An introduction to six-functor formalisms

TL;DR

This paper provides a comprehensive introduction to Grothendieck's six-functor formalism in algebraic geometry, including an overview of its applications to rigid-analytic motives, aimed at educating newcomers and researchers.

Contribution

It offers a self-contained, accessible overview of the six-functor formalism, bridging foundational concepts with applications in motivic homotopy theory.

Findings

Clarifies the structure of six-functor formalism

Connects formalism to rigid-analytic motives

Serves as an educational resource

Abstract

These are notes for a mini-course given at the summer school and conference "The Six-Functor Formalism and Motivic Homotopy Theory" in Milan 9/2021. They provide an introduction to the formalism of Grothendieck's six operations in algebraic geometry and end with an excursion to rigid-analytic motives. The notes do not correspond precisely to the lectures delivered but provide a more self-contained account for the benefit of the audience and others. No originality is claimed.