[Paper Review] Mixed $\ell$-adic complexes for schemes over number fields

Sophie Morel|arXiv (Cornell University)|Jun 8, 2018

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper resolves the absence of a weight filtration in mixed $π$-adic perverse sheaves on schemes over number fields by constructing a derived category of perverse sheaves that admit weight filtrations. Using Deligne's weight-monodromy theorem and a crossed functor formalism, it establishes the full six operations (including tensor products and internal Hom) on this category, enabling generalization of Beilinson's intermediate extension formula to arbitrary finitely generated fields.

ABSTRACT

If $X$ is a variety over a number field, Annette Huber has defined a category of "horizontal" (or "almost everywhere unramified") $\ell$-adic complexes and $\ell$-adic perverse sheaves on $X$. For such objects, the notion of weights makes sense (in the sense of Deligne), just as in the case of varieties over finite fields. However, contrary to what happens in that last case, mixed perverse sheaves (or mixed locally constant sheaves) on $X$ do not have a weight filtration in general, even when $X$ is a point. The goal of this paper is to show how to avoid this problem by working directly in the derived category of the abelian category of perverse sheaves that do admit a weight filtration. As an application, the methods of a previous paper of the author to calculate the intermediate extension of a pure perverse sheaf apply over any finitely generated field, and not just over a finite field.

Motivation & Objective

• To address the failure of mixed perverse sheaves on schemes over number fields to admit weight filtrations, even for a point.
• To construct a category of perverse sheaves that are closed under the six operations and admit weight filtrations.
• To generalize Beilinson's intermediate extension formula from finite fields to arbitrary finitely generated fields.
• To establish a formalism of sheaf operations compatible with weights using Deligne's weight-monodromy theorem.
• To provide a framework for extending results from finite fields to number fields via stable t-structures and Ext vanishing.

Proposed method

• Define the full abelian subcategory $\operatorname{Perv}_{mf}(X)$ of horizontal mixed perverse sheaves on $X$ that admit a weight filtration.
• Use Deligne's weight-monodromy theorem to prove that perverse direct images preserve $\operatorname{Perv}_{mf}(X)$, ensuring stability under key operations.
• Employ the formalism of crossed functors (inspired by Deligne, Voevodsky, and Ayoub) to systematically handle compatibility of sheaf operations.
• Construct the four operations ($f^*, f_*, f_!, f^{!}$) and tensor products/internal Hom on $D^b\operatorname{Perv}_{mf}(X)$ using perverse sheaf-level constructions.
• Prove that the categories ${}^w\!D^{\leq a}(X)$ and ${}^w\!D^{\geq a+1}(X)$ form a t-structure on $D^b\operatorname{Perv}_{mf}(X)$, extending the weight filtration.
• Use Yoneda description of Ext groups to show $\operatorname{Ext}^i(K,L) = 0$ for $K$ of weight $\leq a$, $L$ of weight $\geq a+1$, ensuring orthogonality of weight components.

Experimental results

Research questions

• RQ1 Can the six operations be defined on the derived category of perverse sheaves that admit weight filtrations over schemes over number fields?
• RQ2 Does the category $\operatorname{Perv}_{mf}(X)$ remain stable under perverse direct images when $X$ is defined over a number field?
• RQ3 Can Beilinson's formula for the intermediate extension of a pure perverse sheaf be generalized beyond finite fields?
• RQ4 What t-structure on $D^b\operatorname{Perv}_{mf}(X)$ corresponds to the weight filtration, and how does it interact with sheaf operations?
• RQ5 How can the Ext vanishing between sheaves of disjoint weights be established without assuming semisimplicity of pure objects?

Key findings

• The category $\operatorname{Perv}_{mf}(X)$ is stable under perverse direct images, as shown via Deligne's weight-monodromy theorem.
• The pair $({}^w\!D^{\leq a}, {}^w\!D^{\geq a+1})$ forms a t-structure on $D^b\operatorname{Perv}_{mf}(X)$, generalizing the weight filtration to the derived category.
• Ext groups vanish between perverse sheaves of disjoint weights: $\operatorname{Ext}^i(K,L) = 0$ when $K$ has weight $\leq a$ and $L$ has weight $\geq a+1$, even without semisimplicity.
• The canonical morphisms $w_{\geq a}j_!K \to j_{!*}K \to w_{\leq a}j_*K$ are isomorphisms for pure $K$ of weight $a$, generalizing Beilinson's result to number fields.
• The derived category $D^b\operatorname{Perv}_{mf}(X)$ supports all six operations (including tensor products and internal Hom) via a crossed functor formalism.
• The construction allows the intermediate extension formula from [21] to be extended to schemes over any finitely generated field, not just finite fields.