Let (A,I) be a bounded prism, and X be a smooth p-adic formal scheme over Spf(A/I). We consider the notion of crystals on Bhatt--Scholze's prismatic site (X/A)_{\prism} of X relative to A. We prove that if X is proper over 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 [0,2n]. We also establish a Poincar duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of (X/A)_{\prism}. The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.  

　　Publication:  

　　Journal f r die reine und angewandte Mathematik (Crelles Journal), vol. 2023, no. 800, 2023, pp. 217-257.  

　　https://doi.org/10.1515/crelle-2023-0032  

　　Author:  

　　Yichao Tian  

　　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  

　　Email: yichaot@math.ac.cn