The abelianization of inverse limits of groups
The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all direct limits. It is thus natural to wonder about the behavior of the abelianization functor under inverse limits. There is always a natural map from the abelianization of an inverse limit of groups to the inverse limit of their abelianizations. In this lecture I will present results giving restrictions on the kernel and cokernel of this natural map, in certain cases. These cases include countable directed inverse limits of finite groups, and can thus help in the calculation of the abelianization of certain profinite groups. If time permits I will also consider other families of functors into abelian groups.
This is a joint work with Saharon Shelah.