proof. The action GL_{r} × 𝔸^{r} → 𝔸^{r} composed with ψ_{W,r} gives us the morphism in H_{•}(k)

Ψ_{W} : (W × GL_{r})_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0) → Σ_{ℙ^{1}}^{r}W_{+};

for each section g ∈ GL_{r}(W), composing with the corresponding section s_{g} : W → W × GL_{r} gives the map

Ψ_{W} ∘ s_{g} : W_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0) → Σ_{ℙ^{1}}^{r}W_{+}

which is clearly equal to ψ_{W,r}^{g}. The open immersion j : W × GL_{r} → W × 𝔸^{r^{2}} is by lemma 5.3 an isomorphism in SH_{S^{1}}(k)/f_{1}; as (𝔸^{r}/𝔸^{r} \ 0) ≅ Σ_{ℙ^{1}}^{r} Spec k_{+}, we see that the induced map

j ∧ id : (W × GL_{r})_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0) → (W × 𝔸^{r^{2}})_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0)

is an isomorphism in SH_{S^{1}}(k)/f_{r+1}, and thus the projection

(W × GL_{r})_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0) → W_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0)

is also an isomorphism in SH_{S^{1}}(k)/f_{r+1}. From this it follows that the maps

s_{g} ∧ id, s_{id} ∧ id : W_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0) → (W × GL_{r})_{+} ∧ (𝔸^{r}/𝔸^{r} \ 0)

are equal in SH_{S^{1}}(k)/f_{r+1}, hence ψ_{W,r}^{g} = ψ_{W,r} in SH_{S^{1}}/f_{r+1}. □

As application we have the following result

Proposition 5.5. 1. Let S be in Sm/k. Let x ⊂ ℙ_{S}^{1} \1_{S} be a closed subscheme, smooth over k and finite over S, such that the co-normal bundle m_{x}/m_{x}^{2} is trivial. Then the maps

co-tr_{x,f} : (ℙ_{S}^{1}, 1) → (ℙ_{x}^{1}, 1)

in SH_{S^{1}}(k)/f_{2} are independent of the choice of generator f for m_{x}/m_{x}^{2}. If S = Spec O_{X,x} for x a finite set of points on some X ∈ Sm/k, the analogous independence holds, this time as morphisms in pro-SH_{S^{1}}(k)/f_{2}.

2. Let g : ℙ^{1} → ℙ^{1} be a k-automorphism, with g(1) = 1. Then g : (ℙ^{1}, 1) → (ℙ^{1}, 1) is the identity in SH_{S^{1}}(k)/f_{2}.

3. Take a, b ∈ ℙ^{1}(k), with a ≠ b and a, b ≠ 1. The canonical isomorphism a ⨿ b → Spec k ⨿ Spec k gives the canonical identification (ℙ_{a,b}^{1}, 1) ≅ (ℙ^{1}, 1) ∨ (ℙ^{1}, 1). Then for each choice of generator f for m_{a,b}/m_{a,b}^{2}, the map

co-tr_{a,b,f} : (ℙ^{1}, 1) → (ℙ_{a,b}^{1}, 1) ≅ (ℙ^{1}, 1) ∨ (ℙ^{1}, 1)

is equal in SH_{S^{1}}(k)/f_{2} to the co-multiplication σ_{ℙ^{1}}.

Proof. (1) Suppose that we have generators f, f′ for m_{x}/m_{x}^{2}. There is thus a unit a ∈ O_{x}^{∗} with f′ = af. Note that co-tr_{x,f′} = g ∘co-tr_{x,f}, where g : ℙ_{x}^{1} → ℙ_{x}^{1} is the automorphism τ_{1,∞}^{0}μ(a)τ_{1,∞}^{1}. By lemma 5.4, the map

μ(a) = ψ_{Spec k,1}^{a} ∘ ψ_{Spec k,1}^{−1} : (ℙ^{1}, ∞) → (ℙ^{1}, ∞)

is the identity in SH_{S^{1}}(k)/f_{2}, whence (1).

For (2), we may replace 1 with ∞. The affine group of isomorphisms g : ℙ^{1} → ℙ^{1} with g(∞) = ∞ is generated by the matrices of the form

(\begin{matrix}u & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ λ & 1 \end{matrix}) ,

with u ∈ k^{×} and λ ∈ k. Clearly the automorphisms of the second type act as the identity on (ℙ^{1}, ∞) in H_{•}(k); the automorphisms of the first type act by the identity on (ℙ^{1}, ∞) in SH_{S^{1}}(k)/f_{2} by lemma 5.4. (3). Let g : ℙ^{1} → ℙ^{1} be the automorphism sending (0, 1, ∞) to (a, 1, b). Choose a generator f for m_{a,b}/m_{a,b}^{2}, then g^{∗}f gives a generator for m_{0,∞}/m_{0,∞}^{2}. The automorphism g extends to an isomorphism g̃ : W_{0,∞} → W_{a,b}, giving us a commutative diagram

\xymatrix{(ℙ^{1}, 1) \ar[r]^{i_{1}} \ar[d]_{g} & (W_{0,∞}, W_{0,∞} \ s_{0,∞}) \ar[d]_{g̃} & (ℙ_{0,∞}^{1}, 1) \ar[l]_{i_{0,g^{∗}f}} \ar[d]^{β} \\ (ℙ^{1}, 1) \ar[r]_{i_{1}} & (W_{a,b}, W_{a,b} \ s_{a,b}) & (ℙ_{a,b}^{1}, 1) \ar[l]^{i_{0,f}}}

where β : ℙ_{0,∞}^{1} → ℙ_{a,b}^{1} is canonical isomorphism over (0, ∞) → (a, b). This gives us the identity in H_{•}(k):

co-tr_{a,b,f} ∘ g = β ∘ co-tr_{0,∞,g^{∗}f}.

By (1), the maps co-tr_{a,b,g^{∗}f} and co-tr_{0,∞,f} are independent (in SH_{S^{1}}(k)/f_{2}) of the choice of f and by (2), g is the identity in SH_{S^{1}}(k)/f_{2}. For suitable f, lemma 4.1 tells us co-tr_{0,∞,f} = σ_{ℙ^{1}}, completing the proof of (3). □

As the map

co-tr_{x,f} : (ℙ_{S}^{1}, 1) → (ℙ_{x}^{1}, 1)

in SH_{S^{1}}(k)/f_{2} is independent of the choice of generator f ∈ m_{x}/m_{x}^{2}; we denote this map by co-tr_{x}. We have one additional application of lemma 5.4.

Lemma 5.6. Let W ⊂ U be a codimension ≥ r closed subscheme of U ∈ Sm/k, let w_{1}, …, w_{m} be the generic points of W of codimension = r in U. Then there is a canonical isomorphism of pro-objects in in SH_{S^{1}}/f_{r+1}

(U, U \ W) ≅ ⊕_{i=1}^{m}Σ_{ℙ^{1}}^{r}w_{i+}.

Specifically, letting m_{i} ⊂ O_{U,w_{i}} be the maximal ideal, this isomorphism is independent of any choice of isomorphism m_{i}/m_{i}^{2} ≅ k(w_{i})^{r}.

Proof. Let w = {w_{1}, ….w_{m}} and let O_{U,w} denote the semi-local ring of w in U. Consider the projective system V := {V_{α}} consisting of open subschemes of U of the form V_{α} = U \ C_{α}, where C_{α} is a closed subset of W containing no generic point w_{i} of W.

Take V_{α} ∈ V. By applying remark 5.2, and noting that U \ V_{α} has codimension ≥ r + 1 in U, the argument used in the proof of lemma 5.3 shows that the cofiber of

(V_{α}, V_{α} \ W) → (U, U \ W)

is in Σ_{ℙ^{1}}^{r+1}SH_{S^{1}}(k). On the other hand, the collection of V_{α} ∈ V such that V_{α} ∩ W is smooth and has on each connected component a trivial normal bundle in V_{α} forms a cofinal subsystem V′ in V. For each V_{α} ∈ V′, we have V_{α} ∩ W = ⨿_{i=1}^{m}W_{i}^{α}, with w_{i} the unique generic point of W_{i}^{α}, and we have the isomorphism

(V_{α}, V_{α} \ W) ≅ ⋁_{i=1}^{m} Σ_{ℙ^{1}}^{r}W_{i+}^{α}

in H_{•}(k). Since w_{i} is equal to the projective limit of the W_{i}^{α}, we have the desired isomorphism of pro-objects in SH_{S^{1}}(k)/f_{r+1}. We need only verify that the resulting isomorphism (U, U \ W) ≅ ⊕_{r=1}^{m}Σ_{ℙ^{1}}^{r}w_{i+} is independent of any choices. Let V = Spec O_{U,W}, and let O denote the henselization of w in V. We have the canonical excision isomorphism (of proobjects in H_{•}(k))

(V, V \ V ∩ W) ≅ (Spec O, Spec O \ w).

A choice of isomorphism m_{w}/m_{w}^{2} ≅ k(w)^{r} gives the isomorphism in pro-H_{•}(k)

Σ_{ℙ^{1}}^{r}w_{+} ≅ (Spec O, Spec O \ w);

this choice of isomorphism is thus the only choice involved in constructing our isomorphism (U, U \ W) ≅ ⊕_{i=1}^{m}Σ_{ℙ^{1}}^{r}w_{i+}. Explicitly, the choice of isomorphism m_{w}/m_{w}^{2} ≅ k(w)^{r} is reflected in the isomorphism (Spec O, Spec O \w) ≅ Σ_{ℙ^{1}}^{r}w_{+} through the identification of the exceptional divisor of the blow-up of V × 𝔸^{1} along w×0 with ℙ_{w}^{r}. The desired independence now follows from lemma 5.4. □

The computation which is crucial for enabling us to introduce transfers on the higher slices of S^{1}-spectra is the following:

Lemma 5.7. Let μ_{n} : (ℙ^{1}, ∞) → (ℙ^{1}, ∞) be the map μ_{n}(t_{0} : t_{1}) = (t_{0}^{n} : t_{1}^{n}). Assume the characteristic of k is prime to n!. Then in SH_{S^{1}}(k)/f_{2}, μ_{n} is multiplication by n.

Proof. The proof goes by induction on n, starting with n = 1, 2. The case n = 1 is trivial. For n = 2, lemma 5.6 gives us the canonical isomorphisms in SH_{S^{1}}(k)/f_{2}

(ℙ^{1}, ℙ^{1} \ {±1}) ⟶^{α_{±1}} (ℙ^{1}, ∞) ∨ (ℙ^{1}, ∞); (ℙ^{1}, ℙ^{1} \ {1}) ⟶^{α_{1}} (ℙ^{1}, ∞).

In addition, we have the commutative diagram

\xymatrix{(ℙ^{1}, ∞) \ar[r] \ar[d]_{μ_{2}} & (ℙ^{1}, ℙ^{1} \ \{±1\}) \ar[d]^{μ_{2}} \\ (ℙ^{1}, ∞) \ar[r] & (ℙ^{1}, ℙ^{1} \ \{1\})}

The bottom horizontal arrow is an isomorphism in H_{•}(k). We claim the diagram

\xymatrix{(ℙ^{1}, ℙ^{1} \ \{±1\}) \ar[d]^{μ_{2}} \ar[r]^{α_{±1}} & (ℙ^{1}, ∞) ∨ (ℙ^{1}, ∞) \ar[d]^{(id,id)} \\ (ℙ^{1}, ℙ^{1} \ \{1\}) \ar[r]_{α_{1}} & (ℙ^{1}, ∞)}

commutes in SH_{S^{1}}(k)/f_{2}. Indeed, the isomorphism α_{±1} arises from the Morel-Voevodsky homotopy purity isomorphism identifying (ℙ^{1}, ℙ^{1}\{±1}) canonically with the Thom space of the tangent space T (ℙ^{1})_{±1} of ℙ^{1} at ±1, followed by the isomorphism

T h(T (ℙ^{1})_{±1}) ≅ Σ_{ℙ^{1}}(±1_{+}) = (ℙ^{1}, ∞) ∨ (ℙ^{1}, ∞)

induced by a choice of basis for T (ℙ^{1})_{±1} (which plays no role in SH_{S^{1}}(k)/f_{2}). Similarly the map α_{1} arises from a canonical isomorphism of (ℙ^{1}, ℙ^{1} \{1}) with T h(T (ℙ^{1})_{1}) followed by the isomorphism

T h(T (ℙ^{1})_{1}) → (ℙ^{1}, ∞)

induced by a choice of basis. As the map μ_{2} is étale over 1, the differential

dμ_{2} : T (ℙ^{1})_{±1} → T (ℙ^{1})_{1}

is isomorphic to the sum map

𝔸^{1} ⊕ 𝔸^{1} → 𝔸^{1}.

As this sum map induces (id, id) on the Thom spaces, we have verified our claim. Using proposition 5.5 and we see that this diagram together with the isomorphisms α_{±1} and α_{1} gives us the factorization of μ_{2} (in SH_{S^{1}}(k)/f_{2}) as

(ℙ^{1}, ∞) ⟶^{σ} (ℙ^{1}, ∞) ∨ (ℙ^{1}, ∞) ⟶^{(id,id)} (ℙ^{1}, ∞).

Here σ is the co-multiplication (using ∞ instead of 1 as base-point). Since (id, id) ∘ σ is multiplication by 2, this takes care of the case n = 2. In general, we consider the map ρ_{n} : (ℙ^{1}, ∞) → (ℙ^{1}, ∞) sending (t_{0} : t_{1}) to (w_{0} : w_{1}) := (t_{0}^{n} : t_{1}^{n} − t_{0}t_{1}^{n−1} + t_{0}^{n}). We may form the family of morphisms

ρ_{n}(s) : (ℙ^{1} × 𝔸^{1}, ∞ × 𝔸^{1}) → (ℙ^{1} × 𝔸^{1}, ∞ × 𝔸^{1})

sending (t_{0} : t_{1}, s) to (t_{0}^{n} : t_{1}^{n} − st_{0}t_{1}^{n−1} + st_{0}^{n}). By homotopy invariance, we have ρ_{n}(0) = ρ_{n}(1), and thus ρ_{n} = μ_{n} in H_{•}(k). As above, we localize around w := w_{1}/w_{0} = 1. Note that ρ_{n}^{−1}(1) = {0, 1}. We replace the target ℙ^{1} with the henselization O at w = 1, and see that ℙ^{1} ×_{ρ_{n}} O breaks up into two components via the factorization w − 1 = t(t^{n−1} − 1), t = t_{1}/t_{0}. On the component containing 1, the map ρ_{n} is isomorphic to a hensel local version of μ_{n−1}, and on the component containing 0, the map ρ_{n} is isomorphic to the identity.

Using Nisnevich excision and proposition 5.5(1), we thus have the following commutative diagram (in SH_{S^{1}}(k)/f_{2})

\xymatrix{(ℙ^{1}, ∞) \ar[r] \ar[d]_{ρ_{n}} & (ℙ^{1}, ℙ^{1} \ \{0, 1\}) \ar[d]_{ρ_{n}} \ar[r]^{∼} & (ℙ^{1}, ∞) ∨ (ℙ^{1}, ∞) \ar[d]^{μ_{n−1}∨id} \\ (ℙ^{1}, ∞) \ar[r] & (ℙ^{1}, ℙ^{1} \ \{1\}) \ar[r]_{∼} & (ℙ^{1}, ∞)}

By proposition 5.5(2), the upper row is the co-multiplication (in SH_{S^{1}}(k)/f_{2}), and thus

ρ_{n} = μ_{n−1} + id

in SH_{S^{1}}(k)/f_{2}. As ρ_{n} = μ_{n} in H_{•}(k), our induction hypothesis gives μ_{n} = n·id, and the induction goes through. □

While we are on the subject, we might as well note that

Remark 5.8. The co-group ((ℙ^{1}, 1), σ_{ℙ^{1}}) in SH_{S^{1}}(k)/f_{2} is co-commutative.

As pointed out by the referee, every object in SH_{S^{1}}(k)/f_{2} is a co-commutative co-group, since SH_{S^{1}}(k)/f_{2} is a triangulated category and hence each object is a double suspension. In addition, the co-group structure ((ℙ^{1}, 1), σ_{ℙ^{1}}) is isomorphic in H_{•}(k) to the co-group structure on S^{1} ∧ 𝔾_{m} induced by the co-group structure on S^{1}, so the “triangulated” co-group structure on ℙ^{1} agrees with the one we have given. One should, however, be able to reproduce our entire theory “modulo Σ_{ℙ^{1}}^{2}” in the unstable category. We have not done this here, as we do not at present have available a theory of the motivic Postnikov tower in the H_{•}(k). We expect that, given such a theory, the results of this section would hold in the unstable setting and in particular, that the co-group ((ℙ^{1}, 1), σ_{ℙ^{1}}) would be co-commutative “modulo Σ_{ℙ^{1}}^{2}”.

We now return to our study of properties of the co-transfer map in SH_{S^{1}}(k)/f_{2}. We will find it convenient to work in the setting of smooth schemes essentially of finite type over k; as mentioned at the beginning of §3, we consider schemes Y essentially of finite type over k as pro-objects in H(k), SH_{S^{1}}(k), etc. In the end, we use scheme essentially of finite type over k only as a tool to construct maps in pro-SH_{S^{1}}(k)/f_{n+1} between objects of SH_{S^{1}}(k)/f_{n+1}; this will in the end give us morphisms in SH_{S^{1}}(k)/f_{n+1}, as the functor SH_{S^{1}}(k)/f_{n+1} → pro-SH_{S^{1}}(k)/f_{n+1} is fully faithful, Suppose we have a semi-local smooth k-algebra A, essentially of finite type, and a finite extension A → B, with B smooth over k. Suppose further that B is generated as an A-algebra by a single element x ∈ B:

B = A[x].

We say in this case that B is a simply generated A-algebra. Let f̃ ∈ A[T] be the monic minimal polynomial of x, giving us the point x′ of 𝔸_{A}^{1} = Spec A[T] with ideal (f̃). We identify 𝔸_{A}^{1} with ℙ_{A}^{1} \ {1} as usual, giving us the subscheme x of ℙ_{A}^{1} \ {1}, smooth over k and finite over Spec A, in fact, canonically isomorphic to Spec B over Spec A via the choice of generator x. Let

φ_{x} : x → Spec B

be this isomorphism. We let f be the generator of m_{x}/m_{x}^{2} determined by f̃. Via the composition

(ℙ_{A}^{1}, 1) ⟶^{co-tr_{x,f}} (ℙ_{x}^{1}, 1) ⟶^{φ_{x}×id} (ℙ_{B}^{1}, 1)

we have the morphism

co-tr_{x} : (ℙ_{A}^{1}, 1) → (ℙ_{B}^{1}, 1)

in pro-H_{•}(k).

Lemma 5.9. Suppose that Spec B → Spec A is étale over each generic point of Spec A. Then the map co-tr_{x} : (ℙ_{A}^{1}, 1) → (ℙ_{B}^{1}, 1) in pro-SH_{S^{1}}(k)/f_{2} is independent of the choice of generator x for B over A.

Via this result, we may write co-tr_{B/A} for co-tr_{x}.

Proof. We use a deformation argument; we first localize to reduce to the case of an étale extension A → B. For this, let a ∈ A be a non-zero divisor, and let x be a generator for B as an A-algebra. Then x is a generator for B[a^{−1}] as an A[a^{−1}]-algebra and by lemma 3.8 we have the commutative diagram

\xymatrix{ℙ_{A[a^{−1}]}^{1} \ar[r] \ar[d]_{_{co-tr_{x}}} & ℙ_{A}^{1} \ar[d]^{_{co-tr_{x}}} \\ ℙ_{B[a^{−1}]}^{1} \ar[r] & ℙ_{B}^{1},}

with horizontal arrows isomorphisms in pro-SH_{S^{1}}(k)/f_{2}. Thus, we may assume that A → B is étale. Suppose we have generators x ≠ x′ for B over A; let d = [B : A]. Let s be an indeterminate, let x(s) = sx + (1 − s)x′ ∈ B[s], and consider the extension B̃_{s} := A[s][x(s)] of A[s], considered as a subalgebra of B[s]. Clearly B̃_{s} is finite over A[s]. Let m_{A} ⊂ A be the Jacobson radical, and let A(s) be the localization of A[s] at the ideal (m_{A}A[s] + s(s − 1)). In other words, A(s) is the semi-local ring of the set of closed points {(0, a), (1, a)} in 𝔸^{1} × Spec A, as a runs over the closed points of Spec A. Define B(s) := B ⊗_{A} A(s) and B_{s} := B̃_{s} ⊗_{A} A(s) ⊂ B(s). Let y = (1, a) be a closed point of A(s), with maximal ideal m_{y}, and let x_{y} be the image of x in B(s)/m_{y}B(s). Clearly x_{y} is in the image of B_{s} → B(s)/m_{y}B(s), hence B_{s} → B(s)/m_{y}B(s) is surjective. Similarly, B_{s} → B(s)/m_{y}B(s) is surjective for all y of the form (0, a); by Nakayama’s lemma B_{s} = B(s). Also, B(s) and A(s) are regular and B(s) is finite over A(s), hence B(s) is flat over A(s) and thus B(s) is a free A(s)-module of rank d. Finally, B(s) is clearly unramified over A(s), hence A(s) → B(s) is étale.

Using Nakayama’s lemma again, we see that B(s) is generated as an A(s) module by 1, x(s), x(s)^{2}, …, x(s)^{d−1}. It follows that x(s) satisfies a monic polynomial equation of degree d over A(s), thus x(s) admits a monic minimal polynomial f_{s} of degree d over A(s). Sending T to x(s) defines an isomorphism

φ_{s} : A(s)[T]/(f_{s}) → B(s).

We let x_{s} ⊂ 𝔸_{A(s)}^{1} = ℙ_{A(s)}^{1} \{1} be the closed subscheme of ℙ_{A(s)}^{1} corresponding to f_{s}; the isomorphism φ_{s} gives us the isomorphism

φ_{s} : x_{s} → Spec B(s).

Thus, we may define the map

co-tr_{x(s)} : (ℙ_{A(s)}^{1}, 1) → (ℙ_{B(s)}^{1}, 1)

giving us the commutative diagram

\xymatrix{(ℙ_{A}^{1}, 1) \ar[d]_{_{co-tr_{x′}}} \ar[r]^{i_{0}} & (ℙ_{A(s)}^{1}, 1) \ar[d]_{_{co-tr_{x(s)}}} & (ℙ_{A}^{1}, 1) \ar[d]^{_{co-tr_{x}}} \ar[l]_{i_{1}} \\ (ℙ_{B}^{1}, 1) \ar[r]_{i_{0}} & (ℙ_{B(s)}^{1}, 1) & (ℙ_{B}^{1}, 1) \ar[l]^{i_{1}}}

By lemma 5.3, the map (ℙ_{A(s)}^{1}, 1) → (ℙ_{A[s]}^{1}, 1) is an isomorphism in pro-SH_{S^{1}}(k)/f_{2}. By homotopy invariance, it follows that the maps i_{0}, i_{1} are isomorphisms in pro-SH_{S^{1}}(k)/f_{2}, inverse to the map (ℙ_{A(s)}^{1}, 1) → (ℙ_{A}^{1}, 1) induced by the projection Spec A(s) → Spec A. Therefore co-tr_{x′} = co-tr_{x}, as desired. □

Lemma 5.10. co-tr_{A/A} = id_{(ℙ_{A}^{1},1)}.

Proof. We may choose 0 as the generator for A over A, which gives us the point x = 0 ∈ ℙ_{A}^{1}. The result now follows from lemma 3.7. □

Lemma 5.11. Let A → C be a finite simply generated extension and A ⊂ B ⊂ C a sub-extension, with B also simply generated over A. We suppose that A, B and C are smooth over k, that A → B and A → C are étale over each generic point of Spec A, and B → C is étale over each generic point of Spec B. Then

co-tr_{C/A} = co-tr_{C/B} ∘ co-tr_{B/A}.

Proof. This is another deformation argument. As in the proof of lemma 5.9, we may assume that A → B, B → C and A → C are étale extensions; we retain the notation from the proof of lemma 5.9. Let y be a generator for C over A, x a generator for B over A. These generators give us corresponding closed subschemes y, x ⊂ ℙ_{A}^{1} and y_{B} ⊂ ℙ_{B}^{1}. Let y(s) = sy + (1 − s)x, giving y(s) ⊂ ℙ_{A(s)}^{1}. Note that y(1) = y, y(0)_{red} = x As in the proof of lemma 5.9, the element y(s) of C(s) is a generator over A(s) after localizing at the points of Spec A(s) lying over s = 1. The subscheme y(s) in a neighborhood of s = 0 is not in general regular, hence y(s) is not a generator of C(s) over A(s). However, let μ : W := W_{x} → ℙ^{1} × 𝔸^{1} be the blow-up along {(x, 0)}, and let ỹ ⊂ W_{A(s)} be the proper transform μ^{−1}[y]. An elementary local computation shows that this blow-up resolves the singularities of y(s), and that ỹ is étale over A(s); the argument used in the proof of lemma 5.9 goes through to show that A(s)(ỹ) ≅ C(s). In addition, let C_{0} be the proper transform to W_{A(s)} of ℙ^{1} × 0 and E the exceptional divisor, then ỹ(0) is disjoint from C_{0}. Finally, after identifying E with ℙ_{A[x]}^{1} = ℙ_{B}^{1} (using the monic minimal polynomial of x as a generator for m_{x}), we may consider ỹ(0) as a closed subscheme of ℙ_{B}^{1}; the isomorphism A(s)(ỹ) ≅ C(s) leads us to conclude that A(ỹ(1)) = B(ỹ(0)) = C. By lemma 5.9, we may use ỹ(0) to define co-tr_{C/B}. The map co-tr_{C/A} in pro-SH_{S^{1}}(k)/f_{2} is defined via the diagram

(ℙ_{A}^{1}, 1) → (ℙ_{A}^{1}, ℙ_{A}^{1} \ y) ≅ (ℙ_{C}^{1}, 1)

where the various choices involved lead to equal maps. By lemma 5.3, W_{A(s)} → W_{A[s]} is an isomorphism in pro-SH_{S^{1}}(k)/f_{2}; by homotopy invariance, the projection W_{A(s)} → W is also an isomorphism pro-SH_{S^{1}}(k)/f_{2}. The inclusions i_{1} : ℙ_{A}^{1} → W_{A(s)}, i_{0} : ℙ_{A[x]}^{1} → W_{A(s)} induce isomorphisms (in pro-SH_{S^{1}}(k)/f_{2})

(ℙ_{A}^{1}, ℙ_{A}^{1} \ y) = (ℙ_{A}^{1}, ℙ_{A}^{1} \ y(1)) ≅ (W_{A(s)}, W_{A(s)} \ ỹ(s)) ≅ (ℙ_{A[x]}^{1}, ℙ_{A[x]}^{1} \ ỹ(0)).

As in the proof of lemma 5.9, we can use homotopy invariance to see that co-tr_{C/A} is also equal to the composition

(ℙ_{A}^{1}, 1) → (ℙ_{A}^{1}, ℙ_{A}^{1} \ y) ⟶^{i_{1}} (W_{A(s)}, W_{A(s)} \ ỹ(s))

⟶^{i_{0}^{−1}} (ℙ_{A[x]}^{1}, ℙ_{A[x]}^{1} \ ỹ(0)) ≅ (ℙ_{C}^{1}, 1).

Now let s_{1A(s)} be the transform to W_{A(s)} of the 1-section. By lemma 3.3, the inclusion i_{0} : (ℙ_{A[x]}^{1}, 1) → (W_{A(s)}, C_{0} ∪ s_{1A(s)}) is an isomorphism in pro-H_{•}(k). The above factorization of co-tr_{C/A} shows that co-tr_{C/A} is also equal to the composition

(ℙ_{A}^{1}, 1) ⟶^{i_{1}} (W_{A(s)}, C_{0} ∪ s_{1A(s)}) ⟶^{i_{0}^{−1}} (ℙ_{A[x]}^{1}, 1) → (ℙ_{A[x]}^{1} \ ỹ(0)) ≅ (ℙ_{C}^{1}, 1).

Using remark 3.5, this latter composition is co-tr_{C/B} ∘ (co-tr_{B/A}), as desired. □

Remark 5.12. 1. Suppose we have simply generated finite generically étale extensions A_{1} → B_{1}, A_{2} → B_{2}, with A_{i} smooth, semi-local and essentially of finite type over k. Then

co-tr_{B_{1}×B_{2}/A_{1}×A_{2}} = co-tr_{B_{1}/A_{1}} ∨ co-tr_{B_{2}/A_{2}}

where we make the evident identification (ℙ_{B_{1}×B_{2}}^{1}, 1) = (ℙ_{B_{1}}^{1}, 1) ∨ (ℙ_{B_{2}}^{1}, 1) and similarly for A_{1}, A_{2}.

2. Let B_{1}, B_{2} be simply generated finite generically étale A algebras and let B = B_{1} × B_{2}. As a special case of lemma 5.11, we have

co-tr_{B/A} = (co-tr_{B_{1}/A} ∨ co-tr_{B_{2}/A}) ∘ σ_{ℙ_{A}^{1}}

Indeed, we may factor the extension A → B as A ⟶^{δ} A × A → B_{1} × B_{2} = B. We then use (1) and note that σ_{ℙ_{A}^{1}} = co-tr_{A×A/A} by lemma 4.1.

Next, we make a local calculation. Let (A, m) be a local ring of essentially finite type and smooth over k. Let s ∈ m be a parameter, let B = A[T]/T^{n}−s and let t ∈ B be the image of T. Set Y = Spec B, X = Spec A, Z = Spec A/(s), W = Spec B/(t); the extension A → B induces an isomorphism α : W ⟶^{∼} Z. We write co-tr_{Y/X} for co-tr_{B/A}, etc. This gives us the diagram in pro-SH_{S^{1}}(k)/f_{2}

\xymatrix{ℙ_{Z}^{1} \ar[r]^{i_{Z}} & ℙ_{X}^{1} \ar[d]^{_{co-tr_{Y /X}}} \\ ℙ_{W}^{1} \ar[r]_{i_{W}} \ar[u]^{α} & ℙ_{Y}^{1} .}

Lemma 5.13. Suppose that n! is prime to char k. In pro-SH_{S^{1}}(k)/f_{2} we have

co-tr_{Y/X} ∘ i_{Z} ∘ α = n × i_{W} .

Proof. First, suppose we have a Nisnevich neighborhood f : X′ → X of Z in X, giving us the Nisnevich neighborhood g : Y′ := Y ×_{X} X′ → Y of W in Y. As

co-tr_{Y/X} ∘ f = g ∘ co-tr_{Y′/X′}

we may replace X with X′, Y with Y′. Similarly, we reduce to the case of A a hensel DVR, i.e., the henselization of 0 ∈ 𝔸_{F}^{1} for some field F, Z = W = 0, with s the image in A of the canonical coordinate on 𝔸_{F}^{1} . The map co-tr_{Y/X} is defined by the closed immersion

Y ⟶^{i_{Y}} 𝔸_{X}^{1} = ℙ_{X}^{1} \ 1_{X} ⊂ ℙ_{X}^{1}

where i_{Y} is the closed subscheme of 𝔸^{1} = Spec A[T] defined by T^{n} −s, together with the isomorphism

(ℙ_{X}^{1}, ℙ_{X}^{1} \ Y) ≅ ℙ_{Y}^{1}

furnished by the blow-up μ : W_{Y} → 𝔸_{X}^{1} × 𝔸^{1} of 𝔸_{X}^{1} × 𝔸^{1} along (Y, 0). The composition co-tr_{Y/X} ∘ i_{Z} ∘ α is given by the composition

(ℙ_{W}^{1} , 1) ≅ (ℙ_{W}^{1} , ℙ_{W}^{1} \ 0_{W}) ⟶^{α} (ℙ_{Z}^{1}, ℙ_{Z}^{1} \ 0_{Z})

⟶^{i_{Z}} (ℙ_{X}^{1}, ℙ_{X}^{1} \ 0_{X}) ⟵^{id} (ℙ_{X}^{1}, 1) → (ℙ_{X}^{1}, ℙ_{X}^{1} \ Y) ≅ (ℙ_{Y}^{1} , 1).

In both cases, the isomorphisms (in pro-SH_{S^{1}}(k)/f_{2}) are independent of a choice of trivialization of the various normal bundles. Let U → ℙ_{X}^{1} be the hensel local neighborhood of 0_{Z} in ℙ_{X}^{1}, Spec O_{ℙ_{X}^{1},0_{Z}}^{h}. Let p : U → X be the map induced by the projection p_{X} : ℙ_{X}^{1} → X and let U_{Z} = p^{−1}(Z), with inclusion i_{Z} : U_{Z} → U. We may use excision to rewrite the above description of co-tr_{Y/X} ∘ i_{Z} ∘ α as a composition as

(ℙ_{W}^{1} , 1) ≅ (ℙ_{Z}^{1}, 1) ≅ (U_{Z}, U_{Z} \ 0_{Z}) ⟶^{i_{Z}} (U, U \ Y) ≅ (ℙ_{Y}^{1} , 1).

Similarly, letting i_{0} : X → X × ℙ^{1} be the 0-section, the map i_{W} may be given by the composition

(ℙ_{W}^{1} , 1) ≅ (X, X \ Z) ⟶^{i_{0}} (U, U \ Y) ≅ (ℙ_{Y}^{1} , 1);

again, the isomorphisms in pro-SH_{S^{1}}(k)/f_{2} are independent of choice of trivializations of the various normal bundles. We write (s, t) for the parameters on U induced by the functions s, T on 𝔸_{X}^{1}. We change coordinates in U by the isomorphism (s, t) ↦ (s − t^{n}, t). This transforms Y to the subscheme s = 0, is the identity on the 0-section, and transforms s = 0 to t^{n} + s = 0. Replacing s with −s, we have just switched the roles of Y and U_{Z}. Let

φ : U_{Z} → U

be the map φ(t) = (t^{n}, t). After making our change of coordinates, the map co-tr_{Y/X} ∘ i_{Z} ∘ α is identified with

(ℙ_{W}^{1} , 1) ≅ (U_{Z}, U_{Z} \ 0_{Z}) ⟶^{φ} (U, U \ U_{Z}) ≅ (ℙ_{Y}^{1} , 1)

while the description of i_{W} becomes

(ℙ_{W}^{1} , 1) ≅ (X, X \ Z) ⟶^{i_{0}} (U, U \ U_{Z}) ≅ (ℙ_{Y}^{1} , 1);

here we are using lemma 5.4 to conclude that the automorphism (x_{0} : x_{1}) ↦ (−x_{0} : x_{1}) of ℙ_{W}^{1} induces the identity on (ℙ_{W}^{1} , ∞) in pro-SH_{S^{1}}(k)/f_{2}. We now construct an 𝔸^{1}-family of maps (U_{Z}, U_{Z} \ 0_{Z}) → (U, U \ U_{Z}). Let

Φ : U_{Z} × 𝔸^{1} → U

be the map Φ(t, v) = (t^{n}, vt). Note that Φ defines a map of pairs

Φ : (U_{Z}, U_{Z} \ 0_{Z}) × 𝔸^{1} → (U, U \ U_{Z}).

Clearly Φ(−, 1) = φ while Φ(−, 0) factors as

U_{Z} ⟶^{μ_{n}} U_{Z} ⟶^{β} X ⟶^{i_{0}} U

where μ_{n} is the map t ↦ t^{n} and β is the isomorphism β(t) = s. Thus, we can rewrite co-tr_{Y/X} ∘ i_{Z} ∘ α as

(ℙ_{W}^{1} , 1) ≅ (X, X \ Z) ⟶^{μ_{n}} (X, X \ Z) ⟶^{i_{0}} (U, U \ U_{Z}) ≅ (ℙ_{Y}^{1} , 1)

We identify X with the hensel neighborhood of 0_{Z} in ℙ_{Z}^{1}. Using excision again, we have the commutative diagram in pro-H_{•}(k)

\xymatrix{(X, X \ Z) \ar[r]^{μ_{n}} \ar[d] & (X, X \ Z) \ar[d] \\ (ℙ_{Z}^{1}, ℙ_{Z}^{1} \ 0_{Z}) \ar[r]^{μ_{n}} & (ℙ_{Z}^{1}, ℙ_{Z}^{1} \ 0_{Z}) \\ (ℙ_{Z}^{1}, ∞) \ar[r]^{μ_{n}^{Z}} \ar[u] & (ℙ_{Z}^{1}, ∞) \ar[u]}

where the vertical arrows are all isomorphisms. By lemma 5.7 the bottom map is multiplication by n, which completes the proof. □

Lemma 5.14. Let A → B be a finite simple étale extension, A as above. Let X = Spec A, Y = Spec B, let i_{x} : x → X be the closed point of X and i_{y} : y → Y the inclusion of y := x ×_{X} Y. Then

co-tr_{Y/X} ∘ i_{x} = i_{y} ∘ co-tr_{y/x}.

Proof. Take an embedding of Y in 𝔸_{X}^{1} = ℙ_{X}^{1} \ 1_{X} ⊂ ℙ_{X}^{1}; the fiber of Y → 𝔸_{X}^{1} over x → X is thus an embedding y → 𝔸_{x}^{1} = ℙ_{x}^{1} \ 1_{x} ⊂ ℙ_{x}^{1}. The result follows easily from the commutativity of the diagram

\xymatrix{ℙ_{x}^{1} \ y \ar[r] \ar[d] & ℙ_{x}^{1} \ar[d] \\ ℙ_{X}^{1} \ Y \ar[r] & ℙ_{X}^{1}}

□

Proposition 5.15. Let A → B be a finite generically étale extension, with A a DVR and B a semi-local principal ideal ring. Let X = Spec A, Y = Spec B, let i_{x} : x → X be the closed point of X and i_{y} : y → Y the inclusion of y := x ×_{X} Y. Write y = {y_{1}, …, y_{r}}, with each y_{i} irreducible. Let n_{i} denote the ramification index of y_{i}; suppose that n_{i}! is prime to char k for each i. Then

co-tr_{Y/X} ∘ i_{x} = ∑_{i=1}^{r} n_{i} · i_{y_{i}} ∘ co-tr_{y_{i}/x}.

Proof. We note that every such extension is simple. By passing to the henselization A → A^{h}, we may assume A is hensel. By remark 5.12(2), we may assume that r = 1. Let A → B_{0} ⊂ B be the maximal unramified subextension. As co-tr_{B/A} = co-tr_{B/B_{0}} ∘ co-tr_{B_{0}/B}, we reduce to the two cases A = B_{0}, B = B_{0}. We note that a finite separable extension of hensel DVRs A → B with trivial residue field extension degree and ramification index prime to the characteristic is isomorphic to an extension of the form t^{n} = s for some s ∈ m_{A} \ m_{A}^{2}. Thus, the first case is lemma 5.13, the second is lemma 5.14. □

Consider the functor

(ℙ_{?}^{1}, 1) : Sm/k → SH_{S^{1}}(k)/f_{2}

sending X to (ℙ_{X}^{1}, 1) ∈ SH_{S^{1}}(k)/f_{2}, which we consider as a SH_{S^{1}}(k)/f_{2}-valued presheaf on Sm/k^{op} (we could also write this functor as X ↦ Σ_{ℙ^{1}}^{∞}X_{+}). We proceed to extend (ℙ_{?}^{1}, 1) to a presheaf on SmCor(k)^{op}; we will assume that char k = 0, so we do not need to worry about inseparability. We first define the action on the generators of Hom_{SmCor}(X, Y), i.e., on irreducible W ⊂ X × Y such that W → X is finite and surjective over some component of X. As SH_{S^{1}}(k)/f_{2} is an additive category, it suffices to consider the case of irreducible X. Let U ⊂ X be a dense open subscheme. Then the map (ℙ_{U}^{1}, 1) → (ℙ_{X}^{1}, 1) induced by the inclusion is an isomorphism in SH_{S^{1}}(k)/f_{2}. We may therefore define the morphism

(ℙ_{?}^{1}, 1)(W) : (ℙ_{X}^{1}, 1) → (ℙ_{Y}^{1} , 1)

in SH_{S^{1}}(k)/f_{2} as the composition (in pro-SH_{S^{1}}(k)/f_{2})

(ℙ_{X}^{1}, 1) ≅ (ℙ_{k(X)}^{1}, 1) ⟶^{co-tr_{k(W)/k(X)}} (ℙ_{k(W)}^{1}, 1) ⟶^{p_{2}} (ℙ_{Y}^{1} , 1).

We extend linearly to define (ℙ_{?}^{1}, 1) on Hom_{SmCor}(X, Y). Suppose that Γ_{f} ⊂ X × Y is the graph of a morphism f : X → Y. It follows from lemma 5.10 that co-tr_{k(Γ_{f})/k(X)} is the inverse to the isomorphism p_{1} : (ℙ_{k(Γ_{f})}^{1}, 1) → (ℙ_{k(X)}^{1}, 1). Thus, the composition

(ℙ_{k(X)}^{1}, 1) ⟶^{co-tr_{k(Γ_{f})/k(X)}} (ℙ_{k(Γ_{f})}^{1}, 1) ⟶^{p_{2}} (ℙ_{Y}^{1} , 1)

is the map induced by the restriction of f to Spec k(X). Since (ℙ_{k(X)}^{1}, 1) → (ℙ_{X}^{1}, 1) is an isomorphism in pro-SH_{S^{1}}(k)/f_{2}, it follows that (ℙ_{?}^{1}, 1)(Γ_{f}) = f, i.e., our definition of (ℙ_{?}^{1}, 1) on Hom_{SmCor}(X, Y) really is an extension of its definition on Hom_{Sm/k}(X, Y). The main point is to check functoriality.

Lemma 5.16. Suppose char k = 0. For α ∈ Hom_{SmCor}(X, Y), β ∈ Hom_{SmCor}(Y, Z), we have

(ℙ_{?}^{1}, 1)(β ∘ α) = (ℙ_{?}^{1}, 1)(β) ∘ (ℙ_{?}^{1}, 1)(α)

Proof. It suffices to consider the case of irreducible finite correspondences W ⊂ X × Y, W′ ⊂ Y × Z. If W is the graph of a flat morphism, the result follows from lemma 3.8. As the action of correspondences is defined at the generic point, we may replace X with η := Spec k(X). Then W becomes a closed point of Y_{η} and the correspondence W_{η} : η → Y factors as p_{2} ∘ i_{W_{η}} ∘ p_{1}^{t}, where p_{1} : W_{η} → η and p_{2} : Y_{η} → Y are the projections. Let W′_{η} ⊂ Y_{η} × Z be the pull-back of W′. As we have already established naturality with respect to pull-back by flat maps, we reduce to showing

(ℙ_{?}^{1}, 1)(W′_{η} ∘ i_{W_{η}}) = (ℙ_{?}^{1}, 1)(W′_{η}) ∘ (ℙ_{?}^{1}, 1)(i_{W_{η}}).

Since Y is quasi-projective, we can find a sequence of closed subschemes of Y_{η}

W_{η} = W_{0} ⊂ W_{1} ⊂ … ⊂ W_{d−1} ⊂ W_{d} = Y_{η}

such that W_{i} is smooth of codimension d−i on Y_{η}. Using again the fact the co-tr is defined at the generic point, and that we have already proven functoriality with respect to composition of morphisms, we reduce to the case of Y = Spec O for some DVR O, and i_{η} the inclusion of the closed point η of Y. Let W′′ → W′ be the normalization of W′. Using functoriality with respect to morphisms in Sm/k once more, we may replace Z with W′′ and W′ with the transpose of the graph of the projection W′′ → Y. Changing notation, we may assume that W′ is the transpose of the graph of a finite morphism Z → Y.

This reduces us to the case considered in proposition 5.15; this latter result completes the proof. □

We will collect the results of this section, generalized to higher loops, in theorem 6.1 of the next section.

6. Higher loops

The results of these last sections carry over immediately to statements about the n-fold smash product (ℙ^{1}, 1)^{∧n} for n ≥ 1. For clarity and completeness, we list these explicitly in an omnibus theorem. Let R be a semi-local k-algebra, smooth and essentially of finite type over k, and let x ⊂ ℙ_{R}^{1} and f be as in section 3. For n ≥ 1, define

co-tr_{x,f}^{n} : Σ_{ℙ^{1}}^{n} Spec R_{+} → Σ_{ℙ^{1}}^{n}x_{+}

to be the map Σ_{ℙ^{1}}^{n−1}(co-tr_{x,f}) (in pro-H_{•}(k)). Similarly, let A be a semi-local k-algebra, smooth and essentially of finite type over k. Let B = A[x] be a simply generated finite generically étale A-algebra. For n ≥ 1, define

co-tr_{x}^{n} : Σ_{ℙ^{1}}^{n} Spec A_{+} → Σ_{ℙ^{1}}^{n} Spec B_{+}

to be the map Σ_{ℙ^{1}}^{n−1}(co-tr_{x}) (in pro-SH_{S^{1}}(k)/f_{n+1}).

Theorem 6.1. 1. co-tr_{0,−s}^{n} = id.

2. Let R → R′ be a flat extension of smooth semi-local k-algebras, essentially of finite type over k. Let x be a smooth closed subscheme of ℙ_{R}^{1} \ {1}, finite and generically étale over R. Let x′ = x ×_{R} R′ ⊂ ℙ_{R′}^{1}. Let f be a generator for m_{x}/m_{x}^{2}, and let f′ be the extension to m_{x′}/m_{x′}^{2}. Suppose that either R → R′ is smooth or that x → Spec R is étale. Then the diagram

\xymatrix{Σ_{ℙ^{1}}^{n} Spec R′_{+} \ar[r]^{^{co-tr{n \atop x′,f′}}} \ar[d] & Σ_{ℙ^{1}}^{n}x′_{+} \ar[d] \\ Σ_{ℙ^{1}}^{n} Spec R_{+} \ar[r]_{_{co-tr{n \atop x,f}}} & Σ_{ℙ^{1}}^{n}x_{+}}

is well-defined and commutes.

3. The co-group structure Σ_{ℙ^{1}}^{n−1}(σ_{ℙ^{1}}) on (ℙ^{1}, 1)^{∧n} is given by the map

co-tr_{{0,∞},(−s,s^{−1})}^{n} : (ℙ^{1}, 1)^{∧n} → (ℙ^{1}, 1)^{∧n} ∨ (ℙ^{1}, 1)^{∧n}.

4. The co-group ((ℙ^{1}, 1)^{∧n}, Σ_{ℙ^{1}}^{n−1}(σ_{ℙ^{1}})) in SH_{S^{1}}(k)/f_{n+1} is co-commutative.

5. For an extension A → B as above, the map co-tr_{x}^{n} : Σ_{ℙ^{1}}^{n} Spec A_{+} → Σ_{ℙ^{1}}^{n} Spec B_{+} is independent of the choice of x, and is denoted co-tr_{B/A}^{n}.

6. Suppose that char k = 0. The SH_{S^{1}}(k)/f_{n+1}-valued presheaf on Sm/k^{op}

Σ_{ℙ^{1}}^{n}?_{+} : Sm/k → SH_{S^{1}}(k)/f_{n+1}

extends to an SH_{S^{1}}(k)/f_{n+1}-valued presheaf on SmCor(k)^{op}, by sending a generator W ⊂ X × Y of Hom_{SmCor}(X, Y) to the morphism Σ_{ℙ^{1}}^{n}X_{+} → Σ_{ℙ^{1}}^{n}Y_{+} in SH_{S^{1}}(k)/f_{n+1} determined by the diagram

\xymatrix{Σ_{ℙ^{1}}^{n} Spec k(X)_{+} \ar[r]^{∼} \ar[d]_{_{co-tr{n \atop k(W)/k(X)}}} & Σ_{ℙ^{1}}^{n}X_{+} \\ Σ_{ℙ^{1}}^{n} Spec k(W)_{+} \ar[d]_{p_{2}} \\ Σ_{ℙ^{1}}^{n}Y_{+}}

in pro-SH_{S^{1}}(k)/f_{n+1}. The assertion that

Σ_{ℙ^{1}}^{n} Spec k(X)_{+} → Σ_{ℙ^{1}}^{n}X_{+}

is an isomorphism in pro-SH_{S^{1}}(k)/f_{n+1} is part of the statement. We write the map in SH_{S^{1}}(k)/f_{n+1} associated to α ∈ Hom_{SmCor}(X, Y) as

co-tr^{n}(α) : Σ_{ℙ^{1}}^{n}X_{+} → Σ_{ℙ^{1}}^{n}Y_{+}.

7. Supports and co-transfers

In this section, we assume that char k = 0. We consider the following situation. Let i : Y → X be a codimension one closed immersion in Sm/k, and let Z ⊂ X be a pure codimension n closed subset of X such that i^{−1}(Z) ⊂ Y also has pure codimension n. We let T = i^{−1}(Z), X^{(Z)} = (X, X \Z), Y^{(T)} = (Y, Y \T), so that i induces the map of pointed spaces

i : Y^{(T)} → X^{(Z)}.

Let z be the set of generic points of Z, O_{X,z} the semi-local ring of z in X, X_{z} = Spec O_{X,z} and X_{z}^{(z)} = (X_{z}, X_{z} \ z). We let t be the set of generic points of T, and let O_{X,t} be the semi-local ring of t in X, X_{t} = Spec O_{X,t}. Set Y_{t} := X_{t} ×_{X} Y and let Y_{t}^{(t)} = (Y_{t}, Y_{t} \ t).

Lemma 7.1. There are canonical isomorphisms in pro-SH_{S^{1}}(k)/f_{n+1}

X^{(Z)} ≅ X_{z}^{(z)} ≅ Σ_{ℙ^{1}}^{n}z_{+}; Y^{(T)} ≅ Y_{t}^{(t)} ≅ Σ_{ℙ^{1}}^{n}t_{+}.

Proof. This follows from lemma 5.6. □

Thus, the map i : Y^{(T)} → X^{(Z)} gives us the map in pro-SH_{S^{1}}(k)/f_{n+1}:

(7.1)  i : Σ_{ℙ^{1}}^{n}t_{+} → Σ_{ℙ^{1}}^{n}z_{+}.

On the other hand, we can define a map

(7.2)  i_{co-tr} : Σ_{ℙ^{1}}^{n}t_{+} → Σ_{ℙ^{1}}^{n}z_{+}

as follows: Let Z_{t} = Z ∩ X_{t} ⊂ X_{t}. Since Y has codimension one in X and intersects Z properly, t is a collection of codimension one points of Z, and thus Z_{t} is a semi-local reduced scheme of dimension one. Let p : Z̃_{t} → Z_{t} be the normalization, and let t̃⊂ Z̃_{t} be the set of points lying over t ⊂ Z_{t}. Write t̃ as a union of closed points, t̃= ⨿_{j}t̃_{j}. For each j, we let n_{j} denote the multiplicity at t̃_{j} of the pull-back Cartier divisor Y_{t} ×_{X_{t}} Z̃_{t}, and let t_{j} = p(t̃_{j}). This gives us the commutative diagram

[figure: ⨿_{j}t̃_{j}; t̃; ĩ; p; Z̃_{t}; p; z; j; t; i; Z.]

Note that j is an isomorphism in pro-SH_{S^{1}}(k)/f_{1}. We define i_{co-tr} to be the composition

Σ_{ℙ^{1}}^{n}t_{+} \underrightarrow{∏_{j} n_{j}co-tr_{t̃_{j} /t}^{n}} ⊕_{j}Σ_{ℙ^{1}}^{n} t̃_{j+} = Σ_{ℙ^{1}}^{n} t̃_{+} ⟶^{Σ_{ℙ^{1}}^{n} ĩ} Σ_{ℙ^{1}}^{n} Z̃_{+} ⟶^{Σ_{ℙ^{1}}^{n} j^{−1}} Σ_{ℙ^{1}}^{n}z_{+}

in pro-SH_{S^{1}}(k)/f_{n+1}.

Lemma 7.2. The morphisms (7.1) and (7.2)are equal in pro-SH_{S^{1}}(k)/f_{n+1}.

Proof. Using Nisnevich excision, we may replace X with the henselization of X along t; we may also assume that t is a single point. Via a limit argument, we may then replace X with a smooth affine scheme of dimension n + 1 over k(t); Z is thus a reduced closed subscheme of X of pure dimension one over k(t). We may also assume that Y is the fiber over 0 of a morphism X → 𝔸_{k(t)}^{1} for which the restriction to Z is finite. As we are working in pro-SH_{S^{1}}(k)/f_{n+1}, we may replace (X, Z) with (X′, Z′) if there is a morphism f : X → X′ over 𝔸_{k(t)}^{1} which makes (X, t) a hensel neighborhood of (X′, f(t)) and such that the restriction of f to f_{Z} : Z → Z′ is birational. Using Gabber’s presentation lemma [6, lemma 3.1], we may assume that X = 𝔸_{k(t)}^{n+1}, that t is the origin 0 and that Y is the coordinate hyperplane X_{n+1} = 0. We write F for k(t) and write simply 0 for t. After a suitable linear change of coordinates in 𝔸_{F}^{n+1}, we may assume that each coordinate projection

q : 𝔸_{F}^{n+1} → 𝔸_{F}^{r}
q(x_{1}, …, x_{n+1}) = (x_{i_{1}}, …, x_{i_{r}}),

r = 1, …, n, restricts to a finite morphism on Z, and that Z → q(Z) is birational if r ≥ 2. We now reduce to the case in which Z is contained in the coordinate subspace X′ = 𝔸_{F}^{2} defined by X_{1} = … = X_{n−1} = 0. For this, consider the map

m : 𝔸^{1} × 𝔸_{F}^{n+1} → 𝔸^{1} × 𝔸_{F}^{n+1}
m(t, x_{1}, …, x_{n+1}) = (t, tx_{1}, …, tx_{n−1}, x_{n}, x_{n+1})

Let Z = m(𝔸^{1} × Z) ⊂ 𝔸^{1} × 𝔸_{F}^{n+1}. By our finiteness assumptions, Z is a (reduced) closed subscheme of 𝔸^{1} × 𝔸_{F}^{n+1}, and each fiber Z_{t} ⊂ t × 𝔸_{F}^{n+1} is birationally isomorphic to Z ×_{F} F (t). Consider the inclusion map

(𝔸^{1} × Y)^{(𝔸^{1}×0)} → (𝔸^{1} × X)^{(Z)}

The maps

i_{0}, i_{1} : Y^{(0)} → (𝔸^{1} × Y)^{(𝔸^{1}×0)}

are clearly isomorphisms in pro-H_{•}(k), and the maps

i_{1} : X^{(Z)} → (𝔸^{1} × X)^{(Z)}
i_{0} : X^{(Z_{0})} → (𝔸^{1} × X)^{(Z)}

are easily seen to be isomorphisms in pro-SH_{S^{1}}(k)/f_{n+1}. Combining this with the commutative diagram

\xymatrix{Y^{(0)} \ar[r] \ar[d]_{i_{1}} & X^{(Z)} \ar[d]^{i_{1}} \\ (𝔸^{1} × Y)^{(𝔸^{1}×0)} \ar[r] & X^{(Z)} \\ Y^{(0)} \ar[r] \ar[u]^{i_{0}} & X^{(Z_{0})} \ar[u]_{i_{0}}}

shows that we can replace Z with Z_{0} ⊂ X′. Having done this, we see that the map Y^{(0)} → X^{(Z)} is just the n − 1-fold ℙ^{1} suspension of the map

(Y ∩ X′)^{(0)} → (X′)^{(Z)}

This reduces us to the case n = 1. Since p_{2} : Z → 𝔸_{F}^{1} is finite, we may take X = ℙ^{1} × 𝔸_{F}^{1} instead of 𝔸^{1} × 𝔸_{F}^{1} . Then the map Y^{(0)} → X^{(Z)} is isomorphic to (ℙ^{1} × 0, ∞ × 0) → X^{(Z)}. We extend this to the isomorphic map

(ℙ^{1} × 𝔸_{F}^{1} , ∞ × 𝔸_{F}^{1} ) → X^{(Z)} = (ℙ^{1} × 𝔸_{F}^{1} , ℙ^{1} × 𝔸_{F}^{1} \ Z).

Let s be the generic point of 𝔸_{F}^{1} , Z_{s} the fiber of p_{2} over s. Then the inclusions

(ℙ^{1} × 0, ∞ × 0) ⟶^{j_{0}}(ℙ^{1} × 𝔸_{F}^{1} , ∞ × 𝔸_{F}^{1} ) ⟵^{j_{s}} (ℙ^{1} × s, ∞ × s)

(ℙ^{1} × 𝔸_{F}^{1} , ℙ^{1} × 𝔸_{F}^{1} \ Z) ⟵^{j_{s}} (ℙ^{1} × s, ℙ_{s}^{1} \ Z_{s})

are isomorphisms in pro-SH_{S^{1}}(k)/f_{2}, and thus the map

i_{0} : Y^{(0)} ≅ (ℙ^{1} × 0, ∞ × 0) → X^{(Z)} = (ℙ^{1} × 𝔸_{F}^{1} , ℙ^{1} × 𝔸_{F}^{1} \ Z)

is isomorphic in pro-SH_{S^{1}}(k)/f_{2} to the collapse map

(ℙ^{1} × s, ∞ × s) → (ℙ^{1} × s, ℙ_{s}^{1} \ Z_{s}).

Therefore, the map

i : Σ_{ℙ^{1}}0_{+} → Σ_{ℙ^{1}}z_{+}

we need to consider is equal to the co-transfer map

co-tr_{Z_{s}/s} : Σ_{ℙ^{1}}s_{+} → Σ_{ℙ^{1}}z_{s+}

composed with the (canonical) isomorphisms

Σ_{ℙ^{1}}0_{+} ⟶^{i_{0}} Σ_{ℙ^{1}}𝔸_{+}^{1} ≅ Σ_{ℙ^{1}}s_{+}, Σ_{ℙ^{1}}z_{s+} ≅ Σ_{ℙ^{1}}z_{+};

the latter isomorphism arising by noting that z_{s} is a generic point of Z over F. The result now follows directly from proposition 5.15. □

Definition 7.3. 1. Take X, X′ ∈ Sm/k, and let Z ⊂ X, Z′ ⊂ X′ be pure codimension n closed subsets. Take a generator A ∈ Hom_{SmCor}(X, X′), A ⊂ X × X′. Let q : A^{N} → A be the normalization of A. Let z be the set of generic points of Z, let a be the set of generic points of A ∩ X × Z′ and let a′ = q^{−1}(a). Suppose that

(1) A^{N} → X is étale on a neighborhood of a′

(2) p_{X}(a) is contained in Z. Let O_{A^{N} ,a} be the semi-local ring of a′ in A^{N}, and let A_{a′}^{N} = Spec O_{A^{N} ,a′}; define X_{z} similarly. Define

co-tr^{n}(W) : X^{(Z)} → X^{′(Z′)}

to be the map in SH_{S^{1}}(k)/f_{n+1} given by the following composition:

X^{(Z)} ≅ X_{z}^{(z)} ≅ Σ_{ℙ^{1}}^{n}z_{+} ⟶^{co-tr_{a′/z}^{n}} Σ_{ℙ^{1}}^{n}a′_{+} ≅ A_{a′}^{N(a′)} ⟶^{p_{X′}} X^{′(Z′)}.

2. Let Hom_{SmCor}(X, X′)_{Z,Z′} ⊂ Hom_{SmCor}(X, X′) be the subgroup generated by A satisfying (a) and (b). We extend the definition of the morphism co-tr^{n}(A) to Hom_{SmCor}(X, X′)_{Z,Z′} by linearity.

Note that we implicitly invoke lemma 7.1 to ensure that the isomorphisms used in the definition of co-tr^{n}(A) exist and are canonical; condition (1) implies in particular that A^{N} is smooth in a neighborhood of a′, so we may use lemma 7.1 for the isomorphism Σ_{ℙ^{1}}^{n}a′_{+} ≅ A_{a′}^{N(a′)}.

Lemma 7.4. Take X, X′, X′′ ∈ Sm/k, and let Z ⊂ X, Z′ ⊂ X′ and Z′′ ⊂ X′′ be pure codimension n closed subsets. Take α ∈ Hom_{SmCor}(X, X′)_{Z,Z′}, α′ ∈ Hom_{SmCor}(X′, X′′)_{Z′,Z′′}. Then

1. α′ ∘ α is in Hom_{SmCor}(X, X′′)_{Z,Z′′}

2. co-tr^{n}(α′) ∘ co-tr^{n}(α) = co-tr^{n}(α′ ∘ α).

Proof. For (1), we may assume that α and α′ are generators A and A′. We may replace X, X′ and X′′ with the respective strict henselizations along z, z′ and z′′. Write z = {z_{1}, …, z_{r}}, z′ = {z′_{1}, …, z′_{s}}, z′′ = {z′′_{1}, …, z′′_{t}}. Then the normalizations A^{N} and A^{′N} break up as a disjoint union of graphs of morphisms

f_{jk} : X_{z_{k}} → X′_{z′_{j}}; g_{ij} : X′_{z′_{j}} → X′′_{z′′_{i}}

and A′∘A is thus the sum of the graphs of the compositions g_{ij} ∘f_{jk}. Therefore, each irreducible component of the normalization of the support of A′∘A is étale over X. This verifies condition (1) of definition 7.3; the condition (2) is easy and is left to the reader.

(2) follows directly from theorem 6.1(6). □

Proposition 7.5. Let i : Δ_{1} → Δ be a closed immersion of quasi-projective schemes in Sm/k, take X, X′ ∈ Sm/k and α ∈ Hom_{SmCor}(X, X′). Let Z ⊂ X × Δ, Z′ ⊂ X′ × Δ be closed codimension n subsets. Suppose that

(1) Z_{1} := Z ∩ X × Δ_{1} and Z′_{1} := Z′ ∩ X′ × Δ_{1} have codimension n in

X × Δ_{1}, X′ × Δ_{1}, respectively.
(2) α × id_{Δ} is in Hom_{SmCor}(X × Δ, X′ × Δ)_{Z,Z′}
(3) α × id_{Δ_{1}} is in Hom_{SmCor}(X × Δ_{1}, X′ × Δ_{1})_{Z_{1},Z′_{1}}

Then the diagram in SH_{S^{1}}(k)/f_{n+1}

\xymatrix{(X × Δ_{1})^{(Z_{1})} \ar[d]_{id×i} \ar[r]^{co-tr^{n}(α×id)} & (X′ × Δ_{1})^{(Z′_{1})} \ar[d]^{id×i} \\ (X × Δ)^{(Z)} \ar[r]_{co-tr^{n}(α×id)} & (X′ × Δ)^{(Z′)}}

commutes.

Proof. Since Δ is by assumption quasi-projective, we may factor Δ_{1} → Δ as a sequence of closed codimension 1 immersions

Δ_{1} = Δ^{d} → Δ^{d−1} → … → Δ^{1} → Δ^{0} = Δ

such that each closed immersion Δ^{i} → Δ satisfies the conditions of the proposition. This reduces us to the case of a codimension one closed immersion. We may replace X × Δ, X′ × Δ, etc., with the respective semi-local schemes about the generic points of Z_{1} and Z′_{1}. As Δ_{1} has codimension one on Δ, it follows that the normalizations Z^{N}, Z^{′N} of Z and Z′ are smooth over k. Let ĩ : z̃ → Z^{N}, ĩ′ : z̃′ → Z^{′N} be the points of Z^{N}, Z^{′N} lying over Z_{1}, Z′_{1}, respectively, which we write as a disjoint union of closed points

z̃ = ⨿_{j}z̃_{j}; z̃′ = ⨿_{j}z̃′_{j}.

By lemma 7.1 and lemma 7.2, we may rewrite the diagram in the statement of the proposition as

\xymatrix{Σ_{ℙ^{1}}^{n}Z_{1+} \ar[d]_{∑_{j} m_{j}co-tr_{z̃_{j} /Z_{1}}^{n}} \ar[r]^{co-tr^{n}(α×id_{Z^{N}_{1}} )} & Σ_{ℙ^{1}}^{n}Z′_{1+} \ar[d]^{∑_{j} m′_{j}co-tr_{z̃′_{j} /Z′_{1}}^{n}} \\ Σ_{ℙ^{1}}^{n} z̃_{+} \ar[d]_{ĩ} & Σ_{ℙ^{1}}^{n}z̃′_{+} \ar[d]^{ĩ′} \\ Σ_{ℙ^{1}}^{n}Z^{N} \ar[r]_{co-tr^{n}(α×id_{Z^{N}})} & Σ_{ℙ^{1}}^{n}Z^{′N}}

where α × id_{Z^{N}}, α × id_{Z_{1}} denote the correspondences induced by α × id_{Δ} and α × id_{Δ_{1}}, and the m_{j}, m′_{j} are the relevant intersection multiplicities. The commutativity of this diagram follows from theorem 6.1(6). □

8. Slices of loop spectra

Take E ∈ SH_{S^{1}}(k). Following Voevodsky’s remarks in [22], Neeman’s version of Brown representability [16] gives us the motivic Postnikov tower

… → f_{n+1}E → f_{n}E → … → f_{0}E = E,

where f_{n}E → E is universal for morphisms from an object of Σ_{ℙ^{1}}^{n}SH_{S^{1}}(k) to E. The layer s_{n}E is the nth slice of E, and is characterized up to unique isomorphism by the distinguished triangle

(8.1)  f_{n+1}E → f_{n}E → s_{n}E → Σ_{s}f_{n+1}E.

The fact that this distinguished triangle determines s_{n}E up to unique isomorphism rather than just up to isomorphism follows from

(8.2)  Hom_{SH_{S^{1}} (k)}(Σ_{ℙ^{1}}^{n+1}SH_{S^{1}}(k), s_{n}E) = 0

To see this, just use the universal property of f_{n+1}E → E and the long exact sequence of Homs associated to the distinguished triangle (8.1). In particular, using the description of Hom_{SH_{S^{1}} (k)/f_{n+1}}(−, −) via right fractions we have

Lemma 8.1. For all F, E ∈ SH_{S^{1}}(k) and n ≥ 0, the natural map

Hom_{SH_{S^{1}} (k)}(F, s_{n}E) → Hom_{SH_{S^{1}} (k)/f_{n+1}}(F, s_{n}E)

is an isomorphism.

See also [21, proposition 5-3] We recall the de-looping formula [11, theorem 7.4.2]

s_{n}(Ω_{ℙ^{1}}E) ≅ Ω_{ℙ^{1}}(s_{n+1}E)

for n ≥ 0. Take F ∈ Spc_{•}(k). For E ∈ Spt_{S^{1}}(k), we have Hom^{int}(F, E) ∈ SH, which for F = X_{+} is just E(X), and in general is formed as the homotopy limit associated to the description of F as a homotopy colimit of representable objects.

This gives us the “internal Hom” functor

Hom_{SH_{S^{1}} (k)}(F, −) : SH_{S^{1}}(k) → SH_{S^{1}}(k)

and more generally

Hom_{SH_{S^{1}} (k)/f_{n+1}}(F, −) : SH_{S^{1}}(k)/f_{n+1} → SH_{S^{1}}(k),

with natural transformation

Hom_{SH_{S^{1}} (k)}(F, −) → Hom_{SH_{S^{1}} (k)/f_{n+1}}(F, −).

These have value on E ∈ Spt_{S^{1}}(k) defined by taking a fibrant model Ẽ of E (in SH_{S^{1}}(k) or SH_{S^{1}}(k)/f_{n+1}, as the case may be) and forming the presheaf on Sm/k

X ↦ Hom^{int}(F ∧ X_{+}, Ẽ).

Putting the de-looping formula together with lemma 8.1 gives us

Proposition 8.2. For E ∈ SH_{S^{1}}(k) we have natural isomorphisms

s_{0}(Ω_{ℙ^{1}}^{n}E) ≅ Ω_{ℙ^{1}}^{n}s_{n}E ≅ Hom_{SH_{S^{1}} (k)/f_{n+1}}((ℙ^{1}, 1)^{∧n}, s_{n}E)

Proof. Indeed, the first isomorphism is just the de-looping isomorphism repeated n times. For the second, we have

Ω_{ℙ^{1}}^{n}s_{n}E ≅ Hom_{SH_{S^{1}} (k)}((ℙ^{1}, 1)^{∧n}, s_{n}E)
≅ Hom_{SH_{S^{1}} (k)/f_{n+1}}((ℙ^{1}, 1)^{∧n}, s_{n}E)

the second isomorphism following from lemma 8.1. □

Definition 8.3. Suppose that char k = 0. Take E ∈ SH_{S^{1}}(k), take α ∈ Hom_{SmCor}(X, Y) and let n ≥ 1 be an integer. The transfer

Tr_{Y/X}(α) : (Ω_{ℙ^{1}}^{n}s_{n}E)(Y) → (Ω_{ℙ^{1}}^{n}s_{n}E)(X)

is the map in SH defined as follows:

(Ω_{ℙ^{1}}^{n}s_{n}E)(Y) = Hom_{SH_{S^{1}} (k)}(Y_{+}, Ω_{ℙ^{1}}^{n}s_{n}E)

≅ Hom_{SH_{S^{1}} (k)}(Σ_{ℙ^{1}}^{n}Y_{+}, s_{n}E)

≅ Hom_{SH_{S^{1}} (k)/f_{n+1}}(Σ_{ℙ^{1}}^{n}Y_{+}, s_{n}E)

⟶^{co-tr^{n}(α)^{∗}} Hom_{SH_{S^{1}} (k)/f_{n+1}}(Σ_{ℙ^{1}}^{n}X_{+}, s_{n}E)

≅ Hom_{SH_{S^{1}} (k)}(X_{+}, Ω_{ℙ^{1}}^{n}s_{n}E)

≅ (Ω_{ℙ^{1}}^{n}s_{n}E)(X).

Theorem 8.4. Suppose that char k = 0. For E ∈ SH_{S^{1}}(k), the maps Tr(α) extend the presheaf

Ω_{ℙ^{1}}^{n}s_{n}E : Sm/k^{op} → SH

to an SH-valued presheaf with transfers

Ω_{ℙ^{1}}^{n}s_{n}E : SmCor(k)^{op} → SH

Proof. This follows from the definition of the maps Tr(α) and theorem 6.1, the main point being that the maps Tr(α) factor through the internal Hom in SH_{S^{1}}(k)/f_{n+1}. □

Corollary 8.5. Suppose that char k = 0. For E ∈ SH_{S^{1}}(k), there is an extension of the presheaf

s_{0}Ω_{ℙ^{1}}E : Sm/k^{op} → SH

to an SH-valued presheaf with transfers

s_{0}Ω_{ℙ^{1}}E : SmCor(k)^{op} → SH.

Proof. This is just the case n = 1 of theorem 8.4, together with the de-looping isomorphism

s_{0}Ω_{ℙ^{1}}E ≅ Ω_{ℙ^{1}}s_{1}E.
□

Remark 8.6. The corollary is actually the main result, in that one can deduce theorem 8.4 from corollary 8.5 (applied to Ω_{ℙ^{1}}^{n−1}E) and the de-looping formula

Ω_{ℙ^{1}}^{n}s_{n}E ≅ s_{0}Ω_{ℙ^{1}}^{n}E = s_{0}Ω_{ℙ^{1}}(Ω_{ℙ^{1}}^{n−1}E).

As the maps co-tr^{n}(α) are defined by smashing co-tr^{1}(α) with an identity map, this procedure does indeed give back the maps

Tr(α) : Ω_{ℙ^{1}}^{n}s_{n}E(Y) → Ω_{ℙ^{1}}^{n}s_{n}E(X)

as defined above.

proof of theorem 3. The weak transfers defined above give rise to homotopy invariant sheaves with transfers in the usual sense by taking the sheaves of homotopy groups of the motivic spectrum in question. □

For instance, corollary 8.5 gives the sheaf π_{m}(s_{0}Ω_{ℙ^{1}}E) the structure of a homotopy invariant sheaf with transfers, in particular, an effective motive. In fact, these are birational motives in the sense of Kahn-Huber-Sujatha [7, 10], as s_{0}F is a birational S^{1}-spectrum for each S^{1}-spectrum F. The classical Postnikov tower thus gives us a spectral sequence

E_{p,q}^{2} := H^{−p}(X_{Nis}, π_{q}(s_{0}Ω_{ℙ^{1}}E)) ⟹ π_{p+q}(s_{0}Ω_{ℙ^{1}}E(X))

with E^{2} term a “generalized motivic cohomology” of X. As the sheaves π_{q}(s_{0}Ω_{ℙ^{1}}E) are motives, we may replace Nisnevich cohomology with Zariski cohomology; as the sheaves π_{q}(s_{0}Ω_{ℙ^{1}}E) are birational, i.e., Zariski locally constant, the higher Zariski cohomology vanishes, giving us

π_{n}(s_{0}Ω_{ℙ^{1}}E(X)) ≅ H^{0}(X_{Zar}, π_{n}(s_{0}Ω_{ℙ^{1}}E)) = π_{n}(s_{0}Ω_{ℙ^{1}}E(k(X)).

In short, we have shown that the 0th slice of a ℙ^{1}-loop spectrum has transfers in the weak sense. We have already seen in section 2 that this does not hold for an arbitrary object of SH_{S^{1}}(k); in the next section we will see that the higher slices of an arbitrary S^{1}-spectrum do have transfers, albeit in an even weaker sense than the one used above.

9. Transfers on the generalized cycle complex

We begin by recalling from [11, theorem 7.1.1] models for f_{n}E(X) and s_{n}E(X) that are reminiscent of Bloch’s higher cycle complex [1]. To simplify the notation, we will always assume that we have taken a model E ∈ Spt_{S^{1}}(k) which is quasi-fibrant. For W a closed subset of some Y ∈ Sm/k, E^{(W)}(Y) will denote the homotopy fiber of the restriction map E(Y) → E(Y \ W). We make use of the cosimplicial scheme n ↦ Δ^{n} := Spec k[t_{0}, …, t_{n}]/ ∑_{i} t_{i}−1. A face F of Δ^{n} is a subscheme defined by t_{i_{1}} = … = t_{i_{r}} = 0. For a scheme X of finite type and locally equi-dimensional over k, let S_{X}^{(n)}(m) be the set of closed subsets W of X × Δ^{m} of codimension ≥ n, such that, for each face F of Δ^{n}, W ∩ X × F has codimension ≥ n on X × F (or is empty). We order S_{X}^{(n)}(m) by inclusion. For X ∈ Sm/k, we let

E^{(n)}(X, m) := \underrightarrow{lim}_{W ∈S_{X}^{(n)}(m)} E^{(W)}(X × Δ^{m}).

Similarly, for 0 ≤ n ≤ n′, we define

E^{(n/n′)}(X, m) := \underrightarrow{lim}_{W ∈S_{X}^{(n)}(m),W′∈S_{X}^{(n′)}(m)} E^{(W \W′)}(X × Δ^{m} \ W′)

The conditions on the intersections of W with X × F for faces F means that m ↦ S_{X}^{(n)}(m) form a cosimplicial set, denoted S_{X}^{(n)}, for each n and that S_{X}^{(n′)} is a cosimplicial subset of S_{X}^{(n)} for n ≤ n′. Thus the restriction maps for E make m ↦ E^{(n)}(X, m) and m ↦ E^{(n/n′)}(X, m) simplicial spectra, denoted E^{(n)}(X, −) and E^{(n/n′)}(X, −). We denote the associated total spectra by |E^{(n)}(X, −)| and |E^{(n/n′)}(X, −)|. The inclusion S_{X}^{(n′)}(m) → S_{X}^{(n)}(m) for n ≤ n′ and the evident restriction maps give the sequence

|E^{(n′)}(X, −)| → |E^{(n)}(X, −)| → |E^{(n/n′)}(X, −)|

which is easily seen to be a weak homotopy fiber sequence. We note that |E^{(0)}(X, −)| = E(X × Δ^{∗}); as E is homotopy invariant, the canonical map

E(X) → |E^{(0)}(X, −)|

is thus a weak equivalence. We therefore have the tower in SH

(9.1) … → |E^{(n+1)}(X, −)| → |E^{(n)}(X, −)| → … → |E^{(0)}(X, −)| ≅ E(X)

with nth layer isomorphic to |E^{(n/n+1)}(X, −)|. We call this tower the homotopy coniveau tower for E(X). In this regard, one of the main results from [11] states

Theorem 9.1 ([11, theorem 7.1.1]). There is a canonical isomorphism of the tower (9.1) with the motivic Postnikov tower evaluated at X:

… → f_{n+1}E(X) → f_{n}E(X) → … → f_{0}E(X) = E(X), giving a canonical isomorphism

s_{n}E(X) ≅ |E^{(n/n+1)}(X, −)|.

We can further modify this description of s_{n}E(X) as follows: Since s_{n} is an idempotent functor, we have

s_{n}E(X) ≅ s_{n}(s_{n}E)(X) ≅ |(s_{n}E)^{(n/n+1)}(X, −)|.

Note that |(s_{n}E)^{(n/n+1)}(X, −)| fits into a weak homotopy fiber sequence

|(s_{n}E)^{(n+1)}(X, −)| → |(s_{n}E)^{(n)}(X, −)| → |(s_{n}E)^{(n/n+1)}(X, −)|.

Using theorem 9.1 in reverse, we have the isomorphism in SH

|(s_{n}E)^{(n+1)}(X, −)| ≅ f_{n+1}(s_{n}E)(X).

But as f_{n+1} ∘ f_{n} ≅ f_{n+1}, we see that f_{n+1}(s_{n}E) ≅ 0 in SH_{S^{1}}(k) and thus

|(s_{n}E)^{(n)}(X, −)| ≅ |(s_{n}E)^{(n/n+1)}(X, −)| ≅ s_{n}E(X).

We may therefore use the simplicial model |(s_{n}E)^{(n)}(X, −)| for s_{n}E(X). We will need a refinement of this construction, which takes into account the interaction of the support conditions with a given correspondence.

Definition 9.2. Let A ⊂ Y × X be a generator in Hom_{SmCor}(Y, X); for each m, we let A(m) ∈ Hom_{SmCor}(Y × Δ^{m}, X × Δ^{m}) denote the correspondence A × id_{Δ^{m}}. Let S_{X,A}^{(n)}(m) be the subset of S_{X}^{(n)}(m) consisting of those W′ ∈

S_{X}^{(n)}(m) such that

(1) W := p_{Y ×Δ^{m}}(A × Δ^{m} ∩ Y × W′) is in S_{Y}^{(n)}(m).
(2) A(m) is in Hom_{SmCor}(Y × Δ^{m}, X × Δ^{m})_{W,W′}.

For an arbitrary α ∈ Hom_{SmCor}(Y, X), write

α = ∑_{i=1}^{r} n_{i}A_{i}

with the A_{i} generators and the n_{i} non-zero integers and define

S_{X,α}^{(n)}(m) := ∩_{i=1}^{r}S_{X,A_{i}}^{(n)}(m).

If we have in addition to α a finite correspondence β ∈ Hom_{SmCor}(Z, Y), we let S_{X,α,β}^{(n)}(m) ⊂ S_{X,α}^{(n)}(m) be the set of W ⊂ X × Δ^{m} such that W is in S_{X,α}^{(n)}(m)

and p_{Y ×Δ^{m}}^{Y ×X×Δ^{m}}(Y × W ∩ |α| × Δ^{m}) is in S_{Y,β}^{(n)}(m).

For f : Y → X a flat morphism, one has

S_{X,Γ_{f}}^{(n)} (m) = S_{X}^{(n)}(m)

and for g : Z → Y a flat morphism, and α ∈ Hom_{SmCor}(Y, X), one has

S_{X,α,Γ_{g}}^{(n)}(m) = S_{X,α}^{(n)}(m)

Note that m ↦ S_{X,α}^{(n)}(m) and m ↦ S_{X,α,β}^{(n)}(m) define cosimplicial subsets of

m ↦ S_{X}^{(n)}(m). We define the simplicial spectra E^{(n)}(X, −)_{α} and E^{(n)}(X, −)_{α,β} using the support conditions S_{X,α}^{(n)}(m) and S_{X,α,β}^{(n)}(m) instead of S_{X}^{(n)}(m):

E^{(n)}(X, m)_{α} := \underrightarrow{lim}_{W ∈S_{X,α}^{(n)}(m)} E^{(W)}(X × Δ^{m})

E^{(n)}(X, m)_{α,β} := \underrightarrow{lim}_{W ∈S_{X,α,β}^{(n)}(m)} E^{(W)}(X × Δ^{m}),

giving us the sequence of simplicial spectra

E^{(n)}(X, −)_{α,β} → E^{(n)}(X, −)_{α} → E^{(n)}(X, −).

The main “moving lemma” [12, theorem 2.6.2(2)] yields

Proposition 9.3. For X ∈ Sm/k affine, and E ∈ Spt_{S^{1}}(k) quasi-fibrant, the maps

|E^{(n)}(X, −)_{α,β}| → |E^{(n)}(X, −)_{α}| → |E^{(n)}(X, −)|

are weak equivalences.

We proceed to the main construction of this section. Consider the simplicial model |(s_{n}E)^{(n)}(X, −)| for s_{n}E(X). For each m, we may consider the classical Postnikov tower (or rather, its dual version) for the spectrum (s_{n}E)^{(n)}(X, m), which we write as

… → τ_{≥p+1}(s_{n}E)^{(n)}(X, m) → τ_{≥p}(s_{n}E)^{(n)}(X, m) → … → (s_{n}E)^{(n)}(X, m),

where

τ_{≥p+1}(s_{n}E)^{(n)}(X, m) → (s_{n}E)^{(n)}(X, m)

is the p-connected cover of (s_{n}E)^{(n)}(X, m). The pth layer in this tower is of course the pth suspension of the Eilenberg-Maclane spectrum on π_{p}((s_{n}E)^{(n)}(X, m)). Taking a functorial model for the p-connected cover, we have for each p the simplicial spectrum

m ↦ τ_{≥p+1}(s_{n}E)^{(n)}(X, m)

giving us the tower of total spectra

(9.2)

… → |τ_{≥p+1}(s_{n}E)^{(n)}(X, −)| → |τ_{≥p}(s_{n}E)^{(n)}(X, −)| → … → |(s_{n}E)^{(n)}(X, −)|.

The pth layer in this tower are then (up to suspension) the Eilenberg-Maclane spectrum on the chain complex π_{p}(s_{n}E)^{(n)}(X, ∗), with differential as usual the alternating sum of the face maps. The chain complexes π_{p}(s_{n}E)^{(n)}(X, ∗) are evidently functorial for smooth maps and inherit the homotopy invariance property from (s_{n}E)^{(n)}(X, ∗) (see [12, theorem 3.3.5]). Somewhat more surprising is

Lemma 9.4. The complexes π_{p}(s_{n}E)^{(n)}(X, ∗) satisfy Nisnevich excision.

Proof. Let W ⊂ X × Δ^{m} be a closed subset in S_{X}^{(n)}(m), and let w be the set of generic points of W having codimension exactly n on X × Δ^{m}. Then

s_{n}E^{(W)}(X × Δ^{m}) ≅ s_{n}E(Σ_{ℙ^{1}}^{n}w_{+}) ≅ Ω_{ℙ^{1}}^{n}(s_{n}E)(w) ≅ s_{0}(Ω_{ℙ^{1}}^{n}E)(w).

This gives us the following description of π_{p}((s_{n}E)^{(n)}(X, m)):

π_{p}((s_{n}E)^{(n)}(X, m)) ≅ ⊕_{w}π_{p}(s_{0}(Ω_{ℙ^{1}}^{n}E)(w))

where the direct sum is over the set T_{X}^{(n)}(m) of generic points of the irreducible W ∈ S_{X}^{(n)}(m) having codimension exactly n in X × Δ^{m}. Now let i : Z → X be a closed subset with open complement j : U → X. For each m, we thus have the exact sequence

0 → ⊕_{w∈Z×Δ^{m}∩T_{X}^{(n)}(m)}π_{p}(s_{0}(Ω_{ℙ^{1}}^{n}E)(w)) → ⊕_{w∈T_{X}^{(n)}(m)}π_{p}(s_{0}(Ω_{ℙ^{1}}^{n}E)(w))

→ ⊕_{w∈T_{X}^{(n)}(m)∩U×Δ^{m}}π_{p}(s_{0}(Ω_{ℙ^{1}}^{n}E)(w)) → 0

Define the subcomplex π_{p}(s_{n}E)^{(n)}(X, ∗)_{Z} of π_{p}(s_{n}E)^{(n)}(X, ∗) and quotient complex π_{p}(s_{n}E)^{(n)}(U_{X}, ∗) of π_{p}(s_{n}E)^{(n)}(X, ∗) by taking supports in

{W ∈ S_{X}^{(n)}(m) | W ⊂ Z × Δ^{m}}, resp. {W ∩ U × Δ^{m} | W ∈ S_{X}^{(n)}(m)}.

We thus have the term-wise exact sequence of complexes

0 → π_{p}(s_{n}E)^{(n)}(X, ∗)_{Z} → π_{p}(s_{n}E)^{(n)}(X, ∗) → π_{p}(s_{n}E)^{(n)}(U_{X}, ∗) → 0

Claim. The inclusion

π_{p}(s_{n}E)^{(n)}(U_{X}, ∗) ⟶^{ι} π_{p}(s_{n}E)^{(n)}(U, ∗)

is a quasi-isomorphism.

Proof of the claim. This follows using the localization technique [13, theorem 8.10] (for details, see [11, theorem 3.2.1]). In a few words, one takes an integer N and a W ∈ S_{U}^{(n)}(N). We assume that (id_{U} × Δ(σ))(W) = W for each permutation σ of the vertices of Δ^{N}, where Δ(σ) : Δ^{N} → Δ^{N} is the affinelinear extension of σ. For m ≤ N, let T_{X}^{(n)}(m)_{W} ⊂ T_{X}^{(n)}(m) be the set of points w such that w ∈ (id_{U} × Δ(g))^{∗}(W) for some injective g : [m] → [N], and set

π_{p}(s_{n}E)^{(n)}(U, m)_{W} := ⊕_{w∈T_{X}^{(n)}(m)_{W}} π_{p}(s_{n}E)^{(n)}(U, m) ⊂ π_{p}(s_{n}E)^{(n)}(U, m).

For m > N set π_{p}(s_{n}E)^{(n)}(U, m)_{W} = 0. This gives us the subcomplex

π_{p}(s_{n}E)^{(n)}(U, ∗)_{W} ⊂ π_{p}(s_{n}E)^{(n)}(U, ∗);

clearly π_{p}(s_{n}E)^{(n)}(U, ∗) is the colimit of the subcomplexes π_{p}(s_{n}E)^{(n)}(U, ∗)_{W}. Similarly, we have the subcomplex π_{p}(s_{n}E)^{(n)}(U_{X}, ∗)_{W} of π_{p}(s_{n}E)^{(n)}(U_{X}, ∗) and the inclusion

ι_{W} : π_{p}(s_{n}E)^{(n)}(U_{X}, ∗)_{W} → π_{p}(s_{n}E)^{(n)}(U, ∗)_{W} ,

with π_{p}(s_{n}E)^{(n)}(U_{X}, ∗) the colimit of the π_{p}(s_{n}E)^{(n)}(U_{X}, ∗)_{W}.

In [11, theorem 3.2.1], we have the formal sums of maps of simplices

ψ_{W} (m) = ∑_{i} n_{i}ψ_{W} (m)_{i}; ψ_{W} (m)_{i} : Δ^{m} → Δ^{m}

Ψ_{W} (m) = ∑_{i} m_{i}Ψ_{W} (m)_{i}; Ψ_{W} (m)_{i} : Δ^{m+1} → Δ^{m}

for m = 0, …, N, such that the pull-back by the maps ψ_{W} (m) define a map of complexes

ψ_{W}^{∗} : π_{p}(s_{n}E)^{(n)}(U, ∗)_{W} → π_{p}(s_{n}E)^{(n)}(U_{X}, ∗).

Additionally, the pull-back by the Ψ_{W} (m) define homotopies of the map ι ∘ ψ_{W} with the inclusion π_{p}(s_{n}E)^{(n)}(U, ∗)_{W} → π_{p}(s_{n}E)^{(n)}(U, ∗) and similarly of ψ_{W} ∘ ι_{W} with the inclusion π_{p}(s_{n}E)^{(n)}(U_{X}, ∗)_{W} → π_{p}(s_{n}E)^{(n)}(U_{X}, ∗). The claim follows easily from this. □

We therefore have the quasi-isomorphism (9.3)

π_{p}(s_{n}E)^{(n)}(X, ∗)_{Z} → cone (π_{p}(s_{n}E)^{(n)}(X, ∗) ⟶^{j^{∗}} π_{p}(s_{n}E)^{(n)}(U, ∗)) [−1].

Now let

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

be an elementary Nisnevich square, i.e., the square is cartesian, p is étale and induces an isomorphism p : Z′ := X′ \ U′ → Z. Clearly p induces an isomorphism

p^{∗} : π_{p}(s_{n}E)^{(n)}(X, ∗)_{Z} → π_{p}(s_{n}E)^{(n)}(X′, ∗)_{Z′};

using the localization quasi-isomorphism (9.3), it follows that p^{∗} induces a quasi-isomorphism

cone (π_{p}(s_{n}E)^{(n)}(X, ∗) ⟶^{j^{∗}} π_{p}(s_{n}E)^{(n)}(U, ∗)) [−1]

⟶^{p^{∗}} cone (π_{p}(s_{n}E)^{(n)}(X′, ∗) ⟶^{j^{∗}} π_{p}(s_{n}E)^{(n)}(U′, ∗)) [−1],

proving the lemma. □

We will use the results of section 7 to give X ↦ π_{p}(s_{n}E)^{(n)}(X, ∗) the structure of a complex of homotopy invariant presheaves with transfer on Sm/k, i.e. a motive. For this, we consider the complexes π_{p}(s_{n}E)^{(n)}(X, ∗)_{α}, π_{p}(s_{n}E)^{(n)}(X, ∗)_{α,β} constructed above. The refined support condition are constructed so that, for each W ∈ S_{X,α}^{(n)}(m), α × id_{Δ^{m}} is in Hom_{SmCor}(Y × Δ^{m}, X × Δ^{m})_{W′,W}, where

W′ = p_{Y ×Δ^{m}}(Y × Δ^{m} × W ∩ |α × id_{Δ^{m}}|).

We may therefore use the morphism co-tr^{n}(α × id_{Δ^{m}}) to define the map

Tr_{Y/X}(α)(m) : π_{p}((s_{n}E)^{(n)}(X, m))_{α} → π_{p}(s_{n}E)^{(n)}(Y, m).

By proposition 7.5, the maps Tr_{Y/X}(m) define a map of complexes

Tr_{Y/X}(α) : π_{p}(s_{n}E)^{(n)}(X, ∗)_{α} → π_{p}(s_{n}E)^{(n)}(Y, ∗).

Similarly, given β ∈ Hom_{SmCor}(Z, Y), we have the map of complexes

Tr_{Y/X}(α)_{β} : π_{p}(s_{n}E)^{(n)}(X, ∗)_{α,β} → π_{p}(s_{n}E)^{(n)}(Y, ∗)_{β}.

Note that, due to possible cancellations occurring when one takes the composition α ∘ β, we have only an inclusion

S_{X,α,β}^{(n)}(m) ⊂ S_{X,α∘β}^{(n)}(m)

giving us a natural comparison map

ι_{α,β} : π_{p}(s_{n}E)^{(n)}(X, ∗)_{α,β} →: π_{p}(s_{n}E)^{(n)}(X, ∗)_{α∘β}.

Using our moving lemma again, we see that ι_{α,β} is a quasi-isomorphism in case X is affine.

Lemma 9.5. Suppose char k = 0. For

α ∈ Hom_{SmCor}(Z, Y), β ∈ Hom_{SmCor}(Z, Y),

we have

Tr_{Z/Y} (β) ∘ Tr_{Y/X}(α)_{β} = Tr_{Z/X}(α ∘ β) ∘ ι_{α,β}.

Proof. This follows from lemma 7.4. □

We have already noted that complexes π_{p}(s_{n}E)^{(n)}(X, ∗) are functorial in X for flat morphisms in Sm/k, in particular for smooth morphisms in Sm/k. Let S̃m/k denote the subcategory of Sm/k with the same objects and with morphisms the smooth morphisms. The transfer maps we have defined on the refined complexes, together with the moving lemma 7.4 yield the following result:

Theorem 9.6. Suppose char k = 0. Consider the presheaf

π_{p}(s_{n}E)^{(n)}(−, ∗)) : S̃m/k^{op} → C^{−}(Ab)

on S̃m/k^{op}. Let

ι : S̃m/k → SmCor(k)

be the evident inclusion and let

Q : C^{−}(Ab) → D^{−}(Ab)

be the evident additive functor. There is a complex of presheaves with transfers

π̂_{p}((s_{n}E)^{(n)})^{∗} : SmCor(k)^{op} → C^{−}(Ab)

and an isomorphism of functors from S̃m/k^{op} to D^{−}(Ab)

Q ∘ π_{p}(s_{n}E)^{(n)}(−, ∗)) ≅ Q ∘ π̂_{p}((s_{n}E)^{(n)})^{∗} ∘ ι.

Proof. We give a rough sketch of the construction here; for details we refer the reader to [9, proposition 2.2.3], which in turn is an elaboration of [12, theorem 7.4.1]. The construction of π̂_{p}((s_{n}E)^{(n)})^{∗} is accomplished by first taking a homotopy limit over the complexes π_{p}(s_{n}E)^{(n)}(X, ∗)_{α}. These are then functorial on SmCor(k)^{op}, up to homotopy equivalences arising from the replacement of the index category for the homotopy limit with a certain cofinal subcategory. One then forms a regularizing homotopy colimit that is strictly functorial on SmCor(k)^{op}, and finally, one replaces this presheaf with a fibrant model. The moving lemma for affine schemes (proposition 9.3) implies that the homotopy limit construction yields for each affine X ∈ Sm/k a complex canonically quasi-isomorphic to π_{p}(s_{n}E)^{(n)}(X, ∗); this property is inherited by the regularizing homotopy colimit. As the complexes π_{p}(s_{n}E)^{(n)}(X, ∗) satisfy Nisnevich excision (lemma 9.4) and are homotopy invariant for all X, this implies that π̂_{p}((s_{n}E)^{(n)})^{∗}(X) is canonically isomorphic to π_{p}(s_{n}E)^{(n)}(X, ∗) in D^{−}(Ab) for all X ∈ Sm/k. By construction, this isomorphism is natural with respect to smooth morphisms in Sm/k. □

Corollary 9.7. Suppose char k = 0. π̂_{p}((s_{n}E)^{(n)})^{∗} is a homotopy invariant complex of presheaves with transfer.

Proof. By theorem 9.6, we have the isomorphism in D^{−}(Ab)

π̂_{p}((s_{n}E)^{(n)})^{∗}(X) ≅ π_{p}(s_{n}E)^{(n)}(X, ∗).

for all X ∈ Sm/k, natural with respect to smooth morphisms. As the presheaf π_{p}(s_{n}E)^{(n)}(−, ∗) is homotopy invariant, so is π̂_{p}((s_{n}E)^{(n)})^{∗}. □

proof of theorem 2. As in the proof of theorem 9.6, the method of [12, theorem 7.4.1], shows that the tower (9.2) extends to a tower

(9.4)  … → ρ_{≥p+1}s_{n}E → ρ_{≥p}s_{n}E → … → s_{n}E

in SH_{S^{1}}(k) with value (9.2) at X ∈ Sm/k, and with the cofiber of ρ_{≥p+1}s_{n}E → ρ_{≥p}s_{n}E naturally isomorphic to EM_{𝔸^{1}}^{eff}(π̂_{p}((s_{n}E)^{(n)})^{∗}). By lemma 9.4 and corollary 9.7, the presheaves π̂_{p}((s_{n}E)^{(n)})^{∗} define objects in DM_{−}^{eff}(k). Thus, we have shown that the layers in the tower (9.4) have a motivic structure, proving theorem 2. □

10. The Friedlander-Suslin tower

As the reader has surely noticed, the lack of functoriality for the simplicial spectra E^{(n)}(X, −) creates annoying technical problems when we wish to extend the construction of the homotopy coniveau tower to a tower in SH_{S^{1}}(k). In their work on the spectral sequence from motivic cohomology to K-theory, Friedlander and Suslin [4] have constructed a completely functorial version of the homotopy coniveau tower, using “quasi-finite supports”. Unfortunately, the comparison between the Friedlander-Suslin version and E^{(n)}(X, −) is proven in [4] only for K-theory and motivic cohomology. In this last section, we recall the Friedlander-Suslin construction and form the conjecture that the Friedlander-Suslin tower is naturally isomorphic to the homotopy coniveau tower. Let Q_{X}^{(n)}(m) be the set of closed subsets W of 𝔸^{n} ×X ×Δ^{m} such that, for each irreducible component W′ of W, the projection W′ → X × Δ^{m} is quasi-finite. For E ∈ Spt_{S^{1}}(k), we let

E_{F S}^{(n)}(X, m) := \underrightarrow{lim}_{W ∈Q_{X}^{(n)}(m)} E^{(W)}(𝔸^{n} × X × Δ^{m})

As the condition defining Q_{X}^{(n)}(m) are preserved under maps

id_{𝔸^{n}} × f × g : 𝔸^{n} × X′ × Δ^{m′} → 𝔸^{n} × X × Δ^{m},

where f : X′ → X is an arbitrary map in Sm/k, and g : Δ^{m′} → Δ^{m} is a structure map in Δ^{∗}, the spectra E_{F S}^{(n)}(X, m) define a simplicial spectrum E_{F S}^{(n)}(X, −) and these simplicial spectra, for X ∈ Sm/k, extend to a presheaf of simplicial spectra on Sm/k:

E_{F S}^{(n)}(?, −) : Sm/k^{op} → Δ^{op}Spt.

Similarly, if we take the linear embedding i_{n} : 𝔸^{n} → 𝔸^{n+1} = 𝔸^{n} × 𝔸^{1}, x ↦ (x, 0), the pull-back by i_{n} × id preserves the support conditions, and thus gives a well-defined map of simplicial spectra

i_{n}^{∗} : E_{F S}^{(n+1)}(X, −) → E_{F S}^{(n)}(X, −),

forming the tower of presheaves on Sm/k

(10.1)  … → E_{F S}^{(n+1)}(?, −) → E_{F S}^{(n)}(?, −) → ….

We may compare E_{F S}^{(n)}(X, −) and E^{(n)}(X, −) using the method of [4] as follows: The simplicial spectra E^{(n)}(X, −) are functorial for flat maps in Sm/k, in the evident manner. They satisfy homotopy invariance, in that the pull-back map

p^{∗} : E^{(n)}(X, −) → E^{(n)}(𝔸^{1} × X, −)

induces a weak equivalence on the total spectra. We have the evident inclusion of simplicial sets

Q_{X}^{(n)}(−) ↪ S_{𝔸^{n}×X}^{(n)}(−)

inducing the map

φ_{X,n} : E_{F S}^{(n)}(X, −) → E^{(n)}(𝔸^{n} × X, −).

Together with the weak equivalence p^{∗} : |E^{(n)}(X, −)| → |E^{(n)}(𝔸^{n} × X, −)|, the maps φ_{X,n} induce a map of towers of total spectra in SH

(10.2)  φ_{X,∗} : |E_{F S}^{(∗)}(X, −)| → |E^{(∗)}(X, −)|.

Conjecture 10.1. For each X ∈ Sm/k and each quasi-fibrant E ∈ Spt_{S^{1}}(k), the map (10.2) induces an isomorphism in SH of the towers of total spectra.

Combined with the weak equivalence given by homotopy invariance and the results of [11], this would give us an isomorphism in SH_{S^{1}}(k):

f_{n}E ≅ |E_{F S}^{(n)}(?, −)|.

As transfers in some form or other are used in the arguments relating the Friedlander-Suslin complex to the Bloch-type complexes in the known cases, a weaker form of the conjecture might be more reasonable:

Conjecture 10.2. For each X ∈ Sm/k and each quasi-fibrant E ∈ Spt_{S^{1}}(k) with s_{0}E ≅ 0 in SH_{S^{1}}(k), the map (10.2) induces an isomorphism in SH of the towers of total spectra.

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Marc Levine

Northeastern University Department of Mathematics Boston MA 02115 U.S.A. marc@neu.edu Fakultät Mathematik, Campus Essen Universität Duisburg-Essen 45117 Essen Germany marc.levine@uni-due.de 444

Documenta Mathematica · Extra Volume Suslin (2010)