Prismatic crystals for smooth schemes in characteristic p with Frobenius lifting mod p²

2024/02/03 by Wang, Yupeng
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2402.02109

Abstract

Let (A,(p)) be a crystalline prism with A_n = A/pⁿ⁺¹A for all n≥ 0. Let \frakX₀ be a smooth scheme over A₀. Suppose that \frakX₀ admits a lifting \frakX_n over A_n and the absolute Frobenius \rF_\frakX0:\frakX₀→ \frakX₀ admits a lifting over A₁. Then we show that there is an equivalence between the category of the prismatic crystals of truncation n on (\frakX₀/A)_\Prism and the category of p-connections over \frakX_n, which is compatible with cohomologies. This generalises a previous work of Ogus. We also give some remarks on trivializing the Hodge--Tate gerbe π_\frakX0^{\rm HT}:\frakX₀^{\rm HT}→\frakX₀ introduced by Bhatt--Lurie.

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