Artin perverse sheaves

Raphaël Ruimy (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 over excellent schemes of dimension less than $2$ and provide a counter-example in dimension $3$. Its heart $\mathrm{Perv}^A(S,\mathbb{Z}_\ell)$ can be described explicitely in terms of representations in the case of a $1$-dimensional excellent scheme. Over schemes of finite type over a finite field and with coefficients $\mathbb{Q}_\ell$, we also construct a homotopy perverse t-structure and show that it is final among the t-structures such that the inclusion functor is right t-exact. We describe the simple objects of its heart $\mathrm{Perv}^A(S,\mathbb{Q}_\ell)^\#$ and show that the Artin 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 perverse sheaf. If $S$ is a surface, it is also a perverse sheaf but it need not simple in the category of perverse sheaf.

Domaines: Géométrie algébrique [math.AG]

Raphaël Ruimy. Artin perverse sheaves. 2023. ⟨hal-03669896v2⟩