Affiliation:
1. Morningside Center of Mathematics , Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science , Chinese Academy of Sciences , 55 Zhong Guan Cun East Road, 100190 , Beijing , P. R. China
Abstract
Abstract
Let
(
A
,
I
)
{(A,I)}
be a bounded prism, and let X be a smooth p-adic formal scheme over
Spf
(
A
/
I
)
{\operatorname{Spf}(A/I)}
. We consider the notion of crystals on Bhatt–Scholze’s prismatic site
(
X
/
A
)
Δ
Δ
{(X/A)_{{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}}
of X relative to A. We prove that if X is proper over
Spf
(
A
/
I
)
{\operatorname{Spf}(A/I)}
of relative dimension n, then the cohomology of a prismatic crystal is a perfect complex of A-modules with tor-amplitude in degrees
[
0
,
2
n
]
{[0,2n]}
. We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of
(
X
/
A
)
Δ
Δ
{(X/A)_{{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}}
. The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.
Subject
Applied Mathematics,General Mathematics
Reference13 articles.
1. J. Anschütz and A.-C. Le Bras,
Prismatic Dieudonné theory,
Forum Math. Pi 11 (2023), 123–156.
2. M. Artin, P. Deligne, A. Grothendieck and J. L. Verdier,
Séminaire de géométrie algébrique (1963–64) du Bois Marie: Théorie des topos et cohomologie étale des schémas II,
Lecture Notes in Math. 270,
Springer, Berlin 1972.
3. B. Bhatt and A. J. de Jong,
Crystalline cohomology and de Rham cohomology,
preprint (2011), https://arxiv.org/pdf/1110.5001.
4. B. Bhatt and J. Lurie,
Absolute prismatic cohomology,
preprint (2022), https://arxiv.org/abs/2201.06120.
5. B. Bhatt, M. Morrow and P. Scholze,
Integral p-adic Hodge theory,
Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397.
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