We introduce the concept of quasirational relation modules for discrete (prop) presentations of discrete (pro-p) groups. It is shown, that this class of presentations for discrete groups contains CA-presentations and their subpresentations. For pro-p-groups we see that all presentations of pro-p-groups with a single defining relation are quasirational. We offer definitions of padic G(p)-completion and p-adic rationalization of relation modules which are adjusted to quasirational pro-p-presentations. p-adic rationalizations of quasirational relation modules of pro-p-groups are isomorphic to Q p -points of abelianized p-adic Malcev completions.

The concept of schematization consists in replacing simplicial groups by simplicial affine group schemes. In the case when the coefficient field has a zero characteristic, there is a prominent theory of simplicial prounipotent groups, the origins of which lead to the rational homotopy theory of D. Quillen. The specificity of a prime finite field F p is that the Zariski topology on the n-dimensional affine space F n p turns out to be discrete. Therefore, although finite p-groups are F p -points of F p -unipotent affine group schemes, this observation is not actually used. Nevertheless, schematization reveals the profound properties of F p -prounipotent groups, especially in connection with prounipotent groups in the zero characteristic and in the study of quasirationality. In this paper, using results on representations and cohomology of prounipotent groups in characteristic 0, we prove an analogue of Lyndon Identity theorem for one-relator pro-p-groups (question posed by J.P. Serre) and demonstrate the application to one more problem of J.-P. Serre concerning onerelator pro-p-groups of cohomological dimension 2. Schematic approach makes it possible to consider the problems of pro-p-groups theory through the prism of Tannaka duality, concentrating on the category of representations. In particular we attach special importance to the existence of identities in free pro-p-groups ("conjurings").

We study quasirational presentations (QR-presentations) of (pro-p)groups, which contain aspherical presentations and their subpresentations, and also still mysterious prop-groups with a single defining relation. Using schematization of QR-presentations and embedding of the rationalized module of relations into a diagram related to certain prounipotent crossed module, we study cohomological properties of pro-p-groups with a single defining relation (a question of J.-P. Serre).

For every quasirational (pro-p)relation module R we construct the so called p-adic rationalization, which is the pro-fd-module R ⊗Q p = limstands for the rational points of the abelianization of the continuous p-adic Malcev completion of R. We show how R ∧ w embeds into a sequence which arises from a certain prounipotent crossed module. The latter can be seen as concrete examples of proalgebraic homotopy types.We provide the Identity Theorem for pro-p-groups, giving a positive feedback to the question of Serre.