Crystalline Galois representations acquire mu_p-equivariant Breuil-Kisin module reductions, yielding the ↑ constraint on inertial weights and proving weight elimination for a general Serre weight conjecture.
[BL22b] Bhargav Bhatt and Jacob Lurie
11 Pith papers cite this work. Polarity classification is still indexing.
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abstract
The goal of this paper is to study the absolute prismatic cohomology of $p$-adic formal schemes. We do so by recasting the notion of a prismatic crystal on $\mathrm{Spf}(\mathbf{Z}_p)$ in terms of quasicoherent sheaves on a geometric object we call the Cartier-Witt stack.
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A uniform construction of stacks BT^{G,μ}_n using stacky prismatic technology verifies Drinfeld's algebraicity conjecture and yields a linear-algebraic classification of truncated p-divisible groups over general p-adic bases.
Ogus's conjecture is resolved affirmatively in full generality by constructing the required F-isocrystal via p-adic local systems and prismatic methods, while also introducing a prismatic refinement of the p-adic Riemann-Hilbert functor.
For large n, mod p^n reductions of first syntomic cohomology groups of reflexive F-gauges on O_K are isomorphic iff mod p^{2n} reductions of attached Breuil-Kisin modules with G_K-action and Nygaard filtration are isomorphic.
Authors prove representability of log prismatic and syntomic cohomology in log motives, obtain Gysin maps and comparison theorems, and compute examples like Grassmannians.
Defines prismatic cohomology relative to δ-rings, proves independence from prism structure, and establishes equivalence of three definitions under syntomicity hypotheses.
Introduces F-gauges over prisms, constructs syntomic cycle classes, and proves prismatic Poincaré duality for proper smooth schemes.
Prismatic F-gauges are described for finite flat height one group schemes, yielding the crystalline Dieudonné module of Berthelot-Breen-Messing and flat cohomology results via Hoobler-type sequences.
Proves natural equivalence between analytic prismatic F-crystals on the absolute prismatic site and relative Wach modules via correspondence of Galois action with prismatic stratification, plus new descent results.
Equivalence of reflexive sheaves on syntomic stack X^Syn with Z_p-lattices in crystalline local systems on generic fiber X_η, plus results on etale realization and filtered F-isocrystals for proper smooth X.
Exposition of the sheared Witt vectors on rings with perfect F_p-algebra reduction.
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Congruences of first syntomic cohomology groups
For large n, mod p^n reductions of first syntomic cohomology groups of reflexive F-gauges on O_K are isomorphic iff mod p^{2n} reductions of attached Breuil-Kisin modules with G_K-action and Nygaard filtration are isomorphic.