[Paper Review] On weights for relative motives with integral coefficients

Mikhail V. Bondarko|arXiv (Cornell University)|Apr 8, 2013

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper establishes the existence of a Chow weight structure $ w_{\text{Chow}} $ on the category $ DM(S) $ of Voevodsky's motivic complexes with integral coefficients over any excellent, separated, finite-dimensional scheme $ S $. By constructing this weight structure, the authors obtain functorial Chow-weight spectral sequences and filtrations for any (co)homological functor $ H: DM(S) \to \underline{A} $, extending rational coefficient results to the integral setting and enabling new tools for studying motivic (co)homology with integral coefficients.

ABSTRACT

The goal of this paper is to define a certain Chow weight structure for the category of Voevodsky's motivic complexes with integral coefficients (as described by Cisinski and Deglise) over any excellent finite-dimensional separated scheme $S$. Our results are parallel to (though substantially weaker than) the corresponding 'rational coefficient' statements proved by D. Hebert and the author. As an immediate consequence of the existence of 'weights', we obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of $S$-motives.

Motivation & Objective

• To define a Chow weight structure $ w_{\text{Chow}} $ on $ DM(S) $, the category of Voevodsky’s motivic complexes with integral coefficients, over any excellent, separated, finite-dimensional scheme $ S $.
• To extend the theory of weight structures—previously established for rational coefficients—to the integral coefficient case, which is more technically challenging.
• To construct functorial Chow-weight spectral sequences and filtrations for any (co)homological functor $ H: DM(S) \to \underline{A} $, leveraging the existence of $ w_{\text{Chow}} $.
• To compare the integral $ w_{\text{Chow}} $ with its rational counterpart on $ DM_{\mathbb{Q}}(S) $, showing compatibility via comparison functors.
• To provide a foundation for studying motivic (co)homology with integral coefficients using spectral sequences and filtrations derived from weight structures.

Proposed method

• Constructs the Chow weight structure $ w_{\text{Chow}} $ on $ DM(S) $ using a new existence lemma for weight structures in triangulated categories with countable homotopy colimits.
• Applies the theory of weight structures to the triangulated category $ DM(S) $, defined via sheaf-theoretic methods by Cisinski and Deglise, ensuring compatibility with integral coefficients.
• Uses the Karoubi-closure of the full subcategory of Chow motives over $ S $ as the heart of the weight structure, ensuring the existence of weight truncations.
• Establishes functoriality of $ w_{\text{Chow}} $ under pullbacks $ f^* $, pushforwards $ f_* $, and other standard functors for quasi-projective morphisms $ f $.
• Relies on the existence of a differential graded enhancement of $ DM(S) $, which allows the construction of the weight complex functor $ t_S: DM(S) \to K(\underline{Hw}_{\text{Chow}}(S)) $.
• Compares the integral $ w_{\text{Chow}} $ with the rational version on $ DM_{\mathbb{Q}}(S) $, showing that the comparison functor $ DM(S) \otimes \mathbb{Q} \to DM_{\mathbb{Q}}(S) $ is weight-exact for regular $ S $.

Experimental results

Research questions

• RQ1 Can a Chow weight structure be defined on $ DM(S) $ with integral coefficients for any excellent, separated, finite-dimensional scheme $ S $?
• RQ2 How does the integral Chow weight structure relate to the already known rational version on $ DM_{\mathbb{Q}}(S) $?
• RQ3 What are the implications of the existence of $ w_{\text{Chow}} $ for motivic (co)homology theories with integral coefficients?
• RQ4 To what extent is the functoriality of $ w_{\text{Chow}} $ preserved under standard operations in motivic homotopy theory, such as $ f^*, f_* $?
• RQ5 Can the weight structure be used to construct spectral sequences and filtrations that are intrinsic and independent of choices?

Key findings

• The Chow weight structure $ w_{\text{Chow}} $ exists on $ DM(S) $ for any excellent, separated, finite-dimensional scheme $ S $, extending the theory from rational to integral coefficients.
• The existence of $ w_{\text{Chow}} $ implies the existence of functorial Chow-weight spectral sequences for any homological or cohomological functor $ H: DM(S) \to \underline{A} $, converging to $ H(M) $ when $ M $ is bounded.
• For any $ M \in DM(S) $, there is a spectral sequence $ E_1^{pq} = H_q(M^p) $ converging to $ H_{p+q}(M) $, where $ (M^p) $ are the terms of the weight complex $ t(M) $.
• The weight filtration $ (W^k H)(M) = \operatorname{Im}(H({w_{\text{Chow}}_{\geq k}} M) \to H(M)) $ is independent of choices and functorial in $ M $, providing a canonical filtration on $ H(M) $.
• For regular base schemes $ S $, the comparison functor $ DM(S) \otimes \mathbb{Q} \to DM_{\mathbb{Q}}(S) $ is weight-exact, so the spectral sequences and filtrations for rational coefficients coincide with those constructed via the integral $ w_{\text{Chow}} $.
• The results are compatible with standard motivic functors: $ w_{\text{Chow}} $ is functorial with respect to $ f^*, f_*, f^!, f_! $ for quasi-projective morphisms $ f $.