## Lecture Notes on Motivic CohomologyThe notion of a motive is an elusive one, like its namesake "the motif" of Cezanne's impressionist method of painting. Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related field. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000. In addition, many of the original proofs have been simplified and improved so that this book will also be a useful tool for research mathematicians. This book provides an account of the triangulated theory of motives. Its purpose is to introduce Motivic Cohomology, to develop its main properties, and finally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, etale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The first part presents the definition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (etale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part five. |

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### Contents

The category of finite correspondences | 3 |

Presheaves with transfers | 13 |

Motivic cohomology | 21 |

Relation to Milnor KTheory | 29 |

Etale sheaves with transfers | 37 |

The relative Picard group and Suslins Rigidity Theorem | 47 |

Derived tensor products | 55 |

Appendix 8A Tensor triangulated categories | 63 |

Equidimensional cycles | 125 |

Motives with compact support | 128 |

Part5 Higher Chow Groups | 133 |

Higher Chow groups | 135 |

Appendix 17A Cycle maps | 143 |

Higher Chow groups and equidimensional cycles | 149 |

Appendix 18A Generic equidimensionality | 155 |

Motivic cohomology and higher Chow groups | 159 |

Etale motivic cohomology and algebraic singular homology | 75 |

Standard triples | 83 |

Nisnevich sheaves | 89 |

Nisnevich sheaves with transfers | 99 |

Cdh sheaves with transfers | 105 |

The Triangulated Category of Motives | 107 |

The category of motives | 109 |

Nisnevich A1local complexes | 111 |

Motives with Qcoefficients | 116 |

The complex Zn and Pn | 119 |

Geometric motives | 167 |

Zariski Sheaves with Transfers | 173 |

Covering morphisms of triples | 175 |

Zariski sheaves with transfers | 183 |

Contractions | 191 |

Homotopy invariance of cohomology | 197 |

203 | |

Glossary | 207 |

211 | |

### Other editions - View all

Lecture Notes on Motivic Cohomology Carlo Mazza,Vladimir Voevodsky,Charles A. Weibel Limited preview - 2006 |

Lecture Notes on Motivic Cohomology Carlo Mazza,Vladimir Voevodsky,Charles A. Weibel No preview available - 2011 |

### Common terms and phrases

A1-local A1-weak equivalence abelian group admits resolution afﬁne Al-homotopy assume blow-up canonical chain complex chain homotopy Chow groups closed point closed subscheme codimension cokernel composition Cor(X,Y Cork COROLLARY deﬁned deﬁnition denote derived category dimX DMeff,−Nis DMgm elementary correspondence equidimensional étale etale sheaves exact sequence EXAMPLE EXERCISE ﬁber ﬁeld ﬁnite finite and surjective finite correspondence ﬁrst FNis functor FZar Hence Hensel higher Chow groups homology homotopy invariant homotopy invariant presheaf hypercohomology induces injective intersection irreducible isomorphism lecture LEMMA Let F locally constant map f motivic cohomology natural map Nisnevich cover Nisnevich sheaf Nisnevich topology Pic(X presheaf with transfers projection PROOF PROPOSITION PST(k pullback quasi-isomorphism R-modules with transfers relative cycle resolution of singularities sheaves with transfers simplicial Sm/k smooth scheme Speck spectral sequence split standard triple subcategory sufﬁces surjective tensor product tensor triangulated category theorem torsor triangulated category vector bundle yields Zariski topology zero Ztr(X