Notes on T -delooping

by Vladimir Voevodsky

Dec. 2000

Dear Fabien,

I probably reached some progress on the question of the recognition for T -loop spaces and T -delooping which we discussed recently. Here is how things are.

Observe first that one can study T -loop spaces in three different contexts. First one may consider T -loop spaces in the (pointed) unstable A^{1}-homotopy theory. This is the most interesting but also the most difficult case. Second, one can consider T -loop spaces in the s-stable A^{1}-homotopy theory. Third, one can consider T -loop spaces in the A^{1}-homotopy theory of sheaves of abelian groups.

The diffrence between the second and the third context is not so great, in particular they are equivalent rationally. The diffrence between the fisrt and the second is more serious but it is still a matter of s-stabilization which is in the realm of the ordinary homotopy theory (modulo the A^{1}-locality problem which we talked about?). At least one application of the would-be theory of T-loop spaces requires only the s-stable context - the new way of constructing the motivic spectral sequence for algebraic K-theory outlined in [?] (a more detailed account is being written).

In what follows I want to concentrate on the context of the homotopy theory of sheaves of abelian groups and consider most basic case of the delooping problem. Namely, I would like to find a T-analog of the folowing two theorems in topology:

Theorem 0.1 [th1top] Let X be a discrete space. Then finding a representation of X as a loop space is equivalent to giving a group structure on X.

Theorem 0.2 [th2top] Let X be a group. Then Ω^{−1}(X) has a loop space structure if and only if X is abelian. In that case this loop space structure is unique and is, also, an ∞-loop space structure.

For motivic analogs of these results I want to consider, instead of discrete spaces X, T -rigid, strictly homotopy invariant sheaves of abelian groups F: a sheaf is called T -rigid if F_{−1} = 0. We ask the following question: given such a sheaf F, what structure on F defines a representation of F as a T -loop space? What additional structure or what property of the original one is needed for F to be a 2-loop space (resp. n-loop space).

Here is my conjectural answer. I only know how to formulate it in the case of schemes over a field of characteristic zero.

Conjecture 1 [conj1] A T -loop space structure on F is determined by giving F transfer maps for all finite etale coverings which are additive and commute with the base change.

Conjecture 2 [conj2] Let F be a T -rigid, strictly homotopy invariant sheaf with etale transfers. Then Ω_{T}^{−1}(F) is a T -loop space if and only if for a sequence of finite etale maps X →^{f} Y →^{g} Z one has

[com]tr_{gf} = tr_{g}tr_{f}  (0.1)

In this case the T -loop space structure on Ω_{T}^{−1}(F) is unique and is an ∞-loop space structure.

Here are some arguments towards these conjectures.

Lemma 0.3 [l1] Let G be a an A^{1}-local complex of sheaves and assume that Ω_{T}^{1} (G) is quasi-isomorphic to a rigid sheaf F. Then F has canonical transfers for finite etale maps which are additive and commute with the base change.

Note that under the assumptions of the lemma F is automatically strictly homotopy invariant and that F = \underline{H}_{0}(Ω_{T}^{1} (G)). Proof: We knew it fo a long time.

Lemma 0.4 [l2] In the notations of Lemma 0.3, assume that G = Ω_{T}^{1} (H) for an A^{1}-local H. Then the transfers on F satisfy the condition (0.1)

Proof: ???

Proposition 0.5 [l3] Let F be a strictly homotopy invariant rigid sheaf with etale transfers satisfying (0.1). Then F is a sheaf with transfers.

Proof: ???

Corollary 0.6 Let F be a strictly homotopy invariant rigid sheaf with etale transfers satisfying (0.1). Then F is an ∞-loop space.

These results were, kind of, known. Here comes something new. Let F be as in Conjecture 1. I want to define its T -delooping B_{T} F as follows. For a smooth connected scheme S over k define B_{T} F (S) as the colimit over all closed subsets Z of A_{S}^{1} which are equidimensional of relative dimension 0 over S of the group F (Z_{0}) where Z_{0} is the fiber of Z → S over the generic point of S considered as a smooth “variety” over k. We now need to define the pullback homomorphisms B_{T} F (S) → B_{T} F (S′) for morphisms S′ → S. Since a section of B_{T} F (S) is defined by its restriction to the genric point and since it is clear how to define the pull-back homomorphisms for dominant morphisms it is sufficient to define specialization and check that its is “transitive”.

We first define specialization to a point s ∈ S of codimenison 1. Localizing S in s we may assume that S is the spectrum of a disceret valuation ring. Let Z be our closed subset which we consider as a reduced scheme, Z_{0} the generic fiber of Z and f ∈ F (Z_{0}) the section. Let Z̃ be the normalization of Z. It is a normal scheme of dimension ≤ 1 and, therefore, it is smooth. Since F is rigid the section of F over Z_{0} will extend to a unique section of F over Z̃. Let Z_{cl} be the reduced closed fiber of Z and let Z̃_{cl} = (Z̃ ×_{Z} Z_{cl}). We have an etale map p : Z̃_{cl} → Z_{cl}. Set

(Z, f)_{s} = (Z_{cl}, p_{∗}(f̃_{|Z̃_{cl}}))

Let now s be a point of arbitrary codimension. Let S_{s} be the blow-up of S in s and s′ be the genric point of the special fiber of p : S_{s} → S. Then, using the construction of the previous paragraf we can define the specialization of p^{∗}(Z, f) to any point in an open neighborhood of s′. Let s̃ be any k-point in this open neighborhood lying over s. Define (Z, f)_{s} as (p^{∗}(Z, f))s̃.

Lemma 0.7 [l4] The construction described above define a structure of a presheaf on B_{T} F.

Proof: ???

Example 0.8 [ex1]Let F = Z, then B_{T} F described above is just z_{equi}(A^{1}, 0), which is known to be a model for K(Z(1), 2).

Let us now outline a proof of the following:

Pretheorem 0.9 [prth1] Let G and F be as in Lemma 0.3 and assume in addition that G is T-connected. Then B_{T} F is A^{1}-weakly equivalent to G.

Recall that an object is called T -connected if it belongs to the localizing subcategory generated by T -suspensions of smooth schemes, or, in our context of sheaves of abelian groups, by objects of the form Z(X) ⊗ Z(T). The proof of Pretheorem 0.9 consists of two parts.

Consider first the following construction. For a fibrant A^{1}-local complex K define s_{0}(K) as the complex of sheaves associated with the complex of presheaves given by

S ↦ colim_{Z⊂A_{S}^{1}}K(A_{S}^{1} − Z)

where Z runs through all closed subsets in A_{S}^{1} equidimensional of relative dimension 1 over S. Let further τ_{≥1}^{T}(K) be the fiber of the obvious morphism \underline{Hom}(Z(A^{1}), K) → s_{0}(K) such that we have a distinguiished triangle of complexes of sheaves of the form

[eq1]τ_{≥1}^{T}(K) → \underline{Hom}(Z(A^{1}), K) → s_{0}(K)  (0.2)

Pretheorem 0.9 is a corollary of the following two results.

Proposition 0.10 [pr1] For any A^{1}-fibrant complex G such that Ω_{T}^{1} (G) ≅ \underline{H}_{0}(Ω_{T}^{1} (G)) and Ω_{T}^{1} (G) is rigid, one has a canonical quasi-isomorphism of complexes

B_{T} \underline{H}_{0}(Ω_{T}^{1} (G)) = τ_{≥1}^{T}(G)

Proof: ???

Proposition 0.11 [pr2] Let G be a T -connected fibrant complex. Then s_{0}(G) is A^{1}-weakly equivalent to zero.

Proof: ???

Proposition 0.11 togther with the following easy lemma, implies that (0.2) gives a model for the canonical decomposition of any G into a T -connected and a T -rigid part.

Lemma 0.12 [elemma] If G is rigid then G → s_{0}(G) is a quasi-isomorphism i.e. τ_{≤1}^{T}(G) = 0.

Proof: ???