0000000000850990
AUTHOR
Denis-charles Cisinski
Beilinson Motives and Algebraic K-Theory
Section 12 is a recollection on the basic results of stable homotopy theory of schemes, after Morel and Voevodsky. In particular, we recall the theory of orientations in a motivic cohomology theory. Section 13 is a recollection of the fundamental results on algebraic K-theory which we translate into results within stable homotopy theory of schemes. In particular, Quillen’s localization theorem is seen as an absolute purity theory for the K-theory spectrum. In Section 14, we introduce the fibred category of Beilinson motives as an appropriate Verdier quotient of the motivic stable homotopy category. Using the Adams filtration on K-theory, we prove that Beilinson motives have the properties o…
Motivic Complexes and Relative Cycles
This part is based on Suslin and Voevodsky’s theory of relative cycles that we develop in categorical terms, in the style of EGA. The climax of the theory is obtained in the study of a pullback operation for suitable relative cycles which is the incarnation of intersection theory in this language. Properties of this pullback operation, and on the conditions necessary to its definition, are made again inspired by intersection theory. We study the compatibility of this pullback operation with projective limits of schemes. In Section 9, the theory of relative cycles is exploited to introduce Voevodsky’s category of finite type schemes over an arbitrary base with morphisms finite correspondence…
Construction of Fibred Categories
In Section 5, we introduce methods from classical homological algebra (i.e. using mostly the language of derived categories of abelian categories and their Verdier quotients) to construct the main examples of premotivic categories of interest in this book, while, in Section 6, we study how to check that the localization axiom holds in practice. Section 7 is devoted to the process of obtaining new fibred categories from old ones, by considering homotopy theoretic modules over a ring object.
Fibred Categories and the Six Functors Formalism
In Section 1, we introduce the basic language used in this book, the so-called premotivic categories and their functoriality. This is an extension of the classical notion of fibered categories. They appear with different categorical structures. In Section2, the language of premotivic categories is specialized to that of triangulated categories and to algebraic geometry. We introduce several axioms of such categories which ultimately will lead to the full six functors formalism. An emphasis is given on the study of the main axioms, with a special care about the so-called localization axiom. Then in Section 3, the general theory of descent is formulated in the language of premotivic model cat…