Uniqueness of six-functor formalisms

TL;DR

This paper proves that a six-functor formalism with enough cohomologically proper and étale morphisms is uniquely determined by tensor and inverse image functors, using an alternative framework and confirming Scholze's conjecture.

Contribution

It introduces an alternative formulation of key conditions in six-functor formalisms and proves the uniqueness of such formalisms under these conditions, confirming Scholze's conjecture.

Findings

Scholze's conjecture is proven under certain conditions.

A generalization of the conjecture is shown to fail.

A measure of the failure is proposed in terms of K-theory.

Abstract

We present an alternative formulation of Scholze's notions of cohomologically proper and cohomologically \'etale with respect to an abstract six-functor formalism. These conditions guarantee canonical isomorphisms between the direct and exceptional direct images for certain "proper" morphisms, and between the inverse and exceptional inverse images for certain "\'etale" morphisms. Using this framework, we prove Scholze's conjecture, showing that a six-functor formalism with sufficiently many cohomologically proper and \'etale morphisms is uniquely determined by the tensor product and inverse image functors, and can be obtained by a construction of Liu-Zheng and Mann. Additionally, we show that a generalisation of the conjecture fails, and propose a measure of this failure in terms of K-theory.