Geometrization of six-functor formalisms
Geometrization of six-functor formalisms
Given a cohomology theory, it is sometimes possible to associate to each X a stack X', called the transmutation of X, in such a way that the cohomology of X agrees with the coherent cohomology of X'. In practice, each cohomology theory has an associated six-functor formalism Shv, and transmutation has the feature Shv(X) is equivalent to the derived category of quasi-coherent sheaves on X'.听
Transmutations were first constructed by Simpson for de Rham cohomology in characteristic zero, and have in recent years become available for some (but not all) six-functor formalisms of interest.听 听
The goal of this talk is to explain joint work with Scholze in which we construct an infinity topos of geometric objects which we call Gestalten, with the feature that *every* six-functor formalism gives rise to a transmutation functor into Gestalten.