A Riemann-Hilbert correspondence for Cartier crystals

Abstract

For a variety $X$ separated over a perfect field of characteristic $p>0$ which admits an embedding into a smooth variety, we establish an anti-equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{Z}/p\mathbb{Z}$-sheaves on the étale site $X_{\text{eˊt}}$. 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 $D_c^b(X_{\text{eˊt}},\mathbb{Z}/p\mathbb{Z})$.

The results in this paper are part of the author's PhD thesis

References in corpus (2)

• The D-Module structure of R[F]-modules
• Riemann-Hilbert correspondence for unit $F$-crystals on embeddable algebraic varieties