Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), K-Theory and Homology (math.KT)

journal:

--

date:

2023-09-19 16:00:00

Abstract

In this paper we propose an $\infty$-categorical definition of abstract six-functor formalisms on varieties. Our definition is a variation on Mann's definition, with the additional requirement of having Grothendieck and Wirthm\"uller contexts, and recollements. Using Nagata's compactification theorem, we show that such a six-functor formalism can be given by just specifying adjoint triples on open immersions and on proper maps, satisfying certain compatibilities. Moreover, the existence of recollements is equivalent to a sheaf condition for a Grothendieck topology on the category of "varieties and spans of open immersions and proper maps". We show that our definition of six-functor formalism is equivalent to a full subcategory of lax symmetric monoidal functors from the category of smooth and complete schemes to the category of stable $\infty$-categories and adjoint triples, and characterise which lax monoidal functors on complete varieties extend to six-functor formalisms.