Triangulated Categories of Mixed Motives - Foundational

In this review of Triangulated Categories of Mixed Motives the reviewer finds a rigorous, historically important monograph aimed at researchers and advanced graduate students in algebraic geometry and arithmetic geometry. The single biggest reason to buy is that the book offers the first complete construction of a triangulated category of mixed motives with rational coefficients that satisfies the full Grothendieck six functors formalism, making it an essential reference for anyone working on Beilinson's program or on motivic interpretations of higher Chow groups.

Key Features

• Complete construction: Presents a full construction of a triangulated category of mixed motives with rational coefficients, providing a concrete framework for subsequent research.
• Six functors formalism: Develops the full Grothendieck six functors formalism in the motivic setting, allowing compatibility with standard cohomological tools.
• Connection to Beilinson's program: Explains how rational higher Chow groups are interpreted as extension groups, clarifying important conjectural relationships.
• Foundational sources integrated: Builds on Voevodsky's A1-homotopy and motivic complexes while using Gabber's and Ayoub's contributions to ensure technical completeness.
• Integral coefficient development: Includes a thorough development of motivic complexes with integral coefficients over general bases, useful for work beyond rational coefficients.

Who It's For

This book is primarily for researchers, postdoctoral scholars, and advanced graduate students who are already fluent in homotopical methods and familiar with Voevodsky's A1-homotopy theory and motivic complexes. It is most valuable to those pursuing Beilinson's program, studying mixed motives, or using the six functors formalism in arithmetic geometry.

Those looking for an introductory text or a gentle entry to motives should look elsewhere, since the monograph assumes significant background in algebraic geometry, etale cohomology and the technical foundations laid out in SGA4 and related work.

Pros & Cons

Pros

• Authoritative construction that settles key foundational aspects of mixed motives and rational coefficients.
• Careful integration of Voevodsky's theories with Gabber's and Ayoub's results for technical robustness.
• Useful treatment of motivic complexes with integral coefficients that extends applicability beyond rational cases.

Cons

• Highly technical and assumes deep prior knowledge, limiting accessibility to non-specialists.
• Not intended as an introductory textbook, so readers seeking pedagogical exposition may find it dense.

Specifications

Our Verdict

For specialists in algebraic and arithmetic geometry this monograph is indispensable: it provides a historically significant, technically complete construction of mixed motives that supports ongoing research tied to Beilinson's program. While dense for newcomers, its rigorous treatment and integration of foundational work make it excellent value as a reference and research tool.

Frequently Asked Questions

Does this book construct mixed motives with rational coefficients?
Yes. The book gives a complete construction of a triangulated category of mixed motives with rational coefficients consistent with the six functors formalism.

Is prior background required to read this monograph?
Yes. Readers should be familiar with Voevodsky's A1-homotopy theory, motivic complexes, and foundational materials such as SGA4 to follow the arguments.

Does the book address integral coefficients?
It does: the authors develop the theory of motivic complexes with integral coefficients over general bases alongside the rational theory.