We construct a 'triangulated analogue' of coniveau spectral sequences: the motif of a variety over a countable field is 'decomposed' (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodsky's motives. To this end we construct a 'Gersten' weight structure for a certain triangulated category of 'comotives': the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology of its generic fibre; cohomology of function fields contain twisted cohomology of their residue fields (for all geometric valuations). We also develop further the general theory of weight structures for triangulated categories (independently from the 'motivic' part of the paper). Besides, we develop a certain theory of 'nice' pairings of triangulated categories; this subject seems to be new.
Deep Dive into Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields.
We construct a ’triangulated analogue’ of coniveau spectral sequences: the motif of a variety over a countable field is ‘decomposed’ (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodsky’s motives. To this end we construct a ‘Gersten’ weight structure for a certain triangulated category of ‘comotives’: the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology
be expressed in terms of the (homotopy) t-truncations of Y ; this extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the comotif of a smooth connected semi-local scheme is a direct summand of the comotif of its generic point; comotives of function fields contain twisted comotives of their residue fields (for all geometric valuations). Hence similar results hold for any cohomology of (semi-local) schemes mentioned.
Let k be our perfect base field. We recall two very important statements concerning coniveau spectral sequences. The first one is the calculation of E 2 of the coniveau spectral sequence for cohomological theories that satisfy certain conditions; see [BOg94] and [CHK97]. It was proved by Voevodsky that these conditions are fulfilled by any theory H represented by a motivic complex C (i.e. an object of DM ef f -; see [Voe00a]); then the E 2 -terms of the spectral sequence could be calculated in terms of the (homotopy t-structure) cohomology of C. This result implies the second one: H-cohomology of a smooth connected semilocal scheme (in the sense of §4.4 of [Voe00b], i.e. actually an affine essentially smooth one) injects into the cohomology of its generic point; the latter statement was extended to all (smooth connected) primitive schemes by M. Walker.
The main goal of the present paper is to construct (motivically) functorial coniveau spectral sequences converging to cohomology of arbitrary motives; there should exist a description of these spectral sequences (starting from E 2 ) that is similar to the description for the case of cohomology of smooth varieties (mentioned above).
A related objective is to clarify the nature of the injectivity result mentioned; it turned our that (in the case of a countable k) the cohomology of a smooth connected semi-local (more generally, primitive) scheme is actually a direct summand of the cohomology of its generic point. Moreover, the (twisted) cohomology of a residue field of a function field K/k (for any geometric valuation of K) is a direct summand of the cohomology of K. We actually prove more in §4. 3.
Our main homological algebra tool is the theory of weight structures (in triangulated categories; we usually denote a weight structure by w) introduced in the previous paper [Bon10]. In this article we develop it further; this part of the paper could be interesting also to readers not acquainted with motives (and could be read independently from the rest of the paper). In particular, we study nice dualities (certain pairings) of (two distinct) triangulated categories; it seems that this subject was not previously considered in the literature at all. This allows us to generalize the concept of adjacent weight and t-structures (t) in a triangulated category (developed in §4.4 of [Bon10]): we introduce the notion of orthogonal structures in (two possibly distinct) triangulated categories. If Φ is a nice duality of triangulated C, D, X ∈ ObjC, Y ∈ ObjD, t is orthogonal to w, then the spectral sequence S converging to Φ(X, Y ) that comes from the t-truncations of Y is naturally isomorphic (starting from E 2 ) to the weight spectral sequence T for the functor Φ(-, Y ). T comes from weight truncations of X (note that those generalize stupid truncations for complexes). Our approach yields an abstract alternative to the method of comparing similar spectral sequences using filtered complexes (developed by Deligne and Paranjape, and used in [Par96], [Deg09], and [Bon10]). Note also that we relate t-truncations in D with virtual t-truncations of cohomological functors on C. Virtual t-truncations for cohomological functors are defined for any (C, w) (we do not need any triangulated ‘categories of functors’ or t-structures for them here); this notion was introduced in §2.5 of [Bon10] and is studied further in the current paper. Now, we explain why we really need a certain new category of comotives (containing Voevodsky’s DM ef f gm ), and so the theory of adjacent structures (i.e. orthogonal structures in the case C = D, Φ = C(-, -)) is not sufficient for our purposes. It was already proved in [Bon10] that weight structures provide a powerful tool for constructing spectral sequences; they also relate the cohomology of objects of triangulated categories with t-structures adjacent to them. Unfortunately, a weight structure corresponding to coniveau spectral sequences cannot exist on DM ef f -⊃ DM ef f gm since these categories do not contain (any) motives for function fields over k (as well as motives of other schemes not of finite type over k; still cf. Remark 4.5.4 (5)). Yet these motives should (co)generate the heart of this weight structure (since the objects of this heart should corepresent covariant exact functors from the category of homotopy invariant sheaves with transfers to Ab).
So, we need a category that would contain certain homotopy limits of objects of DM ef f gm . We succeed in constructing a
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