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 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.

Identity Theorem for Pro-p-groups

“…We can always identify G u with the continuous Q p -prounipotent completion of G [23, (4)]. This result (Theorem 1) has been announced in [26,Corollary 12] and used in [23, Proposition 3, Corollary 3] for proof of the criterion of cohomological dimension equal to 2, providing a relation with the known group theory results (see [14,32,37,8] and other papers cited there). Let us recall the results of [23, Proposition 3, Corollary 3], since they shed light on the following Serre's question from [35].…”

Quasirationality and prounipotent crossed modules

“…In this paper we shall give the description, announced in [21], of modules of relations of quasirational pro-p-presentations by means of affine group schemes technique. For these purposes, after recalling necessary constructions in Section 2, in Section 3 we construct a prounipotent presentation (3) from a finite presentation of a pro-p-group (2) by means of Qp-prounipotent completion (Definition 3) of finitely generated free pro-p-groups ("schematization").…”

“…For Fp-prounipotent groups, for which pro-p-groups are their Fp-points, it would be too optimistic to hope for a similar statement, but we shall show that existence of an embedding into a similar type of completion implies that cohomological dimension of such a group is less or equal to 2. Using a description of the relations module of a prounipotent group with one defining relation [21,Corollary 12] we point out (Proposition 3 and Corollary 3), in terms of continuous prounipotent completion, the condition under which a finitely generated pro-p-group with one defining relation has cohomological dimension 2. The proof is based on Theorem 1 and Corollary 2.…”

Stacks

“…Breen had an early paper on the schematic sphere [9], and one should note that his paper on cohomology calculations [8] provides in retrospect the foundation for schematization. More recently, Katzarkov Pantev and Toën [23] [24], Pridham [34] and others [28] defined the schematization in full generality and developed Hodge theory for it.…”