Dimensional homotopy t-structures in motivic homotopy theory

Abstract

International audienceThe aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these t-structures possess many good properties, some analogous to those of the perverse t-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that the hearts of these t-structures provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko

Perfection in motivic homotopy theory

“…Remark As remarked in [, Remark 7.4], Theorem implies the formalism of Grothendieck–Verdier duality for for locally of finite type ‐schemes. In particular, this gives an improvement of [, Theorem 2.4.8].…”

“…Proof If is perfect, the statement is [, Corollary 2.4.7]. In general, the morphism induces an equivalence by Corollary .…”

The homotopy Leray spectral sequence

“…In order to be able to apply the construction of the δ-homotopy t-structure of [BD17], we will require that T satisfies the following assumptions:…”

“…The goal of this section is to review a variant of the homotopy t-structure analyzed in [BD17] that makes sense over rather general base schemes. In more detail, Section 2.1 reviews the necessary theory from [BD17] and fixes additional notations and conventions to be used in the sequel. We highlight here Theorem 2.1.4 which summarizes the required existence result and Point 2.1.8 which describes homological conventions for t-structures that will be in force throughout the paper.…”

“…The jumping off point is the work of the second author exploring the close relationship between the notion of cycle module in the sense of Rost and the heart of a t-structure on the motivic stable homotopy category. We use a relative version of Morel's homotopy t-structure, the so-called perverse homotopy t-structure, defined initially by Ayoub [Ayo07] and developed further in [BD17]. The approach to constructing this t-structure in [BD17] is rather flexible and works in great generality: the basic definition requires only the existence of a dimension function δ on the base scheme; for this reason, the t-structure was called the δ-homotopy t-structure in op.…”