Abstract
We introduce the notion of a prism, which may be regarded as a “deperfection” of the notion of a perfectoid ring. Using prisms, we attach a ringed site — the prismatic site — to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories.
As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishing conjecture for the $p$-adic Tate twists $\mathbf {Z}_p(n)$ introduced in our previous joint work with Morrow.
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[AnschutzLeBrasqLog]
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author = {Anschütz, J. and Le~Bras, Arthur-César},
title = {The $p$-completed cyclotomic trace in degree $2$},
journal = {Ann. K-Theory},
fjournal = {Annals of K-Theory},
volume = {5},
number = {3},
year = {2020},
pages = {539--580},
doi = {10.2140/akt.2020.5.539},
url = {https://doi.org/10.2140/akt.2020.5.539},
mrnumber = {4132746},
zblnumber = {1454.19005},
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title = {Specializing varieties and their cohomology from characteristic 0 to characteristic {$p$}},
booktitle = {Algebraic Geometry: {S}alt {L}ake {C}ity 2015},
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