Sheaves of Profinite Modules

Jiacheng Tang

The category of profinite R-modules (i.e. Inverse limits of finite R-modules) is abelian with nice limits, but coproducts are in general not exact functors. This means that it is difficult to use the coproduct in algebraic or homological settings. Instead of the coproduct, there is a notion of "profinite direct sums" with much better behaviour. Although profinite direct sums were defined algebraically when they first appeared, recent research revealed that they can be equivalently constructed as "global cosections of cosheaves" (dual to global sections of sheaves). In the first part of the talk, we will discuss categorical properties of profinite spaces/groups/modules..., including regularity and (Barr-)exactness. In the second part, we will define profinite direct sums both algebraically and categorically, and see why they are better than coproducts in the category of profinite modules.

Return to events