A commutative \({\mathbb P}^1\)-spectrum representing motivic cohomology over Dedekind domains.

*(English)*Zbl 1408.14081This volume represents an important work on integral motivic cohomology in mixed characteristic. Motivic cohomology groups with rational coefficients appear for the first time in the work of C. Soulé [Can. J. Math. 37, 488–550 (1985; Zbl 0575.14015)] as graded pieces of algebraic \(K\)-theory with respect to the Adams operations; the first definition with integral coefficients is due to S. Bloch [Adv. Math. 61, 267–304 (1986; Zbl 0608.14004)], who defined higher Chow groups using cycle complexes, and this definition is generalized by Levine to smooth schemes over a Dedekind domain [M. Levine, “\(K\)-theory and motivic cohomology of schemes. I”, Preprint, https://faculty.math.illinois.edu/K-theory/0336/]. On the other hand, the theory of mixed motives predicts that motivic cohomology groups can be computed as \(Hom\) groups in a triangulated category of motives; over fields, such an aim is achieved by the fundamental work of V. Voevodsky et al. [Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press (2000; Zbl 1021.14006)]. More generally, when working over regular Noetherian schemes, it is conjectured that the following two descriptions are equivalent, both giving rise to the desired category:

1) The triangulated category \(\mathbf{DM}\) constructed out of Voevodsky’s theory of finite correspondences [F. Déglise, Lond. Math. Soc. Lect. Note Ser. 343, 138–205 (2007; Zbl 1149.14014)], by considering Nisnevich sheaves with transfers applied to \(\mathbb{A}^1\)-localization and \(\mathbb{P}^1\)-stabilization.

2) The homotopy category of modules over the motivic Eilenberg-Mac Lane spectrum \(\mathbf{M}R\) in the stable motivic homotopy category \(\mathbf{SH}\) [V. Voevodsky, in: Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. 3–34 (2002; Zbl 1047.14012)].

Moreover, these categories should satisfy the following functoriality: Voevodsky conjectured that for any morphism of regular schemes \(f:X\to Y\), the canonical map

\[ Lf^*\mathbf{M}R_Y\to\mathbf{M}R_X \] is an isomorphism in \(\mathbf{SH}(X)\). The work of Cisinski-Déglise show these conjectures in the following cases:

1) [D.-C. Cisinski and F. Déglise, Triangulated categories of motives. To appear in the series Springer Monographs in Mathematics. (2019)] When \(R=\mathbb{Q}\), these conjectures hold in mixed characteristic, under mild assumptions related to the existence of de Jong alterations [A. J. de Jong, Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996; Zbl 0916.14005)]. The category \(\mathbf{DM}_\mathbb{Q}\) is equivalent to the category of modules over the Beilinson spectrum, the latter being a graded piece of the algebraic \(K\)-theory spectrum [J. Riou, J. Topol. 3, No. 2, 229–264 (2010; Zbl 1202.19004)].

2) [D.-C. Cisinski and F. Déglise, Doc. Math. Extra Vol., 145–194 (2015; Zbl 1357.19004)] When working in equal characteristic \(p\), these conjectures hold for \(R=\mathbb{Z}[1/p]\).

The main ingredient of this volume is the construction of a motivic spectrum \(\mathbf{M}\mathbb{Z}\) over any Dedekind domain in \(\mathbf{SH}\) with integral coefficients, which satisfies the following properties:

1) (Theorem 7.18) For smooth schemes over a Dedekind ring, it computes the Bloch-Levine motivic cohomology.

2) (Theorem 7.14) The rationalization of \(\mathbf{M}\mathbb{Z}\) is the Beilinson spectrum.

3) (Theorem 8.22) \(\mathbf{M}\mathbb{Z}\) is an \(E_\infty\) spectrum, whose restriction to closed points is Voevodsky’s motivic Eilenberg-Mac Lane spectrum. In particular, when \(f\) is the inclusion of a closed point in a Dedekind domain of mixed characteristic, the map \(Lf^*\mathbf{M}\mathbb{Z}_Y\to\mathbf{M}\mathbb{Z}_X\) is an isomorphism which preserves the \(E_\infty\)-structure.

4) (Section 5.2.3) \(\mathbf{M}\mathbb{Z}\) has an étale cycle class map which is compatible with localization sequences.

In particular, for any scheme \(X\) with structural morphism \(f:X\to\text{Spec}(\mathbb{Z})\), the category of modules over \(\mathbf{M}\mathbb{Z}_X:=f^*\mathbf{M}\mathbb{Z}_{\text{Spec}(\mathbb{Z})}\) in \(\mathbf{SH}(X)\) is a triangulated category in which one can define motivic cohomology with integral coefficients; by the work of J. 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)] and Cisinski-Déglise, such a category satisfies the six functors formalism, which is briefly sketched in Section 9.

In Section 3 the author studies the Bloch-Levine motivic complex in details. Using the Bloch-Kato conjecture (proved in [V. Voevodsky, Ann. Math. (2) 174, No. 1, 401–438 (2011; Zbl 1236.14026)]), the torsion part coprime to the characteristic is identified with truncated étale sheaves (Theorem 3.9).

The main construction is carried out in Section 4, where the spectrum \(\mathbf{M}\mathbb{Z}\) is constructed by gluing the rational part and the \(p\)-completed part for all primes \(p\) (Definition 4.27). The rational part is given by the Beilinson spectrum, and the \(p\)-completed spectrum is constructed out of the spectrum with finite \(p\)-power coefficients, which is given by truncated étale sheaves outside characteristic \(p\) and logarithmic de Rham-Witt complexes at characteristic \(p\).

In Sections 5 through 7 the author identifies motivic cohomology over \(\mathbf{M}\mathbb{Z}\) with the Bloch-Levine motivic cohomology. He constructs a motivic spectrum \(\mathcal{M}\) in Section 5 which represents the Bloch-Levine motivic cohomology, and shows a canonical identification between \(\mathcal{M}\) and \(\mathbf{M}\mathbb{Z}\) in Theorem 7.18.

In Section 8 the author identifies the restriction of \(\mathbf{M}\mathbb{Z}\) to fields with Voevodsky’s motivic Eilenberg-Mac Lane spectrum. The hard part is the \(p\)-torsion in characteristic \(p\) case, where the key ingredient of the proof is the Bloch-Kato filtration on \(p\)-adic vanishing cycles.

In Section 10 the main results are applied to give a generalization of the Hopkins-Morel isomorphism as well as a result on the structure of the dual motivic Steenrod algebra.

1) The triangulated category \(\mathbf{DM}\) constructed out of Voevodsky’s theory of finite correspondences [F. Déglise, Lond. Math. Soc. Lect. Note Ser. 343, 138–205 (2007; Zbl 1149.14014)], by considering Nisnevich sheaves with transfers applied to \(\mathbb{A}^1\)-localization and \(\mathbb{P}^1\)-stabilization.

2) The homotopy category of modules over the motivic Eilenberg-Mac Lane spectrum \(\mathbf{M}R\) in the stable motivic homotopy category \(\mathbf{SH}\) [V. Voevodsky, in: Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. 3–34 (2002; Zbl 1047.14012)].

Moreover, these categories should satisfy the following functoriality: Voevodsky conjectured that for any morphism of regular schemes \(f:X\to Y\), the canonical map

\[ Lf^*\mathbf{M}R_Y\to\mathbf{M}R_X \] is an isomorphism in \(\mathbf{SH}(X)\). The work of Cisinski-Déglise show these conjectures in the following cases:

1) [D.-C. Cisinski and F. Déglise, Triangulated categories of motives. To appear in the series Springer Monographs in Mathematics. (2019)] When \(R=\mathbb{Q}\), these conjectures hold in mixed characteristic, under mild assumptions related to the existence of de Jong alterations [A. J. de Jong, Publ. Math., Inst. Hautes Étud. Sci. 83, 51–93 (1996; Zbl 0916.14005)]. The category \(\mathbf{DM}_\mathbb{Q}\) is equivalent to the category of modules over the Beilinson spectrum, the latter being a graded piece of the algebraic \(K\)-theory spectrum [J. Riou, J. Topol. 3, No. 2, 229–264 (2010; Zbl 1202.19004)].

2) [D.-C. Cisinski and F. Déglise, Doc. Math. Extra Vol., 145–194 (2015; Zbl 1357.19004)] When working in equal characteristic \(p\), these conjectures hold for \(R=\mathbb{Z}[1/p]\).

The main ingredient of this volume is the construction of a motivic spectrum \(\mathbf{M}\mathbb{Z}\) over any Dedekind domain in \(\mathbf{SH}\) with integral coefficients, which satisfies the following properties:

1) (Theorem 7.18) For smooth schemes over a Dedekind ring, it computes the Bloch-Levine motivic cohomology.

2) (Theorem 7.14) The rationalization of \(\mathbf{M}\mathbb{Z}\) is the Beilinson spectrum.

3) (Theorem 8.22) \(\mathbf{M}\mathbb{Z}\) is an \(E_\infty\) spectrum, whose restriction to closed points is Voevodsky’s motivic Eilenberg-Mac Lane spectrum. In particular, when \(f\) is the inclusion of a closed point in a Dedekind domain of mixed characteristic, the map \(Lf^*\mathbf{M}\mathbb{Z}_Y\to\mathbf{M}\mathbb{Z}_X\) is an isomorphism which preserves the \(E_\infty\)-structure.

4) (Section 5.2.3) \(\mathbf{M}\mathbb{Z}\) has an étale cycle class map which is compatible with localization sequences.

In particular, for any scheme \(X\) with structural morphism \(f:X\to\text{Spec}(\mathbb{Z})\), the category of modules over \(\mathbf{M}\mathbb{Z}_X:=f^*\mathbf{M}\mathbb{Z}_{\text{Spec}(\mathbb{Z})}\) in \(\mathbf{SH}(X)\) is a triangulated category in which one can define motivic cohomology with integral coefficients; by the work of J. 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)] and Cisinski-Déglise, such a category satisfies the six functors formalism, which is briefly sketched in Section 9.

In Section 3 the author studies the Bloch-Levine motivic complex in details. Using the Bloch-Kato conjecture (proved in [V. Voevodsky, Ann. Math. (2) 174, No. 1, 401–438 (2011; Zbl 1236.14026)]), the torsion part coprime to the characteristic is identified with truncated étale sheaves (Theorem 3.9).

The main construction is carried out in Section 4, where the spectrum \(\mathbf{M}\mathbb{Z}\) is constructed by gluing the rational part and the \(p\)-completed part for all primes \(p\) (Definition 4.27). The rational part is given by the Beilinson spectrum, and the \(p\)-completed spectrum is constructed out of the spectrum with finite \(p\)-power coefficients, which is given by truncated étale sheaves outside characteristic \(p\) and logarithmic de Rham-Witt complexes at characteristic \(p\).

In Sections 5 through 7 the author identifies motivic cohomology over \(\mathbf{M}\mathbb{Z}\) with the Bloch-Levine motivic cohomology. He constructs a motivic spectrum \(\mathcal{M}\) in Section 5 which represents the Bloch-Levine motivic cohomology, and shows a canonical identification between \(\mathcal{M}\) and \(\mathbf{M}\mathbb{Z}\) in Theorem 7.18.

In Section 8 the author identifies the restriction of \(\mathbf{M}\mathbb{Z}\) to fields with Voevodsky’s motivic Eilenberg-Mac Lane spectrum. The hard part is the \(p\)-torsion in characteristic \(p\) case, where the key ingredient of the proof is the Bloch-Kato filtration on \(p\)-adic vanishing cycles.

In Section 10 the main results are applied to give a generalization of the Hopkins-Morel isomorphism as well as a result on the structure of the dual motivic Steenrod algebra.

Reviewer: Fangzhou Jin (Essen)

##### MSC:

14F42 | Motivic cohomology; motivic homotopy theory |

14C25 | Algebraic cycles |

19E20 | Relations of \(K\)-theory with cohomology theories |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |