The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with $Z[1/p]$-coefficients over a perfect field $k$ of characteristic $p$ generate the category $DM^{eff}_{gm}[1/p]$ (of effective geometric Voevodsky's motives with $Z[1/p]$-coefficients). It follows that $DM^{eff}_{gm}[1/p]$ could be endowed with a Chow weight structure $w_{Chow}$ whose heart is $Chow^{eff}[1/p]$ (weight structures were introduced in a preceding paper, where the existence of $w_{Chow}$ for $DM^{eff}_{gm}Q$ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor $DM^{eff}_{gm}[1/p]\to K^b(Chow^{eff}[1/p])$ (which induces an isomorphism on $K_0$-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on $DM^{eff}_{gm}[1/p]$. We also define a certain Chow t-structure for $DM_{-}^{eff}[1/p]$ and relate it with unramified cohomology. To this end we study birational motives and birational homotopy invariant sheaves with transfers.
It is well known that Hironaka's resolution of singularities is very important for the theory of (Voevodsky's) motives over characteristic 0 fields; see [Voe00a], [GiS96], and also [Bon09a] and [Bon10a].
The purpose of this paper is to derive (as many as possible) ‘motivic’ consequences from the recent resolution of singularities result of Gabber (see Theorem 1.3 of [Ill08]). His result could be called ‘Z (l) -resolution of singularities’ over a perfect characteristic p field k (where l is any prime distinct from p). Surprisingly Gabber’s theorem is sufficient to extend all those properties of Voevodsky’s motives (with integral coefficients, over characteristic 0 fields) that were proved in [Bon10a], to Z[ 1 p ]-motives over k. In particular (in the notation of §1.1) we prove the existence of a conservative exact weight complex functor DM ef f gm
). We also establish the existence of (Chow)-weight spectral sequences for any cohomology theory defined on DM ef f gm [ 1 p ] (those generalize Deligne’s weight spectral sequences). Previously the results mentioned were known to hold only for motives with rational coefficients (in preceding papers we noted that these rational coefficient versions can be proved using de Jong’ s alterations, but did not give detailed proofs). Since the results of this paper also hold for motives with coefficients in any Z[ 1 p ]-algebra, as a by-product we justify these claims (in more detail than before).
Most of the results of this paper are already known for char k = 0 and motives (and cohomology) with integral coefficients. Yet we prove some results on birational motives and birational sheaves (see § §3.3-3.4) that are partially new for this case also; note that our proofs work (without any changes) in this alternative setting.
The central ’technical’ notion of this paper is the one of weight structure. Weight structures are natural counterparts of t-structures for triangulated categories, introduced in [Bon10a] (and independently in [Pau08]). They were thoroughly studied and applied to motives in [Bon10a] and [Bon10b] (see also the survey preprint [Bon09s]). Weight structures allow proving several properties of motives. In particular, most of the results mentioned above follow from the following (central) theorem: DM ef f gm [ 1 p ] can be endowed with a weight structure w Chow whose heart is Chow ef f [ 1 p ]. The language of weight structures is also crucial for our proof of this statement (even though the main difficulty was to prove that Chow ef f [ 1 p ] generates DM ef f gm [ 1 p ] as a triangulated category). In contrast, note that the methods of Gillet and Soulé (whose weight complex functor defined in [GiS96] is the ‘first ancestor’ of ‘our weight complexes’) only allow proving the existence of weight complexes either with values in K b (Chow ef f Q) or in the category of unbounded complexes of Z (l) -Chow motives; cf. Remark 3.2.2 below. Now we list the contents of the paper. More details can be found at the beginnings of sections.
In the first section we recall some basic properties of motives and weight structures. Most of them are just modifications of some of the results of [Voe00a] and [Bon10a]; the only absolutely new result is a new condition for the existence of weight structures. We also recall a recent result on resolution of singularities over characteristic p fields (proved by O. Gabber), and deduce certain (immediate) motivic consequences from it.
In §2 we prove our central theorem on the existence of the Chow weight structure for DM ef f gm [ 1 p ]; we deduce this result from its certain Z (l) -version. §3 is dedicated to the applications of the central theorem (yet we deduce some of the results directly from the Gabber’s one). We prove that the Chow weight structure can be extended to
)). Also, there exists a conservative exact weight complex functor DM gm [
). The existence of the Chow weight structure also implies the existence of canonical DM ef f gm [ 1 p ]-functorial (starting from E 2 ) Chow-weight spectral sequences that express (any) cohomology of objects of DM gm [ 1 p ] in terms of that of Chow motives. As was shown in [Bon10a], these spectral sequences generalize the weight spectral sequences of Deligne (note that one can take any cohomology theory and Z[ 1 p ]-coefficients here). Next we prove that the Chow weight structure yields a weight structure for the category of birational motives i.e. for (the idempotent completion of) the localization of DM ef f gm [ 1 p ] by DM ef f gm 1 p (see [KaS02]); its heart contains birational motives of all smooth varieties. We also study birational sheaves. Next we prove the existence of a certain Chow t-structure for
Our results allow us to express unramified cohomology in terms of the Chow t-structure cohomology of homotopy invariant sheaves with transfers.
Lastly, we recall that a method of M. Levine (described in [HuK06], and combined with the fact that Chow[ The idea to write this pap
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