[Paper Review] On perverse homotopy $t$-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures

Mikhail V. Bondarko|arXiv (Cornell University)|Sep 1, 2014

Algebraic structures and combinatorial models23 references3 citations

TL;DR

This paper introduces a new $t$-structure, $t_{\text{hom}}(S)$, on the triangulated category of motives $DM(S)$ over a general base scheme $S$, generalizing Voevodsky's homotopy $t$-structure over fields and Ayoub's perverse homotopy $t$-structure over discrete valuation rings. The construction relies on Borel-Moore motives and coniveau spectral sequences, and it is conjectured that the heart of $t_{\text{hom}}(S)$ consists of cycle modules over $S$, with a Gersten weight structure on comotives providing an alternative description.

ABSTRACT

We study the category $DM(S)$ of Beilinson motives (as described by Cisinski and Deglise) over a more or less general base scheme $S$, and establish several nice properties for a version $t_{hom}(S)$ of the perverse homotopy $t$-structure (essentially defined by Ayoub) for it. $t_{hom}(S)$ is characterized in terms of certain stalks of an $S$-motif $H$ and its Tate twists at fields over $S$; it is closely related to certain coniveau spectral sequences for the cohomology of (the Borel-Moore motives of) arbitrary finite type $S$-schemes. We conjecture that the heart of $t_{hom}(S)$ is given by cycle modules over $S$ (as defined by Rost); for varieties over characteristic $0$ fields this conjecture was recently proved by Deglise. Our definition of $t_{hom}(S)$ is closely related to a new effectivity filtration for $DM(S)$ (and for the subcategory of Chow $S$-motives in it). We also sketch the construction of a certain Gersten weight structure for the category of $S$-comotives; this weight structure yields one more description of $t_{hom}(S)$ and its heart.

Motivation & Objective

• To define a new $t$-structure $t_{\text{hom}}(S)$ on $DM(S)$ that generalizes known $t$-structures over fields and discrete valuation rings.
• To relate $t_{\text{hom}}(S)$ to coniveau spectral sequences for Borel-Moore motives of finite type $S$-schemes.
• To conjecture that the heart of $t_{\text{hom}}(S)$ is equivalent to the category of cycle modules over $S$, as defined by Rost.
• To establish a connection between $t_{\text{hom}}(S)$ and a Gersten weight structure on the category of $S$-comotives $\mathfrak{D}(S)$.
• To provide a framework for studying motivic cohomology via $\delta$-coniveau filtrations and weight spectral sequences.

Proposed method

• The $t$-structure $t_{\text{hom}}(S)$ is defined via the vanishing of certain stalks of $H(i)[i]$ at all fields over $S$, generalizing Voevodsky’s field-based definition.
• Borel-Moore motives of finite type $S$-schemes are used to define the relevant cohomological data and to construct coniveau spectral sequences.
• The $\delta$-effectivity filtration on $DM(S)$ is introduced, refining the motivic filtration by codimension of support.
• A Gersten weight structure $w_{\text{Ger}}(S)$ is constructed on the category of $S$-comotives $\mathfrak{D}(S)$, cogenerated by shifts of effective Chow motives.
• The cohomology of objects in the heart of $t_{\text{hom}}(S)$ is shown to compute Cousin complexes via the coniveau spectral sequence.
• A comparison is established between $w_{\text{Ger}}(S)$-weight spectral sequences and $\delta$-coniveau spectral sequences, especially under realization functors.

Experimental results

Research questions

• RQ1 How can the homotopy $t$-structure on motives over a field be extended to a relative setting over a general base scheme $S$?
• RQ2 What is the relationship between coniveau spectral sequences for Borel-Moore motives and the $t_{\text{hom}}(S)$-structure on $DM(S)$?
• RQ3 Is the heart of $t_{\text{hom}}(S)$ equivalent to the category of cycle modules over $S$, as conjectured?
• RQ4 How does the Gersten weight structure on $\mathfrak{D}(S)$ relate to the $t_{\text{hom}}(S)$-structure and its heart?
• RQ5 Can the $\delta$-coniveau spectral sequence be interpreted as arising from a weight structure on comotives, and what does this imply for motivic cohomology?

Key findings

• The $t$-structure $t_{\text{hom}}(S)$ is characterized by the vanishing of $H(i)[i]$ at all fields over $S$, generalizing Voevodsky’s field-based criterion.
• For $S$ the spectrum of a perfect field, $t_{\text{hom}}(S)$ coincides with the stable extension of Voevodsky’s homotopy $t$-structure.
• For $S$ of finite type over a discrete valuation ring, $t_{\text{hom}}(S)$ is equivalent to Ayoub’s perverse homotopy $t$-structure.
• The cohomology of $\mathcal{M}^{BM}_S(X)$ with coefficients in an object of the heart of $t_{\text{hom}}(S)$ computes the Cousin complex via the coniveau spectral sequence.
• The collection of stalks of $H(i)[i]$ at fields over $S$ satisfies key axioms of cycle modules, supporting the conjecture that the heart of $t_{\text{hom}}(S)$ is the category of cycle modules.
• A Gersten weight structure on $\mathfrak{D}(S)$ is constructed, and its weight spectral sequences are shown to compute the $\delta$-coniveau spectral sequences under suitable conditions.