Motivic Tubular Neighborhoods

Marc Levine

Received: November 12, 2005 Revised: January 2, 2007

Communicated by Alexander Merkurjev

Abstract. We construct motivic versions of the classical tubular neighborhood and the punctured tubular neighborhood, and give applications to the construction of tangential base-points for mixed Tate motives, algebraic gluing of curves with boundary components, and limit motives.

2000 Mathematics Subject Classification: Primary 14C25; Secondary 55P42, 18F20, 14F42 Keywords and Phrases: Limit motives, moduli of curves, Tate motives

1. Introduction

Let i : A → B be a closed embedding of finite CW complexes. One useful fact is that A admits a cofinal system of neighborhoods T in B with A → T a deformation retract. This is often used in the case where B is a differentiable manifold, showing for example that A has the homotopy type of the differentiable manifold T. This situation occurs in algebraic geometry, for instance in the case of the inclusion of the special fiber in a degeneration of smooth varieties X → C over the complex numbers. To some extent, one has been able to mimic this construction in purely algebraic terms. The rigidity theorems of Gillet-Thomason [14], extended by Gabber (details appearing a paper of Fujiwara [13]) indicated that, at least through the eyes of torsion étale sheaves, the topological tubular neighborhood can be replaced by the Hensel neighborhood. However, basic examples of non-torsion phenomena, even in the étale topology, show that the Hensel neighborhood cannot always be thought of as a tubular neighborhood, perhaps the simplest example being the sheaf 𝔾_{m}.^{1} Our object in this paper is to construct an algebraic version of the tubular neighborhood which has the basic properties of the topological construction, at least for a reasonably large class of cohomology theories. It turns out that a “homotopy invariant” version of the Hensel neighborhood does the job, at least for theories which are themselves homotopy invariant. If one requires in addition that the given cohomology theory has a Mayer-Vietoris property for the Nisnevich topology, then one also has an algebraic version of the punctured tubular neighborhood. We extend these constructions to the case of a (reduced) strict normal crossing subscheme by a Mayer-Vietoris procedure, giving us the tubular neighborhood and punctured tubular neighborhood of a normal crossing subscheme of a smooth k-scheme. Morel and Voevodsky have constructed an algebro-geometric version of homotopy theory, in the setting of presheaves of spaces or spectra on the category of smooth varieties over a reasonable base scheme B; we concentrate on the 𝔸^{1}-homotopy category of spectra, SH_{𝔸^{1}}(B). For a map f : X → Y, they construct a pair of adjoint functors

1If O is a local ring with residue field k and maximal ideal 𝔪, the surjection 𝔾_{m}(O) → 𝔾_{m}(k) has kernel (1 + 𝔪)^{×}, which is in general non-zero, even for O Hensel

Rf_{∗} : SH_{𝔸^{1}}(X) → SH_{𝔸^{1}}(Y)
Lf^{∗} : SH_{𝔸^{1}}(Y) → SH_{𝔸^{1}}(X).

If we have a closed immersion i : W → X with open complement j : U → X, then one has the functor

Li^{∗}Rj_{∗} : SH_{𝔸^{1}}(U) → SH_{𝔸^{1}}(W)

One of our main results is that, in case W is a strict normal crossing subscheme of a smooth X, the restriction of Li^{∗}Rj_{∗}E to a Zariski presheaf on W can be viewed as the evaluation of E on the punctured tubular neighborhood of W in X. Consider a morphism p : X → 𝔸^{1} and take i : W → X to be the inclusion of p^{−1}(0). Following earlier constructions of Spitzweck [43], Ayoub has constructed a “unipotent specialization functor” in the motivic setting, essentially (in the case of a semi-stable degeneration) by evaluating Li^{∗}Rj_{∗}E on a cosimplicial version of the appropriate path space on 𝔾_{m} with base-point 1. Applying the same idea to our tubular neighborhood construction gives a model for this specialization functor, again only as a Zariski presheaf on p^{−1}(0). Ayoub has also defined a motivic monodromy operator and monodromy sequence involving the unipotent specialization functor and the functor Li^{∗}Rj_{∗}, for theories with ℚ-coefficients that satisfy a certain additional condition (see definition 9.2.2). We give a model for this construction by combining our punctured tubular neighborhood with a ℚ-linear version of the 𝔾_{m}-path space mentioned above. We conclude with an application of our constructions to the moduli spaces of smooth curves and a construction of a specialization functor for category of mixed Tate motives, which in some cases yields a purely algebraic construction of tangential base-points. Of course, the construction of Ayoub, when restricted to the triangulated category of Tate motives, also gives such a specialization functor, but we hope that the explicit nature of our construction will be useful for applications. We have left to another paper the task of checking the compatibilities of our constructions with others via the appropriate realization functor. As we have already mentioned, our punctured tubular neighborhood construction is comparable with the motivic version of the functor Li^{∗}Rj_{∗} for the situation of a normal crossing scheme i : D → X with open complement j : X \ D → X; this should imply that it is a model for the analogous functor after realization. Similarly, our limit cohomology construction should transform after realization to the appropriate version of the sheaf of vanishing cycles, at least in the case of a semi-stable degeneration, and should be comparable with the constructions of Rappaport-Zink [37] as well as the limit mixed Hodge structure of Katz [22] and Steenbrink [44]. Our specialization functor for Tate motives should be compatible with the Betti, étale and Hodge realizations; similarly, realization functors applied to our limit motive should yield for example the limit mixed Hodge structure. We hope that our rather explicit construction of the limiting motive will be useful in giving a geometric view to the limit mixed Hodge structure of a semi-stable degeneration but we have not attempted an investigation of these issues in this paper. My interest in this topic began as a result of several discussions on limit motives with Spencer Bloch and Hélène Esnault, whom I would like to thank for their encouragement and advice. I would also like to thank Hélène Esnault for clarifying the role of the weight filtration leading to the exactness of Clemens-Schmidt monodromy sequence (see Remark 9.3.6). An earlier version of our constructions used an analytic (i.e. formal power series) neighborhood instead of the Hensel version now employed; I am grateful to Fabien Morel for suggesting this improvement. Finally, I want to thank Joseph Ayoub for explaining his construction of the nearby cycles functor; his comments suggested to us the use of the cosimplicial path space in our construction of limit cohomology. In addition, Ayoub noticed a serious error in our first attempt at constructing the monodromy sequence; the method used in this version is following his suggestions and comments. Finally, we would like to thank the referee for giving unusually thorough and detailed comments and suggestions, which have substantially improved this paper. In particular, the material in sections 7 comparing our construction with the categorical ones of Morel-Voevodsky, as well as the comparison with Ayoub’s specialization functor and monodromy sequence in section 8.3 and section 9 was added following the suggestion of the referee, who also supplied the main ideas for the proofs.

2. Model structures and other preliminaries

2.1. Presheaves of simplicial sets. We recall some facts on the model structures in categories of simplicial sets, spectra, associated presheaf categories and certain localizations. For details, we refer the reader to [17] and[19]. For a small category I and category C, we will denote the category of functors from I to C by C^{I}. We let Ord denote the category with objects the finite ordered sets [n] := {0, …, n} (with the standard ordering) and morphisms the order-preserving maps of sets. For a category C, the functor categories C^{Ord}, C^{Ord^{op}} are the categories of cosimplicial objects of C, resp. simplicial objects of C. For C = Sets, we have the category of simplicial sets, Spc := Sets^{Ord^{op}}, and similarly for C the category of pointed sets, Sets_{∗}, the category of pointed simplicial sets Spc_{∗} := Sets_{∗}^{Ord^{op}}. We give Spc and Spc_{∗} the standard model structures: cofibrations are (pointed) monomorphisms, weak equivalences are weak homotopy equivalences on the geometric realization, and fibrations are detemined by the right lifting property (RLP) with respect to trivial cofibrations; the fibrations are then exactly the Kan fibrations. We let |A| denote the geometric realization, and [A, B] the homotopy classes of (pointed) maps |A| → |B|. For an essentially small category C, we let Spc(C) be the category of presheaves of simplicial sets on C. We give Spc(C) the so-called injective model structure, that is, the cofibrations and weak equivalences are the pointwise ones, and the fibrations are determined by the RLP with respect to trivial cofibrations. We let HSpc(C) denote the associated homotopy category (see [17] for details on these model structures for Spc and Spc(C)).

2.2. Presheaves of spectra. Let Spt denote the category of spectra. To fix ideas, a spectrum will be a sequence of pointed simplicial sets E_{0}, E_{1}, … together with maps of pointed simplicial sets ϵ_{n} : S^{1} ∧ E_{n} → E_{n+1}. Maps of spectra are maps of the underlying simplicial sets which are compatible with the attaching maps ϵ_{n}. The stable homotopy groups π_{n}^{s}(E) are defined by

π_{n}^{s}(E) := lim_{m→∞}[S^{m+n}, E_{m}].

The category Spt has the following model structure: Cofibrations are maps f : E → F such that E_{0} → F_{0} is a cofibration, and for each n ≥ 0, the map

E_{n+1} ∐_{S^{1}∧E_{n}} S^{1} ∧ F_{n} → F_{n+1}

is a cofibration. Weak equivalences are the stable weak equivalences, i.e., maps f : E → F which induce an isomorphism on π_{n}^{s} for all n. Fibrations are characterized by having the RLP with respect to trivial cofibrations. We write SH for the homotopy category of Spt. For X ∈ Spc_{∗}, we have the suspension spectrum Σ^{∞}X := (X, ΣX, Σ^{2}X, …) with the identity bonding maps. Dually, for a spectrum E := (E_{0}, E_{1}, …) we have the 0-space Ω^{∞}E := lim_{n} Ω^{n}E_{n}. These operations form a Quillen pair of adjoint functors (Σ^{∞}, Ω^{∞}) between Spc_{∗} and Spt, and thus induce adjoint functors on the homotopy categories. Let C be a category. A functor E : C^{op} → Spt is called a presheaf of spectra on C. We use the following model structure on the category of presheaves of spectra (see [19]): Cofibrations and weak equivalences are given pointwise, and fibrations are characterized by having the RLP with respect to trivial cofibrations. We denote this model category by Spt(C), and the associated homotopy category by HSpt(C).

As a particular example, we have the model category of simplicial spectra Spt^{Ord^{op}} = Spt(Ord). We have the total spectrum functor

Tot : Spt(Ord) → Spt

which preserves weak equivalences. The adjoint pair (Σ^{∞}, Ω^{∞}) extend pointwise to define a Quillen pair on the presheaf categories and an adjoint pair on the homotopy categories. Let B be a noetherian separated scheme of finite Krull dimension. We let Sm/B denote the category of smooth B-schemes of finite type over B. We often write Spc(B) and HSpc(B) for Spc(Sm/B) and HSpc(Sm/B), and write Spt(B) and HSpt(B) for Spt(Sm/B) and HSpt(Sm/B). For Y ∈ Sm/B, a subscheme U ⊂ Y of the form Y \ ∪_{α}F_{α}, with {F_{α}} a possibly infinite set of closed subsets of Y, is called essentially smooth over B; the category of essentially smooth B-schemes is denoted Sm^{ess}.

2.3. Local model structure. If the category C has a topology, there is often another model structure on Spc(C) or Spt(C) which takes this into account. We consider the case of the small Nisnevich site X_{Nis} on a scheme X (assumed to be noetherian, separated and of finite Krull dimension), and the big Nisnevich sites Sm/B_{Nis} or Sch/B_{Nis}, as well as the Zariski versions X_{Zar}, Sm/B_{Zar}, etc. We describe the Nisnevich version for spectra below; the definitions and results for the Zariski topology and for spaces are exactly parallel.

Definition 2.3.1. A map f : E → F of presheaves of spectra on X_{Nis} is a local weak equivalence if the induced map on the Nisnevich sheaf of stable homotopy groups f_{∗} : π_{m}^{s}(E)_{Nis} → π_{m}^{s}(F)_{Nis} is an isomorphism of sheaves for all m. A map f : E → F of presheaves of spectra on Sm/B_{Nis} or Sch/B_{Nis} is a local weak equivalence if the restriction of f to X_{Nis} is a local weak equivalence for all X ∈ Sm/B or X ∈ Sch/B. □

The Nisnevich local model structure on the category of presheaves of spectra on X_{Nis} has cofibrations given pointwise, weak equivalences the local weak equivalences and fibrations are characterized by having the RLP with respect to trivial cofibrations. We write Spt(X_{Nis}) for this model category, and HSpt(X_{Nis}) for the associated homotopy category. The Nisnevich local model categories Spt(Sm/B_{Nis}) and Spt(Sch/B_{Nis}), with homotopy categories HSpt(Sm/B_{Nis}) and HSpt(Sch/B_{Nis}), are defined similarly. A similar localization gives model categories of presheaves of spaces Spc(X_{Nis}), Spc(X_{Zar}), Spc(Sm/B_{Nis}), etc., and homotopy categories HSpc(X_{Nis}), HSpc(X_{Zar}), HSpc(Sm/B_{Nis}), etc. We also have the adjoint pair (Σ^{∞}, Ω^{∞}) in this setting. For details, we refer the reader to [19].

Remark 2.3.2. Let E be in Spt(Sm/B_{Nis}), and let

\xymatrix{W′ \ar[r]^{i′} \ar[d] & Y′ \ar[d]^{f} \\ W \ar[r]_{i} & X}

be an elementary Nisnevich square, i.e., f is étale, i : W → X is a closed immersion, the square is cartesian, and W′ → W is an isomorphism, with X and X′ in Sm/B (see [34, Definition 1.3, pg. 96]) . If E is fibrant in Spt(Sm/B_{Nis}) then E transforms each elementary Nisnevich square to a homotopy cartesian square in Spt. Conversely, suppose that E transforms each elementary Nisnevich square to a homotopy cartesian square in Spt. Then E is quasi-fibrant, i.e., for all Y ∈ Sm/B, the canonical map E(Y) → E_{fib}(Y), where E_{fib} is the fibrant model of E, is a weak equivalence. See [19] for details. If we define an elementary Zariski square as above, with X′ → X an open immersion, the same holds in the model category Spt(Sm/B_{Zar}). More precisely, one can show (see e.g. [45]) that, if E transforms each elementary Zarisk square to a homotopy cartesian square in Spt, then E satisfies Mayer-Vietoris for the Zariski topology: if X ∈ Sm/B is a union of Zariski open subschemes U and V, then the evident sequence

E(X) → E(U) ⊕ E(V) → E(U ∩ V)

is a homotopy fiber sequence in SH. □

Remark 2.3.3. Let C be a small category with an initial object ∅ and admiting finite coproducts over ∅, denoted X ⨿ Y. A functor E : C^{op} → Spt is called additive if for each X, Y in C, the canonical map

E(X ⨿ Y) → E(X) ⊕ E(Y)

in SH is an isomorphism. It is easy to show that if E ∈ Spt(Sm/B) satisfies Mayer-Vietoris for the Zariski topology, and E(∅) ≅ 0 in SH, then E is additive. From now on, we will assume that all our presheaves of spectra E satisfy E(∅) ≅ 0 in SH. □

2.4. 𝔸^{1}-local structure. One can perform a Bousfield localization on Spc(Sm/B_{Nis}) or Spt(Sm/B_{Nis}) so that the maps Σ^{∞}X × 𝔸_{+}^{1} → Σ^{∞}X_{+} induced by the projections X × 𝔸^{1} → X become weak equivalences. We call the resulting model structure the Nisnevich-local 𝔸^{1}-model structure, denoted Spc_{𝔸^{1}}(Sm/B_{Nis}) or Spt_{𝔸^{1}}(Sm/B_{Nis}). One has the Zariski-local versions as well. We denote the homotopy categories for the Nisnevich version by H_{𝔸^{1}}(B) (for spaces) and SH_{𝔸^{1}}(B) (for spectra). For the Zariski versions, we indicate the topology in the notation. We also have the adjoint pair (Σ^{∞}, Ω^{∞}) in this setting. For details, see [30, 31, 34].

2.5. Additional notation. Given W ∈ Sm/S, we have restriction functors

Spc(S) → Spc(W_{Zar})
Spt(S) → Spt(W_{Zar});

we write the restriction of some E ∈ Spc(S) to Spc(W_{Zar}) as E(W_{Zar}). We use a similar notation for the restriction of E to Spt(W_{Zar}), or for restrictions to W_{Nis}. More generally, if p : Y → W is a morphism in Sm/S, we write E(Y/W_{Zar}) for the presheaf U ↦ E(Y ×_{W} U) on W_{Zar}. For Z ⊂ Y a closed subset, Y ∈ Sm/S and for E ∈ Spc(S) or E ∈ Spt(S), we write E^{Z}(Y) for the homotopy fiber of the restriction map

E(Y) → E(Y \ Z).

We define the presheaf E^{Z_{Zar}}(Y) by setting, for U ⊂ Z a Zariski open subscheme with closed complement F,

E^{Z_{Zar}}(Y)(U) := E^{U}(Y \ F).

A co-presheaf on a category C with values in A is just an A-valued preheaf on C^{op}. As usual, we let Δ^{n} denote the algebraic n-simplex

Δ^{n} := Spec ℤ[t_{0}, …, t_{n}]/ ∑_{i} t_{i} − 1,

and Δ^{∗} the cosimplicial scheme n ↦ Δ^{n}. For a scheme X, we have Δ_{X}^{n} := X × Δ^{n} and the cosimplicial scheme Δ_{X}^{∗}. Let B be a scheme as above. For E ∈ Spc(B) or in Spt(B), we say that E is homotopy invariant if for all X ∈ Sm/B, the pull-back map E(X) → E(X×𝔸^{1}) is a weak equivalence (resp., stable weak equivalence). We say that E satisfies Nisnevich excision if E transforms elementary Nisnevich squares to homotopy cartesian squares.

3. Tubular neighborhoods for smooth pairs

Let i : W → X be a closed immersion in Sm/k. In this section, we construct the tubular neighborhood τ X̂ ; ϵ(W) of W in X as a functor from W_{Zar} to cosimplicial pro-k-schemes. Given E ∈ Spc(k), we can evaluate E on τ X̂ ; ϵ(W), yielding the presheaf of spaces E(τ X̂ ; ϵ(W)) on W_{Zar}, which is our main object of study.

3.1. The cosimplicial pro-scheme τ X̂ ; ϵ(W). For a closed immersion W → T in Sm/k, let T_{Nis}^{W} be the category of Nisnevich neighborhoods of W in T, i.e., objects are étale maps p : T′ → T of finite type, together with a section s : W → T′ to p over W. Morphisms are morphisms over T which respect the sections. Note that T_{Nis}^{W} is a left-filtering essentially small category. Sending (p : T′ → T, s : W → T′) to T′ ∈ Sm/k defines the pro-object T̂_{W}^{h} of Sm/k; the sections s : W → T′ give rise to a map of the constant pro-scheme W to T̂_{W}^{h}, denoted

î_{W} : W → T̂_{W}^{h} .

Given a k-morphism f : S → T, and closed immersions i_{V} : V → S, i_{W} : W → T such that f ∘ i_{V} factors through i_{W} (by f̄: V → W), we have the pull-back functor

f^{∗} : T_{Nis}^{W} → S_{Nis}^{V},
f^{∗}(T′ → T, s : W → T′) := (T′ ×_{T} S, (s ∘ f̄, i_{V})).

This gives us the map of pro-objects f^{h} : Ŝ_{V}^{h} → T̂_{W}^{h}, so that sending W → T to T̂_{W}^{h} and f to f^{h} becomes a pseudo-functor. We let f^{h} : Ŝ_{V}^{h} → T̂_{W}^{h} denote the induced map on pro-schemes. If f happens to be a Nisnevich neighborhood of W → X (so f̄: V → W is an isomorphism) then f^{h} : Ŝ_{V}^{h} → T̂_{W}^{h} is clearly an isomorphism.

Remark 3.1.1. The pseudo-functor (W → T) ↦ T̂_{W}^{h} can be rectified to an honest functor by first replacing T_{Nis}^{W} with the cofinal subcategory T_{Nis,0}^{W} of neighborhoods T′ → T, s : W → T′ such that each connected component of T′ has non-empty intersection with s(W). One notes that T_{Nis,0}^{W} has only identity automorphisms, so we replace T_{Nis,0}^{W} with a choice of a full subcategory T_{Nis,00}^{W} giving a set of representatives of the isomorphism classes in T_{Nis,0}^{W}. Given a

map of pairs of closed immersions f : (V ⟶^{i_{V}} S) → (W ⟶^{i_{W}} T) as above, we modify the pull-back functor f^{∗} defined above by passing to the connected component of (s ∘ f̄, i_{V})(V) in T′ ×_{T} S. We thus have the honest functor (W → T) ↦ T_{Nis,00}^{W} which yields an equivalent pro-object T̂_{W}^{h}. As pointed out by the referee, one can also achieve strict functoriality by rectifying the fiber product; in any case, we will use a strictly functorial version from now on without comment. □

For a closed immersion i : W → X in Sm/k, set Δ̂_{X,W}^{n} := (Δ̂_{X}^{n})_{Δ_{W}^{n}}^{h}. The cosimplicial scheme

Δ_{X}^{∗} : Ord → Sm/k
[n] ↦ Δ_{X}^{n}

thus gives rise to the cosimplicial pro-scheme

Δ̂_{X,W}^{∗} : Ord → Pro-Sm/k
[n] ↦ Δ̂_{X,W}^{n}

The maps î_{Δ_{W}^{n}} : Δ_{W}^{n} → (Δ̂_{X}^{n})_{Δ_{W}^{n}}^{h} give the closed immersion of cosimplicial pro-schemes

î_{W} : Δ_{W}^{∗} → Δ̂_{X,W}^{∗} .

Also, the canonical maps π_{n} : Δ̂_{X,W}^{n} → Δ_{X}^{n} define the map

π_{X,W} : Δ̂_{X,W}^{∗} → Δ_{X}^{∗}.

Let (p : X′ → X, s : W → X′) be a Nisnevich neighborhood of (W, X). The map

p : Δ̂_{X′,W}^{n} → Δ̂_{X,W}^{n}

is an isomorphism respecting the closed immersions î_{W}. Thus, sending a Zariski open subscheme U ⊂ W with complement F ⊂ W ⊂ X to Δ̂_{X\F,U}^{n} defines a co-presheaf Δ̂_{X̂,W_{Zar}}^{n} on W_{Zar} with values in pro-objects of Sm/k; we write

τ X̂ ; ϵ(W) for the cosimplicial object

n ↦ Δ̂_{X̂,W_{Zar}}^{n}.

We use the X̂ in the notation because the co-presheaf Δ̂_{X̂,W_{Zar}}^{n} is depends only on the Nisnevich neighborhood of W in X. Let Δ_{W_{Zar}}^{∗} denote the co-presheaf on W_{Zar} defined by U ↦ Δ_{U}^{∗}. The closed immersions î_{U} define the natural transformation

î_{W} : Δ_{W_{Zar}}^{∗} → τ X̂ ; ϵ(W).

The maps π_{X\F,W \F} for F ⊂ W a Zariski closed subset define the map

π_{X,W} : τ X̂ ; ϵ(W) → Δ_{X|W_{Zar}}^{∗}

where X|W_{Zar} is the co-presheaf W \ F ↦ X \ F on W_{Zar}. We let

(3.1.1)  π̄_{X,W} : τ X̂ ; ϵ(W) → X|W_{Zar}

denote the composition of π_{X.W} with the projection Δ_{X|W_{Zar}}^{∗} → X|W_{Zar}.

3.2. Evaluation on spaces. Let i : W → T be a closed immersion in Sm/k. For E ∈ Spc(T), we have the space E(T̂_{h}^{W}), defined by

E(T̂_{W}^{h}) := colim_{(p:T′→T,s:W →T′)∈T_{Nis}^{W}} E(T′).

Given a Nisnevich neighborhood (p : T′ → T, s : W → T′), we have the isomorphism

p^{∗} : E(T̂_{W}^{h}) → E(T̂′_{s(W)}^{h}).

Thus, for each open subscheme j : U → W, we may evaluate E on the cosimplicial pro-scheme τ X̂ ; ϵ(W)(U), giving us the presheaf of simplicial spectra

E(τ X̂ ; ϵ(W)) on W_{Zar}:

E(τ X̂ ; ϵ(W))(U) := E(τ X̂ ; ϵ(W)(U)).

Now suppose that E is in Spc(k). The map î_{W} : Δ_{W_{Zar}}^{∗} → τ X̂ ; ϵ(W)) gives us the map of presheaves on W_{Zar}

i_{W}^{∗} : E(τ X̂ ; ϵ(W)) → E(Δ_{W_{Zar}}^{∗}).

Similarly, the map π_{X,W} gives the map of presheaves on W_{Zar}

π_{X,W}^{∗} : E(Δ_{X|W_{Zar}}^{∗}) → E(τ X̂ ; ϵ(W)).

The main result of this section is Theorem 3.2.1. Let E be in Spc(k). Then the map i_{W}^{∗} : E(τ X̂ ; ϵ(W)) → E(Δ_{W_{Zar}}^{∗}) is a weak equivalence for the Zariski-local model structure, i.e., for each point w ∈ W, the map i_{W,w}^{∗} on the stalks at w is a weak equivalence of the associated total space.

3.3. Proof of Theorem 3.2.1. The proof relies on two lemmas.

Lemma 3.3.1. Let i : W → X be a closed immersion in Sm/k, giving the closed immersion 𝔸_{W}^{1} → 𝔸_{X}^{1}. For t ∈ 𝔸^{1}(k), we have the section i_{t} : W → 𝔸_{W}^{1} , i_{t}(w) := (w, t). Then for each E ∈ Spc(k), the maps

i_{0}^{∗}, i_{1}^{∗} : E(Δ̂_{𝔸_{X}^{1},𝔸_{W}^{1}}^{∗}) → E(Δ̂_{X,W}^{∗})

are homotopic.

Proof. This is just an adaptation of the standard triangulation argument. For each order-preserving map g = (g_{1}, g_{2}) : [m] → [1] × [n], let

T_{g} : Δ^{m} → Δ^{1} × Δ^{n},

be the affine-linear extension of the map on the vertices

v_{i} ↦ (v_{g_{1}(i)}, v_{g_{2}(i)}).

id_{X} × T_{g} induces the map

T̂_{g} : Δ̂_{X,W}^{m} → ({ \atop Δ^{1} × Δ_{X}^{n}})_{Δ^{1}×Δ_{W}^{n}}^{h}

We note that the isomorphism (t_{0}, t_{1}) ↦ t_{0} of (Δ^{1}, v_{1}, v_{0}) with (𝔸^{1}, 0, 1) induces an isomorphism of cosimplicial schemes

Δ̂_{𝔸_{X}^{1},𝔸_{W}^{1}}^{∗} ≅ ({ \atop Δ^{1} × Δ_{X}^{∗}})_{Δ^{1}×Δ_{W}^{∗}}^{h} .

The maps

T̂_{g}^{∗} : E(Δ̂_{𝔸_{X}^{1},𝔸_{W}^{1}}^{n}) → E(Δ̂_{X,W}^{m})

induce a simplicial homotopy T between i_{0}^{∗} and i_{1}^{∗}. Indeed, we have the simplicial sets Δ[n] : Hom_{Ord}(−, [n]). Let (Δ^{1} × Δ^{∗})^{Δ[1]} be the cosimplicial scheme

n ↦ (Δ^{1} × Δ^{n})^{Δ[1]([n])} := ∏_{s∈Δ[1]([n])} Δ^{1} × Δ^{n}

where the product is ×_{ℤ}. The inclusions δ_{0}, δ_{1} : [0] → [1] thus induce the maps of cosimplicial schemes

δ_{0}^{∗}, δ_{1}^{∗} : (Δ^{1} ×_{k} Δ^{∗})^{Δ[1]} → Δ^{1} ×_{k} Δ^{∗}.

The maps T_{g} satisfy the identities necessary to define a map of cosimplicial schemes

T : Δ^{∗} → (Δ^{1} × Δ^{∗})^{Δ[1]}.

with δ_{0}^{∗} ∘ T = i_{0}, δ_{1}^{∗} ∘ T = i_{1}. Applying the functor ^{h}, we see that the maps T̂_{g} define the map of cosimplicial schemes

T̂ : Δ̂_{X,W}^{∗} → (Δ̂_{𝔸_{X}^{1},𝔸_{W}^{1}}^{∗})^{Δ[1]},

with δ_{0}^{∗} ∘ T̂ = î_{0}, δ_{1}^{∗} ∘ T̂ = î_{1}; we then apply E. □

Lemma 3.3.2. Take W ∈ Sm_{k}. Let X = 𝔸_{W}^{n} and let i : W → X be the 0-section. Then i_{W}^{∗} : E(Δ̂_{X,W}^{∗}) → E(Δ_{W}^{∗}) is a homotopy equivalence.

Proof. Let p : X → W be the projection, giving the map

p̂: Δ̂_{X,W}^{∗} → Δ̂_{W,W}^{∗} = Δ_{W}^{∗}

and p̂^{∗} : E(Δ_{W}^{∗}) → E(Δ̂_{X,W}^{∗}). Clearly î_{W}^{∗} ∘ p̂^{∗} = id, so it suffices to show that p̂^{∗} ∘ î_{W}^{∗} is homotopic to the identity. For this, we use the multiplication map μ : 𝔸^{1} × 𝔸^{n} → 𝔸^{n},

μ(t; x_{1}, …, x_{n}) := (tx_{1}, …, tx_{n}).

The map μ × id_{Δ^{∗}} induces the map

μ̂ : ({ \atop 𝔸^{1} × 𝔸_{W}^{n} × Δ^{∗}})_{𝔸^{1}×0_{W} ×Δ^{∗}}^{h} → ({ \atop 𝔸_{W}^{n} × Δ^{∗}})_{0_{W} ×Δ^{∗}}^{h}

with μ̂ ∘ î_{0} = î_{W} ∘ p̂ and μ̂ ∘ î_{1} = id. Since î_{0}^{∗} and î_{1}^{∗} are homotopic by Lemma 3.3.1, the proof is complete. □

To complete the proof of Theorem 3.2.1, take a point w ∈ W. Then replacing X with a Zariski open neighborhood of w, we may assume there is a Nisnevich neigborhood X′ → X, s : W → X′ of W in X such that W → X′ is in turn a Nisnevich neighborhood of the zero-section W → 𝔸_{W}^{n}, n = codim_{X}W. Since E(Δ̂_{X,W}^{n}) is thus weakly equivalent to E(Δ̂_{𝔸_{W}^{n} ,0_{W}}^{n}), the result follows from Lemma 3.3.2.

Corollary 3.3.3. Suppose that E ∈ Spc(Sm/k), resp. E ∈ Spt(Sm/k) is homotopy invariant. Then for i : W → X a closed immersion, there is a natural isomorphism in HSpc(W_{Zar}), resp. HSpt(W_{Zar})

E(τ X̂ ; ϵ(W)) ≅ E(τ N̂_{i} ; ϵ(0_{W}))

Here N_{i} is the normal bundle of the immersion i, and 0_{W} ⊂ N_{i} is the 0-section.

Proof. This follows directly from Theorem 3.2.1: Since E is homotopy invariant, the canonical map

E(T) → E(Δ_{T}^{∗})

is a weak equivalence for each T ∈ Sm/k. The desired isomorphism in the respective homotopy category is constructed by composing the isomorphisms

E(τ X̂ ; ϵ(W)) ⟶^{i_{W}^{∗}} E(Δ_{W_{Zar}}^{∗}) ← E(W_{Zar})

= E(0_{W Zar}) → E(Δ_{0_{W Zar}}^{∗}) ⟵^{i_{0_{W}}^{∗}} E(τ N̂_{i} ; ϵ(0_{W})).

□

4. Punctured tubular neighborhoods

Our real interest is not in the tubular neighborhood τ X̂ ; ϵ(W), but in the punctured tubular neighborhood τ X̂ ; ϵ(W)^{0}. In this section, we define this object and discuss its basic properties.

4.1. Definition of the punctured neighborhood. Let i : W → X be a closed immersion in Sm/k. We have the closed immersion of cosimplicial pro-schemes

î: Δ_{W}^{∗} → Δ̂_{X,W}^{∗}

giving for each n the open complement Δ̂_{X\W}^{n} := Δ̂_{X,W}^{n} \ Δ_{W}^{n}. We may pass to the cofinal subcategory of Nisnevich neighborhoods of Δ_{W}^{n} in Δ_{X}^{n},

(p : T → Δ_{X}^{n}, s : Δ_{W}^{n} → T)

for which the diagram

\xymatrix{T \ s(Δ_{W}^{n}) \ar[r] \ar[d] & T \ar[d] \\ Δ_{X}^{n} \ Δ_{W}^{n} \ar[r] & Δ_{X}^{n}}

is cartesian, giving us the cosimplicial proscheme n ↦ Δ̂_{X\W}^{n} and the map

ĵ : Δ̂_{X\W}^{∗} → Δ̂_{X,W}^{∗} ,

which defines the “open complement” Δ̂_{X\W}^{∗} of Δ_{W}^{∗} in Δ̂_{X,W}^{∗}. Extending this construction to all open subschemes of X, we have the co-presheaf on W_{Zar},

U = W \ F ↦ Δ̂_{(X\F)\U}^{∗},

which we denote by τ X̂ ; ϵ(W)^{0}. Let Δ_{(X\W)_{Zar}}^{n} be the constant co-presheaf on W_{Zar} with value Δ_{X\W}^{n}, giving the cosimplicial co-presheaf Δ_{(X\W)_{Zar}}^{∗}. The maps

ĵ_{U} : Δ̂_{(X\F)\U}^{∗} → Δ̂_{(X\F),U}^{∗}

define the map ĵ : τ X̂ ; ϵ(W)^{0} → τ X̂ ; ϵ(W). The maps Δ̂_{U\W ∩U}^{∗} → Δ_{X\W}^{∗} give us the map

π : τ X̂ ; ϵ(W)^{0} → Δ_{X\W}^{∗}

where we view Δ_{X\W}^{∗} as the constant co-sheaf on W_{Zar}.

To give a really useful result on the presheaf E(τ X̂ ; ϵ(W)^{0}), we will need to impose additional conditions on E. These are

(1) E is homotopy invariant

(2) E satisfies Nisnevich. excision One important consequence of these properties is the purity theorem of Morel-Voevodsky:

Theorem 4.1.1 (Purity [34, theorem 2.23]). Suppose E ∈ Spt(k) is homotopy invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k and s : W → N_{i} the 0-section of the normal bundle. Then there is an isomorphism in HSpt(W_{Zar})

E^{W_{Zar}}(X) → E^{W_{Zar}}(N_{i})
□

Let E(X|W_{Zar}) be the presheaf on W_{Zar}

W \ F ↦ E(X \ F)

and E(X \ W) the constant presheaf. Let

res : E(X|W_{Zar}) → E(τ X̂ ; ϵ(W))

res^{0} : E(X \ W) → E(τ X̂ ; ϵ(W)^{0})

be the pull-back by the natural maps τ X̂ ; ϵ(W)(W \ F) → X \ F, τ X̂ ; ϵ(W)^{0} → X\W. Let E^{Δ_{W}^{∗}} (τ X̂ ; ϵ(W)) ∈ Spt(W_{Zar}) be the homotopy fiber of the restriction map

ĵ : E(τ X̂ ; ϵ(W)) → E(τ X̂ ; ϵ(W)^{0}).

The commutative diagram in Spt(W_{Zar})

\xymatrix{E(X|W_{Zar}) \ar[r]^{j^{∗}} \ar[d]_{res} & E(X \ W) \ar[d]^{_{res_{0}}} \\ E(τ X̂ ; ϵ(W)) \ar[r]_{ĵ^{∗}} & E(τ X̂ ; ϵ(W)^{0})}

induces the map of homotopy fiber sequences

\xymatrix{E^{W_{Zar}}(X) \ar[r] \ar[d]_{ψ} & E(X|W_{Zar}) \ar[r]^{j^{∗}} \ar[d]_{res} & E(X \ W) \ar[d]^{_{res_{0}}} \\ E^{Δ_{W}^{∗}} (τ X̂ ; ϵ(W)) \ar[r] & E(τ X̂ ; ϵ(W)) \ar[r]_{j^{∗}} & E(τ X̂ ; ϵ(W)^{0})}

We can now state the main result for E(τ X̂ ; ϵ(W)^{0}).

Theorem 4.1.2. Suppose that E ∈ Spt(k) is homotopy invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k. Then the map ψ is a Zariski local weak equivalence.

Proof. Let i_{Δ^{∗}} : Δ_{W}^{∗} → Δ_{X}^{∗} be the immersion id × i. For U = W \ F ⊂ W, τ X̂ ; ϵ(W)^{0}(U) is the cosimplicial pro-scheme with n-cosimplices

τ X̂ ; ϵ(W)^{0}(U)^{n} = Δ̂_{X\F,U}^{n} \ Δ_{U}^{n}

so by Nisnevich excision we have the natural isomorphism

α : E^{Δ_{W_{Zar}}^{∗}} (Δ_{X|W_{Zar}}^{∗}) → E^{Δ_{W}^{∗}} (τ X̂ ; ϵ(W)),

where E^{Δ_{W_{Zar}}^{∗}} (Δ_{X|W_{Zar}}^{∗})(W \F) is the total spectrum of the simplicial spectrum

n ↦ E^{Δ_{W \F}^{n}} (Δ_{X\F}^{n}).

The homotopy invariance of E implies that the pull-back

E^{W \F} (X \ F) → E^{Δ_{W \F}^{n}} (Δ_{X\F}^{n})

is a weak equivalence for all n, so we have the weak equivalence

β : E^{W_{Zar}}(X) → E^{Δ_{W_{Zar}}^{∗}} (Δ_{X|W_{Zar}}^{∗}).

It follows from the construction that ψ = αβ, completing the proof. □

Corollary 4.1.3. There is a distinguished triangle in HSpt(W_{Zar})

E^{W_{Zar}}(X) → E(W_{Zar}) → E(τ X̂ ; ϵ(W)^{0})

Proof. By Theorem 3.2.1, the map î^{∗} : E(τ X̂ ; ϵ(W)) → E(Δ_{W_{Zar}}^{∗}) is a weak equivalence; using homotopy invariance again, the map

E(W_{Zar}) → E(Δ_{W_{Zar}}^{∗})

is a weak equivalence. Combining this with Theorem 4.1.2 yields the result. □

For homotopy invariant E ∈ Spt(k), we let

ϕ_{E} : E(τ N̂_{i} ; ϵ(0_{W})) → E(τ X̂ ; ϵ(W)).

be the isomorphism in HSpt(W_{Zar}) given by corollary 3.3.3.

Corollary 4.1.4. Suppose that E ∈ Spt(k) is homotopy invariant and satisfies Nisnevich excision. Let i : W → X be a closed immersion in Sm/k and let N_{i}^{0} = N_{i} \ 0_{W}.

(1) The restriction maps

res : E(N_{i}/W_{Zar}) → E(τ N̂_{i} ; ϵ(0_{W}))

res^{0} : E(N_{i}^{0}/W_{Zar}) → E(τ N̂_{i} ; ϵ(0_{W})^{0})

are weak equivalences in Spt(W_{Zar}).

(2) There is a canonical isomorphism in HSpt(W_{Zar})

ϕ_{E}^{0} : E(τ N̂_{i} ; ϵ(0_{W})^{0}) → E(τ X̂ ; ϵ(W)^{0})

(3) Consider the diagram (in HSpt(W_{Zar})):

[figure: E^{0_{W Zar}}(N_{i}); E(N_{i}/W_{Zar}); _{res_{E}}; E(N_{i}^{0}/W_{Zar}); _{res{0 \atop E}}; E^{0_{W Zar}}(N_{i}); π; E(τ N̂_{i} ; ϵ(0_{W})); _{ĵ_{N}_{∗}}; ϕ_{E}; E(τ N̂_{i} ; ϵ(0_{W})^{0}); ϕ_{E}^{0}; E^{W_{Zar}}(X); E(τ X̂ ; ϵ(W)); ĵ^{∗}; E(τ X̂ ; ϵ(W)^{0}); E^{W_{Zar}}(X); E(X|W_{Zar}); _{res_{E}}; j^{∗}; E(X \ W); _{res{0 \atop E}}]

The first and last rows are the homotopy fiber sequences defining the presheaves E^{0_{W Zar}}(N_{i}) and E^{W_{Zar}}(X), respectively, the second row and third rows are the distinguished triangles of Theorem 4.1.2, and π is the Morel-Voevodsky purity isomorphism. Then this diagram commutes and each triple of vertical maps defines a map of distinguished triangles.

Proof. It follows directly from the weak equivalence (in Theorem 4.1.2) of the homotopy fiber of

ĵ^{∗} : E(τ X̂ ; ϵ(W)) → E(τ X̂ ; ϵ(W)^{0})

with E^{W_{Zar}}(X) that the triple (id, res_{E}, res_{E}^{0}) defines a map of distinguished triangles. The same holds for the map of the first row to the second row; we now verify that this latter map is an isomorphism of distinguished triangles. For this, let s : W → N_{i} be the zero-section. We have the isomorphism i_{W}^{∗} : E(τ N̂_{i} ; ϵ(0_{W})) → E(W_{Zar}) defined as the diagram of weak equivalences

E(τ N̂_{i} ; ϵ(0_{W})) ⟶^{i_{Δ{∗ \atop W_{Zar}}}^{∗}} E(Δ_{W_{Zar}}^{∗}) ⟵^{ι_{0∗}} E(W_{Zar}).

From this, it is easy to check that the diagram

\xymatrix{E(N_{i}/W_{Zar}) \ar[r]^{^{res^{E}}} \ar[dr]_{s^{∗}} & E(τ N̂_{i} ; ϵ(0_{W})) \ar[d]^{i_{W}^{∗}} \\  & E(W_{Zar})}

commutes in HSpt(W_{Zar}). As E is homotopy invariant, s^{∗} is an isomorphism, hence res_{E} is an isomorphism as well. This completes the proof of (1). The proof of (2) and (3) uses the standard deformation diagram. Let μ̄ : Ȳ → X × 𝔸^{1} be the blow-up of X × 𝔸^{1} along W, let μ̄^{−1}[X × 0] denote the proper transform, and let μ : Y → X × 𝔸^{1} be the open subscheme Ȳ \ μ̄^{−1}[X × 0]. Let p : Y → 𝔸^{1} be p_{2} ∘ μ. Then p^{−1}(0) = N_{i}, p^{−1}(1) = X × 1 = X, and Y contains the proper transform μ̄^{−1}[W × 𝔸^{1}], which is mapped isomorphically by μ to W × 𝔸^{1} ⊂ X × 𝔸^{1}. We let ĩ : W × 𝔸^{1} → Y be the resulting closed immersion. The restriction of ĩ to W × 0 is the zero-section s : W → N_{i} and the restriction of ĩ to W × 1 is i : W → X. The resulting diagram is

(4.1.1)  \xymatrix{W \ar[r]^{i_{0}} \ar[d]_{s} & W × 𝔸^{1} \ar[d]_{ĩ} & W \ar[l]_{i_{1}} \ar[d]^{i} \\ N_{i} \ar[r]^{i_{0}} \ar[d]_{p_{0}} & Y \ar[d]_{p} & X \ar[l]_{i_{1}} \ar[d]^{p_{1}} \\ 0 \ar[r]_{i_{0}} & 𝔸^{1} & 1 \ar[l]^{i_{1}}}

Together with Theorem 4.1.2, diagram (4.1.1) gives us two maps of distinguished triangles:

[E^{0_{W ×𝔸^{1}Zar}}(Y) → E(τ Ŷ ; ϵ (W × 𝔸^{1})) → E(τ Ŷ ; ϵ (W × 𝔸^{1})^{0})]

⟶^{i_{1}^{∗}}

[E^{W_{Zar}}(X) → E(τ X̂ ; ϵ(W)) → E(τ X̂ ; ϵ(W)^{0})]

and

[E^{0_{W ×𝔸^{1}Zar}}(Y) → E(τ Ŷ ; ϵ (W × 𝔸^{1})) → E(τ Ŷ ; ϵ (W × 𝔸^{1})^{0})]

⟶^{i_{0}^{∗}}

[E^{0_{W Zar}}(N_{i}) → E(τ N̂_{i} ; ϵ(0_{W})) → E(τ N̂_{i} ; ϵ(0_{W})^{0})]

As above, we have the commutative diagram

\xymatrix{E(τ Ŷ ; ϵ (W × 𝔸^{1})) \ar[r]^{i_{0}^{∗}} \ar[d]_{i_{W ×𝔸^{1}}^{∗}} & E(τ N̂_{i} ; ϵ(0_{W})) \ar[d]_{i_{W}^{∗}} \\ E(W × 𝔸^{1}) \ar[r]_{i_{0}^{∗}} & E(W).}

As E is homotopy invariant, the maps i_{W}^{∗}, i_{W ×𝔸^{1}}^{∗} and i_{0}^{∗} : E(W ×𝔸^{1}) → E(W) are isomorphisms, hence

i_{0}^{∗} : E(τ Ŷ ; ϵ (W × 𝔸^{1})) → E(τ N̂_{i} ; ϵ(0_{W}))

is an isomorphism. Similarly,

i_{1}^{∗} : E(τ Ŷ ; ϵ (W × 𝔸^{1})) → E(τ X̂ ; ϵ(W))

is an isomorphism. The proof of the Morel-Voevodsky purity theorem [34, Theorem 2.23] shows that

i_{0}^{∗} : E^{0_{W ×𝔸^{1}Zar}}(Y) → E^{0_{W Zar}}(N_{i})
i_{1}^{∗} : E^{0_{W ×𝔸^{1}Zar}}(Y) → E^{W_{Zar}}(X)

are weak equivalences; the purity isomorphism π is by definition i_{1}^{∗} ∘ (i_{0}^{∗})^{−1}. Thus, both i_{0}^{∗} and i_{1}^{∗} define isomorphisms of distinguished triangles, and

i_{1}^{∗} ∘ (i_{0}^{∗})^{−1} : E(τ N̂_{i} ; ϵ(0_{W})) → E(τ X̂ ; ϵ(W))

is the map ϕ_{E}. Defining ϕ_{E}^{0} to be the isomorphism

i_{1}^{∗} ∘ (i_{0}^{∗})^{−1} : E(τ N̂_{i} ; ϵ(0_{W})^{0}) → E(τ X̂ ; ϵ(W)^{0})

proves both (2) and (3). □

Remarks 4.1.5. (1) It follows from the construction of ϕ_{E} and ϕ_{E}^{0} that both of these maps are natural in E.

(2) The maps ϕ_{E}^{0} are natural in the closed immersion i : W → X in the following sense: Suppose we have closed immersions i_{j} : W_{j} → X_{j}, j = 1, 2 and a morphism f : (W_{1}, X_{1}) → (W_{2}, X_{2}) of pairs of immersions such that f restricts to a morphism X_{1} \ W_{1} → X_{2} \ W_{2}. Fix E and let ϕ_{jE}^{0} be the map corresponding to the immersions i_{j} We have the evident maps

ι : τ X̂_{1} ; ϵ(W_{1})^{0} → τ X̂_{2} ; ϵ(W_{2})^{0}  η : τ N̂_{i_{1}} ; ϵ (W_{1})^{0} → τ N̂_{i_{2}} ; ϵ (W_{2})^{0}

Then the diagram

\xymatrix{E(τ N̂_{i_{2}} ; ϵ (W)^{0}) \ar[r]^{ϕ_{2E}} \ar[d]_{η^{∗}} & E(τ Ŷ ; ϵ (W)^{0}) \ar[d]^{ι^{∗}} \\ E(τ N̂_{i_{1}} ; ϵ (W)^{0}) \ar[r]_{ϕ_{1E}} & E(τ X̂ ; ϵ(W)^{0})}

commutes. Indeed, the map f induces a map of deformation diagrams. □

5. The exponential map

If i : M′ → M is a submanifold of a differentiable manifold M, there is a diffeomorphism exp of the normal bundle N_{M′/M} of M′ in M with the tubular neighborhood τ_{ϵ}^{M}(M′). In addition, exp restricts to a diffeomorphism exp^{0} of the punctured normal bundle N_{M′/M} \ 0_{M′} with the punctured tubular neighborhood τ_{ϵ}^{M}(M′) \ M′. Classically, this has been used to define the boundary map in the Gysin sequence for M′ → M, by using the restriction map exp^{0∗} : H^{∗}(M \ M′) → H^{∗}(N_{M′/M} \ 0_{M′}) followed by the Thom isomorphism H^{∗}(N_{M′/M} \ 0_{M′}) ≅ H^{∗−d}(M′). In this section, we use our punctured tubular neighborhood to construct a purely algebraic version of the classical exponential map, at least for the associated suspension spectra. We will use this in section 11 to define a purely algebraic version of the gluing of Riemann surfaces along boundary components.

5.1. Let i : W → X be a closed immersion in Sm/k with normal bundle p : N_{i} → W. We have the map

exp : N_{i} → X

in Spc(k), defined as the composition N_{i} → W → X. We also have the Morel-Voevodsky purity isomorphism

π : Th(N_{i}) → X/(X \ W)

in H(k). In fact:

Lemma 5.1.1. The diagram

(5.1.1)  \xymatrix{N_{i} \ar[r]^{q′} \ar[d]_{exp} & Th(N_{i}) \ar[d]^{π} \\ X \ar[r]_{q} & X/(X \ W)}

commutes in H(k).

Proof. As we have already seen, the construction of the purity isomorphism π relies on the deformation to the normal bundle; we retain the notation from the proof of corollary 4.1.4. We have the total space Y → 𝔸^{1} of the deformation. The fiber Y_{0} over 0 ∈ 𝔸^{1} is canonically isomorphic to N_{i} and the fiber Y_{1} over 1 is canonically isomorphic to X; the inclusions W × 0 → Y_{0}, W × 1 → Y_{1} are isomorphic to the zero-section s : W → N_{i} and the original inclusion i : W → X, respectively. The proper transform μ^{−1}[W × 𝔸^{1}] is isomorphic to W × 𝔸^{1}, giving the closed immersion ι : W × 𝔸^{1} → Y. The diagram thus induces maps in Spc(k):

ī_{0} : Th(N_{i}) → Y/(Y \ W × 𝔸^{1})
ī_{1} : X/X \ W → Y/(Y \ W × 𝔸^{1})

which are isomorphisms in H(k) (see [34, Thm. 2.23]); the purity isomorphism is by definition π := ī_{1}^{−1} ∘ ī_{0}. We have the commutative diagram in Spc(k):

\xymatrix{W \ar[rr]^{id} \ar[r]^{i_{0}} \ar[d]^{s} & W × 𝔸^{1} \ar[r]^{p_{1}} \ar[d]^{ι} & W \ar[l]^{i_{1}} \ar[d]^{i} \\ N_{i} \ar[r] \ar[u]^{p} \ar[d]_{q′} \ar[u] & Y \ar[d] & X \ar[l] \ar[d]^{q} \\ Th(N_{i}) \ar[r]_{ī_{0}} & Y/(Y \ W × 𝔸^{1}) & X/X \ W \ar[l]^{ī_{1}}}

from which the result follows directly. □

Remark 5.1.2. Since we have the homotopy cofiber sequences:

N_{i} \ 0_{W} → N_{i} → Th(N_{i}) → Σ(N_{i} \ 0_{W})_{+}
X \ W → X → X/(X \ W) → Σ(X \ W)_{+}

the diagram (5.1.1) induces a map

Σ(N_{i} \ 0_{W})_{+} → Σ(X \ W)_{+}

in H(k), however, this map is not uniquely determined, hence is not canonical. □ 5.2. The construction. In this section we define a canonical map

exp^{0} : Σ^{∞}(N_{i} \ 0_{W})_{+} → Σ^{∞}(X \ W)_{+}

in SH_{𝔸^{1}}(k) which yields the map of distinguished triangles in SH_{𝔸^{1}}(k):

\xymatrix{Σ^{∞}(N_{i} \ 0_{W})_{+} \ar[d]_{_{exp_{0}}} \ar[r] & Σ^{∞}N_{i+} \ar[r] \ar[d]_{exp} & Σ^{∞}Th(N_{i}) \ar[d]^{π} \\ Σ^{∞}(X \ W)_{+} \ar[r] & Σ^{∞}X_{+} \ar[r] & Σ^{∞}X/(X \ W)}

To define exp^{0}, we apply Corollary 4.1.4 with E a fibrant model of Σ^{∞}(X\W)_{+}. Denote the composition

E(X \ W) ⟶^{res_{E}^{0}} E(τ X̂ ; ϵ(W)^{0})(W)

⟶^{(ϕ_{E}^{0})^{−1}} E(τ N̂_{i} ; ϵ(0_{W})^{0})(W) ⟶^{(res_{E}^{0})^{−1}} E(N_{i}^{0})

by exp_{E}^{0∗}. Since E is fibrant, we have canonical isomorphisms

π_{0}E(N_{i}^{0}) ≅ Hom_{SH_{𝔸^{1}} (k)}(Σ^{∞}N_{i+}^{0}, E)
≅ Hom_{SH_{𝔸^{1}} (k)}(Σ^{∞}N_{i+}^{0}, Σ^{∞}(X \ W)_{+})

π_{0}E(X \ W) ≅ Hom_{SH_{𝔸^{1}} (k)}(Σ^{∞}(X \ W)_{+}, E)
≅ Hom_{SH_{𝔸^{1}} (k)}(Σ^{∞}(X \ W)_{+}, Σ^{∞}(X \ W)_{+})

so exp_{E}^{0∗} induces the map

Hom_{SH_{𝔸^{1}} (k)}(Σ^{∞}(X \ W)_{+}, Σ^{∞}(X \ W)_{+})

⟶^{exp_{E}^{0∗}} Hom_{SH_{𝔸^{1}} (k)}(Σ^{∞}N_{i+}^{0}, Σ^{∞}(X \ W)_{+}).

We set

exp^{0} := exp_{E}^{0∗}(id).

To finish the construction, we show

Proposition 5.2.1. The diagram, with rows the evident homotopy cofiber sequences,

\xymatrix{Σ^{∞}(N_{i} \ 0_{W})_{+} \ar[d]_{_{exp_{0}}} \ar[r] & Σ^{∞}N_{i+} \ar[r] \ar[d]_{exp} & Σ^{∞}Th(N_{i}) \ar[d]^{π} \ar[r]^{∂} & ΣΣ^{∞}(N_{i} \ 0_{W})_{+} \ar[d]_{Σ exp^{0}} \\ Σ^{∞}(X \ W)_{+} \ar[r] & Σ^{∞}X_{+} \ar[r] & Σ^{∞}X/(X \ W) \ar[r]_{∂} & ΣΣ^{∞}(X \ W)_{+}}

commutes in SH_{𝔸^{1}}(k).

Proof. It suffices to show that, for all fibrant E ∈ Spt(k), the diagram formed by applying Hom_{SH_{𝔸^{1}} (k)}(−, E) to our diagram commutes. This latter diagram is the same as applying π_{0} to the diagram

(5.2.1)  \xymatrix{E(N_{i} \ 0_{W}) & E(N_{i}) \ar[l] & E^{0_{W}} (N_{i}) \ar[l] & ΩE(N_{i} \ 0_{W})) \ar[l]_{∂} \\ E(X \ W) \ar[u]_{_{exp_{0∗}}} & E(X) \ar[u]_{_{exp_{∗}}} \ar[l] & E^{W} (X) \ar[u]^{π^{∗}} \ar[l] & ΩE(X \ W) \ar[l]^{∂} \ar[u]_{Ω exp^{0∗}}}

where the rows are the evident homotopy fiber sequences. It follows by the definition of exp^{0} and exp that this diagram is just the “outside” of the diagram in Corollary 4.1.4(3), extended to make the distinguished triangles explicit. Thus the diagram (5.2.1) commutes, which finishes the proof. □

Remark 5.2.2. The exponential maps exp and exp^{0} are natural with respect to maps of closed immersions f : (W′ ⟶^{i′} X′) → (W ⟶^{i} X) satisfying the cartesian condition of remark 4.1.5(2). This follows from the naturality of the isomorphisms ϕ_{E}, ϕ_{E}^{0} described in Remark 4.1.5, and the functoriality of the (punctured) tubular neighborhood construction. □

6. Neighborhoods of normal crossing schemes

We extend our results to the case of a strict normal crossing divisor W ⊂ X by using a Mayer-Vietoris construction.

6.1. Normal crossing schemes. Let D be a reduced effective Cartier divisor on a smooth k-scheme X with irreducible components D_{1},…, D_{m}. For each I ⊂ {1, …, m}, we set

D_{I} := ∩_{i∈I}D_{i}

We let i : D → X the inclusion. For each I ≠ ∅, we let ι_{I} : D_{I} → D, i_{I} : D_{I} → X be the inclusions; for I ⊂ J ⊂ {1, …, m} we have as well the inclusion ι_{I,J} : D_{J} → D_{I}. Recall that D is a strict normal crossing divisor if for each I, D_{I} is smooth over k and codim_{X}D_{I} = |I|. We extend this notion a bit: We call a closed subscheme D ⊂ X a strict normal crossing subscheme if X is in Sm/k and, locally on X, there is a smooth locally closed subscheme Y ⊂ X containing D such that D is a strict normal crossing divisor on Y

6.2. The tubular neighborhood. Let D ⊂ X be a strict normal crossing subscheme with irreducible components D_{1}, …, D_{m}. For each I ⊂ {1, …, m}, I ≠ ∅, we have the tubular neighborhood co-presheaf τ X̂ ; ϵ(D_{I}) on D_{I}. The various inclusions ι_{I,J} give us the maps of co-presheaves

ι̂_{I,J} : ι_{I,J∗}(τ X̂ ; ϵ(D_{J})) → τ X̂ ; ϵ(D_{I});

pushing forward by the maps ι_{I} yields the diagram of co-presheaves on D_{Zar} (with values in cosimplicial pro-objects of Sm/k)

(6.2.1)  I ↦ ι_{I∗}(τ X̂ ; ϵ(D_{I}))

indexed by the non-empty I ⊂ {1, …, m}. We have as well the diagram of identity co-presheaves

(6.2.2)  I ↦ ι_{I∗}(D_{IZar})

as well as the diagram

(6.2.3)  I ↦ ι_{I∗}(Δ_{D_{IZar}}^{∗})

We denote these diagrams by τ X̂ ; ϵ(D), D_{•} and Δ_{D_{•}}^{∗}, respectively. The projections p_{I} : Δ_{D_{IZar}}^{∗} → D_{IZar} and the closed immersions ι̂_{D_{I}} : Δ_{D_{IZar}}^{∗} → τ X̂ ; ϵ(D_{I}) yield the natural transformations

D_{•} ⟵^{p_{•}} Δ_{D_{•}}^{∗} ⟶^{ι̂_{D•}} τ X̂ ; ϵ(D).

Now take E ∈ Spt(k). Applying E to the diagram (6.2.1) yields the diagram of presheaves on D_{Zar}

I ↦ ι_{I∗}(E(τ X̂ ; ϵ(D_{I})))

Similarly, applying E to (6.2.2) and (6.2.3) yields the diagrams of presheaves on D_{Zar}

I ↦ ι_{I∗}(E(D_{IZar}))

and

I ↦ ι_{I∗}(E(Δ_{D_{IZar}}^{∗})).

Definition 6.2.1. For D ⊂ X a strict normal crossing subscheme and E ∈ Spt(k), set

E(τ X̂ ; ϵ(D)) := holim_{I≠∅} ι_{I∗}(E(τ X̂ ; ϵ(D_{I}))).

Similarly, set

E(D_{•}) := holim_{I≠∅} ι_{I∗}(E(D_{I}))

E(Δ_{D_{•}}^{∗}) := holim_{I≠∅} ι_{I∗}(E(Δ_{D_{I}}^{∗}))

□

The natural transformations ι̂_{D} and p_{•} yield the maps of presheaves on D_{Zar}

E(D_{•}) ⟶^{p_{•}^{∗}} E(Δ_{D_{•}}^{∗}) ⟵^{ι̂_{D}^{∗}} E(τ X̂ ; ϵ(D)).

Proposition 6.2.2. Suppose E ∈ Spt(Sm/k) is homotopy invariant and satisfies Nisnevich excision. Then the maps ι̂_{D}^{∗} and p_{•}^{∗} are Zariski-local weak equivalences.

Proof. The maps p_{I}^{∗} are pointwise weak equivalences by homotopy invariance. By Theorem 3.2.1, the maps ι̂_{D_{I}} are Zariski-local weak equivalences. Since the homotopy limits are finite, the stalk of each homotopy limit is weakly equivalence to the homotopy limit of the stalks. By [8] this suffices to conclude that the map on the homotopy limits is a Zariski-local weak equivalence. □

Remark 6.2.3. One could also attempt a more direct definition of τ X̂ ; ϵ(D) by just using our definition in the smooth case i : W → X and replacing the smooth W with the normal crossing scheme D, in other words, the co-presheaf on D_{Zar}

D \ F ↦ Δ̂_{X\F,D\F}^{∗} .

Labeling this choice τ X̂ ; ϵ(D)_{naive}, and considering τ X̂ ; ϵ(D)_{naive} as a constant diagram, we have the evident map of diagrams

ϕ : τ X̂ ; ϵ(D) → τ X̂ ; ϵ(D)_{naive}

We were unable to determine if ϕ induces a weak equivalence after evaulation on E ∈ Spt(k), even assuming that E is homotopy invariant and satisfies Nisnevich excision. We were also unable to construct such an E for which ϕ fails to be a weak equivalence. □

6.3. The punctured tubular neighborhood. To define the punctured tubular neighborhood τ X̂ ; ϵ(D)^{0}, we proceed as follows: Fix a subset I ⊂ {1, …, m}, I ≠ ∅. Let p : X′ → X, s : D_{I} → X′ be a Nisnevich neighborhood of D_{I} in X, and let D_{X′} = p^{−1}(D). Sending X′ → X to Δ_{D_{X′}}^{n} gives us the pro-scheme Δ̂_{D⊂X,D_{I}}^{n}, and the closed immersion Δ̂_{D⊂X,D_{I}}^{n} → Δ̂_{X,D_{I}}^{n}. Varying n, we have the cosimplicial pro-scheme Δ̂_{D⊂X,D_{I}}^{∗}, and the closed immersion Δ̂_{D⊂X,D_{I}}^{∗} → Δ̂_{X,D_{I}}^{∗}. Take a closed subset F ⊂ D_{I}, and let U := D_{I} \ F. As in the definition of the punctured tubular neighborhood of a smooth closed subscheme in section 4.1, we pass to the appropriate cofinal subcategory of Nisnevich neighborhoods to show that the open complements Δ̂_{X\F,U}^{n} \ Δ̂_{D\F ⊂X\F,U}^{n} for varying n form a cosimplicial pro-scheme

n ↦ Δ̂_{X\F,U}^{n} \ Δ̂_{D\F ⊂X\F,U}^{n}.

Similarly, we set

τ X̂ ; ϵ(D, D_{I})^{0}(U) := Δ̂_{X\F,U}^{∗} \ Δ̂_{D\F ⊂X\F,U}^{∗}.

This forms the co-presheaf τ X̂ ; ϵ(D, D_{I})^{0} on D_{IZar}. The open immersions

ĵ_{I}(U)^{n} : Δ̂_{X\F,U}^{n} \ Δ̂_{D\F ⊂X\F,U}^{n} → Δ̂_{X\F,U}^{n}

define the map

ĵ_{I}(U) : τ X̂ ; ϵ(D, D_{I})^{0}(U) → τ X̂ ; ϵ(D_{I})(U),

giving the map of co-presheaves

ĵ_{I} : τ X̂ ; ϵ(D, D_{I})^{0} → τ X̂ ; ϵ(D_{I}).

For J ⊂ I, we have the map ι̂_{J,I} : Δ̂_{X,D_{I}}^{∗} → Δ̂_{X,D_{J}}^{∗} and

ι̂_{J,I}^{−1}(Δ̂_{D⊂X,D_{J}}^{∗}) = Δ̂_{D⊂X,D_{I}}^{∗} .

Thus we have the map ι̂_{J,I}^{0} : τ X̂ ; ϵ(D, D_{I})^{0} → τ X̂ ; ϵ(D, D_{J})^{0} and the diagram of co-presheaves on D_{Zar}

(6.3.1)  I ↦ ι_{I∗}(τ X̂ ; ϵ(D, D_{I})^{0})

which we denote by τ X̂ ; ϵ(D)^{0}. The maps ĵ_{I} define the map

ĵ : τ X̂ ; ϵ(D)^{0} → τ X̂ ; ϵ(D).

The projection maps π_{I} : τ X̂ ; ϵ(D_{I}) → X (where we consider X as the constant co-presheaf on D_{IZar}) restrict to maps π_{I}^{0} : τ X̂ ; ϵ(D, D_{I})^{0} → X \D, which in turn induce the map

π^{0} : τ X̂ ; ϵ(D)^{0} → X \ D,

where we consider X \ D the constant diagram of constant co-presheaves on D_{Zar}.

Definition 6.3.1. For E ∈ Spt(k), let E(τ X̂ ; ϵ(D)^{0}) be the presheaf on D_{Zar},

E(τ X̂ ; ϵ(D)^{0}) := holim_{∅≠I⊂{1,...,m}} ι_{I∗}E(τ X̂ ; ϵ(D, D_{I})^{0}).

□

The map ĵ defines the map of presheaves

ĵ^{∗} : E(τ X̂ ; ϵ(D)) → E(τ X̂ ; ϵ(D)^{0}).

We let E^{D_{Zar}}(τ X̂ ; ϵ(D)) denote the homotopy fiber of ĵ^{∗}. Via the commutative diagram

\xymatrix{E(X \ F) \ar[d]_{π^{∗}} \ar[r]^{j^{∗}} & E(X \ D) \ar[d]^{π^{0∗}} \\ E(τ X̂ ; ϵ(D))(D \ F) \ar[r]_{ĵ^{∗}} & E(τ X̂ ; ϵ(D))^{0}(D \ F)}

we have the canonical map

π_{D}^{∗} : E^{D_{Zar}}(X) → E^{D_{Zar}}(τ X̂ ; ϵ(D)).

We want to show that the map π_{D}^{∗} is a weak equivalence, assuming that E is homotopy invariant and satisfies Nisnevich excision. We first consider a simpler situation. We begin by noting the following

Lemma 6.3.2. Let □_{0}^{n} denote the category of non-empty subsets of {1, …, n} with maps the inclusions, let C be a small category and let F : C × □_{0}^{n} → Spt^{Ord^{op}} be a functor. Let holim_{□_{0}^{n}} F : C → Spt^{Ord^{op}} be the functor with value the simplicial spectrum m ↦ holim_{□_{0}^{n}} F (i × [m]) at i ∈ C. There is a isomorphism

Tot(holim_{□_{0}^{n}} F) → holim_{□_{0}^{n}} Tot(F).

in HSpt(C^{op}).

Proof. Letting □^{n} be the category of all subsets of {1, …, n} (including the empty set), we may extend F to F ∗ : □^{n} → Spt(Ord^{op}) by F ∗ (∅) = 0. Similarly, given a functor G : □^{n} → Spt, we may extend G to G_{♮} : □_{0}^{n+1} → Spt by G_{♮}(I) = 0, G_{♮}(I ∪{n+1}) = G(I) for I ⊂ {1, …, n}. We define the iterated homotopy fiber of G, fib_{n}G ∈ Spt, by

hofib_{n}(G) := holim_{□_{0}^{n+1}} G_{♮}.

One easily checks that for a map g : A → B of spectra, considered in the evident manner as a functor g_{1} : □^{1} → Spt, we have hofibg = hofib_{1}g_{1}. More generally, if we let i_{−}, i_{+} : □^{n−1} → □^{n} be the inclusions

i_{−}(I) := I, i_{+}(I) := I ∪ {n}

we have the evident natural transformation ω : i_{−} → i_{+} and for G : □^{n} → Spt a functor, we have a natural isomorphism

hofib(hofib_{n−1}G ∘ i_{−} ⟶^{hofib_{n−1}G(ω)} hofib_{n−1}G ∘ i_{+}) ≅ hofib_{n}G,

hence the name iterated homotopy fiber. Finally, one has the natural isomorphism

hofib_{n}G∗ ≅ Ω holim_{□_{0}^{n}} G

for G : □_{0}^{n} → Spt. Since Tot is compatible with suspension we may replace our original functor F with ΣF ≅ Ω^{−1}F; using induction on n, it suffices to show that there is a natural isomorphism in HSpt(C^{op})

Tot(hofibF) → hofibTot(F)

for F : A → B a map in Spt^{C×Ord^{op}}. For this, note that for f : X → Y a map of spectra, there is a natural weak equivalence

a(f) : Σhofibf → hocofibf

Since Tot commutes with suspension and preserves weak equivalences, it suffices to define a natural weak equivalence

Tot(hocofibf) → hocofib(Totf).

In fact, since Tot preserves cofiber squares and is compatible with the wedge action of pointed simplicial sets on Spt^{Ord^{op}} and Spt, there is a natural isomorphism Tot(hocofibf) → hocofib(Totf), completing the proof. □

This lemma allows us to define a simplicial model for E^{D_{Zar}}(τ X̂ ; ϵ(D)), induced by the cosimplicial structure on the co-presheaves τ X̂ ; ϵ(D_{I}) and τ X̂ ; ϵ(D_{I})^{0}. In fact, let

E(τ X̂ ; ϵ(D))_{n} := holim_{I≠∅} ι_{I∗}E(Δ̂_{X̂,D_{IZar}}^{n})
E(τ X̂ ; ϵ(D)^{0})_{n} := holim_{I≠∅} ι_{I∗}E(Δ̂_{X̂,D_{IZar}}^{n} \ Δ̂_{D⊂X,D_{IZar}}^{n})

and set

E^{D_{Zar}}(τ X̂ ; ϵ(D))_{n} := hofib(ĵ_{n}^{∗} : E(τ X̂ ; ϵ(D))_{n} → E(τ X̂ ; ϵ(D)^{0})_{n}).

It follows from lemma 6.3.2 that E(τ X̂ ; ϵ(D)), E(τ X̂ ; ϵ(D)^{0}) and E^{D_{Zar}}(τ X̂ ; ϵ(D)) are isomorphic in the homotopy category to the total presheaves of spectra associated to the simplical presheaves

n ↦ E(τ X̂ ; ϵ(D))_{n}

n ↦ E(τ X̂ ; ϵ(D)^{0})_{n}

n ↦ E^{D_{Zar}}(τ X̂ ; ϵ(D))_{n}

respectively. The map π_{D}^{∗} is defined by considering E^{D_{Zar}}(X) as a constant simplicial object. Let

π_{D,0}^{∗} : E^{D_{Zar}}(X) → E^{D_{Zar}}(τ X̂ ; ϵ(D))_{0}

be the map of E^{D_{Zar}}(X) to the 0-simplices of E^{D_{Zar}}(τ X̂ ; ϵ(D)).

Proposition 6.3.3. Suppose that E satisfies Nisnevich excision and D is a strict normal crossing subscheme of X. Then π_{D,0}^{∗} is a weak equivalence.

Before we give the proof of this result, we prove two preliminary lemmas.

Lemma 6.3.4. Let x be a point on a finite type k-scheme X, let Y = Spec O_{X,x} and Z and W be closed subschemes of Y. Then Ŷ_{Z}^{h} ×_{Y} W ≅ Ŵ_{Z∩W}^{h}.

Proof. Since Y and W are local, the pro-schemes Ŷ_{Z}^{h} and Ŵ_{Z∩W} are represented by local Y -schemes. If Y′ → Y, s : Z → Y′ is a Nisnevich neighborhood of Z in Y, and i : Z ∩ W → W is the inclusion, then Y′ ×_{Y} W → W, (s_{|Z∩W} , i) : Z ∩ W → Y′ ×_{Y} W is a Nisnevich neighborhood of Z ∩ W in W, giving us the W -morphism

f : Ŵ_{Z∩W}^{h} → Ŷ_{Z}^{h} ×_{Y} W.

As W is local, we have a co-final family in the category of all finite type étale morphisms W′ → W of the form W′ = Spec (O_{W} [T]/F)_{G}, i.e., the localization of O_{W} [T]/F with respect to some G ∈ O[T], where (∂F/∂T, F) is the unit ideal in O_{W} [T]_{G}. Those W′ → W of this form which give a Nisnevich neighborhood of Z ∩W are those for which F contains a linear factor, modulo the ideal I_{Z∩W} of Z ∩ W. Each such pair (F, G) lifts to a pair (F̃, G̃) of elements in O_{Y} [T] such that Spec (O_{Y} [T]/F̃)G̃ → Y is étale, and such that the linear factor in F mod I_{Z∩W} lifts to a linear factor of F̃ mod I_{Z}. This easily implies that f is an isomorphism. □

Let i : W → Y be a closed immersion of finite type k-schemes, E ∈ Spt(Y_{Zar}). Define the functor

i^{!} : Spt(Y_{Zar}) → Spt(W_{Zar})

by

(i^{!}E)(W \ F) := hofib(E(Y \ F) → E(Y \ W))

for each F ⊂ W closed. For each I ⊂ {1, …, m}, let ι_{I} : D_{I} → D be the inclusion. For J ⊂ I, and F ⊂ D closed, the diagram of restriction maps

\xymatrix{E(D \ (D_{I} ∩ F)) \ar[r] \ar[d] & E(D \ D_{I}) \ar[d] \\ E(D \ (D_{J} ∩ F)) \ar[r] & E(D \ D_{J})}

gives the map

ι_{I∗}ι_{I}^{!}E → ι_{J∗}ι_{J}^{!}E

Lemma 6.3.5. Suppose E ∈ Spt(D_{Zar}) is satisfies Zariski excision. Then the evident map

hocolim_{I∈□_{0}^{nop}} ι_{I∗}ι_{I}^{!}E → E

is a pointwise weak equivalence.

Proof. Suppose temporarily that D is an arbitrary finite type k-scheme, written as a union of two closed subschemes: D = D^{1} ∪D^{2}, and take an E ∈ Spt(D_{Zar}) which is additive. Let D^{12} := D^{1}∩D^{2}, with inclusions ι^{j} : D^{j} → D, ι^{12} : D^{12} → D, ι^{j,12} : D^{12} → D^{j}. We have the natural map

hocolim [ \xymatrix{ι_{12∗}ι^{!12}E \ar[r]^{ι_{∗}^{1,12}} \ar[d]_{ι_{∗}^{2,12}} & ι_{∗}^{1}ι^{1!}E \\ ι_{∗}^{2}ι^{2!}E} ] ⟶^{α} E

We first show that α is a pointwise weak equivalence. It suffices to show that α is a weak equivalence on global sections, equivalently, that the diagram

\xymatrix{E^{D^{12}}(D) \ar[r] \ar[d] & E^{D^{1}}(D) \ar[d] \\ E^{D^{2}}(D) \ar[r] & E(D)}

is homotopy cocartesian.

The homotopy cofiber of E^{D^{1}}(D) → E(D) is homotopy equivalence to E(D \ D^{1}) and the homotopy cofiber of E^{D^{12}}(D) → E^{D^{2}}(D) is homotopy equivalent to E^{D^{2}\D^{12}}(D \ D^{12}). Since

D \ D^{12} = D^{1} \ D^{12} ⨿ D^{2} \ D^{12}

and E is additive, the map on the homotopy cofibers is a weak equivalence, as desired. The proof of the lemma now follows easily by induction on the number m of irreducible components of D = ∪_{i=1}^{m}D_{i}. Indeed, write D = D^{1}∪D^{2}, with D^{1} = D_{1} and D^{2} = ∪_{i=2}^{m}D_{i}. Note that the Zariski excision property is preserved by the functor i^{!} and that a presheaf that satisfies Zariski excision is additive. By induction the maps

hocolim_{∅≠I⊂{2,...,m}} ι_{I∗}ι_{I}^{!}E → ι_{∗}^{2}ι^{2!}E

hocolim_{{1}⫋I⊂{1,...,m}} ι_{I∗}ι_{I}^{!}E → ι_{∗}^{12}ι^{12!}E

are pointwise weak equivalences. Thus the map

hocolim_{I∈□_{0}^{nop}} ι_{I∗}ι_{I}^{!}E → hocolim [ \xymatrix{ι_{12∗}ι^{!12}E \ar[r]^{ι_{∗}^{1,12}} \ar[d]_{ι_{∗}^{2,12}} & ι_{∗}^{1}ι^{1!}E \\ ι_{∗}^{2}ι^{2!}E} ]

is a pointwise weak equivalence; combined with our previous computation, this proves the lemma. □

Proof of proposition 6.3.3. Write D as a sum, D = ∑_{i=1}^{m} D_{i} with each D_{i} smooth (but not necessarily irreducible), and with m minimal. We proceed by induction on m. For m = 1, Nisnevich excision implies that the natural map

E^{D_{Zar}}(X) → E^{D_{Zar}}(X̂_{D}^{h})

is a weak equivalence in Spt(D_{Zar}). Since D is smooth, the map E^{D_{Zar}}(X̂_{D}^{h}) → E^{D_{Zar}}(τ X̂ ; ϵ(D))_{0} is an isomorphism, which proves the result in this case. By lemma 6.3.5 it suffices to show that ι_{I}^{!}(π_{D,0}^{∗}) is a weak equivalence for all I. More generally, let ι_{I,J} : D_{I} → D_{J} be the inclusion for I ⊂ J. If E satisfies Zariski excision on D_{Zar}, the same holds for ι_{I}^{!}E on D_{I,Zar} and there is a natural weak equivalence

ι_{J,I}^{!}(ι_{I}^{!}E) → ι_{J}^{!}E

Thus it suffices to show that ι_{i}^{!}(π_{D,0}^{∗}) is a weak equivalence for all i ∈ {1, …, m}, e.g., for i = m. In what follows, we will only apply ι_{I}^{!} to presheaves E which satisfy Zariski excision, which suffices for the proof.

We use the following notation: for W ⊂ D_{I} a closed subset, we let E^{W_{Zar}}(X) denote the presheaf on D_{I}

E^{W_{Zar}}(X)(D_{I} \ F) := E^{W \F} (X \ F).

We use the same notation for the presheaf

D \ F ↦ E^{W \F} (X \ F)

on D_{Zar}, relying on the context to make the meaning clear. Clearly ι_{m}^{!}ι_{m∗}E^{D_{mZar}}(X) → E^{D_{mZar}}(X) is a weak equivalence and the map E^{D_{mZar}}(X) → E^{D_{Zar}}(X) induces a weak equivalence ι_{m}^{!}E^{D_{mZar}}(X) → ι_{m}^{!}E^{D_{Zar}}(X), so we need to show that

E^{D_{mZar}}(X) → ι_{m}^{!}E^{D_{Zar}}(τ X̂ ; ϵ(D))_{0} = ι_{m}^{!}(holim_{I≠∅} E^{X̂_{D_{I}} ×_{X}D_{Zar}}(X̂_{D_{I}})

is a weak equivalence. For this, we decompose the set of non-empty I ⊂ {1, …, m} into three sets:

1. I = {m},
2. I with m ∉ I,
3. I with {m} ⫋ I.

Let

E_{1} := ι_{m}^{!}E^{X̂_{Dm}^{h} ×_{X}D_{Zar}}(X̂_{D_{m}}^{h})

E_{2} := holim_{m∉I} ι_{m}^{!}E^{X̂_{D_{I}}^{h} ×_{X}D_{Zar}}(X̂_{D_{I}}^{h})

E_{3} := holim_{{m}⫋I} ι_{m}^{!}E^{X̂_{D_{I}}^{h} ×_{X}D_{Zar}}(X̂_{D_{I}}^{h})

We thus have the isomorphism

ι_{m}^{!} (holim_{I≠∅} E^{X̂_{D_{I}}^{h} ×_{X}D_{Zar}}(X̂_{D_{I}}^{h})) ≅ holim [ \xymatrix{ & E_{1} \ar[d] \\ E_{2} \ar[r] & E_{3}}]

For I of type 2, lemma 6.3.4 says that the natural map

X̂_{D_{I∪{m}}}^{h} ×_{X} D_{m} → X̂_{D_{I}}^{h} ×_{X} D_{m}

is an isomorphism. Since the restriction map

ι_{m}^{!}E^{X̂_{D_{I}}^{h} ×_{X}D_{Zar}}(X̂_{D_{I}}^{h}) → ι_{m}^{!}E^{X̂_{D_{I∪{m}}}^{h} ×_{X}D_{Zar}}(X̂_{D_{I∪{m}}}^{h})

identifies itself with the pull-back

E^{X̂_{D_{I}}^{h} ×_{X}D_{mZar}}(X̂_{D_{I}}^{h}) → E^{X̂_{D_{I∪{m}}}^{h} ×_{X}D_{mZar}}(X̂_{D_{I∪{m}}}^{h})

the Nisnevich excision property of E implies that E_{2} → E_{3} is a weak equivalence. Thus

E_{1} → holim [ \xymatrix{ & E_{1} \ar[d] \\ E_{2} \ar[r] & E_{3}}]

is a weak equivalence, and

E^{D_{mZar}}(X) → ι_{m}^{!}E^{D_{Zar}}(τ X̂ ; ϵ(D))_{0} = ι_{m}^{!} (holim_{I≠∅} E^{X̂_{D_{I}}^{h} ×_{X}D_{Zar}}(X̂_{D_{I}}^{h}))

is identified with

E^{D_{mZar}}(X) → ι_{m}^{!}E^{X̂_{Dm}^{h} ×_{X}D_{Zar}}(X̂_{D_{m}}^{h}) = E^{X̂_{Dm}^{h} ×_{X}D_{mZar}}(X̂_{D_{m}}^{h}),

which is a weak equivalence by Nisnevich excision. □

Proposition 6.3.6. Suppose that E is homotopy invariant and satisfies Nisnevich excision, and D is a strict normal crossing subscheme of X. Then

π_{D}^{∗} : E^{D_{Zar}}(X) → E^{D_{Zar}}(τ X̂ ; ϵ(D))

is a weak equivalence in Spt(D_{Zar}).

Proof. Let p_{n} : Δ_{D}^{n} → D be the projection. Applying Proposition 6.3.3 to the strict normal crossing subscheme Δ_{D}^{n} of Δ_{X}^{n}, the map

π_{Δ_{D}^{n},0} : p_{n∗}E^{Δ_{D_{Zar}}^{n}} (Δ_{X}^{n}) → p_{n∗}E^{Δ_{D_{Zar}}^{n}} (τ Δ̂_{X}^{n} ; ϵ (Δ_{D}^{n}))_{0}

is a weak equivalence for each n. Thus

π_{Δ_{D}^{∗}} : p_{∗}E^{Δ_{D_{Zar}}^{∗}} (Δ_{X}^{∗}) → E^{D_{Zar}}(τ X̂ ; ϵ(D))

is a weak equivalence. Indeed, E^{D_{Zar}}(τ X̂ ; ϵ(D) is a simplicial object with n-

simplices p_{n∗}E^{Δ_{D_{Zar}}^{n}} (τ Δ̂_{X}^{n} ; ϵ (Δ_{D}^{n}))_{0}. Since E is homotopy invariant, the map

p^{∗} : E^{D_{Zar}}(X) → p_{∗}E^{Δ_{D_{Zar}}^{∗}} (Δ_{X}^{∗})

is a weak equivalence, whence the result. □

We can now state and prove the main result for strict normal crossing schemes.

Theorem 6.3.7. Let D be a strict normal crossing scheme on some X ∈ Sm/k and take E ∈ Spt(k) which is homotopy invariant and satisfies Nisnevich excision. Then there is a natural distinguished triangle in HSpt(D_{Zar})

E^{D_{Zar}}(X) ⟶^{α_{D}} E(D_{•}) ⟶^{β_{D}} E(τ X̂ ; ϵ(D)^{0})

Proof. By proposition 6.3.6, we have the weak equivalence

π_{D}^{∗} : E^{D_{Zar}}(X) → E^{D_{Zar}}(τ X̂ ; ϵ(D)).

By Proposition 6.2.2, we have the isomorphism

(p_{•}^{∗})^{−1}î_{D}^{∗} : E(τ X̂ ; ϵ(D)) → E(D_{•}).

in HSpt(D_{Zar}). Since E^{D}(τ X̂ ; ϵ(D)) is by definition the homotopy fiber of the restriction map E(τ X̂ ; ϵ(D)) → E(τ X̂ ; ϵ(D)^{0}), the result is proved. □

7. Comparison isomorphisms

We give a comparison of our tubular neighborhood construction with the categorical version Li^{∗}Rj_{∗} of Morel-Voevodsky.

7.1. Model structure and cross functors. Fix a noetherian separated scheme S of finite Krull dimension, and let Sch_{S} denote the category of finite type S-schemes (for our application, we will take S = Spec k for a field k). Morel-Voevodsky show how to make the catgory SH_{𝔸^{1}}(X) functorial in X ∈ Sch_{S}, defining an adjoint pair of exact functors Lf^{∗}, Rf_{∗} for each morphism f : Y → X in Sch_{S}. Roendigs shows in [39] how to achieve this on the model category level and in addition that this structure extends to give cross functors (f_{∗}, f^{∗}, f^{!}, f_{!}) as defined by Voevodsky and investigated in detail by Ayoub [3]. We begin by describing the model structure used by Roendigs, which is different from the one we have used up to now, and recalling his main results. For B ∈ Sch_{S}, we denote by Spc_{∗mot}(Sm/B) the model structure on Spc_{∗}(Sm/B) described by Roendigs in [39]. To describe this model structure, we first recall the projective model structure Spc(Sm/B)_{proj} on Spc(Sm/B). Here the weak equivalences and fibrations are the pointwise ones and the cofibrations are generated by the maps

Z × ∂Δ^{n} → Z × Δ^{n},

with Z ∈ Sm/B. This induces a model structure Spc_{∗}(Sm/B)_{proj} on Spc_{∗}(Sm/B) by forgetting/adjoing a base-point. One has a functorial cofibrant replacement E^{c} → E defined as in [34, Lemma 1.16]. The model structure Spc_{∗mot}(Sm/B) is defined by Bousfield localization: the cofibrations are the same as in Spc_{∗}(Sm/B)_{proj}. E is fibrant if E(∅) is contractible, E is a fibrant in Spc_{∗}(Sm/B)_{proj}, E transforms elementary Nisnevich squares to homotopy fiber squares and transforms Z ×𝔸^{1} → Z to a weak equivalence. A map A → B is a weak equivalence if Hom(B^{c}, E) → Hom(A^{c}, E) is a weak equivalence for each fibrant E. The fibrations in Spc_{∗mot}(Sm/B) are determined by having the right lifting property with respect to trivial cofibrations. Let f : X → Y be a morphism in Sch_{S}. We have the functor

f_{∗} : Spc_{∗}(Sm/X) → Spc_{∗}(Sm/Y)

defined by pre-composition with the pull-back functor − ×_{Y} X, i.e.

f_{∗}E(Y′ → Y) := E(Y′ ×_{Y} X → X).

f_{∗} has the left adjoint f^{∗} defined as the Kan extension, and characterized by f^{∗}(Y′_{+}) = Y′ ×_{Y} X_{+} for Y′ → Y ∈ Sm/Y. In case f is a smooth morphism, f^{∗} is given by precomposition with the functor

f ∘ − : Sm/X → Sm/Y,

and thus has the left adjoint f_{♯} characterized by

f_{♯}(Z ⟶^{p} X) = Z ⟶^{fp} Y

on the representable presheaves. We have

Proposition 7.1.1 (proposition 2.18 of [39]). Let f : X → Y be a morphism in Sch_{S}. Then (f^{∗}, f_{∗}) is a Quillen adjoint pair Spc_{∗mot}(Sm/X) ↔ Spc_{∗mot}(Sm/Y). If f is smooth, then (f_{♯}, f^{∗}) is a Quillen adjoint pair Spc_{∗mot}(Sm/Y) ↔ Spc_{∗mot}(Sm/X).

For spectra, the projective model structure Spt_{mot}(Sm/B)_{proj} on Spt(Sm/B) is defined as follows: For ϕ : E → F a morphism in Spt(Sm/B), ϕ : E → F is a cofibration if ϕ_{0} : E_{0} → F_{0} is a cofibration in Spc_{∗mot}(Sm/B) and if for each n ≥ 1, the map

ϕ_{n} ∪ Σϕ_{n−1} : E_{n} ∪_{ΣE_{n−1}} ΣF_{n−1} → F_{n}

is a cofibration in Spc_{∗}(Sm/B)_{proj}. Weak equivalences (resp. fibrations) are maps ϕ such that ϕ_{n} is a weak equivalence (resp. fibration) in Spc_{∗mot}(Sm/B) for all n. There is a functorial cofibrant replacement E^{c} → E. Now for the motivic model structure Spt_{mot}(Sm/B): The cofibrations are the same as in Spt_{mot}(Sm/B)_{proj}. ϕ is a fibration if ϕ_{n} is a fibration in Spc_{∗mot}(Sm/B) for all n and the diagram

\xymatrix{E_{n} \ar[r] \ar[d]_{ϕ_{n}} & ΩE_{n+1} \ar[d]^{Ωϕ_{n+1}} \\ F_{n} \ar[r] & ΩF_{n+1}}

is homotopy cartesian in Spc_{∗mot}(Sm/B) for all n. There is a fibrant replacement functor E → E^{f}; ϕ : E → F is a weak equivalence if ϕ^{f} : E^{f} → F^{f} is a weak equivalence in Spt_{mot}(Sm/B)_{proj}. Given f : X → Y in Sch_{S}, define the functors f_{∗} : Spt(Sm/X) → Spt(Sm/Y) and f^{∗} : Spt(Sm/Y) → Spt(Sm/X) by f_{∗}(E)_{n} := f_{∗}(E_{n}), f^{∗}(F)_{n} := f^{∗}(F_{n}). If f is smooth, we have f_{♯} : Spt(Sm/X) → Spt(Sm/Y) defined similarly by f_{♯}(E)_{n} := f_{♯}(E_{n}). We have the following result from [39]:

Proposition 7.1.2 (proposition 2.23 of [39]). Let f : X → Y be a morphism in Sch_{S}. Then (f_{∗}, f^{∗}) is a Quillen adjoint pair Spt_{mot}(Sm/X) ↔ Spt(Sm/Y). If f is smooth, then (f_{♯}, f^{∗}) is a Quillen adjoint pair Spt_{mot}(Sm/Y) ↔ Spt_{mot}(Sm/X). In particular:

(1) f^{∗} preserves cofibrations and trivial cofibration and f_{∗} preserves fibrations and trivial fibrations.

(2) if f is smooth, then f^{∗} preserves fibrations and f_{♯} preserves cofibrations

It is clear that a cofibration in Spt_{mot}(Sm/X) is pointwise a cofibration in Spt, hence a cofibration in Spt_{𝔸^{1}}(Sm/X_{Nis}). As mentioned in [39] a fibrant object in Spt_{mot}(Sm/X) satisfies both Nisnevich excision and is 𝔸^{1}-local, hence the weak equivalences between fibrant objects in Spt_{mot}(Sm/X) are weak equivalences in Spt_{𝔸^{1}}(Sm/X_{Nis}) and are in fact pointwise weak equivalences in Spt(Sm/X); similarly one shows that each fibration in Spt_{𝔸^{1}}(Sm/X_{Nis}) is a fibration in Spt_{mot}(Sm/X) and each (trivial) cofibration in Spt_{mot}(Sm/X) is a (trivial) cofibration in Spt_{𝔸^{1}}(Sm/X_{Nis}). Thus the identity on Spt(Sm/X) defines a (left) Quillen equivalence Spt_{mot}(Sm/X) → Spt_{𝔸^{1}}(Sm/X_{Nis}). In particular, we have the equivalence of the homotopy categories

HSpt_{𝔸^{1}}(Sm/X_{Nis}) ≅ HSpt_{mot}(Sm/X).

We write SH_{𝔸^{1}}(X) for either HSpt_{mot}(Sm/X) or HSpt_{𝔸^{1}}(Sm/X_{Nis}), depending on the context. One main result of [39] is

Theorem 7.1.3 ([39, corollary 3.17]). Sending f : Y → X in Sch_{S} to Lf^{∗} : SH_{𝔸^{1}}(X) → SH_{𝔸^{1}}(X) satisfies the conditions of [3, definition 1.4.1]. In particular, the properties of a “2-foncteur homotopique stable” described in [3] are satisfied for X ↦ SH_{𝔸^{1}}(X).

Remark 7.1.4. Let i : D → X be a closed immersion in Sch_{S} with open complement j : U → X. We have the functor

Li^{∗}Rj_{∗} : SH_{𝔸^{1}}(X \ D) → SH_{𝔸^{1}}(D),

We would like to view our construction E(τ X̂ ; ϵ(D)^{0}) as a weak version of Li^{∗}Rj_{∗}, in case D is a normal crossing divisor on a smooth k scheme X, the input E is the pull-back from Spt(Sm/k), and the output E(τ X̂ ; ϵ(D)^{0}) is in HSpt(D_{Zar}). In particular, E(τ X̂ ; ϵ(D)^{0}) is only defined on Zariski open subsets of D, rather than on all of Sm/D. In this section, we make this statement precise, defining an isomorphism of E(τ X̂ ; ϵ(D)^{0}) with the restriction of Li^{∗}Rj_{∗}(E) to HSpt(D_{Zar}). □

7.2. The smooth case. Let i : W → X be a closed immersion in Sch_{S} with open complement j : U → X. Let

Θ : Spt_{mot}(Sm/U) → Spt_{mot}(Sm/W)

be the functor representing Li^{∗}Rj_{∗}, i.e.

Θ(E) := i^{∗}(j_{∗}(E^{f})^{c})^{f}.

Remark 7.2.1. Even for E ∈ Spt_{mot}(Sm/U) bifibrant, one cannot simplify this expression for Li^{∗}Rj_{∗}E beyond replacing E^{f} with E. The inexplicit nature of the cofibrant and fibrant replacement functors make a concrete determination of Li^{∗}Rj_{∗}E difficult, which is one advantage of our approach using the punctured tubular neighborhood. □

Lemma 7.2.2. Let i : W → X be a closed immersion in Sch_{S} with open complement j : U → X.

(1) For fibrant E ∈ Spt_{mot}(Sm/U) all the maps in the square

\xymatrix{j_{∗}(E)^{c} \ar[r] \ar[d] & Rj_{∗}(E)^{c} \ar[d] \\ j_{∗}(E) \ar[r] & Rj_{∗}(E)}

are pointwise weak equivalences

(2) Let X′ → X be in Sm/X, let W′ := W ×_{X} X′. There is a canonical

map

ν_{X′}^{0} : Rj_{∗}(E)^{c}(X′) → ΘE(W′)

natural in X′.

Proof. (1) Since E is fibrant, the canonical map E → E^{f} is a trivial cofibration of fibrant objects in Spt_{mot}(Sm/U), hence a homotopy equivalence. Thus j_{∗}E → Rj_{∗}E := j_{∗}(E^{f}) is a homotopy equivalence of fibrant objects in Spt_{mot}(Sm/X), hence a pointwise weak equivalence. Applying the cofibrant replacement functor, we see that (j_{∗}E)^{c} → (Rj_{∗}E)^{c} is also a homotopy equivalance and a pointwise weak equivalence. Also the cofibrant replacement maps (j_{∗}E)^{c} → j_{∗}E, (Rj_{∗}E)^{c} → Rj_{∗}E are trivial fibrations between fibrant objects of Spt_{mot}(Sm/X), hence are both pointwise weak equivalences. For (2), the unit id → i_{∗}i^{∗} for the adjunction applied to (Rj_{∗}E)^{c} gives us the map

ν_{X′}^{0} : (Rj_{∗}E)^{c}(X′) → i_{∗}i^{∗}(Rj_{∗}E)^{c}(X′)

natural in X′. As i_{∗}i^{∗}(Rj_{∗}E)^{c}(X′) = i^{∗}(Rj_{∗}E)^{c}(W′), we have the natural transformation

ν_{X′}^{0} : (Rj_{∗}E)^{c}(X′) → i^{∗}(Rj_{∗}E)^{c}(W′)

Composing with the canonical map i^{∗}(Rj_{∗}E)^{c} → (i^{∗}(Rj_{∗}E)^{c})^{f} = Θ(E) gives us the map we want. □

For E ∈ Spt(Sm/B) or in Spt(B_{Nis}), we let E_{Zar} denote the restriction to Spt(B_{Zar}). Identifying SH_{𝔸^{1}}(B) with the homotopy category of bifibrant objects in Spt_{mot}(Sm/B), we have the similarly defined restriction functor SH_{𝔸^{1}}(B) → HSpt(B_{Zar}) sending E to E_{Zar}. Let i : W → X be a closed immersion in Sm/k with open complement j : U → X. We note that the “evaluation” maps

E ↦ E(τ X̂ ; ϵ(W)), E ↦ E(τ X̂ ; ϵ(W)^{0})

are in fact defined for E ∈ Spt(Sm/X). Similarly, the evaluation map E ↦ E(τ X̂ ; ϵ(W)^{0}) is defined for E ∈ Spt(Sm/U). In addition, for E ∈ Spt(Sm/U) we have a canonical isomorphism

(7.2.1)  E(τ X̂ ; ϵ(W)^{0}) ≅ (j_{∗}E)(τ X̂ ; ϵ(W))

since Δ̂_{X,W}^{n} \ Δ_{W}^{n} ≅ Δ̂_{X,W}^{n} ×_{X} U (as a pro-scheme).

Lemma 7.2.3. Let i : W → X be a closed immersion in Sm/k with open complement j : U → X, and let E ∈ Spt_{mot}(Sm/U) be fibrant.

(1) There is a map

η_{E}^{c} : (Rj_{∗}E)^{c}(τ X̂ ; ϵ(W)) → Θ(E)_{Zar}

in HSpt(W_{Zar}), natural in E.

(2) Let

[figure: j_{∗}(E)^{c}(τ X̂ ; ϵ(W)); Rj_{∗}(E)^{c}(τ X̂ ; ϵ(W)); E(τ X̂ ; ϵ(W)^{0}); ϕ; ∼; j_{∗}(E)(τ X̂ ; ϵ(W)); Rj_{∗}(E)(τ X̂ ; ϵ(W))]

be the diagram in Spt(W_{Zar}) formed by evaluating the diagram of lemma 7.2.2(1) at τ X̂ ; ϵ(W), and adding the isomorphism (7.2.1). Then all the maps in this diagram are pointwise weak equivalences.

Proof. By lemma 7.2.2, we have maps

η_{X′}^{0} : (Rj_{∗}E)^{c}(X′) → Θ(E)(X′ ×_{X} W)

natural in X′ ∈ Sm/X. For each open subscheme U = D \ F ⊂ W, the maps ηΔ̂_{X\F,U}^{n} define the map

η_{Δ^{∗}}^{0}(U) : (Rj_{∗}E)^{c}(τ X̂ ; ϵ(W)(U)) → Θ(E)(Δ_{U}^{∗}).

Since Θ(E) is 𝔸^{1}-local, the canonical map Θ(E)(U) → Θ(E)(Δ_{U}^{∗}) is a weak equivalence. This gives us the natural map in HSpt(W_{Zar})

η^{0} : (Rj_{∗}E)^{c}(τ X̂ ; ϵ(W)) → Θ(E)_{Zar},

proving (1).

(2) follows immediately from lemma 7.2.2(1). □

Combining the morphism (1) with the diagram (2) gives us the canonical morphism in HSpt(W_{Zar})

η_{E}^{0} : E(τ X̂ ; ϵ(W)^{0}) → Θ(E)_{Zar}.

Let i : D → X be a closed immersion in Sch_{S}. We have the exact functor i^{!} : SH_{𝔸^{1}}(X) → SH_{𝔸^{1}}(D) which is characterized by the identity for fibrant E ∈ Spt_{mot}(Sm/X):

i_{∗}i^{!}E(X′ → X) := hofib(E(X′) → E(X′ ×_{X} (X \ D)).

In fact, this operation gives the distinguished triangle, natural in fibrant E ∈ Spt_{mot}(Sm/X):

Ri_{∗}i^{!}E → E → Rj_{∗}j^{∗}E → i_{∗}i^{!}E[1].

Applying Li^{∗} (and noting that the counit Li^{∗}Ri_{∗} → id is an isomorphism [3, definition 1.4.1]) gives the distinguished triangle in SH_{𝔸^{1}}(D)

(7.2.2)  i^{!}E → Li^{∗}E → Θ(j^{∗}E) → i^{!}E[1]

We refer the reader to [3, proposition 1.4.9] for the construction of this triangle in the abstract setting.

Proposition 7.2.4. Let E ∈ Spt_{mot}(Sm/k) be fibrant, let f : X → Spec k be in Sm/k and let i : W → X be a closed immersion in Sm/k with open complement j : U → X. Then

η_{E}^{0} : E(τ X̂ ; ϵ(W)^{0}) → Θ(j^{∗}f^{∗}E)_{Zar}

is an isomorphism in HSpt(W_{Zar}).

Proof. Let f_{W} : W → Spec k be the structure morphism. Since f and fi = f_{W} are smooth, we have Lf^{∗} ≅ f^{∗}, f_{W}^{∗} ≅ L(fi)^{∗} ≅ Li^{∗}f^{∗}, so Li^{∗}f^{∗} is isomorphic to the restriction functor for f_{W} ∘− : Sm/W → Sm/k. The definition of i^{!} gives the commutative diagram for each X′ → X in Sm/X (with W′ := X′ ×_{X} W)

[figure: E^{W′}(X′); ϕ_{X′}; E(X′); η_{X′}; i^{!}f^{∗}E(W′); Li^{∗}f^{∗}E(W′); ∼; E(W′)]

where η_{X′} is just the restriction map E(X′) → E(W′) and ϕ_{X′} is the canonical isomorphism given by the definition of i^{!}. Using lemma 7.2.3, this gives us the map of distinguished triangles in SH

\xymatrix{E^{W′}(X′) \ar[r] \ar[d]_{ϕ_{X′}} & E(X′) \ar[r] \ar[d]^{η_{X′}} & E(X′ \ W′) \ar[r] \ar[d]^{η_{X′}^{0}} & E^{W′}(X′)[1] \ar[d] \\ i^{!}f^{∗}E(W′) \ar[r] & E(W′) \ar[r] & Li^{∗}Rj_{∗}j^{∗}f^{∗}E(W′) \ar[r] & i^{!}f^{∗}E(W′)[1]}

Just as for η_{E}^{0}, these give rise to the natural map in HSpt(Sm/W_{Zar})

η_{E} : E(τ X̂ ; ϵ(W)) → Li^{∗}f^{∗}E_{Zar}

and the commutative diagram in HSpt(W_{Zar})

\xymatrix{E^{W_{Zar}}(X) \ar[r] \ar[d]_{ϕ} & E(τ X̂ ; ϵ(W)) \ar[r] \ar[d]^{η} & E(τ X̂ ; ϵ(W)^{0}) \ar[r] \ar[d]^{η^{0}} & E^{W_{Zar}}(X)[1] \ar[d] \\ i^{!}f^{∗}E_{Zar} \ar[r] & Li^{∗}f^{∗}E_{Zar} \ar[r] & Li^{∗}Rj_{∗}j^{∗}f^{∗}E_{Zar} \ar[r] & i^{!}f^{∗}E_{Zar}[1]}

The bottom row is the distinguished triangle (7.2.2) for f^{∗}E, restricted to W_{Zar}, and the top row is the distinguished triangle of corollary 4.1.3, after applying theorem 3.2.1. Similarly, theorem 3.2.1 shows that η is an isomorphism in HSpt(W_{Zar}). Since ϕ is an isomorphism in HSpt(W_{Zar}) η^{0} is an isomorphism as well. □

7.3. The normal crossing case. We fix a reduced strict normal crossing divisor i : D → X on some X ∈ Sm/k. Write D = ∑_{i=1}^{m} D_{i} with the D_{i} smooth. For X′ → X in Sm/X, we write D′ for X′ ×_{X} D and D′_{I} for X′ ×_{X} D_{I} and for I ⊂ {1, …, m}. As in the previous section, we note that our definition of E(τ X̂ ; ϵ(D)) extends without change to E ∈ Spt(Sm/X), and similarly, the construction of E(τ X̂ ; ϵ(D)^{0}) extends without change to E ∈ Spt(Sm/X \ D). The extension of proposition 7.2.4 to the normal crossing case follows essentially the same outline as before, with some additional patching results for the operation Li^{∗}Rj_{∗} that allow us give a description of Li^{∗}Rj_{∗} as a homotopy limit, matching our definition of E(τ X̂ ; ϵ(W)^{0}).

Lemma 7.3.1. Suppose that F ∈ Spt(Sm/D) satisfies Nisnevich excision. For I ⊂ {1, …, m}, I ≠ ∅, let F_{I} be the presheaf on Sm/X

F_{I}(X′) := F (X̂_{D′_{I}}^{′h} ×_{X} D).

Then the canonical map

i_{∗}F → holim_{I≠∅} F_{I}

is a weak equivalence in Spt(Sm/X).

Proof. Let {U_{j} → D′ | j ∈ M} be a Nisnevich cover of D′, with M a finite set. For J ⊂ M, set U_{J} := ∏_{j∈J} U_{j}, where the product is ×_{D′}. Since F satisfies Nisnevich excision, the canonical map

F (D′) → holim_{J≠∅} F (U_{J})

is a weak equivalence. An argument similar to that of lemma 6.3.2 shows that one can replace the U_{i} with a pro-system of Nisnevich covers (with M fixed). Similarly, the Zariski stalk of holim_{I≠∅} F_{I} at x ∈ X′ ∈ Sm/X is weakly equivalent to holim_{I≠∅} F (X̂_{x,D′_{I}}^{′h} ×_{X} D), where X′_{x} = Spec O_{X′,x}. Thus we need

only show that for X′ → X smooth, with X′ local, the schemes U_{i} := X̂_{D′_{i}}^{′h} ×_{X} D form a pro-Nisnevich cover of D′, and that

∏_{i∈I} U_{i} ≅ X̂_{D′_{I}}^{′h} ×_{X} D

for each non-empty I ⊂ {1, …, m}. In fact the pro-schemes X̂_{D′_{i}}^{′h} ×_{X} D, i = 1, …, m, obvioiusly form a pro-Nisnevich cover of D′; it follows from lemma 6.3.4 that for each I ⊂ {1, …, m}, I ≠ ∅, we have natural isomorphisms (where ∏ is ×_{X′})

∏_{i∈I} X̂_{D′_{i}}^{′h} ≅ X̂_{D′_{I}}^{′h} .

Thus (with the product over D′)

∏_{i∈I} X̂_{D′_{i}}^{′h} ×_{X} D ≅ X̂_{D′_{I}}^{′h} ×_{X} D.

□

Lemma 7.3.2. Let i : D → X be a strict normal crossing divisor on some f : X → Spec k in Sm/k, and let E ∈ Spt_{mot}(Sm/U) be fibrant. Then there is a canonical map in HSpt(D_{Zar}),

η_{E}^{0} : E(τ X̂ ; ϵ(W)^{0}) → Θ(E)_{Zar},

natural in E.

Proof. As in the smooth case, we construct η_{E}^{0} using lemmas 7.2.2 and 7.3.1. Indeed, let j : X \ D → X be the inclusion. Let Θ(E)_{IZar} denote the pull-back of Θ(E) to Spt(X̂_{D_{I}}^{h} ×_{X} D_{Zar}). Let Θ(E)_{IZar}^{∗} be the presheaf

Θ(E)_{IZar}^{∗}(U) := Θ(E)_{IZar}(Δ_{U}^{∗}).

Similarly, let Θ(E)_{Zar}^{∗} denote the presheaf on D_{Zar}

Θ(E)_{Zar}^{∗}(U) := Θ(E)(Δ_{U}^{∗})

and let Θ(E)_{Zar} denote the restriction of Θ(E) to D_{Zar}. The construction of lemma 7.2.3 gives us the diagram of maps

η̃_{E,I}^{0} : (Rj_{∗}E)^{c}(τ X̂ ; ϵ(D_{I})) → Θ(E)_{IZar}^{∗}

and thus the map

η̃_{E}^{0} : (Rj_{∗}E)^{c}(τ X̂ ; ϵ(D)) → holim_{I≠∅}(I ↦ Θ(E)_{IZar}^{∗})

By lemma 7.3.1 we have the canonical isomorphism in HSpt(D_{Zar})

holim_{I≠∅}(I ↦ Θ(E)_{IZar}^{∗}) ≅ Θ(E)_{Zar}^{∗}.

Since Θ(E) is 𝔸^{1}-homotopy invariant, the canonical map Θ(E)_{Zar} → Θ(E)_{Zar}^{∗} is a pointwise weak equivalence, giving us the map in HSpt(D_{Zar})

η̃_{E}^{0} : (Rj_{∗}E)^{c}(τ X̂ ; ϵ(D)) → Θ(E)_{Zar}

Using the diagram of lemma 7.2.3, with W = D_{I}, and then taking the appropriate homotopy limit, we arrive at a canonical isomorphism in HSpt(D_{Zar})

(Rj_{∗}E)^{c}(τ X̂ ; ϵ(D)) ≅ E(τ X̂ ; ϵ(D)^{0}).

Combining η̃_{E}^{0} with this isomorphism gives us the desired map η_{E}^{0}. □

Lemma 7.3.3. Let i : W → X be a closed immersion in Sch_{S}. Suppose W is a union of closed subschemes, W = W_{1} ∪ W_{2}. Let W_{12} := W_{1} ∩ W_{2} and let i_{j} : W_{j} → X, j = 1, 2, i_{12} : W_{12} → X be the inclusions. Then for E ∈ SH_{𝔸^{1}}(Sm/X) there is a canonical homotopy cartesian diagram in SH_{𝔸^{1}}(Sm/X)

\xymatrix{Ri_{∗}Li^{∗}E \ar[r] \ar[d] & Ri_{1∗}Li_{1}^{∗}E \ar[d] \\ Ri_{2∗}Li_{2}^{∗}E \ar[r] & Ri_{12∗}L_{12}^{∗}E}

Proof. Throughout the proof we use the canonical lifting of Li^{∗}, Ri_{∗}, etc., to functors on Spt_{mot}(Sm/−) by taking the appropriate cofibrant/fibrant replacement, but we use the same notation to denote these liftings. Let ι : W_{1} → W be the inclusion. The unit id → Rι_{∗}Lι^{∗} gives the map

Li^{∗}E → Rι_{∗}Lι^{∗}Li^{∗}E ≅ Rι_{∗}Li_{1}^{∗}E

in SH_{A^{1}}(Sm/W); applying Ri_{∗} gives the map Ri_{∗}Li^{∗}E → Ri_{1∗}Li_{1}^{∗}E. The other maps in the square are defined similarly; as the two compositions Ri_{∗}Li^{∗}E → Ri_{12∗}L_{12}^{∗}E are likewise defined by the adjoint property, these agree and the diagram commutes. To show that the diagram is homotopy cartesian, let j : U → X be the complement of W, j_{1} : U_{1} → X the complement of W_{1} and j′ : U → U_{1}, i′_{2} : D_{2} ∩ U_{1} → U_{1} the inclusions. We have the distinguished triangles (see [3, Lemme 1.4.6])

Lj_{!}j^{∗}E → E → Ri_{∗}Li^{∗}E → Lj_{!}j^{∗}E[1]
Lj_{1!}j_{1}^{∗}E → E → Ri_{1∗}Li_{1}^{∗}E → Lj_{1!}j_{1}^{∗}E[1]
Lj′_{!}j^{∗}E → j_{1}^{∗}E → Ri′_{2∗}Li_{2}^{′∗}j_{1}^{∗}E → Lj′_{!}j^{∗}E[1]

Applying Lj_{1!} to the last line gives us the distinguished triangle

Lj_{!}j^{∗}E → Lj_{1!}j_{1}^{∗}E → Lj_{1!}Ri′_{2∗}Li_{2}^{′∗}j_{1}^{∗}E → Lj_{!}j^{∗}E[1]

Thus we have the distinguished triangle

Lj_{1!}Ri′_{2∗}Li_{2}^{′∗}j_{1}^{∗}E → Ri_{∗}Li^{∗}E → Ri_{1∗}Li_{1}^{∗}E → Lj_{1!}Ri′_{2∗}Li_{2}^{′∗}j_{1}^{∗}E[1]

The same argument applied to the complement j_{2} : U_{2} → X of W_{2}, the map j′′ : U_{2} → U′′ := U \ W_{12}, j′_{1} : U′′ → X and the inclusion i′′_{2} : D_{2} ∩ U_{1} → U′′ gives the distinguished triangle

Lj′_{1!}Ri′′_{2∗}Li_{2}^{′′∗}j_{1}^{′∗}E → Ri_{2∗}Li_{2}^{∗}E → Ri_{12∗}Li_{12}^{∗}E → Lj′_{1!}Ri′′_{2∗}Li_{2}^{′′∗}j_{1}^{′∗}E[1]

Since D_{2} ∩ U_{1} is closed in U_{1} and in U′′, the natural map

Lj_{1!}Ri′_{2∗}Li_{2}^{′∗}j_{1}^{∗}E → Lj′_{1!}Ri′′_{2∗}Li_{2}^{′′∗}j_{1}^{′∗}E

is an isomorphism. This shows that the diagram is homotopy cartesian. □

Given a strict normal crossing divisor i : D → X, D = ∑_{i=1}^{m} D_{i}, we have the inclusions ι_{I} : D_{I} → D, ι_{I,J} : D_{J} → D_{I} for I ⊂ J and i_{I} : D_{I} → X. For E ∈ Spt(Sm/X) we thus have the presheaves i_{I}^{∗}E ∈ Spt(Sm/D_{I}). The isomorphism ι_{I,J}^{∗}i_{I}^{∗}E ≅ i_{J}^{∗}E gives us the canonical maps i_{I}^{∗}E → ι_{I,J∗}i_{J}^{∗}E; applying ι_{I∗} to this map gives us the natural maps α_{J,I} : ι_{I∗}i_{I}^{∗}E → ι_{J∗}i_{J}^{∗}E. For E ∈ Spt_{mot}(Sm/X), using the cofibrant replacement of E, we see that the same procedure gives us the functor

I ↦ ι_{I∗}(i_{I}^{∗}E^{c})^{f} ∈ Spt(Sm/D)

together with the natural map

α : i^{∗}(E^{c})^{f} → holim_{I≠∅} ι_{I∗}(i_{I}^{∗}E^{c})^{f}.

Lemma 7.3.4. The map α is an isomorphism in SH_{𝔸^{1}}(D).

Proof. As the co-unit Li^{∗}Ri_{∗} → id is an isomorphism, Ri_{∗} is faithful, so it suffices to show that Ri_{∗}(α) is an isomorphism in SH_{𝔸^{1}}(X). This follows from lemma 7.3.3 and induction on m. □

Recall that for E ∈ Spt(Sm/k) and i : D → X a strict normal crossing divisor, D = ∑_{i=1}^{m} D_{i}, we have the presheaf E(D_{Zar}) on D_{Zar} defined by

E(D_{Zar})(U) := holim_{I≠∅} E(D_{I} ∩ U).

Proposition 7.3.5. Let E ∈ Spt_{mot}(Sm/k) be fibrant, i : D → X a strict normal crossing divisor on X ∈ Sm/k, f : X → Spec k the structure morphism. Then we have a natural isomorphism in HSpt(D_{Zar})

E(D_{Zar}) ≅ Li^{∗}(f^{∗}E)_{Zar}

Proof. Let i_{I} : D_{I} → X be the inclusion, f_{I} : D_{I} → Spec k the structure morphism. By theorem 3.2.1, the canonical map

η_{E} : E(τ X̂ ; ϵ(D_{I})) → (f_{I}^{∗}E)_{Zar} ≅ Li_{I}^{∗}(f^{∗}E)_{Zar}

is an isomorphism in HSpt(D_{IZar}). By lemma 7.3.4 the induced map on the holim gives the desired isomorphism. □

Theorem 7.3.6. Let i : D → X be a strict normal crossing divisor on f : X → Spec k in Sm/k, and let E be a fibrant object in Spt_{𝔸^{1}}(Sm/k_{Nis}). Then the map

η_{E}^{0} : E(τ X̂ ; ϵ(D)^{0}) → Θ(f^{∗}E)_{Zar} = [Li^{∗}Rj_{∗}(f^{∗}E)]_{Zar}

is an isomorphism in HSpt(D_{Zar}).

Proof. The proof is the same as the proof of proposition 7.2.4, using the distinguished triangle of theorem 6.3.7 together with the isomorphism of proposition 7.3.5 instead of the triangle of corollary 4.1.3. □

Remark 7.3.7. Fix a fibrant E ∈ Spt_{mot}(Sm/k). Let D′ → D be in Sm/D and suppose we have an X′ → X in Sm/X and a D-isomorphism D′ ≅ X′ ×_{X} D. Then we can replace i : D → X with i′ : D′ → X′ and use theorem 7.3.6 to show that our tubular neighborhood construction gives the model E(τ X̂′ ; ϵ(D′)^{0}) for the restriction of Li^{∗}Rj_{∗}(f^{∗}E) to Sm/D′_{Zar}. If D and D′ are affine, then the theorem of [2] gives the existence of an X′ as above, so our result gives at least a “local” description of the entire presheaf Li^{∗}Rj_{∗}(f^{∗}E). □

8. Limit objects

Let p : X → C be a morphism in Sm/k, with C a smooth curve. Fix a k-point 0 ∈ C(k) and a parameter t ∈ O_{C,0}. Ayoub combines the functor Li^{∗}Rj_{∗} with a cosimplicial version of the classical path space (i.e., the universal cover) construction to define the unipotent specialization functor

sp : SH(X \ p^{−1}(0)) → SH(p^{−1}(0))

Replacing Li^{∗}Rj_{∗} with the punctured tubular neighborhood, the same construction gives a model of this construction as a Zariski presheaf on X_{0}. In particular, we give a description of the “limiting values” lim_{t→0} E(X_{t}) for a semi-stable degeneration X → (C, 0). As we mentioned in the introduction, we expect that this construction, applied to a suitable version of the de Rham complex (with weight and Hodge filtrations) as in [4] would yield the classical limit mixed Hodge structure of a semi-stable degeneration.

Remark 8.0.8. In [3, chapter 3] Ayoub describes a general theory of specialization structures; we concentrate on the unipotent structure, which Ayoub denotes Υ, and describes in [3, §3.4]. □

8.1. Path spaces. Before defining the cosimplicial models for various path spaces and homotopy fibers, we recall some basic operations of simplicial sets on schemes. We let Spc_{f} denote the full subcategory of Spc consisting of simplicial sets S with S([n]) finite for each n. Let Y be a k-scheme. For a finite set S, let Y^{S} := ∏_{s∈S} Y, with the product being over Spec k. This defines the contravariant functor S ↦ Y^{S} from finite sets to k-schemes. In particular, for S ∈ Spc_{f} we have the cosimplicial scheme Y^{S} with Y^{S}([n]) := Y^{S([n])}, giving the functor

Y^{?} : Spc_{f}^{op} → Sch_{k}^{Ord}.

Similarly, if T is a simplicial set, we have the cosimplicial-simplicial set (cosimplicial space) T^{S} and the functor

T^{?} : Spc^{op} → Spc^{Ord}.

Setting Y × S := ⨿_{s∈S}Y, we have the functor S ↦ Y × S from finite sets to k-schemes; if S is a simplicial set as above, we thus have the simplicial scheme Y × S, giving the functor

Y ×? : Spc_{f} → Sch_{k}^{Ord^{op}}.

The adjunction

Hom_{Sch_{k}}(X × S, Y) ≅ Hom_{Sch_{k}}(X, Y^{S})

for S a finite set extends to S a simplicial set as above, giving the adjunction

Hom_{Sch}\begin{matrix}Ord & ^{op} \\ k \end{matrix} (X × S, Y) ≅ Hom_{Sch}{Ord \atop k}(X, Y^{S})

where on the left, we consider Y as a constant simplicial scheme and on the right, X as a constant cosimplicial scheme. This is an analog of the adjunction for spaces

Hom_{Spc}(A × S, T) ≅ Hom_{Spc}_{Ord}^{op} (A × S, T) ≅ Hom_{Spc}_{Ord}(A, T^{S})

where the first isomorphism is the well-known identity relating maps of bisimplicial sets with maps of the corresponding diagonal simplicial sets. For E ∈ Spc(k) and Y a simplicial object in Sm/k, we have the cosimplicial space E(Y) with n cosimplices E(Y ([n])). For s an element of a finite set S, and a scheme Y ∈ Sm/k, we have the inclusion i_{s} : Y = Y × s → Y × S; the inclusions i_{s} : Y → Y × S, s ∈ S induce the canonical natural map

E(Y × S) → E(Y)^{S}

which is an isomorphism if E is additive: E(Y ⨿Y′) ≅ E(Y)×E(Y′). This isomorphsim extends immediately to finite simplicial sets S ∈ Spc_{f} and additive E.

Examples 8.1.1. (1) For a k-scheme Y, the free path space P_{Y} on Y is Y^{[0,1]}, where [0, 1] is just the 1-simplex Δ[1] := Hom_{Ord}(−, [1]). Explicitly, P_{Y} has n-cosimplices Y^{n+2}, with structure maps as follows: Label the factors in Y^{n+2} from 0 to n + 1. Send δ_{i}^{n} : [n] → [n + 1] to the diagonal

(y_{0}, …, y_{n+1}) ↦ (y_{0}, …, y_{i−1}, y_{i}, y_{i}, y_{i+1}, …, y_{n+1})

and send s_{i}^{n} : [n] → [n − 1] to the projection

(y_{0}, …, y_{n+1}) ↦ (y_{0}, …, y_{i−1}, y_{i+1}, …, y_{n+1}).

The inclusion {0, 1} → [0, 1] gives rise to the projection Y^{[0,1]} → Y^{{0,1}}, i.e. π : P_{Y} → Y ×_{k} Y; we thus have two structures of a cosimplicial Y -scheme on P_{Y}: π_{1} : P_{Y} → Y and π_{2} : P_{Y} → Y, with π_{i} := p_{i} ∘ π.

(2) For a pointed k-scheme (Y, y : Spec k → Y), we have the pointed path space

P_{Y} (y) := P_{Y} ×_{(π_{2},y)} Spec k.

(3) Now let p : Y → Y be a Y -scheme,, y : Spec k → Y a point. We have the cosimplicial homotopy fiber of p over y:

P_{Y/Y} (y) := Y ×_{(p,π_{1})} P_{Y} (y)

We extend this definition to cosimplicial Y -schemes in the evident manner: if Y^{•} → Y is a cosimplicial Y -scheme, we have the bi-cosimplicial Y -scheme P_{Y^{•}/Y} (y); the extension to functors from some small category to cosimplicial Y -schemes is done in the same way. □

Denoting the pointed k-scheme (Y, y) by Y_{∗}, we sometimes write P_{Y_{∗}} for P_{Y} (y) and P_{Y^{•}/Y_{∗}} for P_{Y^{•}/Y} (y). For E ∈ Spt(k), we have the simplicial spectrum E(P_{Y/Y_{∗}}). The pointed path space P_{Y} (y) is contractible in the following sense:

Lemma 8.1.2. Let (Y, y) be a pointed smooth k-scheme, U a smooth k-scheme. Then for E ∈ Spt(k), the projection U ×P_{Y} (y) → U induces a weak equivalence

E(U) → E(U × P_{Y} (y)).

Proof. To prove the lemma, it suffices to show that, for E ∈ Spc(k), the projection U × P_{Y} (y) → U induces a homotopy equivalence

E(U) → E(U × P_{Y} (y)).

We first show that U × P_{Y} (y) → U induces a homotopy equivalance of cosimplicial schemes.

The projection [0, 1] → pt gives the map of cosimplicial schemes s : Y = Y^{pt} → Y^{[0,1]}; composing with the k-point y → Y gives the point s_{y} : Spec k → Y^{[0,1]} and thus the section (y, s_{y}) : Spec k → P_{Y} (y) to the projection P_{Y} (y) → Spec k. This induces the section s_{U} : U → U × P_{Y} (y) to the projection p_{U} : U × P_{Y} (y) → U. We proceed to construct a homotopy between p_{U} ∘ s_{U} and the identity on U × P_{Y} (y); it suffices to construct the homotopy for U = Spec k. For this, let σ : Y^{[0,1]} → Y^{[0,1]} be the map induced by the map of simplicial sets [0, 1] → [0, 1] sending [0, 1] to 1. Then p_{Spec k} ∘ s_{Spec k} : P_{Y} (y) → P_{Y} (y) is the map (id_{Spec k}, σ). Let p_{0}, p_{1} : Y^{[0,1]×[0,1]} → Y^{[0,1]} be the maps induced by the inclusions i_{0}, i_{1} : [0, 1] → [0, 1]×[0, 1], i_{0}(x) = x×0, i_{1}(x) = x×1, and let π : Y → Y^{[0,1]×[0,1]} be the map induced by [0, 1]×[0, 1] → pt. Let h : ([0, 1]×[0, 1], 1×[0, 1]) → ([0, 1], 1) be any map of pairs of simplicial sets which is the identity on [0, 1] × 0 and the map to 1 ∈ [0, 1] on [0, 1] × 1. Then h defines a map

H : Y^{[0,1]} → Y^{[0,1]×[0,1]}

with

p_{0} ∘ H = id_{Y^{[0,1]}}
p_{1} ∘ H = σ
H ∘ s = π.

From these identities, it follows that (H, id_{y}) induces a co-homotopy

H_{y} : P_{Y} (y) → P_{Y} (y)^{[0,1]}

with p_{0} ∘ H_{y} = id, p_{1} ∘ H_{y} = p_{Spec k} ∘ s_{Spec k}. Taking the adjoint of H_{y}, we have the homotopy

h_{y} : P_{Y} (y) × [0, 1] → P_{Y} (y); h_{y} ∘ i_{0} = id, h_{y} ∘ i_{1} = p_{Spec k} ∘ s_{Spec k},

where P_{Y} (y) × [0, 1] and P_{Y} (y) are to be considered as cosimplicial-simplicial schemes, with P_{Y} (y) constant in the simplicial direction. Applying E to id_{U} × h_{y} and composing with the canonical map

E(U × P_{Y} (y) × [0, 1]) → E(U × P_{Y} (y))^{[0,1]}

gives us the co-homotopy

E(id_{U} × h_{y}) : E(U × P_{Y} (y)) → E(U × P_{Y} (y))^{[0,1]}

between the identity and E(p_{U} ∘ s_{U}). Thus E(U) → E(U × P_{Y} (y)) is a homotopy equivalence, as desired. □

8.2. Limit structures. For our purposes, a semi-stable degeneration need not be proper, so even if this is somewhat non-standard terminology, we use the following definition:

Definition 8.2.1. A semi-stable degeneration is a flat morphism p : X → (C, 0), where (C, 0) is a smooth pointed local curve over k, C = Spec O_{C,0}, X is a smooth irreducible k-scheme, p is smooth over C \ 0 and X_{0} := p^{−1}(0) is a reduced strict normal crossing divisor on X. □

For the rest of this section, we fix a semi-stable degeneration X → (C, 0). We denote the open complement of X_{0} in X by X^{0}. We write 𝔾_{m} for the pointed k-scheme (𝔸_{k}^{1} \ {0}, 1). Fix a uniformizing parameter t ∈ O_{C,0}, giving the morphism t : (C, 0) → (𝔸_{k}^{1}, 0), which restricts to t : C \0 → 𝔾_{m}. Let p[t] : X → 𝔸^{1} be the composition t ∘ p, and let p[t]^{0} : X^{0} → 𝔾_{m} be the restriction of p[t]. Composing p[t] with the canonical morphism τ X̂ ; ϵ(X_{0})^{0} → X^{0} yields the map

p̂[t]^{0} : τ X̂ ; ϵ(X_{0})^{0} → 𝔾_{m}.

Let X_{0}^{1}, …, X_{0}^{m} be the irreducible components of X_{0}. Recalling the construction of τ X̂ ; ϵ(X_{0})^{0} as a diagram (see (6.3.1)), let us denote, for I ⊂ {1, …, m}, the co-presheaf ι_{I∗}(τ X̂ ; ϵ(X_{0I})^{0}) by τ X̂ ; ϵ(X_{0})_{I}^{0}. The map p̂[t]^{0} makes τ X̂ ; ϵ(X_{0})^{0} into a diagram of co-presheaves (on X_{0Zar}) of cosimplicial pro-schemes over 𝔾_{m}. We thus have the diagram of cosimplicial co-presheaves on X_{0Zar}:

I ↦ P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}}.

We denote this diagram by

(8.2.1)  lim_{t→0} X_{t}.

Now let E be in Spt(k). For each I ⊂ {1, …, m}, we have the presheaf of bisimplicial spectra on X_{0Zar}, E(P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}}), giving us the functor

I ↦ Ẽ(P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}}).

where ˜ means fibrant model. Taking the homotopy limit over I of the associated diagram of presheaves of total spectra gives us the fibrant presheaf of spectra

E(lim_{t→0} X_{t}) := holim_{I≠∅} TotẼ(P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}}).

Taking the global sections gives us the spectrum E(lim_{t→0} X_{t})(X_{0}), which we denote by lim_{t→0} E(X_{t}).

Remark 8.2.2. Suppose E ∈ Spt(k) is homotopy invariant and satisfies Nisnevich excision. We can form the homotopy limit Ē(lim_{t→0} X_{t}) of the diagram of presheaves

I ↦ E(P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}}).

Since E is quasi-fibrant (see remark 2.3.2) the map

E(P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}}) → Ẽ(P_{τ X̂ ; ϵ(X_{0})_{I}^{0}/𝔾_{m}})

is a pointwise weak equivalence, hence the map Ē(lim_{t→0} X_{t}) → E(lim_{t→0} X_{t}) is a pointwise weak equivalence. In particular, Ē(lim_{t→0} X_{t})(X)_{0}) → lim_{t→0} E(X_{t}) is a weak equivalence. In short, if E is homotopy invariant and satisfies Nisnevich excision, then it is not necessary to take the fibrant model Ẽ in the construction of E(lim_{t→0} X_{t}) or lim_{t→0} E(X_{t}). □

We remind the reader of the presheaves E(X_{0•}) and E(Δ_{X_{0}•}) on X_{0Zar} described in definition 6.2.1.

Proposition 8.2.3. Suppose E is homotopy invariant and satisfies Nisnevich excision. Then

(1) There is a canonical map in HSpt(X_{0Zar}):

E(X_{0•}) ⟶^{γ_{X}} E(lim_{t→0} X_{t}).

(2) If X_{0} is smooth, then E(X_{0•}) = E(X_{0Zar}) and γ_{X} is an isomorphism.

Proof. We have the maps

E(X_{0•}) ⟶^{p_{•}^{∗}} E(Δ_{X_{0•}}^{∗}) ⟵^{ι̂^{∗}} E(τ X̂ ; ϵ(X_{0})).

which by proposition 6.2.2 are Zariski-local weak equivalences. Similarly, we have the diagram of open immersions

ĵ : τ X̂ ; ϵ(X_{0})^{0} → τ X̂ ; ϵ(X_{0})

inducing

ĵ^{∗} : E(τ X̂ ; ϵ(X_{0})) → E(τ X̂ ; ϵ(X_{0})^{0}).

Thus we have the map

p_{0}^{∗} : E(X_{0•}) → E(τ X̂ ; ϵ(X_{0})^{0});

p_{0}^{∗} := ĵ^{∗}(ι̂^{∗})^{−1}p_{•}^{∗}. Similarly, we have the projection

P_{τ X̂ ; ϵ(X_{0})^{0}/𝔾_{m}} → τ X̂ ; ϵ(X_{0})^{0},

giving the map

q^{∗} : E(τ X̂ ; ϵ(X_{0})^{0}) → E(P_{τ X̂ ; ϵ(X_{0})^{0}/𝔾_{m}});

we set γ_{X} := q^{∗} ∘ p^{∗}. For (2), the diagram X_{•} is just the identity copresheaf X_{0Zar}, hence E(X_{0•}) = E(X_{0Zar}). To show γ_{X} is an isomorphism, fix a point x ∈ X_{0}. There is a Zariski neighborhood U of x in X_{0} and a Nisnevich neighborhood X′ → X of U in X which is isomorphic to a Nisnevich neighborhood of U in U × 𝔸^{1}. Thus it suffices to prove the result in the case X = X_{0} × 𝔸^{1}, (C, 0) = (𝔸^{1}, 0) and p = p_{2} : X → 𝔸^{1}. For each smooth k-scheme T, it follows from homotopy invariance and theorem 3.2.1 that the canonical map p : τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0) → X_{0} × 𝔸^{1} induces a weak equivalence

p^{∗} : E(T ×_{k} X_{0Zar} × 𝔸^{1}) → E(T ×_{k} τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0))

The Morel-Voevodsky purity theorem [34, theorem 2.23] plus Nisnevich excision and the homotopy property for E implies that p induces a weak equivalence

p^{∗} : E^{T ×X_{0Zar}×0}(T ×_{k} X_{0Zar} × 𝔸^{1}) → E^{T ×Δ_{X_{0Zar}×0}^{∗}}(T ×_{k} τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0)).

This gives us the map of homotopy fiber sequences

\xymatrix{E^{T ×X_{0Zar}×0}(T ×_{k} X_{0Zar} × 𝔸^{1}) \ar[d] \ar[r]^{p^{∗}} & E^{T ×Δ_{X_{0Zar}×0}^{∗}}(T ×_{k} τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0)) \ar[d] \\ E(T ×_{k} X_{0Zar} × 𝔸^{1}) \ar[d]_{j^{∗}} \ar[r]^{p^{∗}} & E(T ×_{k} τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0)) \ar[d]^{ĵ^{∗}} \\ E(T ×_{k} X_{0Zar} × 𝔾_{m}) \ar[r]^{p^{0∗}} & E(T ×_{k} τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0)^{0})}

with p^{0∗} induced by the restriction of p,

p^{0} : τ{ \atop X_{0}×𝔸^{1}} ; ϵ(X_{0} × 0)^{0} → X_{0} × 𝔾_{m}.

Thus p^{0∗} is a weak equivalence. Applying these term-by-term with respect to the cosimplicial schemes defining the respective path spaces, we have the weak equivalence (assuming X = X_{0} × 𝔸^{1})

E(U × P_{𝔾_{m}}) → E(P_{τ X̂ ; ϵ(X_{0})^{0}/𝔾_{m}})(U).

Thus we need only show that the projection U × P_{𝔾_{m}} → U induces a weak equivalence

E(U) → E(U × P_{𝔾_{m}})

for all smooth k-schemes U. This is lemma 8.1.2 □

8.3. Comparison. We conclude this section by connecting our construction with the specialization functor sp for the specialization structure Υ defined by Ayoub [3, chapter 3]. Let E ∈ Spt_{mot}(Sm/k) be fibrant, let p : X → (C, 0) be a semi-stable degeneration and choose a parameter t ∈ O_{C,0}. In this setting, Ayoub’s functor sp applied to some E ∈ Spt(Sm/X^{0}) is defined as follows: First form the presheaf E(P_{−/𝔾_{m}}) on Sm/X^{0} by taking the total spectrum

E(P_{−/𝔾_{m}})(X′ → X^{0}) := Tot(E(P_{X′/𝔾_{m}})).

where we use the composition X′ → X^{0} ⟶^{t} 𝔾_{m} as structure morphism. Then sp(E) ∈ SH_{𝔸^{1}}(Sm/X_{0}) is represented by the presheaf

sp(E) := i^{∗} (j_{∗} (E(P_{−/𝔾_{m}})^{f})^{c}) .

Similarly, we have the simplicial presheaf on Sm/X_{0} with n-simplices

sp(E)_{n} := i^{∗} (j_{∗} (E(P_{−/𝔾_{m}}[n])^{f})^{c}) .

Let Tot(sp(E)_{∗}) denote the presheaf formed by taking the total spectrum of n ↦ sp(E)_{n}.

Lemma 8.3.1. Suppose E is fibrant. Then there is a natural isomorphism in SH_{𝔸^{1}}(X_{0})

Tot(sp(E)_{∗}) ≅ sp(E)

Proof. Since E(P_{−/𝔾_{m}}[n])^{f} is fibrant, the presheaf E(P_{−/𝔾_{m}}[n])^{f} on X^{0} satisfies Nisnevich excision and is 𝔸^{1} homotopy invariant. Thus the same holds for the total spectrum of the simplicial spectrum n ↦ E(P_{−/𝔾_{m}}[n])^{f}, hence

Tot(n ↦ E(P_{−/𝔾_{m}}[n])^{f}) → (Tot(n ↦ E(P_{−/𝔾_{m}}[n])^{f})^{f}

is a pointwise weak equivalence in Spt_{𝔸^{1}}(Sm/X^{0}), and thus we still have a pointwise weak equivalence after applying j_{∗}. Similarly, the evident map

(Tot(n ↦ E(P_{−/𝔾_{m}}[n]))^{f} → (Tot(n ↦ E(P_{−/𝔾_{m}}[n])^{f})^{f}

is a pointwise weak equivalence. Taking cofibrant models and applying i^{∗} gives the isomorphism in SH_{𝔸^{1}}(X_{0})

sp(E) ≅ i^{∗} ((Tot(n ↦ E(P_{−/𝔾_{m}}[n])^{f})^{c}) .

On the other hand, taking the total complex commutes with taking the cofibrant model, and with the functor i^{∗}, so we have the isomorphism in Spt_{mot}(Sm/X_{0})

sp(E) = i^{∗} ((Tot(n ↦ E(P_{−/𝔾_{m}}[n]))^{f})^{c})

≅ Tot (n ↦ i^{∗} ((E(P_{−/𝔾_{m}}[n])^{f})^{c})) = Tot(sp_{∗}(E)).

□

Using the diagram of lemma 7.2.3 for the n-cosimplices τ X̂ ; ϵ(X_{0})^{0} × 𝔾_{m}^{n} of P_{τ X̂ ; ϵ(X_{0})^{0}/𝔾_{m}}, and taking the total spectrum, we arrive at a natural map

E(lim_{t→0} X_{t}) → Tot(sp(E)_{∗})_{Zar}

in HSpt(X_{0Zar}); combining this with lemma 8.3.1 gives us the comparison map

θ_{E} : E(lim_{t→0} X_{t}) → sp(E)_{Zar}

in HSpt(X_{0Zar}).

Proposition 8.3.2. The map θ_{E} : E(lim_{t→0} X_{t}) → sp(E)_{Zar} is an isomorphism in HSpt(X_{0Zar}).

Proof. By theorem 7.3.6, the map

θ_{E}(n) : E(P_{τ X̂ ; ϵ(X_{0})^{0}/𝔾_{m}}([n])) → sp_{n}(E)

is an isomorphism in HSpt(X_{0Zar}) for each n, thus the map θ_{E} on the total spectra is also an isomorphism in HSpt(X_{0Zar}). □

9. The monodromy sequence

In this section, we construct the monodromy sequence for the limit object E(lim_{t→0} X_{t}) (see corollary 9.3.5). As pointed out to us by Ayoub, one needs to restrict E quite a bit. We give here a theory valid for presheaves of complexes of ℚ-vector space on Sm/k which are homotopy invariant and satisfy Nisnevich excision, and satisfy an additional “alternating” property (definition 9.2.2). Ayoub [3, Chap. 3] constructs the monodromy sequence in a more general setting; our construction is based on his ideas applied to our tubular neighborhood construction. In particular, our mondromy sequence agrees with the monodromy sequence of loc. cite.

9.1. Presheaves of complexes. For a noetherian ring R, we let C_{R} denote the category of (unbounded) homological complexes of R-modules, C_{R≥0} the full subcategory of C_{R} consisting of complexes which are zero in strictly negative degrees. By the Dold-Kan equivalence, we may identify C_{R≥0} with the category of simplicial R-modules Spc_{R}. The forgetful functor Spc_{R} → Spc_{∗} allows us to use the standard model structure on Spc_{∗} to induce a model structure on Spc_{R}, i.e., cofibrations are degreewise monomorphisms, weak equivalences are homotopy equivalences on the geometric realization and fibrations are maps with the RLP for trivial cofibrations. This induces a model structure on C_{R≥0} with weak equivalence the quasi-isomorphisms; the suspension functor is the usual (homological) shift operator: ΣC := C[1], C[1]_{n} := C_{n−1}, d_{C[1],n} = −d_{C,n−1}. This model structure is extended to C_{R} by identifying C_{R} with the category of “spectra in C_{R≥0}”, i.e., sequences (C^{0}, C^{1}, …) with bonding maps ϵ_{n} : C^{n}[1] → C^{n+1}. Following Hovey [17], the model structure on Spt induces a model structure on spectra of simplicial R-modules, and thus a model structure on C_{R}, with weak equivalences the quasi-isomorphisms. In particular, the homotopy category HC_{R} is just the unbounded derived category D_{R}. Similarly, for a category C, the model structure for the presheaf category Spt(C) gives a model structure for presheaves of complexes on C, C_{R}(C) with weak equivalences the pointwise quasi-isomorphisms, and homotopy category the derived category D_{R}(C). We may introduce a topology (e.g., the Zariski or Nisnevich topology), giving the model categories C_{R}(X_{Zar}), C_{R}(Sm/B_{Zar}), C_{R}(X_{Nis}), C_{R}(Sm/B_{Nis}). These have homotopy categories equivalent to the derived categories (on the small or big sites) D_{R}(X_{Zar}), D_{R}(Sm/S_{Zar}), D_{R}(X_{Nis}), D_{R}(Sm/S_{Nis}), respectively. Finally, we may consider the 𝔸^{1}- localization, giving the Nisnevich-local 𝔸^{1}-model structure C_{R,𝔸^{1}}(Sm/B_{Nis}) with homotopy category D_{R𝔸^{1}}(B). Let I be a small category, F : I → C_{R} a functor. Since we can consider F as a spectrum-valued functor by the various equivalences described above, we may form the complex holim_{I} F. Explicitly, this is the following complex: One first forms the cosimplicial complex \underline{holim}_{I}F with n-cosimplices

\underline{holim}_{I}F^{n} := ∏_{σ=(σ_{0}→...→σ_{n})∈N (I)_{n}} F (σ_{n}).

For g : [m] → [n], with g(m) = m′ ≤ n, the σ-component of the map \underline{holim}_{I}F^{n}(g) sends ∏ x_{τ} to F (σ_{m′} → σ_{n})(x_{g^{∗}(σ)}). The complex holim_{I} F is then the total complex of the double complex n ↦ \underline{holim}_{I}F^{n}, with second differential the alternating sum of the coface maps. This construction being functorial and preserving quasi-isomorphisms, it passes to the derived category D_{R}(C). If I is a finite category, the construction commutes with filtered colimits, hence passes to the Zariski- and Nisnevich-local derived categories, as well as the 𝔸^{1}-local versions.

Remarks 9.1.1. (1) For a set S, let RS denote the free R-module on S. Sending a pointed space (S, ∗) to the simplicial R-module RS, with RS(n) := RS_{n}/R{∗} defines the R-localization functor Spc_{∗} → Spc_{R}. This extends to the spectrum categories, and gives us the exact R-localization functor on homotopy category ⊗R : SH → D_{R}. The R-localization functor ⊗R extends to all the model categories we have been considering, in particular, we have the R-localization

⊗R : SH_{𝔸^{1}}(B) → D_{R,𝔸^{1}}(B),

For R = ℚ, we can also take the ℚ-localization of SH by performing a Bousfield localization, i.e., define Z ∈ Spt to be ℚ-local if π_{n}(Z) is a ℚ-vector space for all n, and E → F a ℚ weak equivalence if Hom_{Spt}(F, Z) → Hom_{Spt}(E, Z) is an isomorphism for all ℚ-local Z. Inverting the ℚ-weak equivalences defines the ℚ-local homotopy category SH_{ℚ}, and ⊗ℚ : SH → D_{ℚ} identifies SH_{ℚ} with D_{ℚ}. This passes to the other homotopy categories we have defined, in particular, ⊗ℚ : SH_{𝔸^{1}}(B) → D_{ℚ,𝔸^{1}}(B) identifies SH_{𝔸^{1}}(B)_{ℚ} with D_{ℚ,𝔸^{1}}(B).

(2) D_{ℚ,𝔸^{1}}(k) is not the same as the (ℚ-localized) big category of motives over k, DM(k)_{ℚ}; the ℚ-localization does not give rise to transfers. □

9.2. The log complex. Let sgn : S_{n} → {±1} be the sign representation of the symmetric group S_{n}. Consider a presheaf of ℚ-vector spaces E on Sm/k. For X, Y ∈ Sm/k, let alt_{n} : E(Y × X^{n}) → E(Y × X^{n}) be the alternating projector

alt_{n} = \frac{1}{n!} ∑_{σ∈S_{n}} sgn(σ)(id_{Y} × σ)^{∗},

with σ operating on X^{n} by permuting the factors. Let E(Y × X^{n})^{alt} ⊂ E(Y × X^{n}) be the image of alt_{n} and E(Y, X^{n})_{⊥}^{alt} the kernel. We extend these constructions to presheaves of complexes E by operating degreewise. If (X, ∗) is a pointed k-scheme, we have the inclusions i_{j} : Y ×X^{n−1} → Y ×X^{n} inserting the point ∗ in the jth factor. For E a presheaf of ℚ-vector spaces, we let E(Y ∧ X^{∧n}) be the intersection of the kernels of the restriction maps

(id_{Y} × i_{j})^{∗} : E(Y × X^{n}) → E(Y × X^{n−1}).

Letting p_{j} : X^{n} → X^{n−1} be the projection omitting the jth factor, the composition (id − p_{n}^{∗}i_{n}^{∗}) ∘ … ∘ (id − p_{1}^{∗}i_{1}^{∗}) gives a splitting

π_{n} : E(Y × X^{n}) → E(Y ∧ X^{∧n})

to the inclusion E(Y ∧ X^{∧n}) → E(Y × X^{n}). Clearly S_{n} acts on E(Y ∧ X^{∧n}) through its action on X^{n}; we let E(Y, X^{∧n})^{alt} and E(Y ∧ X^{∧n})_{⊥}^{alt} be the image and kernel of alt_{n} on E(Y ∧ X^{∧n}). Let f : X → 𝔾_{m} be a morphism, E a presheaf of ℚ-vector spaces on Sm/k. Let f_{n} : X × 𝔾_{m}^{n} → X × 𝔾_{m}^{n+1} be the morphism

f_{n}(x, t_{1}, …, t_{n}) := (x, f(x), t_{1}, …, t_{n}).

Denote the map alt_{n} ∘ π_{n} ∘ f_{n}^{∗} : E(X ∧ 𝔾_{m}^{∧n+1})^{alt} → E(X ∧ 𝔾_{m}^{∧n})^{alt} by

∪f : E(X ∧ 𝔾_{m}^{∧n+1})^{alt} → E(X ∧ 𝔾_{m}^{∧n})^{alt}

One checks that

Lemma 9.2.1. (∪f)^{2} = 0.

Proof. We work in the ℚ-linear category ℚSm/k, with the same objects as Sm/k, disjoint union being direct sum, and, for X, Y connected, Hom_{ℚSm/k}(X, Y) is the ℚ-vector space freely generated by the set Hom_{Sm/k}(X, Y). Product over k makes ℚSm/k a tensor category. The map ∪f is gotten by applying E to the map ∪f^{∨} : X ×𝔾_{m}^{n} → X ×𝔾_{m}^{n+1} in ℚSm/k:

(x, t_{1}, …, t_{n−1}) ↦ alt[((x, f(x)) − (x, 1)) ⊗ t_{1} − 1 ⊗ … ⊗ t_{n} − 1)]

and restricting to E(X ∧ 𝔾_{m}^{∧n})^{alt}. But (∪f^{∨})^{2} is

(x, t_{1}, …, t_{n}) ↦

alt[((x, f(x), f(x)) − (x, 1, f(x)) − (x, f(x), 1) + (x, 1, 1)) ⊗ … ⊗ t_{n} − 1)

which is evidently the zero map. □

Form the complex E(log_{f}) by

E(log_{f})_{n} := E(X ∧ 𝔾_{m}^{∧n}))^{alt}

with differential ∪f. Since E(log_{f})_{0} = E(X), we have the canonical map ι_{X} : E(X) → E(log_{f}). We extend this definition to an I-diagram of schemes over 𝔾_{m}, f^{•} : X^{•} → 𝔾_{m} (with the X^{n} ∈ Sm/k) by

E(log_{f^{•}}) := holim_{i∈I} E(log_{f^{i}});

similarly, we extend to E a presheaf of complexes on Sm/k by taking the total complex of the double complex n ↦ E_{n}(log_{f^{•}}). The map ι_{X} extends to

ι_{X^{•}} : E(X^{•}) → E(log_{f^{•}}),

where

E(X^{•}) := holim_{i∈I} E(X^{i}).

We consider as well a truncation of E(log_{f}). Recall that the stupid truncation σ_{≥n}C of a homological complex C is the quotient complex of C with

σ_{≥n}C_{m} := \begin{cases}C_{m} & for m ≥ n \\ 0 & for m < n. \end{cases}

For E a presheaf of abelian groups and f : X → 𝔾_{m} a morphism in Sm/k, set

E(σ_{≥1} log_{f}) := σ_{≥1}E(log_{f}).

We have the quotient map N : E(log_{f}) → E(σ_{≥1} log_{f}), natural in f and E. We extend to I-diagrams f^{•} : X^{•} → 𝔾_{m} and to presheaves of complexes as for E(log_{f}). The quotient map N defined above extends to the natural map

N : E(log_{f^{•}}) → E(σ_{≥1} log_{f^{•}}).

for f^{•} : X^{•} → 𝔾_{m} an I-diagram of morphisms in Sm/k, and E ∈ C_{ℚ}(Sm/k). Finally, for E ∈ C_{ℚ}(Sm/k), define E(−1) to be the presheaf of complexes

E(−1)(X) := E(X ∧ 𝔾_{m})[1] := ker (E(X × 𝔾_{m}) ⟶^{i_{1}^{∗}} E(X)) [1].

Definition 9.2.2. Let E be be in C_{ℚ}(Sm/k). Call E alternating if for every X ∈ Sm/k and every n ≥ 0, the alternating projection

alt_{n} : E(X ∧ 𝔾_{m}^{∧n}) → E(X ∧ 𝔾_{m}^{∧n})^{alt}

is a quasi-isomorphism. □

Remarks 9.2.3. (1) Clearly, E is alternating if and only if S_{n} acts via the sign representation on H_{p}E(X ∧ 𝔾_{m}^{∧n}) for all X, n and p.

(2) Fix integers 1 ≤ i ≤ n. We have the split injection ι_{i,i+1} : E(X ∧ 𝔾_{m}^{∧n}) → E((X × 𝔾_{m}^{n−2}) ∧ 𝔾_{m}^{∧2}) by shuffling the i, i + 1 coordinates to position n − 1, n. In particular, we have the injection

H_{p}(ι_{i,i+1}) : H_{p}E(X ∧ 𝔾_{m}^{∧n}) → H_{p}E((X × 𝔾_{m}^{n−2}) ∧ 𝔾_{m}^{∧2}).

Since S_{n} is generated by simple transpositions, this shows that E is alternating if and only if the exchange of factors in 𝔾_{m} ∧ 𝔾_{m} acts by -1 on H_{p}E(X ∧ 𝔾_{m} ∧ 𝔾_{m}) for all X and p.

(3) Suppose that E ∈ C_{ℚ}(Sm/k) is homotopy invariant and satisfies Nisnevich excision. Consider ℙ^{1} as pointed by ∞. Then E(X ∧ ℙ^{1}) is quasi-isomorphic to the suspension E(X ∧ 𝔾_{m})[−1], hence E is alternating if and only if the exchange of factors in ℙ^{1} ∧ ℙ^{1} induces the identity on H_{p}E(X ∧ ℙ^{1} ∧ ℙ^{1}) for all X and p. The homotopy invariance and Nisnevich excision properties of E give a natural quasi-isomorphism of E(X ∧ ℙ^{1} ∧ ℙ^{1}) with E(X ∧ (𝔸^{2}/𝔸^{2} \ {0})), with the exchange of factors in ℙ^{1} ∧ ℙ^{1} going over to the linear transformation (x, y) ↦ (y, x). If the characterstic of k is different from 2, this transformation is conjugate to (x, y) ↦ (−x, y). Thus E is alternating if and only if the map [−1] : ℙ^{1} → ℙ^{1}, [−1](x_{0}, x_{1}) = (x_{0}, −x_{1}), acts by the identity on H_{p}E(X ∧ ℙ^{1}) for all X and p.

(4) Call E oriented if E is an associative graded-commutative ring:

μ : E ⊗_{ℚ} E → E

and (roughly speaking) E admits a natural Chern class transformation

c_{1} : Pic → H^{2}E

satisfying the projective bundle formula: For E → X a rank r vector bundle with associated projective space bundle ℙ(E) → X and tautological line bundle O(1), H^{∗}E(ℙ(E)) is a free H^{∗}E(X)-module with basis 1, ξ, …, ξ^{r−1}, where ξ = c_{1}(O(1)) ∈ H^{2}E(ℙ(E)). We do not assume that c_{1} is a group homomorphism. The projective bundle formula and the fact that [−1]^{∗}O_{ℙ^{1}}(1) ≅ O_{ℙ^{1}}(1) implies that an oriented E is alternating. In particular, rational motivic cohomology, ℚ_{ℓ}(∗) étale cohomlogy, ℚ-singular cohomology (with respect to a chosen embedding k → ℂ) and rational algebraic cobordism MGL_{ℚ}^{∗∗} are all alternating. On the other hand, rational motivic co-homotopy is alternating if −1 is a square in k, but is not alternating for k = ℝ. This is pointed out in [31]: if −1 = i^{2}, [−1] is represented by the 2 × 2 matrix with diagonal entries i and −i. As this is a product of elementary matrices, one has an 𝔸^{1}-homotopy connecting [−1] and id. To see the non-triviality of [−1] for k = ℝ, let [X, Y] denote the set of morphisms X → Y in H_{𝔸^{1}}Spc_{∗}(k). Morel defines a map (of sets) [ℙ^{1}, ℙ^{1}] → ϕ(k), where ϕ(k) is the set of isomorphism classes of quadratic forms over k, and notes that the map [u],

[u](x_{0}, x_{1}) := (x_{0}, ux_{1}),

goes to the class of the form ux^{2}. This map extends to a ring homomorphism

Hom_{SH_{S^{1}} (k)}(ℙ^{1}, ℙ^{1}) → GW(k),

where GW(k) is the Grothendieck-Witt ring (see also [32, Lemma 3.2.4] for details). Identifying GW(ℝ) with ℤ × ℤ by rank and signature, we see that [−1] goes to the non-torsion element (1, −1). The example of motivic (co)homotopy is in fact universal for this phenomenon, so if [−1] vanishes in [ℙ^{1}, ℙ^{1}], then every E ∈ C_{ℚ}(Sm/k) satisfying homotopy invariance and Nisnevich excision is alternating. We are grateful to F. Morel for explaining the computation of the transposition action on ℙ^{1}∧ℙ^{1} in terms of quadratic forms and the Grothendieck-Witt group.

(4) Looking at the 𝔸^{1}-stable homotopy category of T -spectra over k, SH(k), one can decompose the ℚ-linearization SH(k)_{ℚ} into the symmetric part SH(k)_{+} and alternating part SH(k)_{−} with respect to the exchange of factors on 𝔾_{m} ∧ 𝔾_{m}. Morel [33] has announced a result stating that SH(k)_{−} is in general equivalent to Voevodsky’s big motivic category DM(k)_{ℚ}, and that SH(k)_{+} is zero if -1 is a sum of squares. This suggests that the alternating part SH_{S^{1}}(k) of the category of rational S^{1}-spectra SH_{S^{1}}(k)_{ℚ} is closely related to the big category of effective motives (with ℚ-coefficients) DM^{eff}(k)_{ℚ}. □

Proposition 9.2.4. Let E be in C_{ℚ}(Sm/k), f : X → 𝔾_{m} an I-diagram of morphisms in Sm/k.

(1) The sequence

E(X) ⟶^{ι_{X}} E(log_{f}) ⟶^{N} E(σ_{≥1} log_{f})

identifies E(σ_{≥1} log_{f}) with the quotient complex E(log_{f})/E(X).

(2) Suppose E is alternating. Then there is a natural quasi-isomorphism alt : E(−1)(log_{f}) → E(σ_{≥1} log_{f}).

Proof. It suffices to prove (1) for E a presheaf of ℚ-vector spaces, and f : X → 𝔾_{m} a morphism in Sm/k, where the assertion is obvious. Similarly, it suffices to construct a natural map θ_{E,X} : E(−1)(log_{f}) → E(σ_{≥1} log_{f}) for E a presheaf of ℚ-vector spaces, extend as above to a map in general, and show that θ_{E,X} is a quasi-isomorphism for E ∈ C_{ℚ}(Sm/k) alternating and f : X → 𝔾_{m} a morphism in Sm/k. In fact, for E a presheaf of ℚ-vector spaces and n ≥ 1,

E(−1)(log_{f})_{n} = ker[(id_{X} × i)^{∗} : E(X × 𝔾_{m}, 𝔾_{m}^{∧n−1})^{alt} → E(X, 𝔾_{m}^{∧n−1})^{alt}]

so E(−1)(log_{f})_{n} is a subspace of E(X, 𝔾_{m}^{∧n}); thus alt_{n} defines a map E(−1)(log_{f})_{n} → E(log_{f})_{n}. One easily checks that this defines a map of complexes

alt_{∗} : E(−1)(log_{f}) → E(σ_{≥1} log_{f}),

as desired. Now suppose that E is alternating, i.e., that

(a)  E(X ∧ 𝔾_{m}^{∧n})^{alt} → E(X ∧ 𝔾_{m}^{∧n})

is a quasi-isomorphism for all n and X. This implies that the maps

E(X × 𝔾_{m}, 𝔾_{m}^{∧n−1})^{alt} → E(X × 𝔾_{m}, 𝔾_{m}^{∧n−1})
E(X ∧ 𝔾_{m}^{∧n−1})^{alt} → E(X ∧ 𝔾_{m}^{∧n−1})

are quasi-isomorphisms, hence

(b) id_{X∧𝔾_{m}} × alt_{n−1} : E((X ∧ 𝔾_{m}) ∧ 𝔾_{m}^{∧n−1}) → E((X ∧ 𝔾_{m}) ∧ 𝔾_{m}^{∧n−1})^{alt}

is a quasi-isomorphism. Since E((X ∧ 𝔾_{m}) ∧ 𝔾_{m}^{∧n−1}) = E(X ∧ 𝔾_{m}^{∧n}), (a) and (b) imply that

id_{X} × alt_{n} : E((X ∧ 𝔾_{m}) ∧ 𝔾_{m}^{∧n−1})^{alt} → E(X ∧ 𝔾_{m}^{∧n})^{alt}

is a quasi-isomorphism. As alt_{∗} : E(−1)(log_{f}) → E(σ_{≥1} log_{f}) is the map on the total complex of the double complexes

n ↦ id_{X} × alt_{n} : E((X ∧ 𝔾_{m}) ∧ 𝔾_{m}^{∧n−1})^{alt} → E(X ∧ 𝔾_{m}^{∧n})^{alt}

we see that alt_{∗} is a quasi-isomorphism. □

9.3. The log complex and path spaces. Let f : X → 𝔾_{m} be a morphism in Sm/k arising from a semi-stable degeneration X → (C, 0) and choice of parameter in O_{C,0}. The monodromy sequence for E(lim_{t→0} X_{t}) arises from the sequence of Proposition 9.2.4 by comparing the path space E(P_{X/𝔾_{m}}) with E(log_{f}). We use the Dold-Kan correspondence to rewrite E(P_{X/𝔾_{m}}) as a complex, namely: take for each p the associated complex E_{p}(P_{X/𝔾_{m}}^{∗}) of the simplicial abelian group n ↦ E_{p}(P_{X/𝔾_{m}}^{n}), with differential the alternating sum of the face maps, and then take the total complex of the double complex

p ↦ E_{p}(P_{X/𝔾_{m}}^{∗}).

We write this complex as E(P_{X/𝔾_{m}}). We also have the normalized subcomplex NE(P_{X/𝔾_{m}}) of E(P_{X/𝔾_{m}}), quasiisomorphic to E(P_{X/𝔾_{m}}) via the inclusion. Recall that, for a simplicial abelian group n ↦ A_{n}, the normalized complex NA_{∗} has

NA_{n} := ∩_{i=1}^{n} ker d_{i} : A_{n} → A_{n−1}

with differential d_{0} : NA_{n} → NA_{n−1}. We define NE(P_{X/𝔾_{m}}) by first taking the normalized subcomplex NE_{p}(P_{X/𝔾_{m}}) of E_{p}(P_{X/𝔾_{m}}^{∗}) for each p, and then forming the total complex of the double complex p ↦ NE_{p}(P_{X/𝔾_{m}}). In particular, we have the inclusion of double complexes

NE_{∗}(P_{X/𝔾_{m}}^{∗}) ⊂ E_{∗}(P_{X/𝔾_{m}}^{∗});

which gives for each n the inclusion of single complexes

NE_{∗}(P_{X/𝔾_{m}}^{n}) ⊂ E_{∗}(P_{X/𝔾_{m}}^{n});

Recalling that P_{X/𝔾_{m}}^{n} = X × 𝔾_{m}^{n}, we thus have for each n the inclusion of complexes

NE_{∗}(P_{X/𝔾_{m}}^{n}) ⊂ E_{∗}(X × 𝔾_{m}^{n}),

We may therefore apply the projections π_{n} : E_{∗}(X × 𝔾_{m}^{n}) → E_{∗}(X ∧ 𝔾_{m}^{∧n}) and alt_{n}, giving the map

alt_{n} ∘ π_{n} : NE_{∗}(P_{X/𝔾_{m}}^{n}) → E_{∗}(X ∧ 𝔾_{m}^{∧n})^{alt}.

Lemma 9.3.1. Suppose that E is alternating. Then

alt_{n} ∘ π_{n} : NE_{∗}(P_{X/𝔾_{m}}^{n}) → E_{∗}(X ∧ 𝔾_{m}^{∧n})^{alt}

is a quasi-isomorphism.

Proof. The map p_{1}^{∗} : E(X) → E(X × 𝔾_{m}) splits i_{1}^{∗} : E(X × 𝔾_{m}) → E(X), so we have the natural splitting

E(X × 𝔾_{m}) = E(X) ⊕ E(X ∧ 𝔾_{m}).

Extending this to E(X ×𝔾_{m}^{n}) by using the maps i_{j}^{∗} and p_{j}^{∗}, we have the natural splitting

(9.3.1)  E(X × 𝔾_{m}^{n}) = ⊕_{m=0}^{n} ⊕\begin{matrix}I⊂\{1,...,n\} \\ |I|=m \end{matrix} E(X ∧ 𝔾_{m}^{∧I}).

To explain the notation: For I ⊂ {1, …, n}, E(X ∧ 𝔾_{m}^{∧I}) = E(X ∧ 𝔾_{m}^{∧|I|}), included in E(X × 𝔾_{m}^{n}) by the composition

E(X ∧ 𝔾_{m}^{∧|I|}) ⊂ E(X × 𝔾_{m}^{|I|}) ⟶^{(id_{X}×p_{I})^{∗}} E(X × 𝔾_{m}^{n})

where p_{I} : 𝔾_{m}^{n} → 𝔾_{m}^{|I|} is the projection on the factors i_{1}, …, i_{m} if I = {i_{1}, …, i_{m}} with i_{1} < … < i_{m}. The action of S_{n} on E(X × 𝔾_{m}^{n}) preserves this decomposition, with σ ∈ S_{n} mapping E(X ∧ 𝔾_{m}^{∧I}) to E(X ∧ 𝔾_{m}^{∧σ^{−1}(I)}) in the evident manner. Now, for a simplicial abelian group A, the inclusion NA_{n} → A_{n} is split by universal expressions in the face and degeneracy maps. If n ↦ C_{∗n} is a simplicial complex, we can form the complex of normalized subgroups (with respect to the simplicial variable) N_{∗_{2}}(C_{∗_{1},∗_{2}}) and take the homology H_{p}(N_{∗_{2}}(C_{∗_{1},∗_{2}}), d_{1}), or we can form the simplicial abelian group n ↦ H_{p}(C_{∗n}) and take the normalized subgroup N_{∗_{2}}H_{p}(C_{∗_{1},∗_{2}}, d_{1}) ⊂ H_{p}(C_{∗_{1},n}, d_{1}). Using the universal spitting mentioned above, we see that the two are the same:

H_{p}(N_{∗_{2}}(C_{∗_{1},∗_{2}}), d_{1}) = N_{∗_{2}}H_{p}(C_{∗_{1},∗_{2}}, d_{1})

Since S_{n} acts by the sign representation on H_{p}E(X ∧ 𝔾_{m}^{∧n}), it follows that, for 1 ≤ j < n, the diagonal map

δ_{j} : 𝔾_{m}^{n−1} → 𝔾_{m}^{n}
(t_{1}, …, t_{n−1}) ↦ (t_{1}, … t_{j}, t_{j}, t_{j+1}, … t_{n−1})

induces the zero map on H_{p}E(X ∧ 𝔾_{m}^{∧n}). Similarly, the inclusion

i_{n} : 𝔾_{m}^{n−1} → 𝔾_{m}^{n}
(t_{1}, …, t_{n−1}) ↦ (t_{1}, … t_{n−1}, 1)

is the zero map on H_{p}E(X ∧ 𝔾_{m}^{I}) if n ∈ I. From this, it is not hard to see that

NH_{p}E_{∗}(NE_{∗}(P_{X/𝔾_{m}}^{n}) = H_{p}E_{∗}(X ∧ 𝔾_{m}^{∧n}),

with respect to the decomposition of E_{∗}(P_{X/𝔾_{m}}^{n}) = E_{∗}(X × 𝔾_{m}^{n}) given by (9.3.1). Indeed,

ker(H_{p}(d_{n})) = ker(i_{n}^{∗} : H_{p}E_{∗}(X × 𝔾_{m}^{n}) → H_{p}E_{∗}(X × 𝔾_{m}^{n−1})

= ⊕\begin{matrix}I⊂\{1,...,n\} \\ n∈I \end{matrix}H_{p}E_{∗}(X ∧ 𝔾_{m}^{∧I})

It is then easy to show by descending induction on i that

∩_{j=i}^{n} ker H_{p}(d_{j}) = ⊕\begin{matrix}I⊂\{1,...,n\} \\ \{i,...,n\}⊂I \end{matrix}H_{p}E_{∗}(X ∧ 𝔾_{m}^{∧I})

from which our claim follows taking i = 1. Thus the projection

p_{n} : NE_{∗}(P_{X/𝔾_{m}}^{n}) → E_{∗}(X ∧ 𝔾_{m}^{∧n})

is a quasi-isomorphism for each n. As E is alternating, the alternating projection

alt_{n} : E_{∗}(X ∧ 𝔾_{m}^{∧n}) → E_{∗}(X ∧ 𝔾_{m}^{∧n})^{alt} is a quasi-isomorphism as well, completing the proof. □

Lemma 9.3.2. Let E be in C_{ℚ}(Sm/k). Let δ_{0} : E(X × 𝔾_{m}^{n}) → E(X × 𝔾_{m}^{n−1}) be the map [(id_{X}, f), id_{𝔾_{m}^{n−1}}]^{∗}. Then the diagram

\xymatrix{E(X × 𝔾_{m}^{n}) \ar[r]^{δ_{0}} \ar[d]_{alt_{n}∘π_{n}} & E(X × 𝔾_{m}^{n−1}) \ar[d]^{alt_{n−1}∘π_{n−1}} \\ E(X ∧ 𝔾_{m}^{∧n})^{alt} \ar[r]_{∪f} & E(X ∧ 𝔾_{m}^{∧n−1})^{alt}}

commutes.

Proof. This follows directly from the definition of ∪f and the fact that alt_{n} ∘π_{n} is the identity on E(X ∧ 𝔾_{m}^{∧n})^{alt}. □

Proposition 9.3.3. Let E ∈ C_{ℚ}(Sm/k) be alternating. Then the maps alt_{n} ∘ π_{n} : NE_{∗}(P_{X/𝔾_{m}}^{n}) → E_{∗}(X × 𝔾_{m}^{∧n})^{alt} define a quasi-isomorphism of total complexes

alt ∘ π : NE(P_{X/𝔾_{m}}) → E(log_{f})

Proof. That the maps alt_{n} ∘ π_{n} : NE_{∗}(P_{X/𝔾_{m}}^{n}) → E_{∗}(X ∧ 𝔾_{m}^{∧n})^{alt} define a map of total complex NE(P_{X/𝔾_{m}}) → E(log_{f}) follows from Lemma 9.3.2 and the fact that, for each n, the differential d_{0} on NE(P_{X/𝔾_{m}}^{n}) is the restriction of δ_{0} : E(X × 𝔾_{m}^{n}) → E(X × 𝔾_{m}^{n−1}). Lemma 9.3.1 implies that alt ∘ π is a quasi-isomorphism. □

We collect our results in

Theorem 9.3.4. Let E ∈ C_{ℚ}(Sm/k) be alternating, f^{•} : X^{•} → 𝔾_{m} an Idiagram of morphisms in Sm/k. Consider the diagram

[figure: E(P_{X/𝔾_{m}}); E(−1)(P_{X/𝔾_{m}}); E(X^{•}); ι_{X^{•}}; NE(P_{X/𝔾_{m}}); alt∘π; i; NE(−1)(P_{X/𝔾_{m}}); alt∘π; i; E(−1)(log_{f^{•}}); 0; E(X^{•}); ι_{X^{•}}; E(log_{f^{•}}); N; E(σ_{≥1} log_{f^{•}}); alt; 0]

Here the maps i are the canonical inclusions and the maps ι_{X^{•}} are the canonical maps given by the identities E_{n}(P_{X^{i}/𝔾_{m}})_{0} = E_{n}(log_{f^{i}})_{0} = E_{n}(X^{i}). Then

(1) The diagram commutes and is natural in E and f^{•}.

(2) All the maps in the diagram are maps of complexes.

(3) All the vertical maps are quasi-isomorphisms

(4) The bottom sequence is termwise exact.

Proof. The first point follows by construction, the remaining assertions follow from the Dold-Kan correspondence, Proposition 9.3.3 and Proposition 9.2.4.

□

Corollary 9.3.5 (Monodromy sequence). Let W ∈ C_{ℚ}(Sm/k) be alternating, p : X → (C, 0) a semi-stable degeneration, t ∈ O_{C,0} a uniformizing parameter. Then there is a distinguished triangle in D(X_{0Zar})

E(τ X̂ ; ϵ(X_{0})^{0}) → E(lim_{t→0} X_{t}) ⟶^{N} E(−1)(lim_{t→0} X_{t}) → E(τ X̂ ; ϵ(X_{0})^{0})[1],

natural in (p, t) and in E.

Proof. The commutative diagram of Theorem 9.3.4 being natural in the choice of I-diagram and in E, one can extend the diagram directly to the case of a co-presheaf of cosimplicial I-diagrams

U ↦ f^{•}(U) : X^{•}(U) → 𝔾_{m}.

If we take I to be finite, we can extend further to a co-presheaf of cosimplicial I-diagrams f^{•} : X^{•} → 𝔾_{m}, with X^{•}(U) a pro-scheme smooth over k, and still preserve the quasi-isomorphisms and exactness. Feeding the I-diagram

t ∘ p : τ X̂ ; ϵ(X_{0})^{0} → 𝔾_{m}

to this machine and taking the distinguished triangle induced by the exact sequence of log complexes at the bottom of the diagram completes the proof.

□

Remark 9.3.6. If we splice together the long exact homotopy sequence for the monodromy distinguished triangle

E(τ X̂ ; ϵ(X_{0})^{0}) → E(lim_{t→0} X_{t}) ⟶^{N} E(−1)(lim_{t→0} X_{t})

with the localization distinguished triangle of Theorem 6.3.7

E^{D_{Zar}}(X) ⟶^{α_{D}} E(D_{Zar}) ⟶^{β_{D}} E(τ X̂ ; ϵ(D)^{0})

(both evaluated on D = X_{0}), we have the complex

(9.3.2) … → E_{n}^{X_{0Zar}}(X) → E_{n}(X_{0Zar}) → E_{n}(lim_{t→0} X_{t}) ⟶^{N}

E(−1)_{n}(lim_{t→0} X_{t}) → E_{n−2}^{X_{0Zar}}(X) → E_{n−2}(X_{0Zar}) → …

If k = ℂ and E represents singular cohomology (for the classical topology)

E_{n}(Y) = H^{−n}(Y (ℂ), ℚ)

then Steenbrink’s theorem [44] states that the above sequence is exact. The argument uses the mixed Hodge structure on all the terms together with a weight argument.

One should be able to define a natural geometric “weight filtration” on E(lim_{t→0} X_{t}) by using the stratification of X_{0} by faces. However, for general E, this additional structure might not suffice to force the exactness of the above sequence. It would be interesting to give a general additional structure on E that would imply this exactness. □

9.4. Ayoub’s monodromy sequence. The monodromy sequence of corollary 9.3.5 agrees with the monodromy sequence constructed by Ayoub in [3, section 3.6] after making the identification described in proposition 8.3.2 and working throughout in the category of rational motives DM(k)_{ℚ}. Indeed, it is easy to check that our complex E(log_{f^{•}}) agrees with the contruction E⊗f_{η}^{∗}Log^{∨} of [3, section 3.6.3], and that our isomorphism E(P_{X/𝔾_{m}}) ≅ E(log_{f^{•}}) agrees with the map E ⊗ f_{η}^{∗}Log^{∨} → E ⊗ f_{η}^{∗}U induced by the map ℓ : Log^{∨} → U of [3, definition 3.6.42]. From there, one can easily compare with Ayoub’s monodromy sequence [3, definition 3.6.37]. We give a sketch of these comparisons.

Ayoub’s construction begins with the Kummer motive K. We denote the object in DM(S)_{ℚ} represented by a smooth S-scheme X as m_{S}(X) and write 𝟙_{S} for m_{S}(S), the unit for the tensor structure in DM(S)_{ℚ}; we delete the subscript S from the notation for S = Spec k. The 1-section i_{1} : S → 𝔾_{mS} induces the splitting

m_{S}(𝔾_{mS}) = 𝟙_{S} ⊕ 𝟙_{S}(1)[1]

and thus the projection π : m_{S}(𝔾_{mS}) → 𝟙_{S}(1)[1] We take S = 𝔾_{m}. The diagonal Δ : 𝔾_{m} → 𝔾_{m} ×_{k} 𝔾_{m} = 𝔾_{mS} induces m_{S}(Δ) : 𝟙_{S} → m_{S}(𝔾_{mS}); composing with π and twisting and shifting gives the map

∪t_{∗} : 𝟙_{𝔾_{m}}(−1)[−1] → 𝟙_{𝔾_{m}}

The Kummer motive K ∈ DM(𝔾_{m})_{ℚ} is defined as the “cone” of ∪t_{∗}: Ayoub shows there is a canonial distinguished triangle in DM(𝔾_{m})_{ℚ}

𝟙_{𝔾_{m}}(−1)[−1] ⟶^{∪t_{∗}} 𝟙_{𝔾_{m}} → K → 𝟙_{𝔾_{m}}(−1)

Next, Ayoub defines the object Log^{∨} of DM(𝔾_{m})_{ℚ}. Viewing K as the twoterm complex [m_{S}(𝔾_{mS})(−1)[−1] ⟶^{∪t_{∗}} m_{S}(𝔾_{mS})] with m_{S}(𝔾_{mS}) in degree zero, one sees that the nth symmetric product Sym^{n}K is the complex

𝟙_{𝔾_{m}}(−n)[−n] ⟶^{∪t_{∗}} 𝟙_{𝔾_{m}}(−n + 1)[−n + 1] ⟶^{∪t_{∗}} …

⟶^{∪t_{∗}} 𝟙_{𝔾_{m}}(−1)[−1] ⟶^{∪t_{∗}} 𝟙_{𝔾_{m}},

where we write the map ∪t_{∗}(−i)[−i] as ∪t_{∗} for short. The map 𝟙_{𝔾_{m}} → K gives rise to the map Sym^{n}K → Sym^{n+1}K. We can take the limit Log^{∨} in DM(𝔾_{m})_{ℚ}

Log^{∨} := lim_{n} Sym^{n}K

As a complex, Log^{∨} is just

… ⟶^{∪t_{∗}} 𝟙_{𝔾_{m}}(−n)[−n] ⟶^{∪t_{∗}} 𝟙_{𝔾_{m}}(−n + 1)[−n + 1] ⟶^{∪t_{∗}} …

⟶^{∪t_{∗}} 𝟙_{𝔾_{m}}(−1)[−1] ⟶^{∪t_{∗}} 𝟙_{𝔾_{m}}.

Now suppose we have a semi-stable degeneration f : X → 𝔸^{1} and an object E ∈ DM(X^{0})_{ℚ}. Let f^{0} : X^{0} → 𝔾_{m} be the restriction of f; since f^{0} is smooth, we have Lf^{0∗} = f^{∗}. Let i : X_{0} → X, j : X^{0} → X be the inclusions. The logarithmic specialization functor log_{f} is defined by

log_{f}(E) := Li^{∗}Rj_{∗}(E ⊗ f^{0∗}Log^{∨})

Remark 9.4.1. If we replace X with X ×_{𝔸^{1}} Spec O_{𝔸^{1},0}, we have the canonical identification of X^{0} with the generic fiber X_{η} and f^{0} with f_{η}. We avoid doing this to keep with the notation of our earlier sections. □

The first step in our comparison is

Lemma 9.4.2. Take E ∈ DM(X^{0})_{ℚ}, represented by a fibrant object Ẽ ∈ C_{ℚ}(Sm/X^{0}). Then E ⊗ f^{0∗}Log^{∨} is represented by Ẽ(log_{f^{•}}).

Proof. Note that we may assume that Ẽ is alternating, since E is a motive. Letting Hom denote the internal Hom in DM(X^{0})_{ℚ}. We have the distinguished triangle

E(−1)[−1] → Hom(𝔾_{mX_{0}}, E) ⟶^{i_{1}^{∗}} Hom(𝟙, E) = E → E(−1)

Thus E(−1)[−1] is represented by the presheaf

X′ ↦ fib[Ẽ(X′ ×_{k} 𝔾_{m}) ⟶^{id×i_{1}^{∗}} Ẽ(X′)]

Similarly, for n ≥ 1, E(−n)[−n] is represented by the presheaf

X′ ↦ Ẽ(X′ ∧ 𝔾_{m}^{∧n}).

SInce Ẽ is alternating, this latter presheaf is equivalent to

X′ ↦ Ẽ(X′, 𝔾_{m}^{∧n})^{alt}.

Finally, the map ∪f : Ẽ(X′_{+} ∧ 𝔾_{m}) → Ẽ(X′) is just the map induced by the pull-back by f and f × id of the diagonal map 𝔾_{m} → 𝔾_{m} ×_{k} 𝔾_{m}, hence ∪f represents the map f^{0∗}(∪t_{∗}). The comparison follows easily from this. □

Next, Ayoub considers the object U of DM(𝔾_{m})_{ℚ}. Interpreting his general construction in the case of DM(𝔾_{m})_{ℚ}, U is the motive associated to the simplicial object

n ↦ Hom_{DM(𝔾_{m})_{ℚ}}(P_{𝔾_{m}}^{n}, 𝟙_{𝔾_{m}}),

i.e., the homological complex which is Hom_{DM(𝔾_{m})_{ℚ}}(P_{𝔾_{m}}^{n}, 𝟙_{𝔾_{m}}) in degree n, and with differential the alternating sum of the maps induced by the coface maps in P_{𝔾_{m}}. Naturally, to make sense of this, we need to lift this construction to the appropriate category of complexes. In any case, the same proof as for lemma 9.4.2 gives us a canonical isomorphism of E ⊗ f^{0∗}U with E(P_{X^{0}/𝔾_{m}}). Similarly, the uni-potent specialization functor Υ is given by

Υ(E) = Li^{∗}Rj_{∗}(E ⊗ f^{0∗}U).

Finally, we have

Hom_{DM(𝔾_{m})_{ℚ}}(P_{𝔾_{m}}^{n}, 𝟙_{𝔾_{m}}) ≅ (𝟙_{𝔾_{m}}(−1)[−1] ⊕ 𝟙_{𝔾_{m}})^{⊗n}

and the first differential U_{1} → U_{0} is

𝟙_{𝔾_{m}}(−1)[−1] ⊕ 𝟙_{𝔾_{m}} ⟶^{∪t_{∗}+id} 𝟙_{𝔾_{m}}

Thus we have the evident map K → U. The diagonal map on P_{𝔾_{m}} dualizes to make U a commutive ring object in DM(𝔾_{m})_{ℚ}. Ayoub notes that K → Log^{∨} is universal for maps of K to a commutative ring in DM(𝔾_{m})_{ℚ}, hence there is a unique ring map ℓ : Log^{∨} → U making

\xymatrix{K \ar[r] \ar[dr] & Log^{∨} \ar[d]^{ℓ} \\  & U}

commute. It is not hard to see that our map alt ∘ π is induced by a map of complexes in ℚSm/𝔾_{m} which commutes with the co-multiplications dual to the ring multiplications for U and Log^{∨}. Since both

E ⊗ f^{0∗}Log^{∨} ⟶^{id⊗ℓ} E ⊗ f^{0∗}U

and

E(P_{X^{0}/𝔾_{m}}) ⟶^{alt∘π} E(log_{f^{•}})

are isomorphisms in DM(X^{0})_{ℚ}, it follows that these maps are inverse to each other. Once we have the pair of compatible isomorphisms E ⊗ f^{0∗}Log^{∨} ≅ E(log_{f^{•}}) and E ⊗ U ≅ E(P_{X^{0}/𝔾_{m}}), it is easy to see that Ayoub’s construction of the monodromy sequence and ours are compatible: Ayoub’s construction follows from the obvious identification of Log^{∨}(−1) with σ_{≤−1}Log^{∨} (cohomological notation) giving the distinguished triangle

𝟙_{𝔾_{m}} → Log^{∨} ⟶^{N} Log^{∨}(−1) → 𝟙_{𝔾_{m}}[1]

which clearly passes over to our identification E(σ_{≥1} log_{f^{•}}) ≅ E(−1) and the monodromy distinguished triangle of corollary 9.3.5.

10. Limit motives

We use our construction of limit cohomology, slightly modified, to give a construction of the limit motive of a semi-stable degeneration, as an object in the big category of motives DM(k).

10.1. The big category of motives. Voevodsky has defined the category of effective motives as the full subcategory DM_{−}^{eff}(k) of the derived category of Nisnevich sheaves with transfer D_{−}(NST(k)) consisting of those complexes with strictly homotopy invariant cohomology sheaves. In his thesis, Spitzweck [43] defines a “big” category of motives over a field k. Other constructions of a big category of motives over a noetherian base scheme S have been given by Østvar-Röndigs [40] and also by Cisinski-Deglise [9]. To give the reader the main idea of all these constructions, we quote from a recent letter from Röndigs [41]: “One may construct a model category of simplicial presheaves with transfers on Sm/k, in which the weak equivalences and fibrations are defined via the functor forgetting transfers. Via the Dold-Kan correspondence, there is an induced model structure on nonnegative chain complexes of presheaves with transfers. Both may be stabilized with respect to T or ℙ^{1}, in the sense of [18]. The Dold-Kan correspondence extends accordingly. Since T is a suspension already, one can then pass to a model category of 𝔾_{m}-spectra of integer-indexed chain complexes as well. For k a perfect field, results from [46] show that the homotopy category of the latter model category contains Voevodsky’s DM_{gm} as a full subcategory. ” We will use the ℙ^{1}-spectrum model. For details, we refer the reader to [40] and [35].

10.2. The cohomological motive. We start with the category of presheaves with transfer PST(k) on Sm/k, which is defined as in [46] as the category of presheaves on the correspondence category Cor(k). We let C_{≥0}(PST(k)) denote the model category of non-negative chain complexes in PST(k), with model structure induced from simplicial presheaves on Sm/k, as described above. For P ∈ C_{≥0}(PST(k)), let P (−1) denote the presheaf

Y ↦ ker[P (Y × ℙ^{1}) ⟶^{i_{∞}^{∗}} P (Y × ∞)][2].

where “ker” means the termwise kernel of the termwise split surjection i_{∞}^{∗}. Let ℤ_{X}^{tr} denote the presheaf on Cor(k) represented by X ∈ Sm/k, and let

ℤ̃_{ℙ^{1}}^{tr} := coker(ℤ_{Spec k}^{tr} ⟶^{i_{∞∗}} ℤ_{ℙ^{1}}^{tr}).

One has the adjoint isomorphism

Hom_{C_{≥0}(PST(k))}(C ⊗ ℤ̃_{ℙ^{1}}^{tr}, C′) ≅ Hom_{C_{≥0}(PST(k))}(C, C′(−1)[−2])

so the bonding maps for ℙ^{1}-spectra in C_{≥0}(PST(k)) can be just as well defined via maps

C_{n} → C_{n+1}(−1)[−2].

We will use this normalization of the bonding morphisms from now on. For an integer q ≥ 0, we have the (homological) Friedlander-Suslin presheaf ℤ_{F S}(q). To define this, one starts with the presheaf with transfers of quasifinite cycles z_{q.fin}(𝔸^{q}), with value on Y ∈ Sm/k the cycles on Y × 𝔸^{q} which are quasi-finite over Y. One then forms the Suslin complex C_{∗}(z_{q.fin}(𝔸^{q})) and reindexes:

ℤ_{F S}(q)(Y)_{n} := C_{n−2q}(z_{q.fin}(𝔸^{q}))(Y) := z_{q.fin}(𝔸^{q})(Y × Δ^{n−2q}).

(see [26, §2.4] for a precise definition). This represents motivic cohomology Zariski-locally:

H^{p}(X, ℤ(q)) = ℍ^{p}(X_{Zar}, ℤ_{F S}(q)).

More generally, for X ∈ Sm/k, define ℤ_{F S}^{X}(q) by

ℤ_{F S}^{X}(q)(Y) := ℤ_{F S}(q)(X × Y).

We define

δ_{n} : ℤ_{F S}^{X}(n) → ℤ_{F S}^{X}(n + 1)(−1)

by sending a cycle W on X × Y × Δ^{m} × 𝔸^{n} to W × Δ, where Δ ⊂ 𝔸^{1} × ℙ^{1} is the graph of the inclusion 𝔸^{1} ⊂ ℙ^{1}, and then reordering the factors to yield a cycle on X × Y × ℙ^{1} × Δ^{m} × 𝔸^{n+1}.

Definition 10.2.1. Let X be in Sm/k. The cohomological motive of X is the sequence

h̃(X) := (ℤ_{F S}^{X}(0), ℤ_{F S}^{X}(1)[2], …, ℤ_{F S}^{X}(n)[2n], …)

with the bonding morphisms δ_{n}[2n]. □

Remark 10.2.2. One can also define the cohomological motive h(X) ∈ DM_{gm}(k) as the dual of the usual (homological) motive m(X) := C^{Sus}(ℤ_{X}^{tr}). For X of dimension d, h(X)(n) is actually in DM_{−}^{eff}(k) for all n ≥ d, and is represented by ℤ_{F S}^{X}(n). From this, one sees that the image of h̃(X) in DM(k) is canonically isomorphic to h(X). Also, one can work in DM_{−}^{eff}(k) if one wants to define the cohomological motive of a diagram in Sm/k if the varieties involved have uniformly bounded dimension. Since our construction of limit cohomology uses varieties of arbitrarily large dimension, we need to work in DM(k). □

10.3. The limit motive. It is now an easy matter to define the limit motive for a semi-stable degeneration. Let X → (C, 0) be a semi-stable degeneration with parameter t at 0; suppose the special fiber X_{0} has irreducible components X_{0}^{1}, …, X_{0}^{m}. We have the diagram (8.2.1) of cosheaves on X_{0Zar}, lim_{t→0} X_{t}, indexed by the non-empty subsets I ⊂ {1, …, m}, which we write as

I ↦ [lim_{t→0} X_{t}]_{I}.

Taking global sections on X_{0} yields the diagram of cosimplicial schemes

I ↦ [lim_{t→0} X_{t}]_{I}(X_{0}).

Applying h̃ gives us the diagram of ℙ^{1}-spectra in C_{≥0}(PST(k))

I ↦ h̃([lim_{t→0} X_{t}]_{I}(X_{0})).

We then take the homotopy limit over this diagram forming the complex

lim_{t→0} h̃(X_{t}) := holim_{I}{I ↦ h̃([lim_{t→0} X_{t}]_{I}(X_{0}))}.

Definition 10.3.1. Let X → (C, 0) be a semi-stable degeneration with parameter t at 0. The limit cohomological motive lim_{t→0} h(X_{t}) is the image of lim_{t→0} h̃(X_{t}) in DM(k). □

Using the same procedure, we have, for D ⊂ X a normal crossing scheme, the motive of the tubular neighborhood h(τ X̂ ; ϵ(D)) and the motive of the punctured tubular neighborhood h(τ X̂ ; ϵ(D)^{0}). All the general results now apply for these cohomological motives. In particular, from corollary 9.3.5 we have the monodromy distinguished triangle (for the ℚ-motive)

h(τ X̂ ; ϵ(X_{0})^{0})_{ℚ} → lim_{t→0} h(X_{t})_{ℚ} → lim_{t→0} h(X_{t})_{ℚ}(−1)

and theorem 6.3.7 gives the localization distinguished triangle

h^{X_{0}}(X) → h(X_{0}) → h(τ X̂ ; ϵ(X_{0})^{0}),

where h^{X_{0}}(X) is represented by

Cone(h̃(X) ⟶^{j^{∗}} h̃(X \ X_{0}))[−1].

From this latter triangle, we see that h(τ X̂ ; ϵ(X_{0})^{0}) is in DM_{gm}(k).

11. Gluing smooth curves

We use the exponential map defined in §5 to define an algebraic version of gluing smooth curves along boundary components. We begin by recalling the construction of the moduli space of smooth curves with boundary components; for details we refer the reader to the article by Hain [15].

11.1. Curves with boundary components. For a k-scheme Y, a smooth curve over Y is a smooth proper morphism of finite type p : C → Y with geometrically irreducible fibers of dimension one. We say that C has genus g if all the geometric fibers of p are curves of genus g. A boundary component of C → Y consists of a section x : Y → C together with an isomorphism v : O_{Y} → x^{∗}T_{C/Y}, where T_{C/Y} is the relative tangent bundle on C. Equivalently, v is a nowhere vanishing section of T_{C/Y} along x. A smooth curve with n boundary components is (C → Y, (x_{1}, v_{1}), …, (x_{n}, v_{n})) with all the x_{i} disjoint. One has the evident notion of isomorphism of such tuples, so we can consider the functor M_{g}^{n} on Sch_{k}:

M_{g}^{n}(Y)

:= {smooth genus g curves over Y with n boundary components}/ ≅

For n = 0, this is just the well-know functor of moduli of smooth curves, which admits the coarse moduli space M_{g}. For n ≥ 1 and g ≥ 1, it is easy to show that a smooth curve over Y with n boundary components admits no non-identity automorphisms (over Y), from which it follows that M_{g}^{n} is representable; we denote the representing scheme by M_{g}^{n} as well. The same holds for genus 0 if n ≥ 2; in fact the data of a genus zero curve C with two points 0, ∞ together with a tangent vector v ≠ 0 in T_{0}(C) has no non-identity automorphisms.

One can form a partial compactification of M_{g}^{n} by allowing stable curves with boundary components. As we will not require the full extent of this theory, we restrict ourselves to connected curves C with a single singularity, this being an ordinary double point p. We require that the boundary components are in the smooth locus of C. If C is reducible, then C has two irreducible components C_{1}, C_{2}; we also require that both C_{1} and C_{2} have at least one boundary component. As above, such data has no non-trivial automorphisms, which leads to the existence of a fine moduli space M̄_{g}^{n}. We let C_{g}^{n} → M_{g}^{n} be the universal curve with universal boundary components (x_{1}, v_{1}), …, (x_{n}, v_{n}), and C̄_{g}^{n} → M̄_{g}^{n} the extended universal curve. The boundary ∂M̄_{g}^{n} := M̄_{g}^{n} \ M_{g}^{n} is a disjoint union of divisors

∂M̄_{g}^{n} := D_{(g,n)} ⨿ ∐_{(g_{1},g_{2}),(n_{1},n_{2})} D_{(g_{1},g_{2}),(n_{1},n_{2})},

where D_{(g_{1},g_{2}),(n_{1},n_{2})} consists of the curves C_{1} ∪ C_{2} with g(C_{i}) = g_{i}, and with C_{i} having n_{i} boundary components (we specify which component is C_{1} by requiring C_{1} to contain the boundary component (x_{1}, v_{1})) and D_{(g,n)} is the locus of irreducible singular curves. Let (C, (x_{1}, v_{1}), …) be a curve in ∂M̄_{g}^{n} with singular point p. By standard deformation theory, it follows that ∂M̄_{g}^{n} is a smooth divisor in M̄_{g}^{n}; let N_{(g_{1},g_{2}),(n_{1},n_{2})} denote the normal bundle of D_{(g_{1},g_{2}),(n_{1},n_{2})}. Deformation theory gives a canonical identification of the fiber of the punctured normal bundle N_{g_{1},g_{2},n_{1},n_{2}}^{0} := N_{(g_{1},g_{2}),(n_{1},n_{2})} \ 0 at (C, (x_{1}, v_{1}), …) with 𝔾_{m}-torsor of isomorphisms

Λ^{2}T_{C,p} ≅ k(p).

11.2. Algebraic gluing. We can now describe our algebraic construction of gluing curves. Fix integers g_{1}, g_{2}, n_{1}, n_{2} ≥ 1. We define the morphism

μ̄ : M_{g_{1},n_{1}} × M_{g_{2},n_{2}} → D_{g_{1},g_{2},n_{1}−1,n_{2}−1}.

by gluing (C_{1}, (x_{1}, v_{1}), …, (x_{n_{1}}, v_{n_{1}})) and (C_{2}, (y_{1}, w_{1}), …, (y_{n_{2}}, w_{n_{2}})) along x_{n_{1}} and y_{1}, forming the curve C := C_{1} ∪ C_{2} with boundary components (x_{1}, v_{1}), …, (x_{n_{1}−1}, v_{n_{1}−1}), (y_{2}, w_{2}), …, (y_{n_{2}}, w_{n_{2}}) and singular point p. We lift μ̄ to

μ : M_{g_{1},n_{1}} × M_{g_{2},n_{2}} → N_{g_{1},g_{2},n_{1},n_{2}}^{0}

using the isomorphism Λ^{2}T_{C,p} → k(p) which sends v_{n−1} ∧ w_{1} to 1 and the identification of (N_{g_{1},g_{2},n_{1},n_{2}}^{0})_{C_{1}∪C_{2},...} described above. We now pass to the category SH_{𝔸^{1}}(k). Taking the infinite suspension, the map μ defines the map

Σ^{∞}μ : Σ^{∞}M_{g_{1},n_{1}+} ∧ Σ^{∞}M_{g_{2},n_{2}+} → Σ^{∞}N_{g_{1},g_{2},n_{1},n_{2}+}^{0}.

Composing with our exponential map defined in §5 gives us our gluing map

⊕ : Σ^{∞}M_{g_{1},n_{1}+} ∧ Σ^{∞}M_{g_{2},n_{2}+} → Σ^{∞}M_{g_{1}+g_{2},n_{1}+n_{2}−2+}.

Remarks 11.2.1. (1) If one fixes a curve E := (E, (x_{1}, v_{1}), (x_{2}, v_{2})) ∈ M_{1,2}, one can form the tower under E⊕

… → Σ^{∞}M_{g,n} → Σ^{∞}M_{g+1,n} → …,

and form the homotopy colimit Σ^{∞}M_{∞,n}. If E is an object of SH_{𝔸^{1}}(k), one thus has the E-cohomology E^{∗}(M_{∞,n}). For intance, this gives a possible definition of stable motivic cohomology or algebraic K-theory of smooth curves. However, it is not at all clear if this limit is independent of the choice of E. In the topological setting, one notes that the space M_{1,2}(ℂ) is connected, so the limit cohomology, for example, is independent of the choice of E. On the contrary, M_{1,2}(ℝ) is not connected (the number of connected components in the real points of the curve corresponding to a real point of M_{1,2} splits M_{1,2}(ℝ) into disconnected pieces), so even there, the choice of E plays a role. It is also not clear if M_{∞,n} is independent of n (up to isomorphism in SH_{𝔸^{1}}(k)).

(2) In the topological setting, the map ⊕ is the infinite suspension of a map

ϕ : M_{g_{1},n_{1}}(ℂ) × M_{g_{2},n_{2}}(ℂ) → M_{g_{1}+g_{2},n_{1}+n_{2}−2}(ℂ),

making ⨿_{g,n}M_{g,n+2}(ℂ) into a topological monoid; the group completion is homotopy equivalent to the plus construction on the stable moduli space lim_{g→∞} M_{g,1}(ℂ) formed as in (1). Letting M_{∞}(ℂ)^{+} denote this group completion, the group structure induces on Σ^{∞}M_{∞}(ℂ)^{+} the structure of a Hopf algebra (this was pointed out to me by Fabian Morel), the co-algebra structure being the canonical one on a suspension spectrum. The functoriality of the exponential map exp^{0} as described in Remark 5.2.2 shows that the maps ⊕ make ⋁_{g,n} Σ^{∞}M_{g,n+2} into a biaglebra object in SH_{𝔸^{1}}(k). It is not clear if there is an analogous “Hopf algebra completion” of ⋁_{g,n} Σ^{∞}M_{g,n+2} in SH_{𝔸^{1}}(k). □

12. Tangential base-points

Since, by work of Østvar-Röndigs [35], motivic cohomology is represented in SH_{𝔸^{1}}(k), our methods are applicable to this theory. However, one can simplify the construction somewhat, since we are dealing with complexes of abelian groups rather than spectra. One can also achieve a refinement incorporating the multiplicative structure; this allows for a motivic definition of tangential base-points for the category of mixed Tate motives from the point of view of cycle algebras. Of course, the unipotent specialization functor of Ayoub [3], when restricted to the triangulated category of Tate motives in DM(−) also gives tangential base-points for mixed Tate motives, but we hope our construction will be useful for applications of this operation.

12.1. Cubical complexes. If we work with presheaves of complexes rather than presheaves of spectra, we can replace all our simplicial constructions with cubical versions. This enables an easy extension to the setting of differential graded algebras (d.g.a.’s), or even graded-commutative d.g.a.’s (c.d.g.a.’s) if we work with complexes of ℚ-vector spaces. We list the main results without proof here; the methods discussed in [26, §2.5] carry over without difficulty.

For a commutative ring R, we denote the model category of complexes of Rmodules on the big Nisnevich site, C_{R}(Sm/S_{Nis}) by C_{R,Nis}(S) and the derived category by D_{R,Nis}(S). The cubical category Cube has objects \underline{n}, n = 0, 1, . . .. Cube is a subcategory of the category of finite sets, with \underline{n} standing for the set \{0, 1\}^{n}, with morphisms making Cube the smallest subcategory of finite sets containing the following maps:

(1) all inclusions s_{i,n,ϵ} : {0, 1}^{n} → {0, 1}^{n+1}, ϵ ∈ {0, 1}, i = 1, …, n + 1, where s_{i,n,ϵ} is the inclusion inserting ϵ in the ith factor.

(2) all projections p_{i,n} : {0, 1}^{n} → {0, 1}^{n−1}, i = 1, …, n, where p_{i,n} is the projection deleting the ith factor.

(3) all maps q_{i,n} : {0, 1}^{n} → {0, 1}^{n−1}, i = 1, …, n − 1, n ≥ 2, defined by

q_{i,n}(ϵ_{1}, …, ϵ_{n}) := (ϵ_{1}, …, ϵ_{i−1}, δ, ϵ_{i+2}, …, ϵ_{n})

with

δ := \begin{cases}0 & if (ϵ_{i}, ϵ_{i+1}) = (0, 0) \\ 1 & else. \end{cases}

A cubical object in a category C is a functor Cube → C. The basic cubical object in Sch is the sequence of n-cubes □^{∗} : Cube → Sm/k. The operations of the projections p_{i,n} and inclusions s_{i,n} are the evident ones; q_{i,n} acts by

q_{i,n}(x_{1}, …, x_{n}) := (x_{1}, …, x_{i−1}, 1 − (x_{i} − 1)(x_{i+1} − 1), x_{i+2}, …, x_{n}).

Now let P : Cube → Mod_{R} be a cubical R-module. We have the cubical realization |P |^{c} ∈ C_{R} with

|P |_{n}^{c} := P (\underline{n})/ ∑_{i=1}^{n} p_{i,n}^{∗}(P (\underline{n − 1})).

The differential d_{n}^{c} : |P |_{n}^{c} → |P |_{n−1}^{c} is

d_{n}^{c} := ∑_{i=1}^{n}(−1)^{i}s_{i,1}^{∗} − ∑_{i=1}^{n}(−1)^{i}s_{i,0}^{∗}.

| − |^{c} is clearly a functor from the R-linear category of cubical R-modules to C(R); in particular, if we apply | − |^{c} to a complex of R-modules, we end up with a double complex. For a complex C, also write |C|^{c} for the total complex of this double complex, letting the context make the meaning clear.

Example 12.1.1. For a presheaf of abelian groups P on Sm/k, we have cubical presheaf C^{c}(P) with

C^{c}(P)(Y) := P (Y × □^{∗}).

Taking the cubical realization yields the cubical Suslin complex C_{∗}(P)^{c} with

C_{∗}(P)^{c}(Y) := |C^{c}(P)(Y)|^{c}.
□

The symmetric group S_{n} acts on C_{n}(P)^{c}, we let C_{n}(P)_{alt}^{c} denote the subpresheaf of alternating sections. If P is a presheaf of ℚ-vector spaces, C_{n}(P)_{alt}^{c} is a canonical summand of C_{n}(P)^{c}, with projection given by the idempotent Alt_{n} := \frac{1}{n!} ∑_{g} sgn(g)g; one checks that the C_{n}(P)_{alt}^{c} form a subcomplex of C_{∗}(P)^{c}. The main result on these constructions is

Proposition 12.1.2. (1) There is a canonical homotopy equivalence of functors

C_{∗} → C_{∗}^{c} : C(k) → C(k)

(2) If P is a complex of presheaves of ℚ-vector spaces, the inclusion

C_{∗}(P)_{alt}^{c} → C_{∗}(P)^{c}

is a quasi-isomorphism

Sketch of proof; see [27, §5] for details. For (1), one uses the algebraic maps

□^{n} → Δ^{n}

which collapse the faces x_{i} = 1 to the vertex (0, …, 0, 1) to get a map C_{∗} → C_{∗}^{c}. The homotopy inverse is given by triangulating the □^{n}. For (2) one checks that S_{n} acts by the sign representation on the homology sheaves of C_{∗}(P)^{c}. The projections Alt_{n} define a map of complexes Alt_{∗} : C_{∗}(P)^{c} → C_{∗}(P)_{alt}^{c} which thus gives the inverse in homology. □

12.2. Cubical tubular neightborhoods. For a closed immersion i : W → X in Sm/k, set ˆ□_{X,W}^{n} := (ˆ□_{X}^{n})_{□_{W}^{n}}^{h}, giving us the cubical pro-scheme

ˆ□_{X,W}^{∗} : Cube → Pro-Sm/k

We use the same notation for morphisms in the cubical setting as in the simplicial version, e.g., î_{W} : □_{W}^{∗} → ˆ□_{X,W}^{∗}. We have as well the co-presheaf on W_{Zar}

ˆ□_{X,W_{Zar}}^{n}(W \ F) := ˆ□_{X\,W \F}^{n}

and the cubical co-presheaf

τ X̂ ; ϵ(W)^{c} := ˆ□_{X,W_{Zar}}^{∗}.

Now let P be in C(k). We define P (τ X̂ ; ϵ(W)^{c})_{∗} to be the complex of presheaves

P (τ X̂ ; ϵ(W)^{c})_{∗} := |P (τ X̂ ; ϵ(W)^{c})|^{c}.

We also have the alternating subcomplex P (τ X̂ ; ϵ(W)^{c})^{alt} ⊂ P (τ X̂ ; ϵ(W)^{c}). We have as well the punctured tubular neighborhood in cubical form

τ X̂ ; ϵ(W)^{0c} := τ X̂ ; ϵ(W)^{c} \ □_{W_{Zar}}^{∗}

on which we can evaluate P:

P (τ X̂ ; ϵ(W)^{0c})_{∗} := |P (τ X̂ ; ϵ(W)^{0c})|^{c}.

Let P (τ X̂ ; ϵ(W)^{0c})^{alt} ⊂ P (τ X̂ ; ϵ(W)^{0c}) be the alternating subcomplex.

We let EM : C → Spt be a choice of the Eilenberg-Maclane spectrum functor. Our main comparison result is

Theorem 12.2.1. (1) Let i : W → X be a closed immersion in Sm/k. For P ∈ C(k), there are natural isomorphisms in SH(W_{Zar})

EM(P (τ X̂ ; ϵ(W)^{c})) ≅ EM(P)(τ X̂ ; ϵ(W))

EM(P (τ X̂ ; ϵ(W)^{0c})) ≅ EM(P)(τ X̂ ; ϵ(W)^{0})

(2) If P is a presheaf of complexes of ℚ-vector spaces, then the inclusion

P (τ X̂ ; ϵ(W)^{c})^{alt} → P (τ X̂ ; ϵ(W)^{c})

is a quasi-isomorphism.

Proof. Define P (τ X̂ ; ϵ(W)) to be the total complex of the double complex associated to the simplicial complex n ↦ P (τ X̂ ; ϵ(W)^{n}). The homotopy equivalence used in Proposition 12.1.2(1) extends, by the functoriality of the Nisnevich neighborhood, to a homotopy equivalence

P (τ X̂ ; ϵ(W))^{c} ∼ P (τ X̂ ; ϵ(W))

This yields a weak equivalence on the associated Eilenberg-Maclane spectra. Since the functor EM passes to the homotopy category, we have a canonical isomorphism

EM(P)(τ X̂ ; ϵ(W))) ≅ EM(P (τ X̂ ; ϵ(W))).

Putting these isomorphisms together completes the proof of the first assertion for the tubular neighborhood. The proof for the punctured tubular neighborhood is essentially the same. The second assertion follows from Proposition 12.1.2(2). □

12.3. The motivic c.d.g.a. There are a number of different complexes which represent motivic cohomology; we will use the strictly functorial one of Friedlander-Suslin, ℤ_{F S}(q) (see the description in §10.2) reindexed as a cohomological complex:

ℤ_{F S}(q)^{n} := ℤ_{F S}(q)_{−n}.

We will use the cubical version ℤ_{F S}(q)^{c}:

ℤ_{F S}(q)^{c,n}(Y) := C_{2q−n}(z_{q.fin}(𝔸^{q}))^{c}(Y).

By Proposition 12.1.2, ℤ_{F S}(q)^{c} is quasi-isomorphic to ℤ_{F S}(q). Passing to ℚ-coefficients, we have the quasi-isomorphic alternating subcomplex ℚ_{F S}(q)_{alt}^{c} ⊂ ℚ_{F S}(q)^{c}. We may also symmetrize with respect to the coordinates in the 𝔸^{q} in z_{q.fin}(𝔸^{q}); it is shown in [26] that the inclusion

ℚ_{F S}(q)_{alt,sym}^{c} ⊂ ℚ_{F S}(q)_{alt}^{c}

is also a quasi-isomorphism. The product map

z_{q.fin}(𝔸^{q})(□^{n} × Y) ⊗ z_{q.fin}(𝔸^{q′})(□^{n′} × Y) → z_{q.fin}(𝔸^{q+q′})(□^{n+n′} × Y) makes the graded complex

Ñ_{ℤ} := ⊕_{q≥0}ℤ_{F S}(q)^{c}

into a presheaf of Adams-graded d.g.a.’s on Sm/k (with Adams grading q). Passing to ℚ-coefficients, and following the product with the alternating and symmetric projections makes

N := ⊕_{q≥0}ℚ_{F S}(q)_{alt,sym}^{c}

a presheaf of Adams-graded c.d.g.a.’s, the motivic c.d.g.a. on Sm/k. We let N → N^{fib} denote a fibrant model of N in the model category of (Adamsgraded) c.d.g.a.’s on Sm/k, where the weak equivalences are Adams-graded quasi-isomorphisms of c.d.g.a.’s for the Zariski topology.

Remarks 12.3.1. (1) Since N is strictly homotopy invariant [46, Theorem 4.2], N^{fib} is homotopy invariant.

(2) In case k admits resolution of singularities (i.e., chark = 0) the canonical map ℤ_{F S}(q) → ℤ_{F S}(q)^{fib} is a pointwise weak equivalence [46, Theorem 7.4]. Thus, in this case, we can use N instead of N^{fib}. □

12.4. The specialization map. We consider the situation of a smooth curve C over our base-field k with a k-point x. We let O denote the local ring of x in C, K the quotient field of O and choose a uniformizing parameter t, which we view as giving a map

t : Spec O → 𝔸^{1}.

sending x to 0. Letting i_{x} : x → Spec O be the inclusion, we have the restriction map

i_{x}^{∗} : N (O) → N (k(x)),

which is a morphism of Adams-graded c.d.g.a.’s. In this section, we extend i_{x}^{∗} to a map

sp_{t} : N (K) → N (k(x))

in the homotopy category of Adams-graded c.d.g.a.’s over ℚ (denoted H(c.d.g.a._{ℚ})). This is essentially our construction of the tubular neighborhood, where we use cubical constructions throughout to keep track of the multiplication. First, if we apply N to □^{∗} × Y and take the alternating projection again, we have the presheaf of c.d.g.a.’s N (□_{alt}^{∗}) and the quasi-isomorphism of presheaves of c.d.g.a.’s

ι : N → N (□_{alt}^{∗}).

Next, write ˆ□_{C,x}^{m0} for ˆ□_{C,x}^{m} \ □_{x}^{m}, and consider the cubical punctured tubular neighborhood ℤ_{F S}(q)^{c}(τ Ĉ ; ϵ(x)^{0c}). The product map

z_{q.fin}(𝔸^{q})(□^{n} × ˆ□_{C,x}^{m0}) ⊗ z_{q.fin}(𝔸^{q′})(□^{n′} × ˆ□_{C,x}^{m′0}))

→ z_{q.fin}(𝔸^{q+q′})(□^{n+n′} × ˆ□_{C,x}^{m+m′0})) makes ⊕_{q≥0}ℤ_{F S}(q)^{c}(τ Ŝpec O ; ϵ(x)^{0c}) into an Adams-graded d.g.a.; taking the alternating projection in both the □^{n} and ˆ□_{C,x}^{m0} variables, and the symmetric projection in 𝔸^{q} and applying the fibrant model gives a presheaf of Adamsgraded c.d.g.a.’s, denoted N^{fib}(τ Ŝpec O ; ϵ(x)_{alt}^{0c}). Similarly, we perform this construction using the full tubular neighborhood, giving the presheaf N^{fib}(τ Ĉ ; ϵ(x)_{alt}^{c}), and the commutative diagram of Adamsgraded c.d.g.a.’s:

\xymatrix{N (k(x)) \ar[d]_{ι} & N (O) \ar[r]^{res} \ar[d]_{π_{O}^{∗}} \ar[l]_{i_{x}^{∗}} & N (K) \ar[d]^{ι} \ar[d]^{π_{K}^{∗}} \\ N^{fib}(□_{alt}^{∗})(k(x)) & N^{fib}(τ Ĉ ; ϵ(x)_{alt}^{c}) \ar[r]_{res} \ar[l]^{i_{x}^{∗}} & N^{fib}(τ Ĉ ; ϵ(x)_{alt}^{0c})}

Replacing (C, x) with (𝔸^{1}, 0) and using 𝔸^{1} and 𝔾_{m} instead of Spec O and Spec K yields the commutative diagram of Adams-graded c.d.g.a.’s

\xymatrix{N (k(0)) \ar[d]_{ι} & N^{fib}(𝔸^{1}) \ar[r]^{res} \ar[d]_{π_{𝔸^{1}}^{∗}} \ar[l]_{i_{0}^{∗}} & N^{fib}(𝔾_{m}) \ar[d]^{ι} \ar[d]^{π_{𝔾m}^{∗}} \\ N^{fib}(□_{alt}^{∗})(k(x)) & N^{fib}(τ 𝔸̂^{1} ; ϵ(0)_{alt}^{c}) \ar[r]_{res} \ar[l]^{i_{0}^{∗}} & N^{fib}(τ 𝔸̂^{1} ; ϵ(0)_{alt}^{0c}).}

By Corollary 3.3.3 and Corollary 4.1.4, the maps π_{𝔸^{1}}^{∗} and π_{𝔾_{m}}^{∗} are quasiisomorphisms of complexes, hence quasi-isomorphisms of Adams-graded c.d.g.a.’s. Since N^{fib} is homotopy invariant, the maps ι are quasi-isomorphisms of Adams-graded c.d.g.a.’s. Finally, the map t induces the commutative diagram of Adams-graded c.d.g.a.’s

\xymatrix{N (k(x)) & N^{fib}(τ Ĉ ; ϵ(x)_{alt}^{c}) \ar[r]^{res} \ar[l]_{i_{x}^{∗}} & N^{fib}(τ Ĉ ; ϵ(x)_{alt}^{0c}) \\ N (k(0)) \ar[u]^{t^{∗}} & N^{fib}(τ 𝔸̂^{1} ; ϵ(x)_{alt}^{c}) \ar[r]_{res} \ar[u]^{t^{∗}} \ar[l]^{i_{0}^{∗}} & N^{fib}(τ 𝔸̂^{1} ; ϵ(x)_{alt}^{0c}). \ar[u]_{t^{∗}}}

Since t : (C, x) → (𝔸^{1}, 0) is a Nisnevich neighborhood of 0 in 𝔸^{1}, all three maps t^{∗} are isomorphisms. Putting these diagrams together and inverting the quasi-isomorphisms ι, t^{∗}, π_{𝔸^{1}}^{∗} and π_{𝔾_{m}}^{∗} yields the commutative diagram in H(c.d.g.a._{ℚ}):

(12.4.1)  \xymatrix{N (k(x)) & N (O) \ar[l]_{i_{x}^{∗}} \ar[r]^{res} \ar[d]_{ϕ_{O}^{∗}} & N (K) \ar[d]^{ϕ_{K}^{∗}} \\ N (k(0)) \ar[u]^{t^{∗}} & N^{fib}(𝔸^{1}) \ar[r]_{res} \ar[l]^{i_{0}^{∗}} & N^{fib}(𝔾_{m})}

Definition 12.4.1. Let i_{1} : Spec k → 𝔾_{m} be the inclusion. The map sp_{t} : N (K) → N (k(x)) in H(c.d.g.a._{ℚ}) is defined to be the composition

N (K) ⟶^{ϕ_{K}^{∗}} N^{fib}(𝔾_{m}) ⟶^{i_{1}^{∗}} N^{fib}(k) ≅ N (k) = N (k(0)) ⟶^{t^{∗}} N (k(x)).

□

Proposition 12.4.2. The diagram in H(c.d.g.a._{ℚ})

\xymatrix{N (O) \ar[r]^{res} \ar[dr]_{i_{x}^{∗}} & N (K) \ar[d]^{sp_{t}} \\  & N (k(x))}

commutes.

Proof. Since N^{fib} is homotopy invariant, the maps

i_{0}^{∗}, i_{1}^{∗} : N^{fib}(𝔸^{1}) → N (k)

are equal in H(c.d.g.a._{ℚ}). The proposition follows directly from this and a chase of the commutative diagrams defined above. □

Remark 12.4.3. In the situation we are considering, we already have the following diagram:

N (K) → N (lim_{t→0} Spec K) ≅ N (k(0)).

However, the above diagram is only a diagram in the homotopy category of complexes of ℚ-vector spaces, which is thus equivalent to the same diagram for cohomology of the complexes involved. We have gone to the trouble of redoing our theory using cubes throughout because we need to keep track of the multiplication, i.e. our construction lifts the above diagram in D^{b}(ℚ) to one in H(c.d.g.a._{ℚ}). □

12.5. The specialization functor. For a field k, we have the triangulated category DMT(k) of mixed Tate motives over k, this being the full triangulated subcategory of Voevodsky’s triangulated category of motives (with ℚcoefficients), DM_{gm}(k)_{ℚ}, generated by the Tate objects ℚ(n), n ∈ ℤ. We will also use in this section the derived category of finite cell modules over an Adams-graded c.d.g.a. A, DCM(A). This construction was introduced in [23]; we refer the reader to the discussion in [26, §5] for the properties of DCM we will be using below. Let O be as in the previous section the local ring of a k-point x on a smooth curve C over k, with quotient field K. The map sp_{t} : N (K) → N (k(x)) yields an exact tensor functor

sp_{t} : DMT(K) → DMT(k(x))

Indeed, as discussed in [26, §5.5], Spitzweck’s representation theorem gives a natural equivalence of DMT(k) with the derived category DCM(N (k)) of finite cell modules over the Adams-graded c.d.g.a. N (k), as triangulated tensor ℚtensor categories.

The functor DCM associating to an Adams-graded ℚ-c.d.g.a. A the triangulated tensor category DCM(A) takes quasi-isomorphisms to triangulated tensor equivalences, hence DCM descends to a well-defined pseudo-functor on H(c.d.g.a._{ℚ}). Thus, we may make the following

Definition 12.5.1. Let O be the local ring of a k-point x on a smooth curve C over k, with quotient field K and uniformizing parameter t. Let sp_{t} : DMT(K) → DMT(k(x)) be the exact tensor functor induced by DCM(sp_{t}) : DCM(N (K)) → DCM(N (k(x)), using Spitzweck’s representation theorem to identify the derived categories of cell modules with the appropriate category of mixed Tate motives. □

Remark 12.5.2. (1) The discussion in [26, §5.5], in particular, the statement and proof of Spitzweck’s representation theorem, is in the setting of motives over a field. However, we now have available a reasonable triangulated category DM(S) of motives over an arbitrary base-scheme S (see [48]), and we can thus define the triangulated category of mixed Tate motives over S, DMT(S), as in the case of a field. Furthermore, if S is in Sm/k for k a field of characteristic zero, then N (S) has the correct cohomology, i.e.

H^{n}(N (S)) = ⊕_{q≥0}H^{n}(S, ℚ(q)),

and one has the isomorphism

H^{n}(S, ℤ(q)) ≅ Hom_{DM(S)}(ℤ, ℤ(q)).

This is all that is required for the argument in [26, §5.5] to go through, yielding the equivalence of the triangulated tensor category of cell modules DCM(N (S)) with DMT(S).

(2) Joshua [20] has defined the triangulated category of ℚ mixed Tate motives over S as DCM(N (S)); the discussion in (1) shows that this agrees with the definition as a subcategory of DM(S)_{ℚ}. □

With these remarks, we can now state the main compatibility property of the functor sp_{t} : DMT(K) → DMT(k(x)).

Proposition 12.5.3. Let O be the local ring of a k-point x on a smooth curve C over k, with quotient field K and uniformizing parameter t. Let i_{x}^{∗} : DMT(O) → DMT(k) and j^{∗} : DMT(O) → DMT(K) be the functors induced by the inclusions i_{x} : Spec k → Spec O and j : Spec K → Spec O, respectively. Then the diagram

\xymatrix{DMT(O) \ar[r]^{j^{∗}} \ar[dr]_{i_{x}^{∗}} & DMT(K) \ar[d]^{sp_{t}} \\  & DMT(k(x))}

commutes up to natural isomorphism.

Proof. This follows from Proposition 12.4.2 and the functoriality (up to natural isomorphism) of the equivalence DCM(N (S)) ∼ DMT(S). □

12.6. Compatibility with specialization on motivic cohomology. As above, let O be the local ring of a closed point x on a smooth curve C over k, with quotient field K and uniformizing parameter t. We have the localization sequence for motivic cohomology

… → H^{n}(O, ℤ(q)) ⟶^{j^{∗}} H^{n}(K, ℤ(q)) ⟶^{∂} H^{n−1}(k(x), ℤ(q − 1)) ⟶^{i_{x∗}} …

In addition, the parameter t determines the element [t] ∈ H^{1}(K, ℤ(1)). One defines the specialization homomorphism

s̃p_{t} : H^{n}(K, ℤ(q)) → H^{n}(k(x), ℤ(q))

by the formula

s̃p_{t}(α) := ∂([t] ∪ α).

On the other hand, if k(x) = k, we have the specialization functor

sp_{t} : DMT(K) → DMT(k(x))

and the natural identifications

H^{n}(K, ℚ(q)) ≅ Hom_{DMT(K)}(ℚ, ℚ(q)[n])
H^{n}(k, ℚ(q)) ≅ Hom_{DMT(k)}(ℚ, ℚ(q)[n]).

Thus the functor sp_{t} induces the homomorphism

sp_{t} : Hom_{DMT(K)}(ℚ, ℚ(q)[n]) → Hom_{DMT(k)}(ℚ, ℚ(q)[n])

and hence a new homomorphism

sp′_{t} : H^{n}(K, ℚ(q)) → H^{n}(k, ℚ(q)).

Proposition 12.6.1. sp′_{t} agrees with the ℚ-extension of s̃p_{t}.

Proof. Using the equivalence DMT(K) ∼ DCM(N (K)) and the canonical identifications

Hom_{DCM(K)}(ℚ, ℚ(q)[n]) ≅ H^{n}(N (K)) ≅ ⊕_{q≥0}H^{n}(K, ℚ(q))

(and similarly for k) we need to show that the ℚ-linear extension of s̃p_{t} agrees with the map

H^{n}(sp_{t}) : H^{n}(N (K)) → H^{n}(N (k))

induced by sp_{t} : N (K) → N (k). For this, take an element α ∈ H^{n}(K, ℤ(q)) and set

β̄ := ∂α ∈ H^{n−1}(k, ℤ(q − 1)).

Since i_{x} : x → Spec O is split by the structure morphism π : Spec O → Spec k, we can lift β̄ to β : π^{∗}(β̄) ∈ H^{n−1}(O, ℤ(q − 1)). Then

∂([t] ∪ β) = ∂([t]) ∪ i_{x}^{∗}β = β̄,

the first identity following from the Leibniz rule and the second from the fact that ∂([t]) = 1 ∈ H^{0}(k, ℤ(0)). Thus

∂(α − [t] ∪ β) = 0,

hence there is a class γ ∈ H^{n}(O, ℤ(q)) with

j^{∗}γ = α − [t] ∪ β.

We consider γ as an element of H^{n}(N (O)). By Proposition 12.4.2, we have

H^{n}(i_{x}^{∗})(γ) = H^{n}(sp_{t})(α − [t] ∪ β).

By the functoriality of the identification

H^{n}(N (−)) ≅ ⊕_{q≥0}Hom_{DCM}(N (−))(ℚ, ℚ(q))

and Proposition 12.4.2 it follows that

s̃p_{t}(j^{∗}γ) = H^{n}(i_{x}^{∗})(γ) = H^{n}(sp_{t})(j^{∗}γ)

so we reduce to showing

s̃p_{t}([t] ∪ β) = 0 = H^{n}(sp_{t})([t] ∪ β).

The first identity follows from [t] ∪ [t] = 0 in H^{2}(K, ℚ(2)). For the second, because sp_{t} is a morphism in H(c.d.g.a._{q}), the map H^{∗}(sp_{t}) is multiplicative, hence it suffices to show that H^{1}(sp_{t})([t]) = 0. For this, it follows from the constuction of the map sp_{t} : N (K) → N (k(x)) in H(c.d.g.a._{ℚ}) that sp_{t} is natural with respect to Nisnevich neighborhoods f : (C′, x′) → (C, x) of x, i.e.,

sp_{f^{∗}(t)} ∘ f^{∗} = sp_{t}.

Now, the map t : (C, x) → (𝔸^{1}, 0) is clearly a Nisnevich neighborhood of 0 (after shrinking C if necessary) and

[t] = t^{∗}([T])

where 𝔸^{1} = Spec k[T]. Thus, we may assume that C = 𝔸^{1} and t = T. But then [T] is a well-defined element of H^{1}(N (𝔾_{m})) hence

H^{1}(sp_{t})([T]) = i_{1}^{∗}([T]) = [1] = 0

by definition of sp_{t} : N (O_{𝔸^{1},0}) → N (k). This completes the proof. □

Remark 12.6.2. Since sp′_{t} is multiplicative, as we have already remarked, Proposition 12.6.1 gives a rather long-winded re-proof of the multiplicativity of the specialization homomorphism s̃p_{t} □ 12.7. Tangential base-points. As shown in [29], the category DMT(k) carries a canonical exact weight filtration. For an Adams-graded c.d.g.a. A, the derived category of cell modules DCM(A) carries a natural weight filtration as well; the equivalence DCM(N (k)) ∼ DMT(k) given by Spitzweck’s representation theorem is compatible with the weight filtrations [26, Theorem 5.24]. If A is cohomologically connected (H^{n}(A) = 0 for n < 0 and H^{0}(A) = ℚ · id), then DCM(A) carries a t-structure, natural among cohomologically connected A. Finally, if A is 1-minimal then DCM(A) is equivalent to the derived category of the heart of this t-structure (see [26, §5]). Thus, if N (F) is cohomologically connected, then DMT(F) has a t-structure; the heart is called the category of mixed Tate motives over F, denoted MT(F). In fact, MT(F) is a Tannakian category , with natural fiber functor given by the weight filtration; let Gal_{μ}(F) denote the pro-algebraic group scheme over ℚ associated to MT(F) by the Tannakian formalism. If N (F) is 1-minimal, then DMT(F) is equivalent to D^{b}(MT(F)), but we won’t be using this. Now let x be a k-point on a smooth curve C over k, and t a parameter in O_{C,x}. The specialization functor

sp_{t} : DMT(k(C)) → DMT(k(x))

arises from the map sp_{t} : N (k(C)) → N (k(x)) in H(c.d.g.a._{ℚ}), hence sp_{t} is compatible with the weight filtrations. When N (k(C)) and N (k(x)) are cohomologically connected, sp_{t} is compatible with the t-structures, hence induces an exact functor of Tannakian categories

sp_{t} : MT(k(C)) → MT(k(x))

compatible with the fiber functors gr^{W}. By Tannakian duality, sp_{t} is equivalent to a homomorphism

\frac{∂}{∂t}_{∗} : Gal_{μ}(k(x)) → Gal_{μ}(k(C)),

which is the tangential base-point associated to the parameter t. This gives a purely “motivic” construction of the tangential base-point construction of Deligne-Goncharov [10]; the construction in [10] relies on realization functors.

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Marc Levine^{∗}

Department of Mathematics Northeastern University Boston, MA 02115 USA

∗The author gratefully acknowledges the support of the NSF via grants DMS 0140445 and DMS-0457195.