Finiteness and duality for the cohomology of prismatic crystals

Abstract

Abstract Let ( A , I ) {(A,I)} be a bounded prism, and let X be a smooth p-adic formal scheme over Spf ⁡ ( A / I ) {\operatorname{Spf}(A/I)} . We consider the notion of crystals on Bhatt–Scholze’s prismatic site ( X / A ) Δ ⁢ Δ {(X/A)_{{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}} of X relative to A. We prove that if X is proper over Spf ⁡ ( A / I ) {\operatorname{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 , 2 ⁢ n ] {[0,2n]} . We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of ( X / A ) Δ ⁢ Δ {(X/A)_{{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}} . The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.