Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Six-functor formalisms are compactly supported

In this talk I will discuss a version of Mann’s infinity-categorical definition of six-functor formalisms. With Nagata's compactifiction theorem, we can show that any six-functor formalism with Grothendieck and Wirthmüller contexts, can be given by just specifying adjoint triples on open immersions and on proper maps, satisfying certain compatibilities. Moreover, for such a six-functor formalism, having recollements and descent for abstract blowups is equivalent to a sheaf condition for a Grothendieck topology on the category of “varieties and spans with an open immersion and a proper map”. We use this to show that a six-functor formalism with recollements and descent for abstract blowups, is uniquely determined by the restriction of the inverse image (upper star) to smooth and complete varieties. Moreover we can characterize which lax symmetric monoidal functors from the category of complete varieties to the category of stable infinity-categories and adjoint triples (encoding upper star), extend to six-functor formalisms.