Local Rigidity and Six Functor Formalisms

TL;DR

This paper demonstrates that in six functor formalisms with locally rigid coefficient categories, certain pushforward and pullback formulas can be derived formally for proper maps and open embeddings.

Contribution

It shows that well-known pushforward formulas in six functor formalisms can be deduced formally under local rigidity conditions.

Findings

Formal derivation of pushforward formulas for proper maps and open embeddings.

Local rigidity of coefficient categories enables formal definitions of adjunctions.

Extension of known formulas to broader contexts via formal methods.

Abstract

The coefficient categories of six functor formalisms are often locally rigid, and when this is the case, the exceptional pushforward and pullback adjunctions may be defined formally. In this short note it is shown that for f a proper map resp. an open embedding the well known formulas f_! = f_* resp. f_! = f_# may likewise be deduced formally.