[Paper Review] Riemann-Hilbert correspondence for unit $F$-crystals on embeddable algebraic varieties

Sachio Ohkawa|arXiv (Cornell University)|Jan 7, 2016

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes a Riemann-Hilbert correspondence for unit $F$-crystals on $W_n$-embeddable algebraic varieties over a perfect field of positive characteristic, proving an anti-equivalence between the bounded derived category of locally finitely generated unit $F$-crystals with finite Tor-dimension and the bounded derived category of constructible étale sheaves of $\mathbb{Z}/p^n\mathbb{Z}$-modules. The construction is independent of the choice of immersion into a proper smooth $W_n$-scheme, generalizing Emerton-Kisin's theory beyond smooth liftings.

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 $p>0$.

Motivation & Objective

• To generalize Emerton-Kisin's Riemann-Hilbert correspondence for unit $F$-crystals from smooth $W_n$-schemes to $W_n$-embeddable algebraic varieties over a perfect field of characteristic $p>0$.
• To define a derived category of unit $F$-crystals on $W_n$-embeddable varieties that is independent of the choice of immersion into a proper smooth $W_n$-scheme.
• To prove that the Riemann-Hilbert correspondence respects Grothendieck's six operations, particularly $f_+$, $f^!$, and $τ$-product.
• To establish a $t$-structure on the $\mathcal{D}$-module side that corresponds to Gabber's perverse $t$-structure on the étale side for $W_n$-embeddable varieties.
• To show that the constructible $t$-structure on the $\mathcal{D}$-module side corresponds to the standard $t$-structure on the étale side via the Riemann-Hilbert equivalence.

Proposed method

• Uses Kashiwara's theorem and its positive characteristic analogue (Emerton-Kisin) to define the derived category of unit $F$-crystals on $X$ as the full subcategory of complexes in $D^b_{\rm lfgu}(\mathcal{D}_{F,P})^\circ$ supported on $X$, for any immersion $X \hookrightarrow P$ with $P$ proper smooth over $W_n$.
• Proves that this subcategory is independent of the choice of immersion $X \hookrightarrow P$, denoted $D^b_{\rm lfgu}(X/W_n)^\circ$.
• Constructs the Riemann-Hilbert correspondence as an anti-equivalence $D^b_{\rm lfgu}(X/W_n)^\circ \xrightarrow{\cong} D^b_{\rm ctf}(X_{\rm\acute{e}t}, \mathbb{Z}/p^n\mathbb{Z})$.
• Introduces Grothendieck's operations ($f_+$, $f^!$, $τ$-product) on the $\mathcal{D}$-module side and proves they correspond to $f_!$, $f^{-1}$, and $τ$-product on the étale side.
• Defines a $t$-structure on the $\mathcal{D}$-module side, called the constructible $t$-structure, using the structure of the ambient $P$, and proves its independence from the choice of $P$.
• Shows that the constructible $t$-structure corresponds to the standard $t$-structure on $D^b_{\rm c}(X_{\rm\acute{e}t}, \mathbb{Z}/p\mathbb{Z})$ via the Riemann-Hilbert correspondence.

Experimental results

Research questions

• RQ1 Can the Riemann-Hilbert correspondence for unit $F$-crystals be extended from smooth $W_n$-schemes to more general $W_n$-embeddable varieties over a perfect field of characteristic $p>0$?
• RQ2 Is the derived category of unit $F$-crystals on a $W_n$-embeddable variety independent of the choice of immersion into a proper smooth $W_n$-scheme?
• RQ3 Does the Riemann-Hilbert correspondence respect Grothendieck's six operations, particularly $f_+$, $f^!$, and $τ$-product, in the $W_n$-embeddable setting?
• RQ4 Can a $t$-structure on the $\mathcal{D}$-module side be defined that corresponds to the standard $t$-structure on the étale side for $W_n$-embeddable varieties?
• RQ5 Does Gabber's perverse $t$-structure on the étale side correspond to the standard $t$-structure on the $\mathcal{D}$-module side in the $W_n$-embeddable case?

Key findings

• The derived category $D^b_{\rm lfgu}(X/W_n)^\circ$ of locally finitely generated unit $F$-crystals with finite Tor-dimension on a $W_n$-embeddable variety $X$ is independent of the choice of immersion into a proper smooth $W_n$-scheme $P$.
• The Riemann-Hilbert correspondence induces an anti-equivalence $D^b_{\rm lfgu}(X/W_n)^\circ \xrightarrow{\cong} D^b_{\rm ctf}(X_{\rm\acute{e}t}, \mathbb{Z}/p^n\mathbb{Z})$, generalizing Emerton-Kisin's result to non-smoothly liftable varieties.
• The Riemann-Hilbert correspondence respects Grothendieck's operations: $f_+$ corresponds to $f_!$, $f^!$ to $f^{-1}$, and $τ$-product to $τ$-product.
• A $t$-structure on the $\mathcal{D}$-module side, called the constructible $t$-structure, is defined and shown to be independent of the choice of $P$, via the structure of the ambient $P$.
• The constructible $t$-structure on $D^b_{\rm lfgu}(X/W_n)^\circ$ corresponds to the standard $t$-structure on $D^b_{\rm c}(X_{\rm\acute{e}t}, \mathbb{Z}/p\mathbb{Z})$ via the Riemann-Hilbert correspondence.
• For $n=1$, the constructible $t$-structure on the $\mathcal{D}$-module side corresponds to Gabber's perverse $t$-structure on the étale side, extending Emerton-Kisin's result to $W_n$-embeddable varieties.