[Paper Review] Cartier crystals and perverse constructible étale $p$-torsion sheaves

Tobias Schedlmeier|arXiv (Cornell University)|Mar 24, 2016

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper establishes a Riemann-Hilbert correspondence for Cartier crystals on singular schemes in positive characteristic p, linking them to perverse constructible étale p-torsion sheaves via the solution functor Solκ. It constructs intermediate extensions for Cartier crystals and proves that Solκ preserves these extensions, thereby extending classical D-module duality to the positive characteristic setting using locally finitely generated unit modules and recollement formalism.

ABSTRACT

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 étale site $X_{ ext{ét}}$. 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 $D_c^b(X_{ ext{ét}},\mathbb{Z}/p\mathbb{Z})$. Furthermore, we define intermediate extensions for Cartier crystals and show that our equivalence between Cartier crystals and perverse constructible étale sheaves commutes with the intermediate extension functor.

Motivation & Objective

• To extend the Riemann-Hilbert correspondence from smooth complex varieties to singular schemes in positive characteristic p.
• To define and study intermediate extensions of Cartier crystals using recollement formalism.
• To establish a compatibility between the solution functor Solκ and intermediate extensions, ensuring preservation of perverse structures.
• To generalize Emerton-Kisin's adjunction and Riemann-Hilbert correspondence to singular schemes via locally finitely generated unit modules.
• To provide a derived category equivalence between Cartier crystals and perverse constructible étale p-torsion sheaves on singular schemes.

Proposed method

• Uses Cartier crystals as the algebraic counterpart to D-modules in positive characteristic, defined via Frobenius pullbacks.
• Introduces locally finitely generated unit (lfgu) modules as a generalization of Cartier modules to singular schemes.
• Applies recollement formalism to the derived categories of étale sheaves on open-closed decompositions X = U ∪ Z.
• Constructs intermediate extensions j!∗M as the image of the canonical morphism pj! → pj∗ in the derived category.
• Employs the solution functor Solκ: D⁺_crys(κ(X)) → D⁺_c(Ué t, ℤ/pℤ) to relate Cartier crystals to perverse sheaves.
• Proves compatibility of Solκ with intermediate extensions via adjunction and universal property arguments in the recollement setting.

Experimental results

Research questions

• RQ1 Can a Riemann-Hilbert correspondence be established for Cartier crystals on singular schemes in positive characteristic?
• RQ2 How do intermediate extensions of Cartier crystals behave under the solution functor Solκ?
• RQ3 Does the solution functor Solκ commute with intermediate extensions in the derived category of étale sheaves?
• RQ4 Can the adjunction formalism of Emerton-Kisin be generalized to singular schemes using lfgu modules?
• RQ5 What is the precise relationship between perverse constructible étale p-torsion sheaves and Cartier crystals on singular schemes?

Key findings

• The solution functor Solκ induces an equivalence between the derived category of Cartier crystals and the derived category of perverse constructible étale p-torsion sheaves on singular schemes.
• Intermediate extensions j!∗M of Cartier crystals are characterized as the unique extensions that admit no nontrivial subcrystals or quotients supported on the closed complement.
• The intermediate extension functor j!∗ on Cartier crystals preserves monomorphisms and epimorphisms, ensuring compatibility with exact sequences.
• Solκ commutes with intermediate extensions: there is a natural isomorphism j!∗∘Solκ ≅ Solκ∘j!∗, which establishes the compatibility of the solution functor with the intermediate extension construction.
• The Riemann-Hilbert correspondence for Cartier crystals on singular schemes is realized via the solution functor Solκ, extending the classical correspondence to the singular and positive characteristic setting.
• The formalism of recollement applies to the derived categories of étale sheaves with constructible cohomology, allowing the construction of intermediate extensions and the verification of their universal properties.