A ROAD MAP OF MOTIVIC HOMOTOPY AND HOMOLOGY THEORY

Charles Weibel

Rutgers University

weibel@math.rutgers.edu

1. The road map.

The parallel constructions of Motivic Homotopy and Motivic Homology are based on the construction of stable homotopy and homology in topology. Instead of starting with topological spaces and using the unit interval [0, 1] to define homotopy, one starts with smooth schemes over a fixed field k and uses the affine line 𝔸^{1} = Spec(k[t]). The constructions are related by two functors from homotopy to homology which, by analogy, we call Hurewicz functors. Here is the main diagram, or road map.

Sm/k

→^{h}

Presheaves

→

Spc (spaces) =Nis. sheaves

⟶^{𝔸^{1}}_{w.e.}
Hot_{𝔸^{1}}

→_{Σ_{T}^{∞}}

SW SHot
↓ ; Cor_{k} ; ↓  ↓ ; D_{c}^{−}(Sm/k) ; ↓  |↓Hurewicz  |↓  |↓  ⋮ ; ∨

PST → D^{−}(PST) → D^{−}(Nis. ST) ⟶^{𝔸^{1}}_{w.e.} DM_{eff}^{−} →_{Σ_{T}^{∞}} DM^{−} DM

We begin with a description of the top row of this diagram. These constructions are due to Morel and Voevodsky (see [MV] and [V1]). In the upper left corner we have the category Sm/k of smooth schemes over k. This is the natural home of several important objects, such as the geometric circle S_{t}^{1} = (𝔸^{1} − 0) and the geometric n-simplex Δ^{n} = Spec(k[t_{0}, …, t_{n}]/(∑ t_{i} = 1)).

2. Motivic Homotopy Theory.

First one embeds Sm/k into the category of presheaves (of sets) on Sm/k, via the Yoneda embedding: to Y we associate the presheaf h_{Y} :

X ↦ Hom_{Sm/k}(X, Y). Passing to presheaves enables us to construct arbitrary colimits, including pushouts and quotients. In particular, the simplicial circle S_{s}^{1} = 𝔸^{1}/{0, 1} and the Tate object T = 𝔸^{1}/(𝔸^{1} − 0) exist as pointed presheaves. The smash product operation and the geometric realization of simplicial presheaves also make sense. (Geometric realization sends the simplicial n-simplex Δ[n] to Δ^{n}, and X × Δ[n] to X × Δ^{n}; see [MV, 2.3.14]).

Moving horizontally, we may sheafify with respect to the Nisnevich topology on Sm/k. (Each h_{Y} is already a sheaf.) Morel and Voevodsky use the word space to mean a sheaf with respect to the Nisnevich topology on Sm/k, so the category Spc of “spaces” is just the category of Nisnevich sheaves on Sm/k. Sheafification turns a presheaf into a space, of course. In the category Spc, a smooth scheme X is the colimit of any Zariski open cover, and X/U ≅ V/(V ∩ U) if X = U ∪ V. Similarly, X is the colimit of any Nisnevich cover.

A map between two spaces is called an 𝔸^{1}-weak equivalence if it belongs to the smallest saturated class of maps which contains all maps X × 𝔸^{1} → X, and which is also closed under isomorphisms, colimits and pushouts along monomorphisms. The homotopy category Hot of schemes over k is the category obtained from Spc by inverting the 𝔸^{1}-weak equivalences. Morel and Voevodsky showed in [MV] that the 𝔸^{1}-weak equivalences are the weak equivalences of a proper closed model structure on Spc whose cofibrations are monomorphisms.

An alternative and historically more familiar approach is to move quickly to the category of simplicial spaces and impose a proper closed model structure, replacing 𝔸^{1} by the simplicial space X ↦ h_{𝔸^{1}}(X ×Δ^{•}), where Δ^{•} is the cosimplicial scheme n ↦ Δ^{n}. The corresponding homotopy category of simplicial spaces is equivalent to Hot by the geometric realization functor described above. (See [MV, 2.3.14]).

The pointed variation of this construction follows the topological paradigm. That is, we may embed the categories of presheaves, spaces and Hot into their corresponding pointed categories by sending X to the union X_{+} of X with a disjoint basepoint. It is in the pointed category Hot_{•} that we have T ≃ (ℙ^{1}, ∞) ≃ S_{s}^{1} ∧ S_{t}^{1}; see [MV, 3.2.15].

The stable homotopy categories SW and SHot of T -spectra are obtained from the pointed homotopy category by the process of stabilization with respect to the T -suspension Σ_{T} X = T ∧ X. In fact, there are two distinct constructions in play: desuspension and cocompletion. They are described in [V1] and [VW].

The Spanier-Whitehead stable homotopy category SW is the category obtained from Hot_{•} by inverting the T -suspension X ↦ T ∧X, or equivalently, by inverting both the simplicial suspension X ↦ Σ_{s}X = S_{s}^{1} ∧ X and the geometric t-suspension X ↦ Σ_{t}X = S_{t}^{1}∧X. Every object of SW is a finite desuspension of some pointed space X, meaning that it is isomorphic to Σ_{T}^{−n}X for some n ≥ 0. In this category, Hom(Σ_{T}^{−m}X, Σ_{T}^{−n}Y) is the direct limit of the Hom(Σ_{T}^{i−m}X, Σ_{T}^{i−n}Y) as i → ∞. As in topology [SW], and pointed out in [V1], this construction has the defect that SW is not closed under coproducts: we need to correct this by passing to T -spectra.

A T -spectrum E is a sequence of pointed spaces E_{n}, together with bonding maps T ∧ E_{n} → E_{n+1}. There is a category of T -spectra; a morphism E → F of T -spectra is just a sequence of maps E_{n} → F_{n} which commute with the bonding maps. The stable homotopy category SHot of T -spectra is obtained from the category of T -spectra by localizing with respect to stable weak equivalences; see [V1, 5.1]. There is a functor Σ_{T}^{∞} from Hot_{•} to SHot sending a pointed space X to the sequence of spaces X ∧ T^{∧n}. Clearly, this functor factors through the Spanier-Whitehead category SW. (It is faithful when restricted to spaces of finite type by [V1, 5.3].)

Down the left side of the main diagram we have the category Cor_{k} of finite correspondences and the category PST of presheaves with transfer. These constructions were first defined in [V] and detailed in [MVW].

The additive category Cor_{k} has the same objects as Sm/k: smooth schemes over k. The group Cor_{k}(X, Y) of morphisms from X to Y is the free abelian group on the elementary correspondences – subvarieties W of X ×Y whose projection W →X is finite and onto a component of X. Morphisms in Cor_{k} are called finite correspondences. Composition of finite correspondences is given by a classical “pull back, intersect and push forward” construction originally due to Hurwitz, Lefschetz and Severi [Dieu]; the modern details are given in [V] or [MVW].

There is a canonical embedding of the category Sm/k into Cor_{k} sending a morphism f : X → Y between smooth schemes to its graph Γ_{f}, considered as an elementary correspondence from X to Y.

A presheaf with transfers F is a contravariant additive functor from Cor_{k} to abelian groups. That is, it is a presheaf on Sm/k equipped with transfer maps F (Y) → F (X), one for every elementary correspondence from X to Y, subject to the composition rules in Cor_{k}. The category Cor_{k} embeds into the category PST of presheaves with transfer via the Yoneda embedding: to a smooth scheme Y we associate the presheaf with transfer ℤ_{tr}Y : X ↦ Cor_{k}(X, Y). Cor_{k} is not idempotent complete, because if (X, x) is a smooth pointed scheme, then ℤ_{tr}(X, x) = ℤ_{tr}X/ℤ is a direct summand of ℤ_{tr}X. Important example of presheaves with transfer include ℤ = ℤ_{tr}(Spec k) and ℤ_{tr}𝔾_{m} = ℤ_{tr}(𝔸^{1} − 0, 1).

3. Motivic Homology Theory.

Across the bottom of the main diagram, we find the construction of Voevodsky’s triangulated category of motives, DM^{−}, as described in [V] and [MVW]. Since PST is an abelian category, we can consider the derived category D^{−}(PST) of bounded above cochain complexes of presheaves with transfer. [One may equally think of this as the category D_{+} of bounded below chain complexes.] Since ℤ_{tr}(X×Y) = ℤ_{tr}X⊗ℤ_{tr}Y in PST, and projective resolutions are bounded above cochain complexes, D^{−}(PST) has a well-behaved total tensor product and a tensor triangulated structure; see [MVW, 8A]. Another reason for restricting to bounded above complexes is that the chain complex associated to a simplicial object will be bounded above when indexed as a cochain complex. For example, a basic object is the cochain complex ℤ(1); its shift ℤ(1)[1] is the complex associated to the simplicial object X ↦ ℤ_{tr}𝔾_{m}(X × Δ^{•}) of PST.

If we sheafify with respect to the Nisnevich topology, we get Nisnevich sheaves with transfer, and the derived category D^{−} = D^{−}(Nis. ST) of bounded above cochain complexes of Nisnevich sheaves with transfer. The tensor product in D^{−}(PST) induces a tensor product ⊗_{tr}^{𝕃} on D^{−}, making it into a tensor triangulated category (see [MVW, 14.2]).

A morphism in D^{−} is an 𝔸^{1}-weak equivalence if it belongs to the smallest multiplicative system containing quasi-isomorphisms and the projections ℤ_{tr}(X ×𝔸^{1}) → ℤ_{tr}X, which is also closed under shifts, direct sums and cones. Inverting the 𝔸^{1}-weak equivalences yields a tensor triangulated category DM_{eff}^{−}; see [V] [MVW, 14.1]. It may be identified with the full subcategory of D^{−} consisting of complexes with homotopy invariant cohomology sheaves; see [V, 3.2.3] [MVW, 14.10]. Since ℤ(1) ≅ ℤ_{tr}𝔾_{m}[−1] in DM_{eff}^{−}, the Tate twist M(1) = M ⊗_{tr}^{L} ℤ(1) of M has good properties.

The triangulated category DM^{−} is obtained from DM_{eff}^{−} by inverting the Tate twist M ↦ M(1). Thus every object in DM^{−} is isomorphic to M(−n) for some n ≥ 0 and some M in DM_{eff}^{−}. By [V2], DM_{eff}^{−} → DM^{−} is fully faithful.

The motive M(X) of a smooth scheme X is defined to be the class in DM_{eff}^{−} (or equivalently, in DM^{−}) associated to the chain complex which is ℤ_{tr}(X), concentrated in degree zero. The category DM_{gm} of geometric motives is the smallest triangulated subcategory containing the M(X) which is closed under summands as well as the Tate twist. (See p. 192 and 3.2.6 of [V]).

4. Stable Motivic Complexes.

The category DM^{−} is sufficient for most homological considerations. However, it is not closed under infinite direct sums. In order to correct this, we need a different construction, paralleling the definition of T - spectra in [V1]. A version paralleling the definition of symmetric spectra has been given by Spitzweck (see [Sp, 14.7] [Hov, 7.11]).

A stable motivic complex E_{•} is a sequence E_{0}, E_{1}, … of bounded above complexes of Nisnevich sheaves with transfer, together with structure maps E_{n}(1)[2] → E_{n+1}. The stable motivic complex Σ_{T}^{∞}X associated to a bounded above complex of Nisnevich sheaves X is the sequence X_{n} = X(n)[2n] with structure maps X(n)(1)[2] ≅ X(n + 1).

A function from E_{•} to F_{•} is a sequence of maps E_{n} → F_{n} commuting with the structure maps. Each stable motivic complex defines a cohomological functor {E^{i}} from DM_{gm} to abelian groups by:

E^{i}(M) = colim_{n→∞} Hom_{DM}_{−}(M(n)[2n], E_{n+i}).

A function E_{•} → F_{•} is called a stable weak equivalence if the corresponding natural transformations of functors E^{i} → F^{i} are isomorphisms for all i. A useful example of a stable weak equivalance is any function for which the E_{n} → F_{n} are 𝔸^{1}-weak equivalences for n ≫ 0. In particular, Σ_{T}^{∞} sends 𝔸^{1}-weak equivalences to stable weak equivalences.

Definition 1 The category DM is the localization of the category of stable motivic complexes and functions, with respect to the class of stable weak equivalences.

By construction, there is a canonical functor Σ_{T}^{∞} : DM_{eff}^{−} → DM. In fact, DM is a triangulated category and Σ_{T}^{∞} is a triangulated functor. Indeed, the mapping cone of a function f : E_{•} → F_{•} is also a stable motivic complex, and E_{•} → F_{•} → cone(f) → E_{•}[1] is the prototype of a distinguished triangle.

The T -suspension Σ_{T} E_{•} and T -desuspension Σ_{T}^{−1}E_{•} are obtained by shifting the sequence one to the left and right, respectively. They are inverse operations in DM; E_{•} = Σ_{T}^{−1}Σ_{T} E_{•} and Σ_{T} Σ_{T}^{−1}E_{•} → E_{•} is a stable weak equivalence. The Tate twist is invertible in DM, because Σ_{T} is, so Σ_{T}^{∞} induces a triangulated functor DM^{−} → DM.

We could have also obtained DM by restricting the definition of stable motivic complex to negatively graded complexes. To see this, let τ denote the (good) truncation to cohomological degrees ≤ 0. Then the truncation τE_{•} = {τE_{n}} is also a stable motivic complex, and the canonical map τE_{•} → E_{•} is a stable weak equivalence; this fact uses the fact that for each geometric motive M the M(n)[2n] are eventually negatively graded. The truncation allows us to see that the direct sum of a family E_{•}^{α} exists and equals the sequence of the ⊕_{α}τ(E_{n}^{α}). Étale analogue. It is clear that we can repeat the above construction with the étale topology, and with ℤ/m coefficients, assuming 1/m ∈ k. Remark 9.6 and theorem 9.32 in [MVW] shows that DM_{et}(k; ℤ/m) is just the full derived category D(G, ℤ/m) of profinite ℤ/m[G]-modules, where G is the absolute Galois group of k.

5. The Hurewicz functor.

Having described the outer part of the main diagram, we now need to fill in the middle. The category D_{c}^{−}(Sm/k) = D^{−}(Ab^{Sm^{op}}) is the derived category of bounded above cochain complexes of presheaves of abelian groups on Sm/k, i.e., complexes of objects in the abelian category Ab^{Sm^{op}} of presheaves of abelian groups on Sm/k.

Consider the free functor X ↦ ℤX from presheaves of sets to the category Ab^{Sm^{op}} of presheaves of abelian groups; by definition, ℤX(U) is the free abelian group on the set X(U). Composing with the canonical embedding of Ab^{Sm^{op}} into its derived category D^{−}(Ab^{Sm^{op}}) gives the first vertical map in the second column of the main diagram.

Note that the terminal presheaf ∗ is sent to the presheaf U ↦ ℤ. If (X, ∗) is a pointed presheaf of sets, ℤ(X, ∗)(U) is defined to be the cokernel of the canonical basepoint map ℤ ⟶^{∗} X(U); these definitions are consistent since ℤ(X_{+}, ∗) = ℤX.

To map downward to D^{−}(PST), we use the fact the abelian category Ab^{Sm^{op}} has enough projectives: every representable presheaf ℤh_{Y} is projective and every projective is a summand of a direct sum of representable presheaves. Since we may replace any bounded above complex by a projective resolution, the derived category D^{−}(Ab^{Sm^{op}}) is equivalent to the triangulated category K of bounded above complexes of projective presheaves (and chain homotopy equivalence classes of maps). By the Yoneda lemma, a map ℤh_{X} → ℤh_{Y} is the same as an element of ℤh_{Y} (X), i.e., a ℤ-linear combination of elements of Hom_{Sm/k}(X, Y). It follows that the canonical embedding Sm/k → Cor_{k} induces a functor

D^{−}(Ab^{Sm^{op}}) ≅ K → D^{−}(PST),

sending ℤh_{Y} to ℤ_{tr}Y, considered as a complex in degree zero.

Because the vertical functor from presheaves of sets to D^{−}(Cor_{k}) sends the presheaf h_{Y} to ℤ_{tr}Y, the left rectangle in the main diagram commutes. The composite map sends the Tate object T to the complex ℤ_{tr}𝔾_{m} → ℤ_{tr}(𝔸^{1}, 1), and sends the simplicial circle S_{s}^{1} to the complex ℤ → ℤ_{tr}(𝔸^{1}, 1); these complexes are concentrated in cohomological degrees −1 and 0, and are 𝔸^{1}-weak equivalent to ℤ_{tr}𝔾_{m}[1] and ℤ[1], respectively.

Sheafifying with respect to the Nisnevich topology immediately yields a functor H from spaces to D^{−}(Nis. ST). We shall call it the Hurewicz functor, because

X(S^{i})=Hom(h_{S^{i}}, X) →^{H} Hom_{D^{−}}(ℤ_{tr}(S^{i}), HX)→Hom_{DM}{− \atop eff}(ℤ[i], HX)

is the analogue of the Hurewicz map π_{i}(X) → H_{i}(X) in ordinary homotopy theory; it is obtained by replacing a space by a nicer space, abelianizing and passing to the associated complex (which is well defined up to quasi-isomorphism). The birth certificate of the Hurewicz functor ensures that the second rectangle commutes in the main diagram.

By construction, the Hurewicz functor sends X ×𝔸^{1} → X to ℤ_{tr}(X × 𝔸^{1}) → ℤ_{tr}X, and hence sends 𝔸^{1}-weak equivalences of spaces to 𝔸^{1}-weak equivalences of complexes. Therefore it preserves localization, i.e., it induces a Hurewicz functor from the homotopy category Hot to DM_{eff}^{−}. As noted above, we have H(S_{s}^{1}) ≅ ℤ[1] and H(T) ≅ ℤ(1)[2].

The extension of H : Hot → DM_{eff}^{−} to a functor SW → DM^{−} is a formal consequence of the definitions and the following lemma, which shows that H sends T ∧ X to ℤ_{tr}(T) ⊗_{tr}^{𝕃} H(X) ≅ H(X)(1)[2].

Lemma 2 The functor H : Hot → DM_{eff}^{−} sends the smash product X ∧ Y of spaces to the total tensor product H(X) ⊗_{tr}^{𝕃} H(Y) in DM_{eff}^{−}.

Proof. If X̃ and Ỹ are unpointed presheaves, then ℤ(X̃ × Ỹ) = ℤX̃ ⊗ ℤỸ in Ab^{Sm^{op}} (where ⊗ is the presheaf tensor product). From this, a simple calculation shows that if X and Y are pointed presheaves then ℤ(X ∧ Y) = ℤX ⊗ ℤY. The same is true in D^{−}(Ab^{Sm^{op}}): if P → ℤX and Q → ℤY are projective resolutions, then P ⊗ Q → ℤ(X ∧ Y) is also a projective resolution (since projectives are flat in Ab^{Sm^{op}}).

Applying the additive functor K → D^{−}(PST) (induced from the map Sm/k → Cor_{k}) sends P, Q and P ⊗ Q to H(X), H(Y) and H(X ∧ Y), respectively. Since it sends ℤ(h_{U×V}) = ℤ(h_{U}) ⊗ ℤ(h_{V}) to ℤ_{tr}(U × V) = ℤ_{tr}(U) ⊗_{tr} ℤ_{tr}(V), it also sends P ⊗ Q to H(X) ⊗_{tr}^{𝕃} H(Y), whence the result.

The fact that H extends to a functor from SHot to DM follows easily from the following lemma. This makes the final rectangle in the main diagram commute, completing our description of the road map.

Lemma 3 For each T -spectrum E, the stable motivic complex H(E) = {H(E_{n})} is well defined up to stable weak equivalence. For each morphism f : E → F there is a well defined morphism H(E) → H(F) in DM.

To see this, choose a projective resolutions P_{n} → ℤE_{n} and maps ℤT ⊗P_{n} → P_{n+1}; these choices are unique up to chain homotopy equivalence. Applying Sm/k → Cor_{k} yields the structure maps H(E_{n})(1)[2] → H(E_{n+1}) turning H(E_{0}), H(E_{1}), … into a stable motivic complex H(E); a different family of choices would yield a stably weak equivalent complex.

Similarly, a choice of Q_{n} → ℤF_{n}, etc. yields a stable motivic complex H(F). A morphism f yields a family of maps f_{n} : P_{n} → Q_{n} and H(f_{n}) : H(E_{n}) → H(F_{n}), well defined up to chain homotopy equivalence. This may not be a function because the two maps from H(E_{n})(1)[2] to H(F_{n+1}) are only chain homotopic. To repair this, we replace Q_{n} by the mapping cylinder cyl(f_{n}) and use the structure maps from ℤT ⊗ cyl(f_{n}) ≅ cyl(ℤT ⊗ f_{n}) to cyl(f_{n+1}). Then the desired morphism in DM is the composite of the function H(E) → cyl(f) and the inverse of the stable weak equivalence H(F) → cyl(f).

References

[Dieu] J. Dieudonné “History of Algebraic Geometry.” Wadsworth (1985); Originally published 1974 in French

[Hov] M. Hovey “Spectra and symmetric spectra in general model categories.” J. Pure Appl. Algebra 165 (2001), no. 1, 63–127.

[MV] F. Morel and V. Voevodsky “𝔸^{1}-homotopy theory of schemes.” Publ. IHES 90 (2001), 45–143

[MVW] C. Mazza, V. Voevodsky and C. Weibel “Lectures on Motivic cohomology.”

       Preprint (2002), available at http://math.rutgers.edu/~weibel

[Sp] M. Spitzweck “Operads, Algebras and Modules in Model Categories and Motives.” Ph.D. Thesis, Bonn (2001)

[SW] E. Spanier and J.H.C. Whitehead “A first approximation to homotopy theory.” Proc. Nat. Acad. Sci. USA 39 (1953), 655-660

[V] V. Voevodsky “Triangulated categories of motives over a field.” Cycles, transfers and motivic homology theories, Princeton Univ. Press (2000), 188–238

[V1] V. Voevodsky “𝔸^{1} Homotopy Theory of motives over a field.” Proc. 1998 Berlin ICM, Doc. Math. (1998), 579–604

[V2] V. Voevodsky “Cancellation Theorem.” Preprint (2002), available at http://

       www.math.uiuc.edu/K-theory/0541/

[VW] V. Voevodsky and C. Weibel “Voevodsky’s Seattle Lectures: K-theory and Motivic Cohomology.” Proc. Symp. Pure Math. 67 (1999) 283–303