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2 results

  • Completed prismatic F-crystals and crystalline Zp-local systems

  • Part of
  • Heng Du, Tong Liu, Yong Suk Moon, Koji Shimizu
  • Journal:
    Compositio Mathematica / Volume 160 / Issue 5 / May 2024
    Published online by Cambridge University Press:
    18 April 2024, pp. 1101-1166
    Print publication:
    May 2024
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  • We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.

  • Duality in the cohomology of crystalline local systems

  • WIESŁAWA NIZIOL
  • Journal:
    Compositio Mathematica / Volume 109 / Issue 1 / October 1997
    Published online by Cambridge University Press:
    04 December 2007, pp. 67-97
    Print publication:
    October 1997
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  • Let $k$ be a perfect field of a positive characteristic $p, K$ – the fraction field of the ring of Witt vectors $W(k)$. Let $X$ be a smooth and proper scheme over $W(k)$. We present a candidate for a cohomology theory with coefficients in crystalline local systems: $p$-adic étale local systems on $X_K$ characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber $X_k$. We show that this cohomology satysfies a duality theorem.