Riemann-Hilbert correspondence for unit $F$-crystals on embeddable algebraic varieties

Abstract

For a separated scheme $X$ of finite type over a perfect field $k$ of characteristic $p>0$ which admits an immersion into a proper smooth scheme over the truncated Witt ring $W_n$, we define the bounded derived category of locally finitely generated unit $F$-crystals with finite Tor-dimension on $X$ over $W_n$, independently of the choice of the immersion. Then we prove the anti-equivalence of this category with the bounded derived category of constructible étale sheaves of $\mathbb{Z}/p^n\mathbb{Z}$-modules with finite Tor dimension. We also discuss the relationship of $t$-structures on these derived categories when $n=1$. Our result is a generalization of the Riemann-Hilbert correspondence for unit $F$-crystals due to Emerton-Kisin to the case of (possibly singular) embeddable algebraic varieties in characteristic $p>0$.

This is the final version, to appear in Annales de l'Institut Fourier

Cited by in corpus (3)

• Cartier crystals and perverse constructible étale $p$-torsion sheaves
• A motivation for some local cohomologies
• A Riemann-Hilbert correspondence for Cartier crystals