$K$-theory and logarithmic Hodge-Witt sheaves of formal schemes  in characteristic $p$

Morrow, Matthew

doi:10.24033/asens.2415

K

-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic

p

[

K

-théorie et faisceaux de Hodge-Witt logarithmiques de schémas formels en caractéristique

p

]

Morrow, Matthew

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1537-1601.

Résumé
Abstract

Nous décrivons les $K$ -groups ${K_{n} (A / I^{s}) / p^{r}}_{s}$ modulo $p^{r}$ d'une $𝔽_{p}$ -algèbre régulière locale $A$ modulo les puissances d'un idéal approprié $I$ en termes des groupes de Hodge-Witt logarithmique, en démontrant des analogues pro des théorèmes de Geisser-Levine et Bloch-Kato-Gabber. Ceci est accompli en utilisant le théorème d'Hochschild-Kostant-Rosenberg pro en homologie cyclique topologique et le développement de la théorie des complexes de de Rham-Witt et de Hodge-Witt logarithmique sur les $𝔽_{p}$ -schémas formels.

Des applications incluent les suivants : la partie infinitésimale de la conjecture de Lefschetz faible pour les groupes de Chow ; une version $p$ -adique de la conjecture de Kato-Saito que leurs groupes des classes de dimension supérieure Zariski et Nisnevich sont isomorphes ; des résultats de continuité en $K$ -théorie ; et des conditions, en termes des classes de cycles motiviques étales entières ou torsions, pour que les cycles algébriques sur un schéma formel admettent des déformations infinitésimales.

De plus, dans le cas où $n = 1$ nous comparons la cohomologie étale de $W_{r} Ω_{log}^{1}$ et la cohomologie fppf de $μ_{p^{r}}$ sur un schéma formel, et ainsi présentons des conditions équivalentes pour que les fibres en droites déforment en termes de leurs classes dans chacune de ces cohomologies.

We describe the mod $p^{r}$ pro $K$ -groups ${K_{n} (A / I^{s}) / p^{r}}_{s}$ of a regular local $𝔽_{p}$ -algebra $A$ modulo powers of a suitable ideal $I$ , in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of Geisser-Levine and Bloch-Kato-Gabber. This is achieved by combining the pro Hochschild-Kostant-Rosenberg theorem in topological cyclic homology with the development of the theory of de Rham-Witt complexes and logarithmic Hodge-Witt sheaves on formal schemes in characteristic $p$ .

Applications include the following: the infinitesimal part of the weak Lefschetz conjecture for Chow groups; a $p$ -adic version of Kato-Saito's conjecture that their Zariski and Nisnevich higher dimensional class groups are isomorphic; continuity results in $K$ -theory; and criteria, in terms of integral or torsion étale-motivic cycle classes, for algebraic cycles on formal schemes to admit infinitesimal deformations.

Moreover, in the case $n = 1$ , we compare the étale cohomology of $W_{r} Ω_{log}^{1}$ and the fppf cohomology of $μ_{p^{r}}$ on a formal scheme, and thus present equivalent conditions for line bundles to deform in terms of their classes in either of these cohomologies.

Publié le : 2019-11-18

MR Zbl

DOI : 10.24033/asens.2415

@article{ASENS_2019__52_6_1537_0,
     author = {Morrow, Matthew},
     title = {$K$-theory and logarithmic {Hodge-Witt} sheaves of formal schemes  in characteristic~$p$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1537--1601},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {6},
     year = {2019},
     doi = {10.24033/asens.2415},
     mrnumber = {4061020},
     zbl = {1440.19004},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2415/}
}

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Morrow, Matthew. $K$-theory and logarithmic Hodge-Witt sheaves of formal schemes  in characteristic $p$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1537-1601. doi : 10.24033/asens.2415. https://www.numdam.org/articles/10.24033/asens.2415/

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