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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@article{AnschutzLeBrasqLog,
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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author = {Anschütz, J. and Le~Bras, Arthur-César},
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author = {André,
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title = {La conjecture du facteur direct},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
volume = {127},
year = {2018},
pages = {71--93},
issn = {0073-8301},
mrclass = {13D22 (13A35 13B40 13D09 18A99)},
mrnumber = {3814651},
mrreviewer = {Marcel Morales},
doi = {10.1007/s10240-017-0097-9},
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} -
[AndreAbhyankar]
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author = {André,
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title = {Le lemme d'{A}bhyankar perfectoide},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
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pages = {1--70},
issn = {0073-8301},
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author = {Bhatt, Bhargav},
title = {Specializing varieties and their cohomology from characteristic 0 to characteristic {$p$}},
booktitle = {Algebraic Geometry: {S}alt {L}ake {C}ity 2015},
series = {Proc. Sympos. Pure Math.},
volume = {97},
pages = {43--88},
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[BMS1]
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author = {Bhatt, Bhargav and Morrow, Matthew and Scholze, Peter},
title = {Integral {$p$}-adic {H}odge theory},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
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year = {2018},
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mrreviewer = {Daniel Robert Gulotta},
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[BMS2]
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