Wildly Compatible Systems and Six Operations

TL;DR

This paper introduces wildly compatible systems of constructible sheaves over schemes, demonstrating their stability under Grothendieck's six operations and showing that classical compatible systems are examples of these systems.

Contribution

It defines wildly compatible systems for schemes over regular bases and proves their stability under key operations, extending the understanding of compatible systems in algebraic geometry.

Findings

Wildly compatible systems are preserved by six operations for schemes over bases of dimension ≤ 1.

All classical $\,\ell$-adic compatible systems are examples of wildly compatible systems.

The paper establishes a new framework linking compatible systems with six operations stability.

Abstract

For a scheme $X$ separated and of finite type over an excellent regular scheme $S$, we define wildly compatible systems of constructible sheaves of modules over finite fields on $X$ for certain vector spaces $V$. The main result is that for $\dim S \leq 1$, wildly compatible systems are preserved by Grothendieck's six operations and Verdier's duality. Finally, for a smooth integral scheme $X$ over a finite field, we prove that all $\ell$-adic compatible systems gives wildly compatible systems.