Topological Hochschild homology and integral $p$-adic Hodge theory

Bhatt, Bhargav; Morrow, Matthew; Scholze, Peter

doi:10.1007/s10240-019-00106-9

Article

Topological Hochschild homology and integral

p

-adic Hodge theory

Bhatt, Bhargav ¹ ; Morrow, Matthew ; Scholze, Peter

¹

Publications Mathématiques de l'IHÉS, Tome 129 (2019), pp. 199-310

Résumé

In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$ -theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $A Ω$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $THH$ is via flat descent to semiperfectoid rings.

As one application, we refine the construction of the $A Ω$ -complex by giving a cohomological construction of Breuil–Kisin modules for proper smooth formal schemes over $𝒪_{K}$ , where $K$ is a discretely valued extension of $𝐐_{p}$ with perfect residue field. As another application, we define syntomic sheaves $𝐙_{p} (n)$ for all $n \geq 0$ on a large class of $𝐙_{p}$ -algebras, and identify them in terms of $p$ -adic nearby cycles in mixed characteristic, and in terms of logarithmic de Rham-Witt sheaves in equal characteristic $p$ .

Reçu le : 2018-03-10
Accepté le : 2019-04-09
Première publication : 2019-04-17
Publié le : 2019-06-01

MR Zbl

DOI : 10.1007/s10240-019-00106-9

Affiliations des auteurs :

Bhatt, Bhargav ¹ ; Morrow, Matthew ; Scholze, Peter

¹

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     author = {Bhatt, Bhargav and Morrow, Matthew and Scholze, Peter},
     title = {Topological {Hochschild} homology and integral $p$-adic {Hodge} theory},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {199--310},
     year = {2019},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {129},
     doi = {10.1007/s10240-019-00106-9},
     mrnumber = {3949030},
     zbl = {1478.14039},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-019-00106-9/}
}

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Bhatt, Bhargav; Morrow, Matthew; Scholze, Peter. Topological Hochschild homology and integral $p$-adic Hodge theory. Publications Mathématiques de l'IHÉS, Tome 129 (2019), pp. 199-310. doi: 10.1007/s10240-019-00106-9

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