This is a much-welcomed text on the motivic cohomology theory of Suslin and Voevodsky, based on lectures given by Voevodsky. Its main contents are the definition of motivic cohomology, the (derived) category of motives, and the proofs of several basic theorems on motivic cohomology. The first two-thirds of the book is self-contained and should be accessible to graduate students with some background in algebraic geometry, intersection theory, and homological algebra. This part begins with the definition of presheaves with transfers and motivic cohomology, and the proof that ${\bf Z}(1)\cong O^\times[-1]$Z(1)≅O×[−1], and that $H^{n,n}(K,{\bf Z})\cong K^M_n(F)$Hn,n(K,Z)≅KMn(F). Then, étale sheaves with transfers, Nisnevich sheaves with transfers, the tensor triangulated categories $DM^{\rm eff,-}_{\rm Nis}(k)$DMeff,−Nis(k) and $DM^{\rm eff,-}_{\text{\'et}}(k)$DMeff,−ét(k) and their properties are discussed. As an application, étale motivic cohomology with finite coefficients prime to the characteristic of the base field is identified with usual étale cohomology, and shown to be the dual of Suslin's singular homology. Supplemental material on finite correspondences, the Nisnevich topology, and tensor triangulated categories is also provided. The last third of the book is more technical. It is devoted to the proof that motivic cohomology agrees with higher Chow groups for smooth schemes, and to the proof of the basic theorem (used before) that over a perfect field, a homotopy invariant presheaf with transfers has homotopy invariant Nisnevich cohomology, and admits a Gersten resolution.
Lectures notes on motivic cohomology / C. Mazza, V. Voevodsky, C. Weibel. - United States of America : American Mathematical Society, Providence, RI, 2006. - ISBN 978-0-8218-3847-1.
Lectures notes on motivic cohomology
C. MazzaPrimo
;
2006
Abstract
This is a much-welcomed text on the motivic cohomology theory of Suslin and Voevodsky, based on lectures given by Voevodsky. Its main contents are the definition of motivic cohomology, the (derived) category of motives, and the proofs of several basic theorems on motivic cohomology. The first two-thirds of the book is self-contained and should be accessible to graduate students with some background in algebraic geometry, intersection theory, and homological algebra. This part begins with the definition of presheaves with transfers and motivic cohomology, and the proof that ${\bf Z}(1)\cong O^\times[-1]$Z(1)≅O×[−1], and that $H^{n,n}(K,{\bf Z})\cong K^M_n(F)$Hn,n(K,Z)≅KMn(F). Then, étale sheaves with transfers, Nisnevich sheaves with transfers, the tensor triangulated categories $DM^{\rm eff,-}_{\rm Nis}(k)$DMeff,−Nis(k) and $DM^{\rm eff,-}_{\text{\'et}}(k)$DMeff,−ét(k) and their properties are discussed. As an application, étale motivic cohomology with finite coefficients prime to the characteristic of the base field is identified with usual étale cohomology, and shown to be the dual of Suslin's singular homology. Supplemental material on finite correspondences, the Nisnevich topology, and tensor triangulated categories is also provided. The last third of the book is more technical. It is devoted to the proof that motivic cohomology agrees with higher Chow groups for smooth schemes, and to the proof of the basic theorem (used before) that over a perfect field, a homotopy invariant presheaf with transfers has homotopy invariant Nisnevich cohomology, and admits a Gersten resolution.File | Dimensione | Formato | |
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