Mixed motives and their realization in derived categories.

*(English)*Zbl 0938.14008
Lecture Notes in Mathematics. 1604. Berlin: Springer Verlag. xv, 207 p. (1995).

The main goal of the still conjectural theory of motives is to construct a category of motivic sheaves \({\mathcal M}{\mathcal M}(V)\) for any algebraic variety \(V\). This construction should carry the standard six functorial operations introduced by Grothendieck into any cohomology theory. Also it should be supplied by a derived category with heart \({\mathcal M}{\mathcal M}(V)\) and a canonical filtration by pure motives of some weight.

This construction must lead to two kinds of cohomology theories: geometric cohomologies and absolute motivic cohomologies connected by a spectral sequence. The values of the geometric cohomologies belong to the category \({\mathcal M}{\mathcal M}(\text{Spec}(k))\) of motivic sheaves over a point and the values for the absolute theory lie in abelian groups.

Any known cohomology theory can be considered as a realization (some kind of truncation) of this universal construction. Among the well established realizations we can see étale cohomologies and Saito-Hodge cohomologies. In particular, for the latter theory geometric cohomologies are the De Rham theory (with their mixed Hodge structure) and absolute cohomologies will be the Deligne cohomologies. U. Jannsen [“Mixed motives and algebraic K-theory”, Lect. Notes Math. 1400 (1990; Zbl 0691.14001)] has introduced some category \({\mathcal M}{\mathcal R}(V)\) served as an approximation to the category \({\mathcal M}{\mathcal M}(V)\) and constructed the geometric cohomologies in this case.

The author returns back to Jannsen’s construction and defines the absolute cohomology theory in this case. She proves all the properties which the motivic people expect including the spectral sequence connecting the both theories and the Bloch-Ogus axioms for cohomologies. She goes even further defining the Chern classes with values in the new cohomologies and proving relations between the elements of higher K-groups with vanishing Chern classes and the elements of the motivic extension group in the category \({\mathcal M}{\mathcal R}(V)\).

The book consists of five parts. The first part contains the simplicial techniques and, in particular, Beilinson gluing of categories which is the basis of the next part where the category \({\mathcal M}{\mathcal M}(V)\) is redefined and the absolute cohomologies are introduced. The careful clear description in the first part has an independent interest. The second part also contains a detailed exposition of the étale and Hodge realizations of the motivic category. The next part develops the six operations formalism and the theory of Chern classes. The book is written rather clear and even transparent and can serve as an introduction to the main problems connected with the motivic “world”.

This construction must lead to two kinds of cohomology theories: geometric cohomologies and absolute motivic cohomologies connected by a spectral sequence. The values of the geometric cohomologies belong to the category \({\mathcal M}{\mathcal M}(\text{Spec}(k))\) of motivic sheaves over a point and the values for the absolute theory lie in abelian groups.

Any known cohomology theory can be considered as a realization (some kind of truncation) of this universal construction. Among the well established realizations we can see étale cohomologies and Saito-Hodge cohomologies. In particular, for the latter theory geometric cohomologies are the De Rham theory (with their mixed Hodge structure) and absolute cohomologies will be the Deligne cohomologies. U. Jannsen [“Mixed motives and algebraic K-theory”, Lect. Notes Math. 1400 (1990; Zbl 0691.14001)] has introduced some category \({\mathcal M}{\mathcal R}(V)\) served as an approximation to the category \({\mathcal M}{\mathcal M}(V)\) and constructed the geometric cohomologies in this case.

The author returns back to Jannsen’s construction and defines the absolute cohomology theory in this case. She proves all the properties which the motivic people expect including the spectral sequence connecting the both theories and the Bloch-Ogus axioms for cohomologies. She goes even further defining the Chern classes with values in the new cohomologies and proving relations between the elements of higher K-groups with vanishing Chern classes and the elements of the motivic extension group in the category \({\mathcal M}{\mathcal R}(V)\).

The book consists of five parts. The first part contains the simplicial techniques and, in particular, Beilinson gluing of categories which is the basis of the next part where the category \({\mathcal M}{\mathcal M}(V)\) is redefined and the absolute cohomologies are introduced. The careful clear description in the first part has an independent interest. The second part also contains a detailed exposition of the étale and Hodge realizations of the motivic category. The next part develops the six operations formalism and the theory of Chern classes. The book is written rather clear and even transparent and can serve as an introduction to the main problems connected with the motivic “world”.

Reviewer: A.N.Parshin (Moskva)

##### MSC:

14F42 | Motivic cohomology; motivic homotopy theory |

19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |

14F40 | de Rham cohomology and algebraic geometry |

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |