[Paper Review] A DG guide to Voevodsky's motives

Alexander Beilinson, Vadim Vologodsky|ArXiv.org|Apr 1, 2006

Advanced Topics in Algebra5 references4 citations

TL;DR

This paper provides a concise, technical guide to Voevodsky's motivic homotopy theory, focusing on the construction of the category of mixed motives via differential graded (DG) categories. It explains how Voevodsky's approach uses finite correspondences and localization to define motives, establishing a tensor triangulated category that realizes motivic cohomology and connects to singular cohomology through a derived functorial framework.

ABSTRACT

We give a concise exposition of Voevodsky's theory of motives.

Motivation & Objective

• To provide a clear, accessible exposition of Voevodsky's construction of the category of mixed motives using DG categories.
• To clarify the role of finite correspondences in replacing algebraic maps, enabling a homotopical framework for motives.
• To establish a tensor triangulated structure on the category of motives that captures motivic cohomology and connects to classical cohomology via the singular chains functor.
• To formalize the construction at the DG level, enabling future developments such as motivic descent and higher categorical structures.

Proposed method

• Use of DG categories as a foundational framework to define motives, replacing the more abstract triangulated category approach.
• Construction of a DG category freely generated by topological manifolds, then quotienting by relations modeling excision and homotopy invariance.
• Formal imitation of this topological construction in the algebro-geometric setting using finite correspondences instead of morphisms.
• Application of Keller’s homotopy DG quotient construction to define the derived category of motives as a quotient of a DG category.
• Use of Verdier localization and right admissibility to ensure the existence of localization triangles and adjoint functors in the derived category.
• Lifting of Gysin isomorphisms and duality pairings to the DG level to prove non-degeneracy of motivic duality.

Experimental results

Research questions

• RQ1 How can Voevodsky’s motivic category be constructed rigorously at the DG level rather than just the triangulated level?
• RQ2 What is the role of finite correspondences in replacing morphisms in the construction of motives?
• RQ3 How does the singular chains functor induce an equivalence between topological motives and the derived category of abelian groups?
• RQ4 What conditions ensure that the subcategory of motives is right admissible, allowing for well-defined localization and adjoint functors?
• RQ5 How can motivic duality pairings be lifted to the DG level and shown to be non-degenerate?

Key findings

• The category of motives $\mathcal{D}_{\mathcal{M}}$ is constructed as a homotopy DG quotient of a DG category generated by algebraic varieties with finite correspondences.
• The singular chains functor induces an equivalence between the DG category of topological motives and $D_{ab}$, the derived category of finitely generated abelian groups.
• The construction ensures that the category of motives is a rigid tensor triangulated category, compatible with motivic cohomology and duality.
• Non-degeneracy of the motivic duality pairing is established by showing that each graded piece of the pairing is non-degenerate via Gysin isomorphisms and compatibility with the diagonal class.
• The use of DG categories allows for a more refined structure than triangulated categories, enabling future work on motivic descent and higher categorical properties.
• The theory realizes motivic cohomology as the cohomology of the motive of a variety, with the pairing $\epsilon_{FG}$ inducing a perfect duality on the level of resolutions.