Triangulated Categories of Mixed Motives

By Denis-Charles Cisinski, Frédéric Déglise. Springer Monographs in Mathematics, 2009

291 citations recorded by the source corpus.

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b0f18cff2cf792a06688678aad9e460aaa7ddc60

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Citations in the corpus

Fundamental classes in motivic homotopy theory (2018, background): “See [CD12] or [Ayo07], and [Kha16] or [Hoy15, Appendix C] for the extension to quasi-compact quasi-separated base schemes.”
Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, background): “ categories (e.g., the Grothendieck construction) or using a good theory of 2- functors. Following these outlines, a great deal of hard work in the motivic context has developed this coherence theory [9, 19]. In the case of parametrized spectra, eﬀorts in this direction can be found in [44, §13,§17]”
Motivic and real étale stable homotopy theory (2016, background): “The six functors formalism [CD12, § A.5] is a very strong, and very general, duality theory.”