Topological Hochschild homology and integral $p$-adic Hodge theory
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\...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
09.02.2018
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Subjects | |
Online Access | Get full text |
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Summary: | 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 $A\Omega$
constructed in our previous work, and in equal characteristic $p$ to
crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is
via flat descent to semiperfectoid rings.
As one application, we refine the construction of the $A\Omega$-complex by
giving a cohomological construction of Breuil--Kisin modules for proper smooth
formal schemes over $\mathcal O_K$, where $K$ is a discretely valued extension
of $\mathbb Q_p$ with perfect residue field. As another application, we define
syntomic sheaves $\mathbb Z_p(n)$ for all $n\geq 0$ on a large class of
$\mathbb Z_p$-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 $p$. |
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DOI: | 10.48550/arxiv.1802.03261 |