Proalgebraic crossed modules of quasirational presentations

Abstract

For every quasirational (pro-p) relation module R, we construct the so called p-adic rationalization, which is the pro-fd module (Formula Presented), and prove the isomorphism (Formula Presented), where (Formula Presented) stands for the rational points of the abelianization of the continuous p-adic Malcev completion of R. We show how (Formula Presented) embeds into a sequence which arises from a certain prounipotent crossedmodule. The latter can be seen as concrete examples of proalgebraic homotopy types. We provide the Identity Theorem for pro-p-groups, answering a question of Serre.