Duality between Cartier crystals and perverse $\mathbb{F}_p$-sheaves, and application to generic vanishing

We show that on any Noetherian $F$-finite $\mathbb{F}_p$-scheme, there is an anti-equivalence of categories between Cartier crystals and \'etale perverse $\mathbb{F}_p$-sheaves, commuting with derived proper pushforwards. We use this duality to construct an upper shriek functor for Cartier crystals, and give new proofs of Kashiwara's equivalence and the finite length of Cartier crystals. Finally, we deduce a generic vanishing statement for perverse $\overline{\mathbb{F}}_p$-sheaves on abelian varieties of characteristic $p > 0$, reminiscent of the characteristic zero and $l$-adic statements.