Actions of pro-groups and pro-rings

E. Yu. Voronetskii

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 40, Tome 531 (2024), pp. 53-70

Résumé

The notion of pro-groups, i.e. formal projective limits of groups, is quite useful in algebraic geometry, algebraic topology, and algebraic $\mathrm K$-theory. Such objects may be considered as pro-sets with a group structure, namely, the category of pro-groups is a full subcategory of the category of pro-sets. It is known that the category of pro-groups is semi-Abelian, i.e. it admits the notions of internal actions and semi-direct products. This paper is devoted to the natural problem of explicit description of pro-group actions on each other. It is proved that such actions are given by ordinary pro-set morphisms satisfying certain axioms as in the case of group actions by automorphisms. This result is also generalized to several categories of non-unital pro-rings. Finally, a counterexample is given showing that a similar description does not hold for Lie pro-algebras.