On the conservativity of the functor assigning to a motivic spectrum its motive

Abstract

Given a 0-connective motivic spectrum E ∈ SH(k) over a perfect field k, we determine h 0 of the associated motive M E ∈ DM(k) in terms of π 0 (E). Using this we show that if k has finite 2-étale cohomological dimension, then the functor M : SH(k) → DM(k) is conservative when restricted to the subcategory of compact spectra, and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-étale cohomological dimension by considering what we call "real motives". h 0 (E) ∼ = π 0 (E)/η.Here we use implicitly that the heart of DM(k) can be identified with a subcategory of the heart of SH(k). This theorem shows that we lose some information in passing to the motive, and so cannot expect a perfect analogy with the classical situation. Nonetheless we can prove the following.Theorem (Conservativity Theorem I; see Theorem 16). Let k be a perfect field of finite 2-étale cohomological dimension and E ∈ SH(k) be compact. If M E ≃ 0 then E ≃ 0.(Recall that an object is called compact if Hom SH(k) (E, •) commutes with arbitrary sums.) Combined with the Motivic Hurewicz Theorem, one easily obtains the following.Corollary (Pic-injectivity Theorem; see Theorem 18). In the situation of the Theorem, the natural homomorphism P ic(SH(k)) → P ic(DM(k)) is injective.This result was one of the main motivations for our investigations. See later in this introduction for more details on applications. (We do not know if P ic(SH(k)) → P ic(DM(k)) might also be surjective.)The Conservativity Theorem (and by extension the Pic-injectivity Theorem) stated here is not optimal. Let us discuss to what extent the assumptions can be weakened.In characteristic zero, the compactness assumption can be replaced by the technical assumptions of connectivity and "slice-connectivity" (see Section 4 for a definition of this term). In characteristic p > 0 we can also replace compactness by connectivity and slice-connectivity, but in this case we must additionally assume that the natural map E p − → E is an isomorphism. If k is a non-orderable field (i.e. one in which -1 is a sum of squares; we still assume k perfect), then one may present k as a colimit of perfect subfields with finite 2-étale cohomological dimension, and the theorem can then be deduced for such k, see [5]. Compactness instead of connectivity and slice-connectivity is then essential, however.The author currently does not know how to weaken the perfectness assumption on k.A further weakening is to consider orderable k, i.e. fields where -1 is not a sum of squares. (Such fields necessarily have characteristic zero.) Assume for simplicity that for any ordering of k, there is an order preserving embedding of k into R. Write Sper(k) for the set of orderings of k. (Equivalently in this case, embeddings k ֒→ R.) If σ ∈ Sper(k), then there is a so-called real realisation functor R R σ : SH(k) → SH, related to considering the real points of a smooth variety, with their strong topology. We compose this with the singular chain complex functor to obtain the real ...

Perfection in motivic homotopy theory

“…Remark Using Theorem , we can deduce the Pic‐injectivity result of [, Theorem 18]. This extends Bachmann's results on Po Hu's conjecture on invertibility of the suspension spectra of affine quadrics to imperfect fields (see [4] for details).…”

“…In view of the cartesian square as in [, (3.1)] (cf. [, Lemma 17; , Lemma 1.16]), it will in fact suffice to only check invertibility in and in . The former is obvious.…”

“…We also have the following variant of Bachmann's conservativity theorem . Theorem Let be a field with finite 2‐étale cohomological dimension and exponential characteristic .…”

“…Proof Using Corollary and the analogous result for mixed motives [, Lemma 3.15], we may replace by its perfection. Then the result is proven in [, Theorem 16].…”

“…We also have the following variant of Bachmann's conservativity theorem [4]. is conservative on compact objects.…”