Local and stable homological algebra in Grothendieck abelian categories (original) (raw)
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Integral mixed motives in equal characteristic
EMS Press eBooks, 2014
For noetherian schemes of finite dimension over a field of characteristic exponent p, we study the triangulated categories of Z[1/p]-linear mixed motives obtained from cdh-sheaves with transfers. We prove that these have many of the expected properties. In particular, the formalism of the six operations holds in this context. When we restrict ourselves to regular schemes, we also prove that these categories of motives are equivalent to the more classical triangulated categories of mixed motives constructed in terms of Nisnevich sheaves with transfers. Such a program is achieved by comparing these various triangulated categories of motives with modules over motivic Eilenberg-MacLane spectra. CONTENTS Conventions 3 1. Motivic complexes and spectra 4 2. Modules over motivic Eilenberg-MacLane spectra 6 2.a. Symmetric Tate spectra and continuity 6 2.b. Motivic Eilenberg-MacLane spectra over regular k-schemes 9 3. Comparison theorem: regular case 11 3.a. Some consequences of continuity 13 3.b. Motives over fields 15 3.c. Proof in the regular case 18 4. More modules over motivic Eilenberg-MacLane spectra 22 5. Comparison theorem: general case 24 6. Finiteness 29 7. Duality 31 8. Bivariant cycle cohomology 33 9. Realizations 39 References 42