For an $F$-finite scheme $X$ separated over a perfect field $k$ of characteristic $p>0$ which admits an embedding into a smooth $k$-scheme, we establish an equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{ Z}/p\mathbb{ Z}$-sheaves on the \'{ e}tale site $X_{ \text{ \'{ e}t}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal{ O}_{ F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by using Emerton-Kisin's Riemann-Hilbert correspondence, we show that it is equivalent to Gabber's category of perverse sheaves in $D_c^b(X_{ \text{ \'{ e}t}},\mathbb{ Z}/p\mathbb{ Z})$. Furthermore, we define intermediate extensions for Cartier crystals and show that our equivalence between Cartier crystals and perverse constructible \'{ e}tale sheaves commutes with the intermediate extension functor.
Categories: math.AG math.AC math.NT