Artin perverse sheaves

TL;DR

This paper explores the structure of Artin perverse sheaves, establishing t-structures on Artin $ ext{l}$-adic complexes for schemes of dimension less than 2, and analyzing their properties over finite fields.

Contribution

It introduces a t-structure on Artin $ ext{l}$-adic complexes for low-dimensional schemes and describes the heart explicitly in the 1-dimensional case, also constructing a perverse homotopy t-structure over finite fields.

Findings

The perverse t-structure induces a t-structure on Artin $ ext{l}$-adic complexes for schemes of dimension less than 2.

The heart of this t-structure can be explicitly described in the 1-dimensional case.

The perverse homotopy t-structure over finite fields is the best approximation of the perverse t-structure.

Abstract

We show that the perverse t-structure induces a t-structure on the category $\mathcal{D}^A(S,\mathbb{Z}_\ell)$ of Artin $\ell$-adic complexes when $S$ is an excellent scheme of dimension less than $2$ and provide a counter-example in dimension $3$. The heart $\mathrm{Perv}^A(S,\mathbb{Z}_\ell)$ of this t-structure can be described explicitly in terms of representations in the case of $1$-dimensional schemes. When $S$ is of finite type over a finite field, we also construct a perverse homotopy t-structure over $\mathcal{D}^A(S,\mathbb{Q}_\ell)$ and show that it is the best possible approximation of the perverse t-structure. We describe the simple objects of its heart $\mathrm{Perv}^A(S,\mathbb{Q}_\ell)^\#$ and show that the weightless truncation functor $\omega^0$ is t-exact. We also show that the weightless intersection complex $EC_S=\omega^0 IC_S$ is a simple Artin homotopy perverse sheaf. If $S$ is a surface, it is also a perverse sheaf but it need not be simple in the category of perverse sheaves.