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.
[IKY24] Naoki Imai, Hiroki Kato, and Alex Youcis,A tannakian framework for prismaticF-crystals, Preprint,https://arxiv.org/abs/2406.08259
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Syntomification and crystalline local systems
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.