Polynomial functors on categories of finitely generated modules

A strict polynomial functor is a functor from the category of finitely generated projective modules to the category of modules which acts on morphisms via polynomials. The latter condition may be seen as a compatibility with base changes. If we extend the domain to the categories of all finitely generated modules over varying rings, we arrive at the notion of an operation. Exterior, symmetric and divided powers provide typical examples of operations. In characteristic zero, we give a full classification of the category of operations in terms of Schur functors associated to representations of symmetric groups. We will also explain the relationship to tensor categorical algebraic geometry. This framework leads to conjectures about operations on other categories as well, for example about operations from commutative algebras to modules.