In this talk, I will explain a theorem of Bhatt-Scholze: the equivalence between prismatic $F$-crystal and $\mathbb Z_p$-lattices inside crystalline representation, and how to extend this theorem to allow more general types of base ring like Tate algebra ${\mathbb Z}_p\langle t^{\pm 1}\rangle$.  This is a joint work with Heng Du, Yong-Suk Moon and Koji Shimizu.

This is a talk in integral $p$-adic Hodge theory.  So in the pre-talk, I will explain the motivations and base ideas in integral $p$-adic Hodge theory.