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.