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In this paper, we prove that for any $p$-adic smooth separated formal scheme$\mathfrak X$, the category of prismatic $F$-crystals with $I$ inverted isequivalent to the category of \'etale $\mathbb Z_p$-local systems on thegeneric fiber of $\mathfrak X$. We also compare the cohomology of thecorresponding coefficients.

Relative $(\varphi,\Gamma)$-modules and prismatic $F$-crystals