Integral mixed motives in equal characteristic.(English)Zbl 1357.19004
The theory of mixed motives over a base with rational coefficients, has now all the main properties one would like to have, namely six functors formalism, localisation exact sequences and comparison with higher Chow groups. This has been settled in a series of works byJ. Ayoub [Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I. Paris: Société Mathématique de France (2007;Zbl 1146.14001); Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II. Paris: Société Mathématique de France (2007;Zbl 1153.14001)] and the authors [Compos. Math. 152, No. 3, 556–666 (2016;Zbl 1453.14059)].
With integral coefficients the situation is subtler, even over a field. There are (at least) two possible theories, the étale and the Nisnevich one; they have different advantages (the first should have a \(t\)-structures, the second has a weight structure and computes Chow groups) and coincide only rationally.
Over a field admitting resolution of singularities the two integral theories has been extensively studied byV. Voevodsky [Ann. Math. Stud. 143, 188–238 (2000;Zbl 1019.14009)]. As S. Kelly showed in his thesis, these results are also true over a field of characteristic \(p>0\) if one allows to invert \(p\) in the coefficients.
Over a base, the étale integral theory as been settled in [J. Ayoub, Ann. Sci. Éc. Norm. Supér. (4) 47, No. 1, 1–145 (2014;Zbl 1354.18016)] and the authors [Compos. Math. 152, No. 3, 556–666 (2016;Zbl 1453.14059)]. Note that this theory does not compute higher Chow groups, hence the importance of defining the Nisnevich theory.
In the paper under review, the authors define a good category of Nisnevich motives over a regular scheme of characteristic \(p\), with coefficients in \(\mathbb{Z}[1/p]\). These categories verify the six functors formalism, localization, duality and compute higher Chow groups.
As the authors explain very clearly in the introduction, a key idea is to work with cdh-topology. For such a topology they can define a category of motives over a general base (possibly singular). They can show localization in this more general context. The six functor formalism is then a consequence, using results of Ayoub. Afterwards, the authors show that these functors respect constructible motives. For Nisnevich motives, all these results are then deduced by a comparison theorem (where the regular hypothesis on the base is needed).
The last section of the paper is devoted to the construction of realizations. Some deep \(\ell\)-adic independence results are deduced as an easy consequence of the machinery.
With integral coefficients the situation is subtler, even over a field. There are (at least) two possible theories, the étale and the Nisnevich one; they have different advantages (the first should have a \(t\)-structures, the second has a weight structure and computes Chow groups) and coincide only rationally.
Over a field admitting resolution of singularities the two integral theories has been extensively studied byV. Voevodsky [Ann. Math. Stud. 143, 188–238 (2000;Zbl 1019.14009)]. As S. Kelly showed in his thesis, these results are also true over a field of characteristic \(p>0\) if one allows to invert \(p\) in the coefficients.
Over a base, the étale integral theory as been settled in [J. Ayoub, Ann. Sci. Éc. Norm. Supér. (4) 47, No. 1, 1–145 (2014;Zbl 1354.18016)] and the authors [Compos. Math. 152, No. 3, 556–666 (2016;Zbl 1453.14059)]. Note that this theory does not compute higher Chow groups, hence the importance of defining the Nisnevich theory.
In the paper under review, the authors define a good category of Nisnevich motives over a regular scheme of characteristic \(p\), with coefficients in \(\mathbb{Z}[1/p]\). These categories verify the six functors formalism, localization, duality and compute higher Chow groups.
As the authors explain very clearly in the introduction, a key idea is to work with cdh-topology. For such a topology they can define a category of motives over a general base (possibly singular). They can show localization in this more general context. The six functor formalism is then a consequence, using results of Ayoub. Afterwards, the authors show that these functors respect constructible motives. For Nisnevich motives, all these results are then deduced by a comparison theorem (where the regular hypothesis on the base is needed).
The last section of the paper is devoted to the construction of realizations. Some deep \(\ell\)-adic independence results are deduced as an easy consequence of the machinery.
Reviewer: Giuseppe Ancona (Paris)
MSC:
| 19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14C25 | Algebraic cycles |
| 14F20 | Étale and other Grothendieck topologies and (co)homologies |
| 14F42 | Motivic cohomology; motivic homotopy theory |
