Motivic Homotopy

Oct. 14, 2004

Examples of Quillen’s homology for model categories.

• In the category of spaces. Given a space X, we produce the free abelian topological group on X, denoted by ℤX. (If X is not cofibrant we should replace X by a cofibrant approximation first.) The Dold-Thom theorem says that π_{∗}(ℤX) ≅ H_{∗}(X).

• In the category of simplical spaces, a simplical set X_{·} goes to the simplicial abelian group ℤX_{·} whose n’th group is ℤX_{n}. There is a Dold-Kan correspondence between simplicial abelian groups and nonnegatively graded chain complexes. Given a simplicial abelian group A_{·} one can form the chain complex N(A_{·}), whose n’th term is ∩_{i=1}^{n}ker(d_{i} : A_{n} → A_{n−1}), with boundary operator d_{0}. In the case A_{·} = ℤX_{·}, this gives a chain complex chain equivalent to the standard chain complex used to compute the homology H_{∗}(X; ℤ).

• In the category of augmented simplicial commutative rings over a fixed base ring R, we have the following model structure. The fibrations are levelwise surjections of simplicial rings, and the weak equivalences are maps which induce a homotopy equivalence of underlying spaces (or, by the Dold-Kan correspondence, a homology equivalence between the corresponding chain complexes). An abelian group object in this category is a simplicial augmented R-algebra A whose augmentation ideal I is squarezero; in other words, the category of abelian group objects is equivalent to the category of R-modules. For a general ring A over R, the “abelianization” functor from simplicial rings to R-modules is the functor A ↦ Ω_{A} = I/I^{2}, where I is the kernel of the map A → R. Quillen homology is gotten by first forming a resolution of A by a cofibrant simplicial ring over R and then applying Ω levelwise.

Definition. A sheaf M of abelian groups on V_{Nis} (the site of smooth varieties in the Nisnevich topology) is 𝔸^{1}-invariant if M(X × 𝔸^{1}) → M(X) is an iso for all objects X. (This is similar to being 𝔸^{1}-local.) M is strictly 𝔸^{1}-invariant if H^{n}(X × 𝔸^{1}, M) ← H^{n}(X, M) is an isomorphism for all n and X.

Examples.

• Constant sheaves A are strictly 𝔸^{1} invariant.

• (Theorem of Quillen) The sheaf associated to the presheaf U ↦ K_{i}(U), algebraic K-theory.

• (Theorem of Voevodsky) If M is 𝔸^{1}-invariant and has transfers for branched covers, them M is strictly 𝔸^{1}-invariant. (This is hard to prove.) Note that these transfers aren’t necessarily smooth, and so these don’t really live on the site V_{Nis}; to get at them, one defines a category of correspondences from X to Y as a subset of the free abelian group on cycles of X × Y that are finite over X.

• The Voevodsky motivic complexes ℤ(n).

Aside. One model (due to Suslin, from around 1986 or so) for ℤ[Y] (or Sym^{∞}(Y)) is to consider the free abelian group on all correspondences (“multivalued functions”) X → Y, which is not quite homotopy invariant. We get a functor Corr(−, Y)_{+}. To make this homotopy invariant, consider the complex Corr(− × Δ^{·}, Y)_{+}, where Δ^{·} is the cosimplicial scheme given by

Δ^{n} = Spec ℤ[t_{0}, …, t_{n}]/(Σt_{i} − 1).

This is a simplicial abelian group, and this functor automatically forms a sheaf. Taking the normalized chain complex gives us a chain complex of sheaves which we call the “homology type of Y ”, or ℤ_{M}Y. This is not the same thing as the Quillen homology.

Voevodsky used Y = 𝔸^{n} \ {0} to define Σ^{n−1}ℤ(n) = ℤ_{M}(𝔸^{n} \ {0}).

This definition gives H^{n}(Spec(k), ℤ(n)) = K_{n}^{M}(k), the n’th Milnor K-group.

Morel shows us that if we replace ℤ_{M} by ℤ, the actual Quillen homology, then K_{n}^{M}(k) is replaced by the Grothendieck-Witt ring when n = 0. (We haven’t yet defined what the higher Grothendieck-Witt ring is.)

Morel’s theorem shows that the sheaves π_{k}^{𝔸^{1}}(X_{·}), for X_{·} any simplicial Nisnevich sheaf, are always strictly 𝔸^{1}-invariant sheaves.

Example. Balmer shows 𝕎(−) (the Witt ring sheaf) is an 𝔸^{1} homotopy group for an analogue of L-theory. Morel’s theorem then implies that it is strictly 𝔸^{1}-invariant. Review. V_{Nis} is the site of smooth varieties in the Nisnevich topology over a fixed field k. Δ^{op}Sh(V_{Nis}) is the category of simplicial sheaves on V_{Nis}.

There are two homotopy theories that go along with this category.

• Combinatorial homotopy theory. Here cofibrations are monomorphisms, and weak equivalences are stalkwise weak equivalences. This is the Joyal-Jardine model structure. (As usual, the fibrations are forced.)

• The 𝔸^{1} model category structure. Here cofibrations are still monomorphisms, but the weak equivalences are 𝔸^{1}-homotopy equivalences.

For each of these, there is a homotopy category. We denote by H_{s} the homotopy category of the first model category - the s is for simplicial. If the field is ambiguous, denote it by H_{s}(k). Denote the homotopy category of the 𝔸^{1} model category by H.

We now recall the notion of an 𝔸^{1}-weak equivalence. Conceptually, this is formed by adding in all weak equivalences generated by using 𝔸^{1} weak equivalences, local weak equivalences, etc. The following is a shortcut method for defining the 𝔸^{1} weak equivalences.

Definition. Z ∈ H_{s} is 𝔸^{1}-local if H_{s}(X × 𝔸^{1}, Z) ← H_{s}(X, Z) is a bijection for all X.

Definition. X → Y is an 𝔸^{1}-equivalence if H_{s}(Y, Z) → H_{s}(X, Z) is bijective for all 𝔸^{1}-local Z.

However, this definition gives us very little control over the 𝔸^{1} equivalences.

Example. Suppose M is a strictly 𝔸^{1}-invariant sheaf of abelian groups. We can form the simplicial sheaf K(M, n) which corresponds to the chain complex with M in degree n and 0 otherwise. Then there exists a tautological isomorphism, for U a smooth scheme,

H_{s}(U, K(M, n)) ≅ H^{n}(U; M).

Here U is viewed as a constant simplicial sheaf.

For a general simplicial smooth variety X_{·}, H_{s}(X_{·}, K(M, n)) can be computed with a hypercohomology spectral sequence with terms H^{n−i}(X_{i}, M). Since M is strictly 𝔸^{1}-invariant, H_{s}(X_{·}, K(M, n)) ← H_{s}(X_{·} ×𝔸^{1}, K(M, n)) is an isomorphism for all simplicial smooth varieties X_{·}.

Every sheaf is a direct limit of representable sheaves, so K(M, n) is an 𝔸^{1}-local object. Now we will discuss two theorems of Morel.

More generally than the previous case, suppose C_{∗} is a chain complex with associated simplicial sheaf K(C_{∗}). (Note that if

C_{∗} = {… C_{2} → C_{1} → C_{0} → C_{−1} → …},

K(C_{∗}) only uses C_{0}, C_{1}, . . ..

Theorem. (Morel) If C_{∗} is 0-connected, i.e. C_{i} = 0 for i ≤ 0, then K(C_{∗}) is 𝔸^{1}-local if and only if C_{∗} is strictly 𝔸^{1}-invariant. (Here 𝔸^{1}-invariance for C_{∗} is defined in terms of hypercohomology.)

So: If we want to check M is strictly 𝔸^{1}-invariant, consider the chain complex … 0 → 0 → M → 0, which corresponds to K(M, 1). We can check the 𝔸^{1}- invariance of M by checking only if K(M, 1) is 𝔸^{1}-local, which requires only looking at H^{0}(−; M) and H^{1}(−; M).

Now, let H_{s}^{𝔸^{1}} ⊂ H_{s} be the full subcategory of 𝔸^{1}-local objects. The composition H_{s}^{𝔸^{1}} → H_{s} → H is an equivalence of categories. (This is a formal result.) There exists a “localization” functor L^{𝔸^{1}} : H_{s} → H_{s} which is idempotent, left adjoint to the inclusion of H_{s}^{𝔸^{1}} in H_{s}.

There is an analagous story for the derived category D of complexes of Nisnevich sheaves of abelian groups, with the subcategory D^{𝔸^{1}} of strictly 𝔸^{1}-invariant complexes. There is a localization functor, also denoted by L^{𝔸^{1}}, from D to D^{𝔸^{1}}, with similar properties.

Theorem. (Morel, generalizing a theorem of Voevodsky for k perfect) If C_{∗} ∈ D is (−1)-connected, then L_{𝔸^{1}}C_{∗} is weakly equivalent to a complex D_{∗} which is also (−1)-connected.

(Equivalently, L^{𝔸^{1}} preserves connectivity. This is one of a series of connectivity results of Morel’s.)

Homotopy sheaves.

Suppose X ∈ Δ^{op}Sh(V_{Nis}). (or: X is a “space”.)

For every smooth U, we can associate the simplicial set X(U). Without loss of generality, we can alter X by an equivalence so that X(U) is fibrant for all U. Given x ∈ X(k), restriction gives a point x ∈ X(U) for all U, and then π_{n}(X(U), x) is a presheaf of sets. Call the associated sheaf π_{n}(X). (This is really a notion belonging to combinatorial homotopy theory.)

Definition. π_{n}^{𝔸^{1}}(X) = π_{n}(L^{𝔸^{1}}X).

Definition. X is (n − 1)-connected if π_{i}(X) = 0 for i < n, and all base fields.

Theorem. (Morel) If X is (n−1)-connected, then L^{𝔸^{1}}(X) is also. Furthermore, π_{n}(L^{𝔸^{1}}X) = H_{n}^{𝔸^{1}}X - the Quillen homology. H_{n}^{𝔸^{1}}(X) = π_{n}(ℤ^{𝔸^{1}}X).

Morel explicitly calculates this for 𝔸^{n+1} \ {0}.

Morel shows that strictly 𝔸^{1}-invariant sheaves form an abelian category - the heart of a t-structure - and so Postnikov towers work in the 𝔸^{1}-homotopy category. His program is an analogue of Serre’s theory of classes of finite groups. One main obstacle remaining in the theory is that we don’t yet know finite generation of basic 𝔸^{1} homotopy groups.