A prismatic approach to (, G) -modules and F-crystals

We give a new construction of (, G) -modules using the theory of prisms developed by Bhatt and Scholze. We give two applications of our results. Firstly, we provide a new proof for the equivalence between the category of prismatic F -crystals in finite locally free O*-₃. ₉₏ₓ{} -modules over (O₊) *-₃. ₉₏ₓ{} and the category of lattices in crystalline representations of G₊, where K is a complete discretely valued field of mixed characteristic with perfect residue field. Moreover, we generalize this result to semistable representations using the absolute logarithmic prismatic site defined by Koshikawa.