The localization spectral sequence in the motivic setting

We construct and study a motivic lift of a spectral sequence associated to a stratified scheme, recently discovered by Petersen in the context of mixed Hodge theory and ℓ-adic Galois representations.The original spectral sequence expresses the compactly supported cohomology of an open stratum in terms of the compactly supported cohomology of the closures of strata and the combinatorics of the poset underlying the stratification.Some of its special cases are classical tools in the study of arrangements of subvarieties and configuration spaces.Our motivic lift lives in the triangulated category of étale motives and takes the shape of a Postnikov system.We describe its connecting morphisms and study some of its functoriality properties.We also study a dual version of our main theorem (Theorem 3.8) where we are interested in describing the object j * j * F .Due to the lack of duality in the general setting of algebraic derivators, we cannot simply repeat the proof.Instead, we rely on applying Poincaré-Verdier duality, but the latter is available at the motivic level only under certain assumptions (see §3.4).Note that, if we gave up on functoriality, then we would not need to work in the setting of algebraic derivators and could prove the dual statement (without functoriality) in full generality.This strongly suggests that the dual statement (with functoriality) is true in full generality, even though we are not able to prove it with our methods.In any case, if one is only interested in working with realizations, one can first apply a realization functor to the main theorem and then dualize.Perspectives.A natural direction of research would be to try and apply our main theorem to prove motivic representation stability results in the spirit of the homological representation stability results of Petersen Pet17.Also, it would be desirable to clarify the general functoriality properties of our construction, beyond those already explored here.A motivation for this project is the possibility to study a more general geometric setting mixing j ! and j * extensions, depending on the strata.The corresponding motives can be viewed as relative cohomology motives on some blow-up of the ambient variety and are ubiquitous in the geometric study of periods (see, e.g., Gon02 and the introduction of Dup17).Outline.In §1 we review classical definitions and properties of poset (co)homology; the only original (to the best of our knowledge) content is the introduction of connecting morphisms relating poset (co)homology complexes of different intervals in a poset.In §2 we work in the setting of triangulated derivators and collect some tools to produce and study functorial Postnikov systems.In §3 we apply those tools to our geometric setting and prove the main results.