A note on purity of crystalline local systems

In this short note, we prove a purity result for crystalline local systems on a smooth p p -adic affine formal scheme. Our method is based on the prismatic description of crystalline local systems (see Du et al. [Compos. Math. 160 (2024), pp. 1101–1166]; cf. Guo and Reinecke [Invent. Math. 236 (2024), pp. 17–164]), which is different from the approach by Tsuji [Crystalline sheaves and filtered convergent F F -isocrystals on log schemes, preprint].