| Izvestiya: Mathematics |
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Rational and A. M. Mikhovich Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, Moscow
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    § 1. IntroductionJ. H. C. Whitehead’s conjecture (also known as Whitehead’s asphericity problem) is a claim in algebraic topology. It states that every connected subcomplex of a two-dimensional aspherical $\mathrm{CW}$-complex is aspherical. The question was formulated by Whitehead [52] in 1941 and is still an open problem with many implications in topology, non-commutative geometry and group theory [5], [7]. Despite the long history of the question, which originally arose in relation with the asphericity of the knot complement in the 3-sphere, the hypothesis is still open today, as “the level of our ignorance” [3] in this area remains extremely high after the last 40 years. If Whitehead’s asphericity conjecture is true, then there is a counterexample to the Eilenberg–Ganea conjecture [6], [22] (the existence of a group of cohomological dimension 2 that does not have a 2-dimensional Eilenberg–Maclane space), promising negative solutions to a bunch of other important open problems in algebraic topology and non-commutative geometry. Collapsing the maximal subtree of the 1-skeleton of a 2-dimensional $\mathrm{CW}$-complex $K$ into a point does not change the homotopy type of $K$ and therefore the statement of J. H. C. Whitehead’s asphericity conjecture is equivalent to the statement that each subpresentation of the aspherical presentation of a group is aspherical. A presentation $(X\,|\,R)$ of a discrete group $G$ is by definition an exact sequence where $\Phi=\Phi(X)$ is a free discrete group with set $X$ of generators, and $R$ is a normal subgroup in $\Phi$ normally generated by a set $Y\subset R$ of defining relations. Our use of the term “presentation” is not standard and somewhat misleading. It should be clearly understood that we are considering the presentation of a group, and not the extension of the group that it defines. We leave the standard notation $(X\,|\,Y)$, where $Y$ is a set of relations for simplicial presentations, which appear below (3). By a subpresentation of $(X\,|\,R)$, we mean a presentation $(X\,|\,R_0)$, $R_0\subset R$, where $R_0$ is normally generated by a subset $Y_0\subset Y$. A group presentation $(X\,|\,R)$ is called aspherical if the two-dimensional $\mathrm{CW}$-complex $ K(X\,|\,R)$ associated with this presentation is aspherical, that is, $\pi_2K(X\,|\,R)=0$. The exact construction of $ K(X\,|\,R)$ will appear in § 2, we also refer the reader to standard homotopy theory textbooks as [20].A pro-$ p $-group is a group isomorphic to an inverse limit of finite $p$-groups. This is a topological group (with inverse limit topology), which is compact and totally disconnected. For such groups, one has a presentation theory similar in many aspects to the combinatorial theory of discrete groups. The origin of cohomological and combinatorial theory of pro-$p$-groups lies in the early papers of J.-P. Serre and J. Tate. This theory took its modern form in the monograph [46]. Several important results in discrete group theory arose as analogues of similar statements on pro-$p$-groups. For example, the celebrated Stallings theorem, stating that a discrete group is free if and only if its cohomological dimension equals one, arose from the analogy proposed by J.-P. Serre, with the known result from pro-$p$-group theory. Such parallelism arises again and again, especially in problems of (co)homological origin. For instance, Shafarevich’s question (see [47], § 18, C) asking why Demushkin pro-$p$-groups have similar defining relations as discrete presentations of surface groups is solved cohomologically [14] because, as in both discrete and pro-$p$ cases, these groups are Poincaré duality groups of dimension 2. Serre’s philosophy began to gain ground in Sullivan (see [48], Chap. 3) for simple spaces of finite type in the form of the so-called “arithmetic squares”. Sullivan uses completions or localizations in order to “fracture” a homotopy type into “mod-p components” together with coherence information over the rationals. These ideas were further developed by Bousfield and Kan for nilpotent simplicial spaces (see [8], Lemma 8.1) and it turned out that such a space $X$ (we refer the reader to [32] concerning simplicial methods) can be obtained as a pull-back of the diagram where $\pi$ is a set of all primes, the top map is induced by $\kappa_p$, which are the so-called $(\mathbb{F}_p)_{\infty}$-completions (that are the $p$-adic completions [8]), the bottom map is induced by $\mathbb{Q}_{\infty}(\kappa_p)$, which are completions of the top maps. $\mathbb{Q}_{\infty}$ (it is often called rationalization or localization at $(0)$; see [8]) and $(\mathbb{F}_p)_{\infty}$-completion functors are related to the so-called $p$-adic and $k$-prounipotent completions of discrete groups (the exact Definition 8 of prounipotent groups will appear in § 4.1, where we will also see that pro-$p$-groups can be regarded as prounipotent affine group schemes over $\mathbb{F}_p$). We recall the definition.Definition 1 (see [18], Appendix A.2, [33], § 2). Let us fix a group $G$ and a field $k$ of characteristic zero, define the prounipotent completion of $G$ as the following universal diagram, where $\rho$ is a Zariski dense homomorphism from $G$ to the group of $k$-points of a prounipotent affine group scheme $G_k^{\wedge}$, We require that, for each Zariski dense homomorphism $\chi$, there exists a unique homomorphism $\tau$ of prounipotent groups making the diagram commutative. When $k=\mathbb{F}_p$, we define the prounipotent completion of $G$ as the pro-$p$-completion, which is a pro-$p$-group $G^{\wedge}_p$ that obeys the same universal diagram. Here, $H(k)$ is a finite $p$-group and $G^{\wedge}_p\cong\varprojlim_{|G/U_{\lambda}|=p^{k_{\lambda}}} G/U_{\lambda}$, see Example 2.1.6 in [42]. According to Bousfield and Kan (see [8], Proposition 4.1), for each simplicial reduced space $X$ of finite type, one can construct an $R_{\infty}$-completion up to homotopy ($R\,{=}\,\mathbb{Q},\mathbb{F}_p$) in three steps: (i) replace $X$ with the so-called “Kan’s loop group” $\mathrm{GX}$ (see [32], § 26); this is a simplicial group that has the homotopy type of “loops on $X$”; (ii) apply the $R$-prounipotent completions functor dimension-wise, thereby obtaining a simplicial prounipotent group $(\mathrm{GX})^{\wedge}_R$; (iii) take the classifying space $\overline{W}(\mathrm{GX})^{\wedge}_R$ of $(\mathrm{GX})^{\wedge}_R$. The Kan loop-group functor $G$ and the classifying space functor $\overline{W}$ are the adjoint functors (see [32], Theorem 27.1) and, in addition, they establish the so-called Quillen equivalence of the corresponding homotopy categories. Thus, if we understand $(\mathrm{GX})^{\wedge}_R$, then we understand $R_{\infty}X=\overline{W}(\mathrm{GX})^{\wedge}_R$, and therefore, by rational and $p$-adic analogues of a discrete problem we mean the corresponding problems concerning prounipotent and pro-$p$ presentations (in particular, Whitehead’s conjecture). Bousfield and Kan (see [8], p. 2) claim that the rational case “often take care of itself”. In contrast, recent results [33], [35] show that this is not the case in the two-dimensional homotopy theory. For example, one-relator presentations over fields of characteristic zero have prounipotent cohomological dimension two, that is, they behave as their discrete analogues. In contrast, there are uncountably many non-isomorphic finitely generated pro-$p$-groups with a single defining relation [43] (this conjecture is commonly known as the Lubotzky conjecture) and it seems that they are much more complicated (see [35], § 4) than their discrete one-relator cousins. Berrick and Hillman (see [5], Corollary 4.8) proved that Whitehead’s asphericity conjecture is equivalent to the statement about the rational cohomological dimension; as a result, characteristic zero is also important for aspherical presentations. Cohomological and presentation theories are similar for pro-$p$-groups and for prounipotent groups in characteristic zero (see § 4). Moreover, there is a connection (see § 4.2) between zero and positive characteristics. A short but rather imprecise description of the main results of this paper is the following theorem (see Theorem 2 in § 4.7). Theorem 1. Let $(X\,|\,R_0)$ be a subpresentation of an aspherical prounipotent presentation $(X\,|\,R)$ (if the base field $k$ has a positive characteristic, we assume that $(X\,|\,R)$ is a pro-$p$-presentation). Then $(X\,|\,R_0)$ is also aspherical. Anick’s paper [2] contains a simply connected version of the rational Whitehead’s problem, that is, it is assumed that the fundamental group is trivial: “In rational homotopy, we generally consider simply connected spaces only and tensor all homotopy groups with $\mathbb{Q}$”. At the time of writing, it was known that the rational homotopy theory generalizes at least to the case of non-simply connected nilpotent spaces [8], and so the restriction rather reflects the inability to extend the methods of [2] to the non-simply connected case. The paper is organized as follows. In § 2, we give a concise introduction to two-dimensional complexes and their second homotopy groups to explain that Whitehead’s original statement is equivalent to the purely algebraic problem of crossed modules arising from simplicial presentations. In order to formulate correctly the main theorem, it is necessary to define $\pi_2$ for a prounipotent presentation. For a discrete presentation, $\pi_2$ is the second homotopy group of the geometric realization and can be calculated as $\pi_1$ of the reduced simplicial group (3). Thus, the homotopy groups of the simplicial group (3) are homotopy groups of the geometric realization with index minus one. The above approach allows us to define $\pi_2$ for prounipotent presentations in § 4.5, just as for discrete presentations, as $\pi_1$ of the corresponding prounipotent simplicial group as in Proposition 1. It follows from the Bousfield–Kan result mentioned above that if we take a finite discrete presentation (3) and construct its prounipotent (pro-$p$) completion (that is, replace discrete free groups with free prounipotent (pro-$p$) groups and the homomorphisms in (3) by those induced by the prounipotent (pro-$p$) completion), then the classifying space of such a simplicial construction is homotopy equivalent to the corresponding Bousfield–Kan completion (which is actively exploited in [36]). In § 2.2, we show that the simplicial presentation is a combinatorial model of based loops $\Omega K(X\,|\,R)$ on the standard 2-dimensional complex $K(X\,|\,R)$. Section 3 is dedicated to affine group schemes, their representations and cohomology. We have followed Jantzen’s account [24] closely, as it perfectly fits our purposes. In § 3.3, we derive the Lyndon–Hochschild–Serre spectral sequence “by hand”, since it is the form that it will be useful for the proof in § 5.4. Section 3.4 contains a new key concept of a topological module, extending the definition given in the Hain’s pioneering work [19]. In the pro-$p$-group theory, it is usually overlooked that complete group algebras equipped with cocommutative comultiplication arise by continuous duality from the only commutative multiplication in the algebra of locally-constant functions on a compact group. Indeed, such a multiplication in itself does not carry additional information. However it allows one to endow complete pro-$p$-group algebras with the structure of a complete Hopf algebra, which makes possible to explain why the topological module definition is exactly what it is, and not something else (see Remark 1). Note that, in characteristic zero, we are dealing with commutative shuffle algebra multiplication. In § 4, we have diligently unified the presentation of the theory of prounipotent groups for characteristics zero and for pro-$p$-groups (simplicial prounipotent presentations, subpresentations and prounipotent crossed modules); as a consequence, thanks to this approach, in the pro-$p$-case, we get all proofs for free. The special feature of our approach is the systematic use of the one-point compactification of some discrete basis as a topological space generating a prounipotent group. This discrete basis, as we show, corresponds to a certain system of generators in the tensor algebra, which, in turn, is an algebra of regular functions for the corresponding free prounipotent group. This is done in particular to make sure that J.-P. Serre’s parallelism can equally arise from the theory of prounipotent groups over a field of characteristic zero. Sometimes the pro-$p$-case, which is technically more simple, fits perfectly into the general theory developed in the language of group schemes. The generality of our approach is due to applications [36]; in fact, it is a variation of Grothendieck’s concept of schematization [17], our cohomological proof automatically works also in the pro-$p$-case. This section also contains the exact statement of Theorem 2, which is the main result of the present paper. In § 5, we define relation modules of prounipotent presentations for the first time in such a generality (we recommend the reader to check the difference with the pro-$p$-case [42]) and establish a criterion of asphericity in the relation module language. Usually, such a theory of presentations is developed under the minimality assumption. It is required that the systems of generators in a prounipotent group and in the corresponding free prounipotent group of a presentation are “cohomologically” isomorphic. We reduce the theory of “non-proper” (that is, non-minimal) presentations to “proper” (that is, minimal) presentations; this paves the way for obtaining final result in the required generality. This part contains a proof of a criterion of cohomological dimension 2 for a prounipotent group, defined by a subpresentation, adapting Tsvetkov’s ideas for pro-$p$-groups. In [4], [12], it was shown that any homotopy type $X$ can be represented by a pair of discrete groups $(G_X,P_X)$, where $P_X$ is a normal perfect subgroup in $G_X$ (a group $P$ is said to be perfect if $P=[P,P]$). In more detail, $X$ can be recovered in the homotopy category from the plus-construction $(K(G_X,1),P_X)^+$. Thus, the category of connected $\mathrm{CW}$-complexes and pointed homotopy classes of maps is equivalent to the category of fractions of the category of pairs $(G,P)$, where $G$ is a group and $P$ is a perfect normal subgroup. It is noteworthy that, in the category of prounipotent groups, where there are no non-trivial perfect subgroups, asphericity is completely determined, as it should be, with the help of $K(G,1)$ (see Propositions 15, 16) by the cohomological dimension. Section 6 contains a discussion of applications from [36]: Theorem 3, which states that Bousfield–Kan completions are aspherical for finite subcomplexes of contractible 2-complexes (that is the critical case of the discrete Whitehead’s conjecture [5]), an explanation of Serre’s philosophy to the effect that the homotopy types of pro-$p$-presentations determine the behavior of discrete presentations in problems of (co)homological nature. In contrast to our previous works [35], [33], we use right topological modules, as they are better compatible with the existing topological literature [24], [9]. We use the standard notation: $\mathbb{Z}_p$ for $p$-adic integers; $\mathbb{Q}_p$ for rational $p$-adic numbers; $\mathbb{F}_p$ for a prime field of positive characteristic, that is, $\operatorname{char}(k)=p\geqslant 2$ is a prime number. § 2. Homotopy theory of discrete presentationsSection 2.1 contains a summary of homotopy theory of discrete group presentations. We closely follow the Brown–Hübschmann [9] approach to make our presentation as consistent as possible with references. In Lemma 1, we show that a simplicial purely algebraic crossed module definition coincides with the classical ones going back to Whitehead and Reidemeister. We have also included Proposition 1 in the Brown–Loday’s description, which is convenient in the prounipotent case. In § 2.2, we follow the ideas of Kan, Milnor and Gabriel–Zisman to prove that the simplicial group presentation is indeed a combinatorial model of based loops on $K(X\,|\,R)$ (the required definitions are given below). 2.1. Presentations, crossed modules, and second homotopy groupsGiven a presentation $(X\,|\,R)$ of a group $G\cong \Phi(X)/R$, let $K(X\,|\,R)$ be the standard 2-dimensional $\mathrm{CW}$-complex of $(X\,|\,R)$ (sometimes called the geometric realization of $(X\,|\,R)$). This $\mathrm{CW}$-complex is constructed as follows: it has one 0-cell $*$; 1-cell $e^1_x$ for each element $x\in X$ (hence $\pi_1(K^1,*)$ is the free discrete group $\Phi(X)$ on $X$), and a 2-cell $e^2_y$ for each defining relation $y\in R$ attached by a representative $r_y$ of the relator $y\in R\lhd\Phi(X)$. It is known that the homotopy type of $K(X\,|\,R)$ does not depends on the choice of representative attaching maps for 2-cells. By $\widetilde{K}$ we denote the universal cover of $ K(X\,|\,R)$. Now (since $\pi_1 \widetilde{K}\cong 1$ and fibers are discrete) the long exact sequence of homotopy groups of the covering implies the isomorphism $\pi_2\widetilde{K}\cong \pi_2 K(X\,|\,R)$. A direct application of Hurewicz’s theorem (see [20], Theorem 4.37) leads to a homological description of the second homotopy group where $C_*(\widetilde{K},\partial)$ is the chain complex of the universal cover $\widetilde{K}$ of $K(X\,|\,R)$ (see [9], Corollary 2, p. 167).It turns out that the chain complex $C_*(\widetilde{K})$ of the universal cover $\widetilde{K}$ is $G$-isomorphic to the chain complex $C(X\,|\,R)$ of free $G$-modules associated with the presentation $(X\,|\,R)$ of the group $G$ (see [9], Proposition 9). It is constructed as follows:
$$
\begin{equation*}
C(X\,|\,R)\colon C_2(X\,|\,R)\xrightarrow{d_2} C_1(X\,|\,R)\xrightarrow{d_1} C_0(X\,|\,R),
\end{equation*}
\notag
$$
where $C_0(X\,|\,R)=\mathbb{Z}G, C_1(X\,|\,R)=\bigoplus_X \mathbb{Z}G, C_2(X\,|\,R)=\bigoplus_Y \mathbb{Z}G$ with bases as for right $\mathbb{Z}G$-modules, respectively, 1; $e^1_x$, $x\in X;$ and $e^2_y$, $y\in Y$. Let $\pi\colon \Phi\to G$ and $\mathbb{Z}\Phi\to\mathbb{Z}G$ be the projections determined by the presentation. The boundaries are given by $d_1(e^1_x)=1-\pi x$, $x\in X$, $d_2(e^2_y)=\sum_X e^1_x\cdot\pi(\partial r_y/\partial x)$, $y\in Y$, where $\partial r_y/\partial x$ are the so-called Fox derivations of $r_y$ (see [9], Chap. 4). We need several definitions in order to obtain a purely algebraic definition of the second homotopy group. Definition 2. By a pre-crossed module one calls a triple $(G_2,G_1,\partial)$, where $G_1$, $G_2$ are groups, $\partial\colon G_2 \to G_1$ is a homomorphism of groups, $G_1$ acts on $G_2$ from the right and satisfies the identity $\mathrm{CM}\,1)$ $\partial(g_2^{g_1}) = g_1^{-1} \,\partial(g_2) g_1$, where the action is written in the form $(g_2,g_1) \to g_2^{g_1}$, $g_2 \in G_2$, $g_1 \in G_1$. Definition 3. A pre-crossed module is called crossed if, in addition, the Peiffer identity $\mathrm{CM}\,2)\,g_2^{\partial(g_1)}= g_1^{-1} g_2 g_1$ holds for all $g_1, g_2\in G_2$. Definition 4. A (pre-)crossed module $(G_2,G_1,d)$ is called a free (pre-)crossed module with base $Y\in G_2$ if $(G_2,G_1,d)$ has the following universal property with respect to maps $\nu\colon Y\to G'_2$, that is, for any (pre-)crossed module $(G'_2,G'_1,d')$ and a map $\nu\colon Y\to G'_2$ and for any homomorphism of groups $f\colon G_1\to G'_1$ such that $fd(Y)=d'\nu(Y)$, there exists a unique homomorphism of groups $h\colon G_2\to G'_2$ such that $h(Y)=\nu(Y)$ and the pair $(h,f)$ is a homomorphism of (pre-)crossed modules. J. H. C. Whitehead’s (1949) description of the second homotopy group of a presentation $(X\,|\,R)$ uses the notion of a free crossed module of a presentation (1) in the context of relative homotopy group. In our particular case, consider a pair of spaces $(K,L)$ arising from the standard 2-dimensional $\mathrm{CW}$-complex $K=K(X\,|\,R)$ of the presentation, as indicated above $K(X\,|\,R)=L\cup \{e^2_y\}_{y\in Y}$, where $L=K^1(X\,|\,R)$ is the 1-dimensional skeleton with the base point $*$. We can take the elements $a_y\in \pi_2(K,L)$ as homotopy classes of characteristic maps $h_y\colon (E^2,S^1)\to (K,L)$ of the 2-cells $e^2_y$ together with a choice of paths in $L$, one for each $y$, joining $h_y(1)$ to $*$. Whitehead’s important discovery is that $\pi_2(K,L)\xrightarrow{\partial}\pi_1(L)$ is a free crossed module on the elements $a_y\in \pi_2(K,L)$, where $\pi_1(L)$ acts in the standard way (see [23], Corollary 14.2, and [20], § 4.1). The Whitehead’s free crossed module of a presentation $(X\,|\,R)$ also arises as the second step on a way of constructing a free simplicial resolution from the given presentation. By definition, a simplicial resolution of a group $G$ is a free simplicial group $F_{\bullet}$ (that is, $F_n$ is a free group) such that $\pi_0F_{\bullet}\cong G$ and $\pi_n F_{\bullet}=0$ for $n>0$, this concept is analogous to the Eilenberg–MacLane space $K(G,1)$ in the category of simplicial groups. By homotopy groups of a simplicial group $F_{\bullet}$, we understand the homology groups of its Moore complex $(NF_n=\bigcap^{n-1}_{i=0}\operatorname{Ker}d_i^n,\, d_n^n|_{NF_n})$. There is a “pas-à-pas” method for constructing a free simplicial resolution, which goes back to André (see [39], § 3, and the references given there) and is similar to gluing $ n $-dimensional cells along mappings representing non-zero elements of homotopy groups in the process of constructing $K(G,1)$. Let $F^{(1)}_{\bullet}$ be the result of the first two steps; thus, we get a truncated simplicial group degenerated in dimensions greater than one here, $d_0^1$, $d_1^1$, $s_0^0$ for $x \in X$, $y \in Y$, $r_y \in R$ are defined by the identities $d_0^1(x)=x$, $d_0^1(y)=1$, $d_1^1(x)=x$, $d_1^1(y)=r_y$, and we will identify the image of $s_0^0(X)$ with $X$, that is, $s_0^0(x)=x$.The generators $s_1^1 s_0^0(X)\cup s_0^1(Y)\cup s_1^1(Y)$ of $\Phi(s_1^1 s_0^0(X)\cup s_0^1(Y)\cup s_1^1(Y))$ of $F^{(1)}_{\bullet}$ are degenerate by construction. Setting $R=\operatorname{Im}d^1_1(\operatorname{Ker}d_0^1)$, we have $G\cong \Phi(X)/R$. A pair of free groups $\Phi(s_0^0(X)\cup Y)$ and $\Phi(X)$ connected by the homomorphisms $d_0^1$, $d_1^1$, $s_0^0$ as in (3) is called a a simplicial presentation of the group $G$. Let us build a free pre-crossed module, starting with (3). To this end, we first go to the Moore complex $N(F^{(1)}_{\bullet})_1\xrightarrow{d_1^1} N(F^{(1)}_{\bullet})_0$ of (3), where $N(F^{(1)}_{\bullet})_0=\Phi(X)$, $N(F^{(1)}_{\bullet})_1=\operatorname{Ker}d_0^1$. We have $N(F^{(1)}_{\bullet})_1=\operatorname{Ker} d_0\cong \Phi(Y\times \Phi(X))$ (see [9], Proposition 1), and so we get the so-called free pre-crossed module of a discrete presentation (3) on the set $Y$ as a pair $\Phi(Y\times \Phi(X))\xrightarrow{d_1} \Phi(X)$, where the right action of $\Phi(X)$ is given by $a^u=s_0^0(u)^{-1}as_0^0(u)$, $u \in \Phi(X)$, $a \in \Phi(Y \times \Phi(X))$. From now on, we will omit the superscripts in $d^1_i$ and $s^0_0$, since they are already obvious in the 2-dimensional case. We use the notation $P$ for the Peiffer normal subgroup of $H=\operatorname{Ker} d_0$, normally generated by Peiffer commutators $\langle u,a\rangle = u^{-1}au(a^{d_1(u)})^{-1}$, where $u, a\in H$ (it was proved in [9] that $(H/P, \Phi(X),\overline{d_1})$ is a free crossed module, which coincides with the aforementioned Whitehead crossed module). Lemma 1. Given a simplicial presentation (3), there is an isomorphism of crossed modules arising from the coincidence of the Peiffer subgroup $P$ and $[\operatorname{Ker} d_0, \operatorname{Ker} d_1]$ in $H=\operatorname{Ker} d_0$. Proof (sketch). For any Peiffer commutator where $u, a\in \operatorname{Ker}d_0$, we denote $b=s_0 d_1(u)$, $\widetilde{a}^{-1}=u^{-1}au$. Then $\langle u,a\rangle =\widetilde{a}^{-1}b^{-1}u\widetilde{a}u^{-1}b=[u^{-1}b,\widetilde{a}]$, and therefore, the Peiffer subgroup is generated by the elements $[u^{-1} s_0 d_1(u), a]$.
Any element $x \in \operatorname{Ker} d_1$ can be written as $s_1 d_1(y) \cdot y^{-1}$, where $y \in \operatorname{Ker} d_0$. Indeed, $G_1 \cong NG_1 \leftthreetimes s_0 G_0$, and so $x=y \cdot s_0(y_0)$, with $y_0 \in G_0$, $y \in NG_1$. Since $x \in \operatorname{Ker} d_1$, we have $1 = d_1(x) = d_1(y) \cdot d_1 s_0(y_0)$. Hence $y_0 = d_1(y^{-1})$, and so $x=y \cdot s_0 d_1(y^{-1})$. Therefore, $P=[\operatorname{Ker} d_1, \operatorname{Ker} d_0]$. This proves the lemma. By the Brown–Loday lemma (see [10], § 5.7, and [39]), in any simplicial group degenerate in dimension two, in particular, in (3), the subgroups $[\operatorname{Ker} d_0, \operatorname{Ker} d_1]$ and $\operatorname{Im}\overline{d_2}$ coincide, where $\overline{d_2}\colon N(F^{(1)}_{\bullet})_2\to N(F^{(1)}_{\bullet})_1$ is the restriction of $d^2_2$ to $N(F^{(1)}_{\bullet})_2$. Let $C=\operatorname{Ker} d_0/\operatorname{Im}\overline{d_2}$. It is easy to check that $(C, \Phi(X),\overline{d_1})$ is a crossed module (see [40] for details). The following result summarizes the above findings. Proposition 1. Given a simplicial presentation (3), there is an isomorphism of crossed modules arising from the coincidence of the subgroups $[\operatorname{Ker} d_0, \operatorname{Ker} d_1]=\operatorname{Im}\overline{d_2}$ in $\operatorname{Ker} d_0$ and hence the isomorphism which allow us to identify $\pi_2K(X\,|\,R)$ with $\pi_1F^{(1)}_{\bullet}$. 2.2. Simplicial presentation as a model of $\Omega K(X\,|\,R)$The simplicial presentation (3) coincides in dimension 2 with the free simplicial group $B_{K(X\,|\,R)}$ constructed by Kan (see [25], Definition 5.2); $B_{K(X\,|\,R)}$ is loop homotopy equivalent (see [25], Theorem 5.5) to the “Kan loop group” $\mathrm{GS_1K}(X\,|\,R)$ of the first Eilenberg subcomplex $S_1K(X\,|\,R)$ (see [32], Definition 8.3) of the total singular complex $\mathrm{SK}(X\,|\,R)$ of $K(X\,|\,R)$ (actually, $S_1K(X\,|\,R)$ is homotopy equivalent to $\mathrm{SK}(X\,|\,R)$ (see [32], Theorem 8.4), but it is necessary in order to apply the Kan loop group functor). Recall that the “Kan loop group functor” $G$ associates with each reduced simplicial set $X$ a free simplicial group $\mathrm{GX}$ subject to the “principal twisted cartesian product” (see [32], Theorem 26.6) $\mathrm{GX}\to \mathrm{GX} \times_{\tau} X \xrightarrow{p} X$, where $|\mathrm{GX} \times_{\tau} X|$ is a contractible space, $p$ is a Kan fibration and therefore $\mathrm{GX}$ is a simplicial “loop space” for $X$. By Theorem 10.10 in [15], the geometric realization $|p|$ of the Kan fibration $p$ is a Serre fibration. Next, according to Theorem 10.9 in [15], $|\mathrm{GX}|$ is the fiber of $|\mathrm{GX}|\to |\mathrm{GX} \times_{\tau} X| \xrightarrow{|p|} |X|$ and, therefore, by [20], Proposition 4.66, there is a weak homotopy equivalence $\beta\colon |\mathrm{GS_1K}(X\,|\,R)|\simeq \Omega K(X\,|\,R)$. Since $K(X\,|\,R)$ is a $\mathrm{CW}$-complex, it follows from Corollary 2 in [37] that $\Omega K(X\,|\,R)$ is homotopy equivalent to a $\mathrm{CW}$-complex. Next, being a geometric realization is a $\mathrm{CW}$-complex (see [32], Theorem 14.1) by Whitehead’s theorem, $\beta$ is a homotopy equivalence. Finally, we have a homotopy equivalence $|B_{K(X\,|\,R)}|\simeq \Omega K(X\,|\,R)$, and therefore, $B_{K(X\,|\,R)}$ is a combinatorial model of based loops on $K(X\,|\,R)$. § 3. Representations and cohomology of affine group schemesThe most convenient presentation of affine group schemes for our purposes can be found in the books [51], [24]; for a detailed account of the general theory of Hopf algebras, see Chap. 1 in [38] and [11]. We have specially included some useful facts about regular representations and induced modules as they arise in the study of prounipotent presentations. A modern exposition of cohomology theory is contained in [24]; nevertheless, to construct the spectral sequence, we follow classical approach [21] to obtain an explicit double complex. The category of topological modules is defined in § 3.4 by continuous duality. This generality leads automatically to the fact that the introduced category is abelian, since the axioms of abelian category are self-dual (see [16], § 1.4). 3.1. Affine group schemes and their representationsBy an affine group scheme over a field $k$, we mean a representable functor $G$ from the category $\mathrm{Alg}_k$ of commutative $k$-algebras with unit to the category of groups. If $G$ is representable by the Hopf algebra $\mathcal{O}(G)$, ]then, as a functor, $G$ is given for any commutative $k$-algebra $A$, by We assume that the homomorphisms $Hom_{\mathrm{Alg}_k}$ send the unit of the algebra $\mathcal{O}(G)$ to the unit of a $k$-algebra $A$. The Hopf algebra $\mathcal{O}(G)$ representing the functor $G$ is usually called the algebra of regular functions of $G$. We recall that the composition law in $G(A)$ is given by the formula $(g_1\cdot g_2)(x)=m_A(g_1\otimes g_2)\Delta(x)$, where $g_1, g_2\in G(A)$, $x\in \mathcal{O}(G)$, $\Delta\colon \mathcal{O}(G)\to\mathcal{O}(G)\otimes \mathcal{O}(G)$, is the coalgebra map, $m_A$ is the multiplication in $A$; the inverse of $g\in G(A)$ is a $k$-algebra homomorphism given as the composition $g\circ s$, where $s\colon \mathcal{O}(G)\to \mathcal{O}(G)$ is the antipod. The Yoneda lemma implies the anti-equivalence of the categories of affine group schemes and commutative Hopf algebras (see [51], § 1.3). Let us say that an affine group scheme $G$ is algebraic if its Hopf algebra of regular functions $\mathcal{O}(G)$ is finitely generated as a commutative $k$-algebra. We call the Zariski closure of a subset $S\subseteq G(k)$ the smallest affine subgroup $H$ in $G=\varprojlim G_{\alpha}$ (where $G_{\alpha}$ are algebraic affine group schemes) such that $S\subseteq H(k)$, it is $\varprojlim H_{\alpha}$, where $H_{\alpha}$ is the closure of the image of $S$ in $G_{\alpha}(k)$. Let $M$ be a $k$-module. It defines a $k$-group functor $\mathrm{GL}(M)$ of invertible endomorphisms by the formulas $\mathrm{GL}(M)(A)=\mathrm{End}_A(M\otimes A)^{\times}$ called a general linear group of $M$. If $G$ is an affine $k$-group scheme, by a representation of $G$ on $M$ we understand a homomorphism of group functors $G\to \mathrm{GL}(M)$. This is equivalent to saying that each $G(A)$ acts from the left on $M(A)=M\otimes A$ through $A$-linear maps, so representation provides for each $A$ a group homomorphism $G(A)\to \mathrm{End}_A(M\otimes A)^{\times}$. For any $k$-algebra $A$ and $g\in G(A)$, we have the commutative diagram Let us take a look at $\operatorname{id}_{\mathcal{O}(G)}\times (m\otimes 1) \in G(\mathcal{O}(G))\times (M\otimes \mathcal{O}(G))$. By commutativity of the diagram, we have the identity
$$
\begin{equation*}
f(A)\circ G(g)(\operatorname{id}_{\mathcal{O}(G)})(m\otimes 1)=(\operatorname{id}_M\otimes\, g)\circ \Delta_M (m\otimes 1),
\end{equation*}
\notag
$$
where $\Delta_M\colon M\otimes k\to M\otimes \mathcal{O}(G)$ (a comodule map (see [51], § 3.2) is a restriction to $M$ of $f(\operatorname{id}_{\mathcal{O}(G)})\colon M\otimes \mathcal{O}(G)\to M\otimes \mathcal{O}(G)$). As $g=G(g)(\operatorname{id}_{\mathcal{O}(G)})$, we have, for all $m\in M$
$$
\begin{equation}
g(m\otimes 1)=(\operatorname{id}_M\otimes\, g)\circ \Delta_M (m\otimes 1).
\end{equation}
\tag{4}
$$
Example 1 (the left and right regular representations). Let $\mathbb{A}^1_k$ be the affine line, as $\mathcal{O}(\mathbb{A}^1_k)\cong k[X]$ is a free polynomial algebra, by Yoneda’s lemma, there is an isomorhism $\mathrm{Mor}(G,\mathbb{A}^1_k)\cong\mathcal{O}(G)$ of $k$-vector spaces; the left $\Delta_l$ and right $\Delta_r$ regular actions of $G$ on $M=\mathcal{O}(G)$ are given for $\phi_A\in \mathrm{Mor}(G(A),\mathbb{A}^1_k(A))$ and $y_A,x_A\in G(A)$ by the formulas, respectively, $(y_A\cdot \phi_A)(x_A)=\phi_A(y_A^{-1} x_A)$ and $(y_A\cdot\phi_A)(x_A)=\phi_A(x_A y_A)$. It can be derived using (4) that $\Delta_r$ coincides with the comultiplication in $\mathcal{O}(G)$, that is, $\Delta_r=\Delta_G$ (see also [38], Lemma 1.6.4, Example 1.6.5). The $\mathcal{O}(G)$-comodule structure, which corresponds to the left regular representation, can be written by $\Delta_l=t\circ(s\otimes \operatorname{id}_{\mathcal{O}(G)})\circ \Delta_G$, where $s$ is antipod and $t(a\otimes b)=b\otimes a$. There is a natural notion of a $G$-module homomorphism, which is perfectly explained in [51], § 3.2; the corresponding category $\mathrm{Rep}(G)$ of the left $G$-representations is equivalent to the category of right $\mathcal{O}(G)$-comodules. Let $H$ be a closed subgroup of $G$, then $\mathcal{O}(H)=\mathcal{O}(G)/I_H$, where $I_H$ is the Hopf ideal [51] defining the subgroup $H$, and let $M$ be a $G$-module. hence we have the $k$-linear map
$$
\begin{equation*}
\mu\colon M\to M\otimes\mathcal{O}(G)\to M\otimes\mathcal{O}(H),
\end{equation*}
\notag
$$
which defines left $H$-module structure on $M$ (that is the restriction functor $M\downarrow^G_H$). In order to construct injective hulls, we need the concept of an induced module. So, let $H$ be a closed subgroup of an affine group scheme $G$. For each $H$-module $M$, the induced module $M\uparrow^G_H$ is defined (see [24], § 3.3) as follows: $M\uparrow^G_H=\{ f\in \mathrm{Mor}(G,M_a)\mid f(gh)=h^{-1}f(g)\}$ for $g\in G(A)$ and $h\in H(A)$, $A\in \mathrm{Alg}_k$, with the left regular action of $G$. Proposition 2 (see [24], Part I, § 3.4). Let $H$ be a closed subgroup of an affine group scheme $G$ and $M$ be an $H$-module. Then a) $\varepsilon_M\colon M\uparrow^G_H\to M$ is a homomorphism of $H$-modules; b) For each $G$-module $N$, the map $\varphi\mapsto \varepsilon_M\circ \varphi $ defines an isomorphism Proposition 3 (see [24], Part I, § 3.6). Let $H$ be a closed subgroup of an affine group scheme $G$ and $M$ be an $H$-module. If $N$ is a $G$-module, then there is a canonical isomorphism of $G$-modules Following [24], Part I, § 3.7, we discuss some useful implications of the above propositions. Let $H=1$. hen, for all $k$-modules, $M\uparrow^G_1=M\otimes \mathcal{O}(G)$, where $M$ is considered as a trivial $G$-module; in particular, $k\uparrow^G_1=\mathcal{O}(G)$. Combining the latter identity with assertion b) of Proposition 2, we obtain, for each $G$-module $M$, an isomorphism of $k$-linear spaces (it is actually an isomorphism of topological modules, see § 3.4). Putting $M=k_a$ in the tensor identity, we have, for each $G$-module $N$, a remarkable isomorphism
$$
\begin{equation}
N\otimes \mathcal{O}(G)\cong N\uparrow^G_1=N_{\mathrm{tr}}\otimes \mathcal{O}(G),
\end{equation}
\tag{6}
$$
given by $x\otimes f\mapsto (1\otimes f)\cdot(\operatorname{id}_N\otimes\, s)\circ\Delta_N(x)$, where $N_{\mathrm{tr}}$ denotes the $k$-module $N$ with the trivial action of $G$ and $s$ is the antipode in $\mathcal{O}(G)$. We next consider
$$
\begin{equation*}
M^G=\{m\in M\mid g(m\otimes1)=m\otimes\, 1 \text{ for all } g\in G(A)\text{ and all }A\in \mathrm{Alg}_k\},
\end{equation*}
\notag
$$
which is the submodule of fixed points of a $G$-module $M$. Let $g=\operatorname{id}_{\mathcal{O}(G)}\in G(\mathcal{O}(G))$ in (4), then $M^G=\{m\in M\mid \Delta_M(m)=m\otimes1\}$ and, therefore,
$$
\begin{equation}
M^G=\operatorname{Ker}(\Delta_M-\operatorname{id}_M\otimes\, 1).
\end{equation}
\tag{7}
$$
3.2. Cohomology of affine group schemesIt follows from Proposition 2 that the induction functor $M\uparrow^G_H$ is a right adjoint to the restriction functor $M\downarrow^G_H$, and, therefore, $M_{\mathrm{tr}}\uparrow^G_1\cong M_{\mathrm{tr}}\otimes \mathcal{O}(G)$ is an injective $G$-module, since $M_{\mathrm{tr}}$ is obviously injective as a module over the trivial group. It follows from the unit diagram $\operatorname{id}_M=(\operatorname{id}_M\otimes\, \varepsilon_M)\circ\Delta_M$ (see [51], § 3.2) that the map $M \xrightarrow{\Delta_M} M\otimes\mathcal{O}(G)$ is an inclusion, and, therefore, composing $\Delta_M$ with the isomorphism (6), we embed $M\hookrightarrow M_{\mathrm{tr}}\otimes \mathcal{O}(G)$ into the injective module. The category of $G$-modules is abelian (see [24], Part I, § 2.9) and, as we have just seen, contains enough injective objects. So, we can define the cohomology groups $H^n(G, M)$ of the affine group scheme $G$ with coefficients in $G$-module $M$ as the $n$th derived functors of the fixed points functor $M^G$. $H^*(G,M)$ can be computed explicitly using the Hochschild complex, which is the standard resolution (see [42], § 6.2)
$$
\begin{equation}
C^*(G,M)=M\otimes^n \mathcal{O}(G),\qquad \partial^n\colon C^n(G,M)\to C^{n+1}(G,M),
\end{equation}
\tag{8}
$$
where $\partial^n=\sum^{n+1}_{i=0}(-1)^i\, \partial_i^n$ for $n\in \mathbb{N}$ is defined by
$$
\begin{equation*}
\begin{aligned} \, \partial_0^n(m\otimes f_1\otimes\dots\otimes f_n) &=\Delta_M(m)\otimes f_1\otimes\dots\otimes f_n, \\ \partial_i^n(m\otimes f_1\otimes\dots\otimes f_n) &=m\otimes f_1\otimes\dots\otimes f_{i-1}\otimes\Delta_G(f_i) \\ &\qquad \otimes f_{i+1}\otimes \dots\otimes f_n\quad \text{for }\ 1\leqslant i \leqslant n, \\ \partial_{n+1}^n(m\otimes f_1\otimes\dots\otimes f_n) &=m\otimes f_1\otimes\dots\otimes f_n\otimes 1. \end{aligned}
\end{equation*}
\notag
$$
We can identify $C^n(G,M)$ with the “inhomogeneous bar complex” $\mathrm{Mor}(G^n,M_a)$ (see [21], § I.2), where $G^n$ is the direct product of $n$ copies of $G$. Indeed, by Yoneda’s lemma, $(M\otimes \mathcal{O}(G)^{n})\otimes A\cong \mathrm{Mor} (G_A^n, (M\otimes A)_a)$, and therefore, $\partial_i^n$ is expressed as
$$
\begin{equation*}
\begin{aligned} \, \partial_0^nf(g_1,g_2,\dots,g_{n+1}) &=g_1f(g_2,\dots,g_{n+1}), \\ \partial_i^nf(g_1,g_2,\dots,g_{n+1}) &=f(g_1,g_2,\dots,g_{i-1},g_{i},g_{i+1},\dots,g_{n+1})\quad \text{for } \ 1\leqslant i \leqslant n, \\ \partial_{n+1}^n &=f(g_1,g_2,\dots,g_{n}). \end{aligned}
\end{equation*}
\notag
$$
The Hochschild complex arises from the exact sequence of $G$-modules
$$
\begin{equation*}
0\to k_a\to \mathcal{O}(G)\to \mathcal{O}(G)^{\otimes 2}\to\cdots,
\end{equation*}
\notag
$$
which is simply the injective resolution $0\to k_a\to C^n(G,\mathcal{O}(G))$ of $k_a$ regarding $M=\mathcal{O}(G)$ as $G$-module via $\rho_r$ and $k_a$ as the trivial $G$-module (see [24], § 4.15, (1)). This sequence is in fact a sequence of $G$-module homomorphisms if we assume that $G$ acts on $\mathcal{O}(G)^{\otimes n}$ via $\rho_l$ on the first factor and trivially on all other factors, for this action the map $\mathcal{O}(G)\to \mathcal{O}(G)\otimes \mathcal{O}(G)$, $f\mapsto \Delta_G(f)- f\otimes1$ is $G$-equivariant. We tensor $0\to k_a\to C^n(G,\mathcal{O}(G))$ with $M$ on the left and use (6) to get an injective resolution of $M$
$$
\begin{equation}
0\to M\to M_{\mathrm{tr}}\otimes \mathcal{O}(G)\to M_{\mathrm{tr}}\otimes \mathcal{O}(G)^{\otimes 2}\to\cdots
\end{equation}
\tag{9}
$$
and therefore, $H(G,M)$ is the cohomology of the complex
$$
\begin{equation*}
\begin{aligned} \, 0 &\to(M_{\mathrm{tr}}\otimes \mathcal{O}(G))^G\to (M_{\mathrm{tr}}\otimes \mathcal{O}(G)^{\otimes 2})^G \\ &\to\dots\to \mathrm{Now}(M_{\mathrm{tr}}\otimes \mathcal{O}(G)^{\otimes n+1})^G\cong M_{\mathrm{tr}}\otimes \mathcal{O}(G)^{\otimes n}\cong C^n(G,M), \end{aligned}
\end{equation*}
\notag
$$
as $G$ acts trivially on all but one factor, and since $\mathcal{O}(G)^G=k_a$. Remark 1. Let $G\cong\varprojlim G_{\gamma}$ be a pro-$p$-group (where $G_{\gamma}$ are finite $p$-groups), by Example 5 below, $G$ may be viewed as a group scheme with the algebra of regular functions $\mathcal{O}(G)= \mathbb{F}_pG^{\vee}$. Now, the right $\mathcal{O}(G)$-comodule $M$ over the constant group scheme represented by $\mathcal{O}(G)$ is a left discrete $G$-module (see [46], § 2.1) of exponent equal to $p$. By Lemma 2 (see § 4.3), $M\cong\varinjlim M_{\gamma}$, where $ M_{\gamma}$ are finite-dimentional $G_{\gamma}$-submodules, and $G_{\gamma}$ are taken from the decomposition of $G$. Conversely, if $M$ is a left discrete $G$-module of exponent equal to $p$, then it is a $G$-module in the schematic sense. We have $(M\otimes \mathcal{O}(G)^{\otimes^n})^G\cong \varinjlim (M_{\gamma}\otimes \mathcal{O}(G_{\gamma})^{\otimes^n})^{G_{\gamma}}$, and, since homology commutes with direct limits, we have $H^n(G,M)\cong \varinjlim H^n(G_{\gamma}, M_{\gamma}) $, and the schematic cohomology of pro-$p$-groups with $G$-module coefficients coincide with the cohomology groups of a pro-$p$-group with coefficients in discrete $G$-modules of exponent equal to $p$ (see [42], § 6.2, and [46], § 2.2). 3.3. The Lyndon–Hochschild–Serre spectral sequenceDefinition 5. A double complex $E$ is a collection $E^{p,q}_0$, $p,q\geqslant 0$, of abelian groups and maps $d_0^{p,q}\colon E^{p,q}_0\to E^{p,q+1}_0$, $d_1^{p,q}\colon E^{p,q}_0\to E^{p+1,q}_0$ arranged in the diagram such that the following conditions are satisfied: (i) each row satisfies $d_1\circ d_1=0$; (ii) each column satisfies $d_0\circ d_0=0$; (iii) $d_0\circ d_1+d_1\circ d_0=0$. The total complex of a double complex $X^n=\mathrm{Tot}(E)=\bigoplus_{i+j=n}E^{i,j}_0$ is given with the differential $d=d_0+d_1\colon X^n\to X^{n+1}$. The double complex inherits the filtration $D_0^{p,q}=F^pX^{p+q}=\bigoplus_{i+j=p+q}^{i\geqslant p} E^{i,j}_0$, each quotient of the filtration $F^pX^{p+q}/F^{p+1}X^{p+q}\cong E^{p,q}_0$ should be treated as a single column of $E$ with the differential $d_0\colon E_0^{p,q}\to E_0^{p,q+1}$, since $d_1$ maps into a lower layer. So, we get the derived couple [23]
$$
\begin{equation*}
E_1^{p,q}=H(E^{p,q}_0,d_0),\qquad D_1^{p,q}=H(E_0^{p,q}\oplus E_0^{p+1,q-1}\oplus\cdots, d_0+d_1).
\end{equation*}
\notag
$$
Let $x\in E_0^{p,q}$ and $d_0(x)=0$, so $x$ represents a class $[x]\in E_1^{p,q}$, then $k_1$ in the derived couple is defined as a boundary homomorphism associated with the short exact sequence of chain complexes Since $d_0(x)=0$, we can calculate $k_1[x]=[d_1(x)]\in D_1^{p+1,q}$, that is, the differential $j_1\circ k_1[x]=[d_1(x)]$ of the derived pair coincides with the horizontal differential $d_1$ of $E$ and we obtain the following proposition.Remark 2. It turns out that, in a double complex $E_0^{p,q}$, the differentials $d_2^{p,q}$ may be described using the differentials $d_0$ and $d_1$ as follows. For each class $e\in E_2^{p,q}$, there are elements $x\in E_0^{p,q}$ and $y\in E_0^{p+1,q-1}$ such that: (i) $d_0(x)=0$, $d_1(x)=-d_0(y)$; (ii) $d_2^{p,q}(e)=[d_1(y)]$, where by $[d_1(y)]$ we denote the class of $d_1(y)$ in $E_2^{p+2,q-1}$. Proposition 5. Let $G$ be an affine group scheme, $H\lhd G$ be a closed normal subgroup of $G$ and $M$ is $G$-module. Then there is a cohomological Lyndon–Hochschild–Serre spectral sequence Proof. Using the Hochschild resolution (9), we construct
$$
\begin{equation*}
K^{p,q}=(M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H\otimes \mathcal{O}(G/H)^{\otimes p},
\end{equation*}
\notag
$$
which is the double complex with differentials $d_0^{p,q}, d_1^{p,q}$ given by the formulas for $a\in (M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H$ and $b\in\mathcal{O}(G/H)^{\otimes p}$ as follows:
$$
\begin{equation*}
d_0^{p,q}(a\otimes b)=(-1)^p \widetilde{d}_0^{p,q}(a)\otimes b, \qquad d_1^{p,q}(a\otimes b)=a\otimes \widetilde{d}_1^{p,q}(b),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\widetilde{d}_0^{p,q}\colon (M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H\to (M\otimes \mathcal{O}(G)^{\otimes(q+2)})^H
\end{equation*}
\notag
$$
is the differential extracted from the Hochschild complex (9) and
$$
\begin{equation*}
\widetilde{d}_1^{p,q}\colon (M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H\otimes \mathcal{O}(G/H)^{\otimes p}\to (M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H\otimes \mathcal{O}(G/H)^{\otimes (p+1)}
\end{equation*}
\notag
$$
is the differential in the Hochschild complex $C(G/H,(M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H)$ (see (8)). Hence $(E_0^{p,q},d_0,d_1)$ is a double complex and we define the Lyndon–Hochschild–Serre spectral sequence as the spectral sequence from Proposition 4. Indeed, since $\bigotimes\mathcal{O}(G/H)^{\otimes p}$ is an exact functor, we have $E_1^{p,q}\cong H^q(H,M)\otimes \mathcal{O}(G/H)^{\otimes p}$, and therefore, $E_2^{p,q}=H^p(G/H,H^q(H,M))$.
Now we consider the row filtration of $(E^{q,p},d_1,d_0)$. For any $G$-module $A$, from (6) we have $(A\otimes \mathcal{O}(G))^H\cong A_{\mathrm{tr}}\otimes \mathcal{O}(G)^H$ and $A_{\mathrm{tr}}\otimes \mathcal{O}(G)^H\cong A_{\mathrm{tr}}\otimes \mathcal{O}(G/H)$, and hence $(M\otimes \mathcal{O}(G)^{\otimes(q+1)})^H$ are injective $G/H$-modules, hence for $q>0$ $E_2^{q,p}=H^p(G/H,(M\otimes \mathcal{O}(G)^{\otimes (q+1)})^H)=0$. But since $A^G=(A^H)^{G/H}$ for any $G$-module $A$, it follows that $E_2^{0,p}\cong H^p(G,M)$ and therefore $H^p(\mathrm{Tot}(E))\cong H^p(G,M)$. Proposition 5 is proved. 3.4. Topological modules as a dual categoryDefinition 6. A linearly compact vector space over a field $k$ (we always assume that $k$ is taken with the discrete topology) is a topological vector space $V$ over $k$ satisfying the conditions: 1) the topology is linear: the open affine subspaces form a basis for the topology; 2) any family of closed affine subspaces with finite intersection property has non-empty intersection; 3) the topology is Hausdorff. Linearly compact vector spaces were introduced in order to preserve the duality that holds for finite-dimensional vector spaces. This can be achieved by introducing a topology on a dual space $V^*$, taking
$$
\begin{equation*}
U^{\perp}=\{\phi \in A^{\ast} \mid \phi(U)=0, \text{ where $U$ is a finite dimensional $k$-subspace of }V\}
\end{equation*}
\notag
$$
as a basis of the system of neighborhoods of zero in $V^{\ast}$. It turns out that $V^*\cong \varprojlim_{\dim_k(U)<\infty } V^*/U^{\perp}$, and $V^*$ is a complete topological vector space. For any linearly compact vector space $V$, it makes sense to speak about the dual space $V^\vee$ of continuous linear functions, and the evaluation map defines the continuous duality (see [13], § 1.2; here, $V$ is discrete and $W$ is linearly compact)
$$
\begin{equation*}
e\colon V\to V^{*\vee}, \quad v\mapsto(\phi \to \phi(v)),\qquad \widetilde{e}\colon W\to W^{\vee *}, \quad v\mapsto(\phi \to \phi(v)).
\end{equation*}
\notag
$$
Recall that the completed tensor product $E \mathbin{\widehat{\otimes}_k} F$ of topological $k$-vector spaces $E$ and $F$ is the completion of $E \otimes_k F$ with respect to the topology (called the topology of tensor product) given by the fundamental system of neighborhoods of 0 consisting of the sets $V \otimes_k F + E\otimes_k W$, where $V$ (respectively, $W$) is an arbitrary element of the fundamental system of neighborhoods of $0$ consisting of vector subspaces of $E$ (respectively, $F$); see [13], § 1.2.4. It follows directly from the definition of the completed tensor product that $(E \mathbin{\widehat{\otimes}_k} F)^{\vee}\cong E^{\vee} \otimes_k F^{\vee}$. Let $A(m,\Delta,s,e,\varepsilon)$ be a commutative Hopf algebra with multiplication $m\colon A\otimes A\to A$, comultiplication $\Delta\colon A\to A\otimes A$, antipod $s\colon A\to A$, unit $e\colon k\to A$ and counit $\varepsilon\colon A\to k$, this $k$-linear maps are included in standard diagrams of the Hopf algebra definition (see [38], §§ 1.1, 1.5). Let $A^*=Hom_k(A,k)$ be the dual and let $m_*=\Delta^*\colon A^*\mathbin{\widehat{\otimes}} A^*\to A^*$ be (the dual) multiplication, $\Delta_*=m^*\colon A^*\to A^*\mathbin{\widehat{\otimes}} A^*$ be the comultiplication, $\varepsilon_*=e^*$ and $e_*=\varepsilon^*$, $s_*=s^*$. These $k$-linear maps are also included in standard diagrams of Hopf algebra definition and we come to the following. Definition 7. We say that a linearly compact $k$-vector space $A^*$ with continuous maps $(m_*,\Delta_*,s_*,e_*,\varepsilon_*)$ of linearly compact vector spaces is a complete Hopf algebra (CHA for short) if $A^*(m_*,\Delta_*,s_*,e_*,\varepsilon_*)$ is included in the standard diagrams from the cocommutative Hopf algebra definition, where the usual tensor products are replaced by the completed ones. Remark 3. Our definition is equivalent to the definition of a CHA in Quillen’s paper [41] if and only if a CHA in the Quillen’s sense is finitely generated. We additionally require linear compactness, bearing in mind that CHAs arise as dual of a Hopf algebra. Recollecting what was said above, we obtain the following result. Corollary 1 (see [13], § 2). The association $A\mapsto A^*$ generates a one-one correspondence between the structures of commutative Hopf algebras on a vector $k$-space $V$ with discrete topology and the structures of cocommutative linearly compact complete Hopf algebras on $A^*$. Example 2 (Quillen’s formulas). Let $G$ be a finitely generated discrete group, $k[G]$ be a group ring with coefficients in the field $k$ of characteristic zero, $I$ be the augmentation ideal. Then $\dim_k I/I^2<\infty$, hence the factors $k[G]/I^n$ are of finite dimension and, therefore, $\widehat{k}G=\varprojlim k[G]/I^n$ is a linearly compact vector space. For $g\in G$ a diagonal coproduct $\Delta(g)=g\,{\otimes}\, g$ is given, by linearity we obtain a continuous in the $I$-adic topology map $\Delta\colon kG\to kG\,{\otimes}\, kG$ and, therefore, $\widehat{\triangle}\colon \widehat{k}G\to \widehat{k}G\mathbin{\widehat{\otimes}}\widehat{k}G$ is the continuous coproduct, which, together with completed multiplication and antipod, induced by the map $s(g)=g^{-1}$ for $g\in G$, define CHA-structure on $\widehat{k}G$. It turns out (see [41], A.3 and [26], § 2) that the prounipotent group $G^{\wedge}_k$ with the algebra of regular functions $\mathcal{O}(G^{\wedge}_k)\cong(\widehat{k}G)^\vee$ is the prounipotent completion of $G$ of Definition 1. The right $ \mathcal{O}(G) $-comodule structure on $M$ is determined by the comodule map $\Delta_M\colon M \to M \otimes \mathcal{O}(G)$ and hence, by duality, we obtain a continuous mapping $\Delta_M^*\colon M^* \mathbin{\widehat{\otimes}} \mathcal{O}(G)^* \to M^*$ included in the associativity and unit diagrams Definition 8. Given an affine group scheme $G$, we say that $(M^*,\Delta_M^*)$ is a right topological $\mathcal{O}(G)^*$-module over the complete Hopf algebra $\mathcal{O}(G)^*$ if $M^*$ is a linearly compact topological vector space with an action of $\mathcal{O}(G)^*$, that is, $M^*$ and continuous $k$-linear mapping $\Delta_M^*$ are included in the associativity and unit diagrams (10) or equivalently $(M^{\vee}, \Delta_M^{*\vee})$ is a right $\mathcal{O}(G)$-comodule. We say that a continuous $k$-linear mapping $\phi\colon M_1^*\to M_2^*$ is a homomorphism of right topological $\mathcal{O}(G)^*$-modules if the following diagram is commutative: By Lemma 2 below, any comodule $M$ over $\mathcal{O}(G)$ can be decomposed into a direct limit $M\cong\varinjlim M_{\lambda}$ of finite dimensional “rational” comodules $M_{\lambda}$ over $\mathcal{O}(G_{\lambda})$, where $\mathcal{O}(G_{\lambda})$ are finitely generated (as algebras) Hopf subalgebras of $\mathcal{O}(G)$. By duality, any topological $\mathcal{O}(G)^*$-module $M^*$ can be decomposed into the inverse limit $M^*\cong\varprojlim M^*_{\lambda}$ of finite dimensional topological $\mathcal{O}(G_{\lambda})^*$-modules.Our concept generalizes Hain’s notion (see [19], Theorem 3.4). The topological modules form an abelian category since this category is dual to the category of $G$-modules, which is known to be abelian (see [24], Part I, § 2.9). § 4. Prounipotent groups and their presentationsThe purpose of this section is to introduce prounipotent groups, their presentations, and prounipotent crossed modules. After Definition 8 of a prounipotent group in § 4.1, we give examples. In addition to the survey of prounipotent groups in characteristic zero in Example 3, we discuss in Example 4 free prounipotent groups and their Hopf algebras. Example 5 explains how to look at finite $p$-groups and pro-$p$-groups as prounipotent group schemes over $\mathbb{F}_p$. We also extract information about the algebra of regular functions of a free pro-$p$-group in Example 6. An example of computations cohering zero and positive characteristic is discussed in § 4.2. In § 4.3, the duality between fixed points of $ G $-modules and coinvariants of their dual topological modules is proved. Presentation theory of prounipotent groups is contained in Sections 4.4–4.6. The special feature of our approach lies in the fact that we consider not only “proper” presentations. On simplicial presentations we impose an important “linear independance” constraint on the choice of defining relations. The concept of a prounipotent crossed module of a prounipotent presentation is introduced in § 4.7, this section also contains the statement of the main result. Besides homotopy theory importance prounipotent groups are especially nice since they have presentations theory perfectly similar to the discrete case. Modules over such group schemes are simple enough to get deep results of cohomological nature. We refer the reader to [29]–[31] for pioneering papers on prounipotent groups presentations, to classical results on presentations of pro-$p$-groups [46], [27], [42], [40]. Recent result of the author are scattered across the articles [33], [35], [34]. 4.1. Prounipotent groupsDefinition 9. A unipotent group over a field $k$ is an affine algebraic group scheme $G$ for which every non-zero linear representation $V$ has a non-zero fixed vector $v\in V$. We call $v$ fixed if $G$ acts trivially on the subspace $kv$, that is, $\rho(v)=v\otimes 1$ in $V$. When $k=\mathbb{F}_p$ we restrict ourselves to considering constant group schemes of finite $ p $-groups (see [51], § 2.3). A group scheme $G$ is called a prounipotent group if there is a decomposition of $G$ into an inverse limit $G=\varprojlim G_{\alpha}$ of unipotent groups $G_{\alpha}$. Example 3 (prounipotent groups in characteristic zero). There is also a well-known correspondence between unipotent groups over a field $k$ of characteristics $0$ and nilpotent Lie algebras over $k$, which associates with a unipotent group its nilpotent Lie algebra. This correspondence extends easily to the correspondence between prounipotent groups over $k$ and pronilpotent Lie algebras over $k$. Functoriality of the correspondence enables one, when it is convenient, to interpret problems on unipotent groups in the language of Lie algebras. Prounipotent groups are well behaved in the sense that we can work with them as they are ordinary groups: the image of a closed subgroup of a prounipotent group under a homomorphism of prounipotent groups is always a closed subgroup (see [51], Theorem 15.3), the group of rational points of a factor is isomorphic to the factor of groups of rational points (see [51], § 18.2 (e)). Theorems on the structure of normal series, non-triviality of the center of a unipotent group are transferred from the corresponding statements for Lie algebras. Recall also that $G(k)\cong \mathcal{G}\mathcal{O}(G)^*$ (see [50], Proposition 18), where $\mathcal{G}$ is the functor of group-like elements in the complete Hopf algebra $\mathcal{O}(G)^*$. Let $A$ be a Hopf algebra over a field $k$ of characteristic zero, in which: 1) the product is commutative; 2) the coproduct is conilpotent or, equivalently, coconnected in terminology of [51], Theorem 8.3. Then, as an algebra, $A$ is isomorphic to a free commutative algebra (see [11], Theorem 3.9.1). Therefore, each group of $k$-points $G_{\alpha}(k)= Hom_{\mathrm{Alg}_k}(\mathcal{O}(G),k)$ is isomorphic as an algebraic variety to a certain $n_{\alpha}$-dimensional affine space $\mathbb{A}^{n_{\alpha}}_k$ (see [51], Theorem 4.4), and hence it is an affine algebraic group, and consequently, (see [51], Corollary 4.4) is a linear algebraic group. Thus, we freely use results and methods from the theory of linear algebraic groups in characteristic $0$. Remark 4. Let $G$ be a prounipotent group and let $\widetilde{G}\,{\subset}\, G(k)$ be a discrete finitely generated Zariski dense subgroup of $G(k)$. Then $\widetilde{G}^{\wedge}_k{\cong}\, G$. Indeed, by Quillen’s formulas (Example 2), $\mathcal{O}(\widetilde{G}^{\wedge}_k)^*\cong\widehat{k}\widetilde{G}$. Since group-like elements are linearly independent in CHA (the proof is the same as in Theorem 2.1.2 of [1]) and $\bigcap \widehat{I}^m\,{=}\,0$, where $\widehat{I}$ is the augmentation ideal of $\mathcal{O}(G)^*$, we have $\mathcal{O}(\widetilde{G}^{\wedge}_k)^*\subseteq \mathcal{O}(G)^*$. By the universal property of $\widetilde{G}^{\wedge}_k$, $\widetilde{G}^{\wedge}_k(k)$ is embedded as a closed subgroup of $G(k)$, and therefore, $\widetilde{G}^{\wedge}_k(k)\cong G(k)$, because $\widetilde{G}$ is dense. Example 4 (regular functions of free $k$-prounipotent groups $\operatorname{char}(k)=0$). The concept of a free pro-unipotent group of finite rank, say $n$, over a field of characteristic zero is well-known and is beautifully presented in [50]. Its algebra of regular functions is the tensor algebra $T(V)$ of a $k$-vector space $V$, $\dim_k(V)=n$ with the so-called deconcatenation coproduct defined by the rule $\Delta(e_I)=\sum_{JK=I}e_J\otimes e_K$ on products $e_I=e_{i_1}\cdots e_{i_k}$, where $I=(i_1,\dots,i_k)$ is a multi-index and $e_{i_j}\in Z$, $Z$ is a chosen basis of $V$. The so-called shuffle product defines a commutative composition law in $T(V)$, that is defined on words of lengths $m$ and $n$ as a sum over $(m+n)!/(m!\,n!)$ ways of interleaving the two words, as shown in the following examples: $ab \circ xy = abxy + axby + xaby + axyb + xayb + xyab$, $aaa \,{\circ}\, aa = 10aaaaa$. It may be defined inductively by the formulas $u \circ \omega = \omega \circ u = u$, $ua \circ vb = (u \circ vb)a + (ua \circ v)b$, where $\omega$ is the empty word, $a$ and $b$ are single elements, and $u$ and $v$ are arbitrary words. The shuffle product, the deconcatenation coproduct, the obvious unit and counit maps and the universal enveloping algebra antipod (see [1], Example 1.8) endow the algebra $T(V)$ with the structure of a commutative conilpotent Hopf algebra, and its dual coincides with the algebra of non-commutative power series in $n$ variables $k\langle\langle X_1,\dots,X_n\rangle\rangle$. There are isomorphisms of CHAs (see [41], Example 2.11, p. 271) where $\widehat{\mathcal{U}}\mathrm{LV}$ is the completion of the universal enveloping algebra of the free Lie algebra $\mathrm{LV}$ on $V$ with respect to the augmentation ideal, and $\widehat{k}\Phi(Z)$ is the completion of the group algebra $k[\Phi(Z)]$ of the free discrete group $\Phi(Z)$ with respect to the augmentation ideal.Example 5 (finite $p$-groups as unipotent groups $\operatorname{char}(k)=p>0$). Let $G$ be a finite $p$-group. Then the group algebra $\mathbb{F}_p[G]$ is a cocommutative Hopf algebra with coproduct defined on $g\in G$ by the rule $\Delta (g)=g\otimes g$, an antipod $s(g)=g^{-1}$, the obvious unit, and the augmentation as a counit. We set $\mathcal{O}(G)^*=\mathbb{F}_p[G]$ (the Hopf algebra of a finite group). It is easy to check that $\mathcal{O}(G)\cong(\mathcal{O}(G)^*)^{\vee} \cong \mathbb{F}_p^G$ is a commutative Hopf algebra of functions from $G$ to $\mathbb{F}_p$ with commutative multiplication and addition given by the rules $(f_1\cdot f_2)(x)=f_1(x)\cdot f_2(x)$, $(f_1+ f_2)(x)=f_1(x)+ f_2(x)$, where $f_i\in \mathcal{O}(G)$, $x\in G$, and a (dual) coproduct, given on the dual basis $\{e_g, g\in G\mid e_g(h)\,{=}\,1$ if $h=g$ and $e_g(h)=0$ if $h\neq g\}$ by the rule $\Delta (e_g)=\sum_{g_1g_2=g} e_{g_1}\otimes e_{g_2}$, where $g_1, g_2\in G$. The corresponding group scheme (see [51], § 2.3) with the algebra of regular functions $\mathcal{O}(G)$ is unipotent. Indeed, given an arbitrary representation $\rho\colon M\to M\otimes \mathcal{O}(G)$, we claim that there is a subspace $\mathbb{F}_p\cdot v$, $v\in M$, fixed under the action of $G$, that is, $\rho(v)=v\otimes 1$. By Lemma 2, this is equivalent to saying that $M^*_G\neq 0$. We argue by induction on the rank $n$ of $M^*$. If $n=1$, then $M^*=\mathbb{Z}/p\mathbb{Z}$ is a trivial $G$-module, since $(|\mathrm{Aut}(\mathbb{Z}/p\mathbb{Z})|,p)=(|\mathbb{Z}/(p-1)\mathbb{Z}|,p)=1$, so $M^*_G=M^*\neq0$. Let $n=k$, then the submodule $M^*$ of $G$-fixed elements is not trivial and contains $M^*_0=\mathbb{Z}/p\mathbb{Z}$ with trivial $G$-action. Let $M^*_1=M^*/M^*_0$ and $\psi\colon M^*\to M^*_1$ be the corresponding factorization. Since $\psi(M^*(g-1))=M^*_1(g-1)$, $\psi$ induces the surjection $M^*_G \to (M^*_1)_G$. But $(M^*_1)_G\neq 0$ by induction. Hence $M^*_G\neq0$. It is also clear that $G(\mathbb{F}_p)=Hom_{\mathrm{Alg}_k}(\mathcal{O}(G),\mathbb{F}_p)\cong \mathcal{G}\mathcal{O}(G)^*\cong G$. Passing to the inverse limit of group-valued functors, we see that pro-$p$-groups are naturally $\mathbb{F}_p$-points of prounipotent groups. Example 6 (regular functions of free pro-$p$-groups $\operatorname{char}(k)=p>0$). Let $F(Z)$ be a free pro-$p$-group over a finite set $Z$. The complete group ring $\mathbb{F}_pF(Z)$ can be endowed with the structure of a complete Hopf algebra. Consider the decomposition $F(Z)\cong \varprojlim G_{\alpha}$ into the inverse limit of finite $p$-groups $|G_{\alpha}|=p^l$, then $\mathbb{F}_pF(Z)\cong\varprojlim \mathbb{F}_p[G_{\alpha}]$ (see [27], § 7). Each $\mathbb{F}_p[G_{\alpha}]$ inherits the Hopf algebra structure as in Example 5. Taking $\widehat{\Delta}=\varprojlim \Delta_{\alpha}$, we obtain the complete coproduct $\widehat{\Delta}\colon \mathbb{F}_pF(Z)\to \mathbb{F}_pF(Z)\mathbin{\widehat{\otimes}} \mathbb{F}_pF(Z)$. By Proposition 7.16 in [27], there is an isomorphism of complete algebras $\mathbb{F}_p\langle\langle X_i\rangle\rangle_{i\in Z}\cong \mathbb{F}_pF(Z)$ given on generators $s_i$ of $F(Z)$ by the rule $s_i\mapsto X_i+1$, $s_i^{-1}\mapsto \sum_{n=0}^{\infty}(-X_i)^n$, $X_i\mapsto s_i-1$. The continuous dual is the tensor algebra $\mathbb{F}_p\langle\langle X_i\rangle\rangle_{i\in Z}^{\vee}\cong T(\mathbb{F}_p^Z)$ of the vector space with the basis $e_i$, $i\in Z$, and the deconcatenation coproduct (see [50], Example 29). By duality, as in Example 5, we get a commutative product in $\mathcal{O}(F(Z))$ as follows $(f_1*f_2)(g)=m(f_1\mathbin{\widehat{\otimes}} f_2)(\widehat{\Delta}(g))=m(f_1\mathbin{\widehat{\otimes}} f_2)(g\mathbin{\widehat{\otimes}} g)=f_1(g)\cdot f_2(g)$ and the dual antipod. We set $\mathcal{O}(F(Z))=T(\mathbb{F}_p^Z)$ with just defined Hopf algebra structure. Since $f_1$, $f_2$ are continuous functions, there is some $\alpha$, that $f_1$, $f_2$ can be factored as compositions $f_1,f_2\colon G\xrightarrow{\mathrm{pr}_{\alpha}} G_{\alpha}\xrightarrow{\widetilde{f}_1,\widetilde{f}_2} \mathbb{F}_p$ (see Lemma 1.1.16 in [42]). Similarly to the situation with $\operatorname{char}(k)=0$, in the pro-$p$-case, the continuous functions are also embedded $\mathrm{pr}_{\alpha}^*\colon \mathcal{O}(G_{\alpha})\to \mathcal{O}(G)$ as “regular” algebraic functions, that is, $\mathcal{O}(F(Z))$ is the so-called Hopf algebra of locally constant functions on the free pro-$p$-group $F(Z)$ (see [34]). 4.2. Cohering positive characteristics and characteristic zeroFirst, we obtain a more detailed description of mappings in the arithmetic square (2) in the case of free simplicial groups finitely generated in each dimension. It is straightforward to note that $\kappa_p $ is a dimensionwise embedding of finitely generated discrete groups into their pro-$p$-completions (free pro-$p$-groups). An induction on the degree of nilpotency shows that $\mathbb{Q}_{\infty}(H^{\wedge}_p)\cong H^{\wedge}_{\mathbb{Q}_p}$, the $\mathbb{Q}$-prounipotent completion of a nilpotent finitely generated pro-$p$-group $H^{\wedge}_p$ is indeed the $\mathbb{Q}_p$-prounipotent completion of the underlying discrete nilpotent group $H$. Passing to the inverse limit of nilpotent quotients of a free discrete group $\Phi(m)$ of a finite rank $m$, we see that $\mathbb{Q}_{\infty}( \Phi(m)^{\wedge}_p)\cong \Phi(m)^{\wedge}_{\mathbb{Q}_p}$ is the free prounipotent group over $\mathbb{Q}_p$ of the same rank as $\Phi(m)$. It is similarly verified that $\mathbb{Q}_{\infty}(\kappa_p)$ is the canonical rational points embedding $\Phi(m)^{\wedge}_{\mathbb{Q}}\to \Phi(m)^{\wedge}_{\mathbb{Q}_p}$ of a finitely generated free prounipotent group over $ \mathbb{Q} $ into the corresponding finitely generated free prounipotent group over $\mathbb{Q}_p$ of the same rank (see also [33]). Sometimes it can be useful to cohere information between positive and characteristic zero; the emergence of constant unipotent group schemes over $\mathbb{F}_p$ (that is, of finite $p$-groups) in this case is inevitable. Let us illustrate this with an example of calculations used in Proposition 3.19 of [33] to solve specific problems. Let $F$ be a free prounipotent group of rank $m$ over $\mathbb{Q}_p$. From Quillen’s formulas, we have the decomposition
$$
\begin{equation*}
\mathcal{O}(F)^*\cong \varprojlim \mathbb{Q}_p[\Phi(m)]/I^n_{\mathbb{Q}_p}\cong \varprojlim \mathbb{Z}_p[\Phi(m)]/I^n_{\mathbb{Z}_p}\otimes_{\mathbb{Z}_p}\mathbb{Q}_p,
\end{equation*}
\notag
$$
where $\mathbb{Q}_p[\Phi(m)]$ is the group ring of $\Phi(m)$ and $I_{\mathbb{Z}_p}$ is the augmentation ideal of $\mathbb{Z}_p[\Phi(m)]$. It follows from Lazard’s theorem (see Proposition 7 in [46]) that
$$
\begin{equation*}
\varprojlim\mathbb{Z}_p[\Phi(m)]/I^n_{\mathbb{Z}_p}\cong \varprojlim A(m)/\mathbb{I}^n\cong\varprojlim \mathbb{Z}_pF_p(m)/\widehat{I}^n_{\mathbb{Z}_p},
\end{equation*}
\notag
$$
where $\mathbb{Z}_pF_p(m)$ is the complete group ring of the free pro-$p$-group $F_p(m)$ of rank $m$, $\widehat{I}^n_{\mathbb{Z}_p}$ is the $n$th power of its augmentation ideal, $A(m)\cong \mathbb{Z}_p\langle\langle m\rangle\rangle$ is the non-commutative power series of $m$ variables over $\mathbb{Z}_p$, and $\mathbb{I}$ is its augmentation ideal. The $\mathrm{mod}(p)$-reduction of group rings coefficients
$$
\begin{equation*}
\varprojlim \mathbb{Z}_pF_p(m)/\widehat{I}^n_{\mathbb{Z}_p}\, \overset{\mathrm{mod}(p)}{\rightsquigarrow\,\,}\,\, \varprojlim \mathbb{F}_pF_p(m)/\widehat{I}^n_{\mathbb{F}_p}\cong \mathcal{O}(F_p(m))^*
\end{equation*}
\notag
$$
leads to the complete $\mathbb{F}_p$-Hopf algebra of the free pro-$p$-group of rank $m$. The $\widehat{I}_{\mathbb{F}_p}$-adic filtration induces the so-called Zassenhaus filtration $\mathcal{M}_{n,p}= \{f\in F\mid f-1 \in \widehat{I}_{\mathbb{F}_p}^n,\, n\in \mathbb{N}\}$ of $F_p(m)$ and it follows from [27], § 7.4, that $F_p(m)/\mathcal{M}_{n,p}\cong \Phi(m)/\Phi(m)\cap\mathcal{M}_{n,p}$ are finite $p$-groups. Hence
$$
\begin{equation*}
\begin{aligned} \, \mathcal{O}(F_p(m))^* &\cong \varprojlim\mathbb{F}_p[\Phi(m)]/I^n_{\mathbb{F}_p}\cong \varprojlim \mathbb{F}_p[\Phi(m)/\Phi(m)\cap\mathcal{M}_{n,p}] \\ &\cong \varprojlim \mathcal{O}(\Phi(m)/\Phi(m)\cap\mathcal{M}_{n,p})^*. \end{aligned}
\end{equation*}
\notag
$$
The Hopf algebra structure on $((\mathcal{O}(\Phi(m)/\Phi(m)\cap\mathcal{M}_{n,p})^*)^{\vee}$ is uniquely defined, since there is the only structure of a commutative algebra with unit for the function space of a finite set (analogously to Gelfand’s result [44], § 11.13 (a)) and the “constant” comultiplication is given by duality (Example 5). Mimicking Corollary 1, we may think that complete group algebras with $\mathbb{Z}_p$-coefficients are $\mathbb{Z}_p$-dual to the distribution algebras (see [34], Proposition 1). 4.3. Fixed points and coinvariantsFrom now on, let $G\cong\varprojlim G_{\beta}$ be a prounipotent group decomposed into an inverse limit of unipotent groups and let $M$ be a right topological $\mathcal{O}(G)^*$-module. By $M_G=M/M(g-1)$ we denote the module of $G$-coinvariants. This is a quotient of $M$ by a closed submodule generated by $\{m(g-1),\, g\in G(k),\, m\in M\}$. Lemma 2. Let $G\cong \varprojlim G_{\beta}$ be a prounipotent group and $V$ be a $G$-module. Then $V$ has a decomposition $V=\varinjlim V_{\alpha}$, where $V_{\alpha}$ are $G_{\alpha}$-modules and $G_{\alpha}$ could be chosen from the given decomposition $G\cong \varprojlim G_{\beta}$. Dually, there is a decomposition $V^*=\varinjlim V^*_{\alpha}$ into the inverse limit of finite dimensional $\mathcal{O}(G_{\alpha})^*$-modules. In particular, $V=0$ if and only if $V^G=0$, or, equivalently, $V^*=0$ if and only if $(V^*)_G=0$. Proof. According to Theorem (1) in [51], § 3.3, any finite subset $X$ of $V$ lies in a $k$-subspace of finite dimension $W\subseteq V$ such that $\Delta(W)\subset W\otimes \mathcal{O}(G)$. Let $\{w_i\}$ be a basis of $W$ and $\Delta(w_i)=\sum w_j\otimes a_{ij}$. We fix a basis $\{v_k\}$ of $\mathcal{O}(G)$ and let $\Delta(a_{ij})=\sum v_k\,{\otimes}\, a_{ijk}$. Theorem (2) in [51], § 3.3, says that $k[a_{ij},a_{ijk},Sa_{ij},Sa_{ijk}]\subseteq\mathcal{O}(G)$ is a finitely generated (as algebra) Hopf subalgebra of $ \mathcal{O}(G)$.
It is an intrinsic property of a unipotent group to have a fixed point in any non-trivial representation and, therefore, $V=0$ if and only if $V^G=0$, the rest follows from Lemma 3 below. This proves the lemma. Lemma 3. Let $M$ be an $\mathcal{O}(G)$-comodule over the Hopf algebra of regular functions of a prounipotent group $G$ and let $M^*=Hom_k(M,k)$ be the dual topological $\mathcal{O}(G)^*$-module. Then the linearly compact topological spaces $(M^G)^*\cong (M^*)_G$ are naturally isomorphic. Proof. We prove the claim in the case of characteristic zero; the arguments in the pro-$p$-case are similar. Let us start with (7), the short exact sequence $0\to M^G\xrightarrow{i} M\otimes k \xrightarrow{\Delta_M-\operatorname{id}_M\otimes 1} M\otimes \mathcal{O}(G)$, where $i\colon M^G\to M\cong M\otimes k$ is the embedding $i(m)=m\otimes 1$. Consider the dual exact sequence of linearly compact topological $k$-vector spaces
We have $\mathcal{O}(G)^*=k\cdot 1+ I$, as topological vector spaces, and, therefore, $\operatorname{Im}(\Delta_M-\operatorname{id}_M\otimes 1)^*=M^*\cdot I$, where $I=\operatorname{Ker} (\mathcal{O}(G)^*\to k)$ is the augmentation ideal. Lemma 2 gives the decomposition $M^*\cong \varprojlim M^*_{\alpha}$, where $M^*_{\alpha}$ are finite dimensional topological $\mathcal{O}(G_{\alpha})^*$-modules (in the notation of Lemma 2) and, therefore, $M^*\cdot I\cong \varprojlim (M^*_{\alpha}\cdot I_{\alpha})$, where $I_{\alpha}$ is the augmentation ideal of $\mathcal{O}(G_{\alpha})^*$. Now from Quillen’s formulas (Examples 2 and 3) it follows that $M^*_{\alpha}\cdot I_{\alpha}$ are generated by the elements $\{m\cdot g-m\mid m\in M_{\alpha},\, g\in G_{\alpha}(k)\}$, and the result follows. This proves the lemma. 4.4. Presentations of prounipotent groupsIn § 4.1, we saw that a prounipotent group, both in characteristic zero and in the pro-$p$-case, can be defined as group-valued functors representable by means of commutative Hopf algebras. The notion of a free prounipotent group $F(Z)$ over a finite set $Z$ is well-known and nicely explained, as mentioned in [50], § 3, and [18]. The group of rational points $F(Z)(k)$ posseses a universal property with respect to mappings of $Z$ into rational points of prounipotent groups, since it is the prounipotent completion of the free discrete group $\Phi(Z)$ on $Z$. We extend the notion of a free prounipotent group to an arbitrary set $Z$; in the case of an algebraically closed ground field of characteristic zero it coincides with Definition 2.1 in [29]. Let $Z$ be a set and let $\{Z_i\mid i\in J\}$ be the collection of all finite subsets of $Z$. Defining $i\preceq j$ if $Z_i\subseteq Z_j$, we make $J$ into a poset. For $i\preceq j$, let $\psi^j_i\colon \mathcal{O}(F(Z_i))\to \mathcal{O}(F(Z_j))$ be the tensor algebras embedding of Examples 4, 6, induced by the inclusion of finite sets $\psi^j_i\colon Z_i\to Z_j$; we recall that in both cases, in the case of $ \operatorname{char}(k) = 0 $ and in the pro-$p$-case, we have $\mathcal{O}(F(Z_i)(k))=T(k^{Z_i})$ as vector spaces. We define the Hopf algebra of regular functions $\mathcal{O}(F(Z))$ of a free prounipotent group over an arbitrary set $Z$ as $\mathcal{O}(F(Z))\cong \varinjlim_{i\in J}\mathcal{O}(F(Z_i))$. Thus a free prounipotent group with an infinite number of generators is obtained from its finitely generated free quotient groups that are also closed subgroups. In the pro-$p$-case, for $f_1\in \mathcal{O}(F(Z_1)), f_2\in \mathcal{O}(F(Z_2))$, their product $f_1\cdot f_2$ is properly defined as an element of $\mathcal{O}(F(Z_1\cup Z_2))$ as in Example 6. The following definition is slightly tautological in the pro-$p$-case. Definition 10. By a free $k$-prounipotent $(\operatorname{char}(k)=0)$ group $F(Z)$ over a set $Z$, we mean a $k$-prounipotent group with the Hopf algebra of regular functions $\mathcal{O}(F(Z))\cong T(k^Z)$ isomorphic to the tensor algebra of a vector space $k^Z$ with the shuffle product multiplication, the deconcatenation coproduct and the universal enveloping algebra antipod. By a free pro-$p$-group $(k=\mathbb{F}_p)$ $F(Z)$ over a set $Z$, we mean a $\mathbb{F}_p$-prounipotent group scheme with the Hopf algebra of $\mathbb{F}_p$-valued locally-constant functions, isomorphic as $\mathbb{F}_p$-vector space to the tensor algebra $\mathcal{O}(F(Z))\cong T(k^Z)$ of $k^Z$ with the deconcatenation coproduct, the product and addition of locally-constant functions (through their agreed finite quotient (see [51], § 2.3) and antipod, induced, by duality, from group inversion. Lemma 4. Let $X$, $Y$ be sets. Then $F(X\cup Y)$ is the direct sum of $F(X)$ and $F(Y)$: $F(X\cup Y)\cong F(X)\star F(Y)$. Proof. Using Yoneda’s lemma, we see that $\mathcal{O}(F(X\cup Y))$ is the direct product of $\mathcal{O}(F(X))$ and $\mathcal{O}(F(Y))$ with the natural projections $p_1\colon \mathcal{O}(F(X\cup Y))\to \mathcal{O}(F(X))$ and $p_2\colon \mathcal{O}(F(X\cup Y))\to \mathcal{O}(F(Y))$, that is, we need show that, for any $A=\mathcal{O}(G)$, where $G$ is a prounipotent group, and any Hopf algebra homomorphisms $\phi_1\colon A\to \mathcal{O}(F(X))$ and $\phi_2\colon A\to \mathcal{O}(F(Y))$, there is a unique $\psi\colon A\to \mathcal{O}(F(X\cup Y))$ such that $p_1=i_1 \psi$ and $p_2=i_2 \psi$. Since $A\cong \varinjlim A_{\alpha}$, where $A_{\alpha}$ are finitely generated (as algebras) Hopf algebras (see [51], Theorem 3.3) and since $\phi_1(A_{\alpha})\subseteq T(k^{X_{\alpha}})$, $\phi_2(A_{\alpha})\subseteq T(k^{Y_{\alpha}})$, where $X_{\alpha}\subseteq X$, $Y_{\alpha}\subseteq Y$ are finite subsets of $X$, $Y$, we have $F(X_{\alpha}\cup Y_{\alpha})\cong F(X_{\alpha})\star F(Y_{\alpha})$ by the universal property of finitely generated free prounipotent groups, which holds since they are prounipotent completions of finitely generated free discrete groups on the same generating sets. Therefore, there is a unique $\psi_{\alpha}\colon A_{\alpha}\to T(k^{X_{\alpha}\cup Y_{\alpha}})$ with the required property, and we define $\psi=\varinjlim \psi_{\alpha}$. This proves the lemma. Remark 5 (duality between tensor algebra generators and free prounipotent group generators). Let $Z$ be a finite set and $\mathcal{O}(F(Z))\cong T(k^Z)$ be the Hopf algebra of regular functions of a free prounipotent (pro-$p$-group) group $F(Z)$. It follows from the definition of the deconcatenation coproduct that the subspace of primitive elements $\mathcal{P}\mathcal{O}(F(Z))=\{x\in \mathcal{P}\mathcal{O}(F(Z)) \mid \Delta(x)=1\otimes x + x\otimes 1, \, \varepsilon(x)=0\}$, where $\varepsilon$ is the counit, coincides with $k^Z$. On the other hand, $\mathcal{P}\mathcal{O}(F(Z))\cong C_1/C_0$, where $C_k=\operatorname{Ann}_{\mathcal{O}(F(Z))} I^{k+1}$ is the annihilator of $(k+1)$th power of the augmentation ideal $I$ of $\mathcal{O}(F(Z))^*$, as $C_0=\operatorname{Ann}_{\mathcal{O}(F(Z))} I=k\cdot 1$ and $C_1=k\cdot 1\oplus \mathcal{P}\mathcal{O}(F(Z))$. Indeed, $x\in \operatorname{Ann}_{\mathcal{O}(F(Z)} I^2$ if and only if $\Delta(x)=y\otimes1+ 1\otimes z$, where $y, z\in \mathcal{O}(F(Z))$ are indecomposable elements ($x(i_1\cdot i_2)=m(i_1\otimes i_2)\Delta(x)$ and $1\cdot k$ is the annihilator of $I$). Now, from the commutative Hopf algebra counit axiom (see [51], § 1.4), if $\varepsilon(x)=0$, we have $\Delta(x)=1\otimes x+1\otimes x$. By duality, $\operatorname{Ann}_{\mathcal{O}(F(Z))} I^k\cong (\mathcal{O}(F(Z))/I^k)^{\vee}$ and hence $I/I^2\cong \operatorname{Ker}(\mathcal{O}(F(Z))/I^2\to \mathcal{O}(F(Z))/I)\cong (C_1/C_0)^*\cong (\mathcal{P}\mathcal{O}(F(Z)))^*$. Therefore, $I/I^2\cong (k^Z)^*$. According to [27], § 7.4, Lemma 4.10 (the proof of (I)), for a free pro-$p$-group $F=F(Z)$, the formulas $(X_i-1) +I^2\mapsto s_i\cdot[F,F]F^p$, $s_i\in Z$, give the isomorphism $I/I^2\cong F/[F,F]F^p$. Proceeding as in Example 4, we can also deduce that $I/I^2\cong F/[F,F](k)$ for a free prounipotent group over the base field $k$ of zero characteristic, where, in both cases, $I$ is the augmentation ideal of $\mathcal{O}(F(Z))^*$. By Proposition 25 in [46], in the pro-$p$-case and by Lemma 2.7 in [29], when $\operatorname{char}(k)= 0$, the Zariski generators of $F(Z)(k)$ are in a one-one correspondence with lifts of bases of $F/F^p[F,F]$, in the pro-$p$-case, and of $F/[F,F](k)$, when $\operatorname{char}(k)=0$, under the abelianization homomorphisms $F\to F/F^p[F,F]$, in the pro-$p$-case, and $F(k)\to F/[F,F](k)$, when $\operatorname{char}(k)=0$. As a consequence, we have a right to consider generators of the free prounipotent (pro-$p$) group $F(Z)(k)$ as elements of $Z^*$ dual to the basis $Z$ of $k^Z$ which generate $T(k^Z)\cong \mathcal{O}(F(Z))$. Such liftings can be visualized using Quillen’s formula, since $\Phi(Z)\subset F(Z)(k)$, as in Example 3, and $\Phi(Z)$ maps to $\Phi(Z)/[\Phi(Z),\Phi(Z)]$ under the abelianization homomorphism $F(k)\to F/[F,F](k)$ (see [18], Appendix A.1). We characterize free prounipotent groups as groups with a universal property (Definition 11 and Remark 8 below) and introduce by duality of Remark 5 an inverse system of pointed finite sets, which arises from the direct system of finite subsets of $Z$, and which will be convenient when working with free topological modules in § 5.1. Remark 6 (pointed profinite spaces of generators). Let $Z$ be a set and let $\{Z_i\mid i\in J\}$ be a collection of all finite subsets of $Z$. We make $J$ into a poset by defining $i\preceq j$ if $Z_i\subseteq Z_j$. If $i\preceq j$, we define $\phi^j_i\colon F(Z_j^*,*)\to F(Z_i^*,*)$ as a homomorphism of free finitely generated prounipotent groups induced by the map of punctured finite spaces $\phi^j_i\colon (Z_j^*,*)\to (Z_i^*,*)$ (as in Remark 5 we consider generators $Z_i^*\hookrightarrow F(Z_i)(k)$ corresponding to a dual basis of $(k^{Z_i})^*$, where $\mathcal{O}(F(Z_i)(k))=T(k^{Z_i})$): We always identify $(Z_i^*,1)$ with the subspace of rational points $F(Z_i)(k)$ and assume that the specified space is embedded $(Z_i^*,*)\hookrightarrow F(Z_i)(k)$ by the rule $z \mapsto z$, $z\in Z_i$, and $*\mapsto 1\in F(Z_i)(k)$, where $i\in J$. Left exactness of $\varprojlim$ implies that $(\widehat{Z}^*,*)=\varprojlim_{i\in J} (Z_i^*,*)$ is actually a profinite subspace of $F(Z)(k)$ (this subspace is actually a one-point compactification of $Z$). We will write $F=F(\widehat{Z}^*,*)$, emphasizing that $F(k)$ is generated by $(\widehat{Z}^*,*)$, this basis can be identified with some lifting one-point compactification of the dual basis. Thus, in the notation $(\widehat{Z}^*,*)$, we emphasize three points at once: $(\widehat{Z}^*,*)$ is obtained as a dual basis, the space has the topology of the one-point compactification, we select $*$ (the punctured point). Now we show that the group of Definition 10 possesses a universal property with respect to “convergent” maps of the “pointed profinite basis” $(\widehat{Z}^*,*)$ of $F(\widehat{Z}^*,*)(k)$ to the rational points $G(k)$ of a unipotent group $G$. Proposition 6. Let $G$ be a unipotent group. Then there is a bijection between homomorphisms $f\colon F(\widehat{Z}^*,*)\to G$ and continuous maps $f'\colon (\widehat{Z}^*,*)\to G(k)$ such that $f'(*)=e$ and $f'(\widehat{Z}^*,*)$ has a finite cardinality. Proof. Indeed, suppose we are given a homomorphism $f\colon F(\widehat{Z}^*,*)\to G$ of a free prounipotent group $F(\widehat{Z}^*,*)$ into a unipotent group $G$. Since $\mathcal{O}(G)$ is a finitely generated Hopf algebra, it follows that the image $f^o(\mathcal{O}(G))$ lies in the Hopf subalgebra $\mathcal{O}(F(Z_i^*,*))=T(k^{Z_i})$ for some finite subset $Z_i\subseteq Z$. $\mathcal{O}(F(Z))$ is faithfully flat over $\mathcal{O}(F(Z_i^*,*))$ (see [51], Theorem 14.1) and, therefore (see [51], Theorem 13.2) $f$ is factored via an epimorphism onto a free prounipotent group over $Z_i^*$, which corresponds to a continuous mappings of pointed profinite spaces $g'\colon (\widehat{Z}^*,*)\to (Z_i^*,*)$ and, therefore, taking the composition with the homomorphism $h'\colon F(Z_i^*,*)\to G(k)$, we obtain the required $f'=h'g'$.
A continuous map $f'\colon (\widehat{Z}^*,*)\to G(k)$ with finite image factors through some projection $\phi_i\colon (\widehat{Z}^*,*)\to (Z^*_i,*)$ (see [42], Lemma 1.1.16). This projection corresponds to the homomorphism of prounipotent groups $\phi_i\colon F(\widehat{Z}^*,*)\to F(Z^*_i,*)$ as above. But $F(Z^*_i,*)$ is the prounipotent completion of $\Phi(Z_i^*)$,, the free discrete group on $Z_i^*$ (see [50], § 3). Now the result follows from the universal properties of the free discrete group over $Z_i^*$ and its prounipotent completion. This proves the proposition. Remark 7. It turns out that the decomposition $F(\widehat{Z}^*,*)\cong \varprojlim F(Z_i^*,*)$ is actually the inverse limit of proper closed subgroups, a similar statement is known in the pro-$p$-case (see [42], Corollary 3.3.10). Indeed, the natural maps of pointed profinite spaces $i\colon (Z_i^*,*)\to (\widehat{Z}^*,*)$ and $\operatorname{pr}_i\colon (\widehat{Z}^*,*)\to (Z_i^*,*)$ induce, by the universal property, the homomorphisms $i\colon F(Z_i^*,*)\to F(\widehat{Z}^*,*)$, $\operatorname{pr}_i\colon (\widehat{Z}^*,*)\to (Z_i^*,*)$ of free prounipotent groups and, since $\operatorname{pr}_i\circ i=\operatorname{id}_{F(Z_i^*,*)}$, $i$ establishes an isomorphism of $F(Z_i^*,*)$ with the closed subgroup in $F(\widehat{Z}^*,*)$ generated by $(Z_i^*,*)$. Therefore, $\operatorname{pr}_i$ are actually projections onto closed proper subgroups determined by the Hopf ideals of $\mathcal{O}(F(X))$ generated by $k^{X\setminus X_i}$. As a consequence, once chosen generators in $F(Z_i^*,*)(k)$ are actually generators in $F(\widehat{Z}^*,*)(k)$, as expected. Proposition 6 shows that free prounipotent groups in Definition 10 coincide with the Lubotzky–Magid definition (see [29], Definition 2.1). These groups can be characterized as groups with the so-called lifting property: for any exact sequence of prounipotent groups and homomorphism $\rho\colon F\to G$, there is a homomorphism $h\colon F\to E$ such that $\pi h=\rho$.According to Theorem 2.4 in [29], the lifting property characterizes free prounipotent groups over algebraically closed fields of characteristic zero. However, the proof depends only on standard results on normal subgroups of unipotent groups, which are easy to follow from the corresponding results on nilpotent Lie algebras [45]; thus, free prounipotent groups over nonalgebraically closed fields of characteristic zero are also characterized as groups with the lifting property. In fact, in the pro-$p$-case, the indicated characterizations of free pro-$p$-groups were obtained already in [46], § 3.4. Lemma 5. Let $F$ be a free prouniotent group. Then any epimorphism $\phi\colon G\to F$ of prounipotent groups has a splitting, which is a homomorphism of prounipotent groups. Proof. We take $G=F$, $E=G$ and $\rho=\operatorname{id}_F$ in the diagram of lifting property above. Now $h$ is the desired splitting. This proves the lemma. We say that a prounipotent group $G$ has cohomological dimension $n$ (written $\operatorname{cd}(G)=n$) if, for every $G$-module $A$ and every $i>n$, $H^i(G,A)=0$ and $H^n(G,V)\neq 0$, for some $G$-module $V$. Prounipotent groups of cohomological dimension one are free prounipotent groups; this follows from Proposition 6 and [29], formula (1.18). An important feature of prounipotent groups (which makes their cohomological study sometimes easier than in the discrete case) is that $\operatorname{cd}(G)\leqslant n$ if and only if $H^{n+1}(G,k_a)=0$, because $k_a$ is the only simple $G$-module. This fact is easily deduced with the “dévissage” argument [46], § 3.1. The following propositions makes possible to introduce a concept of prounipotent presentation in the form (1). Proposition 7. Let $G$ be a prounipotent group over a base field $k$ of characteristic zero or $k=\mathbb{F}_p$ and $G$ be a pro-$p$-group. Then there is a free prounipotent group $F(X)$ over a set $X$ of cardinality $\leqslant d(G)$ (the cardinality of a basis of $Hom(G,k_a)$) and an epimorphism $f\colon F(X)\to G$. Proof. The proof uses the lifting property of free prounipotent groups and Lemma 5 (for details, which are similar to the pro-$p$-case, see [29], Proposition 2.8). Cohomology and “proper” presentation theory of prounipotent groups in characteristic zero closely parallels thos of the cohomology of pro-$p$-groups [46]: the free prounipotent groups turn out to be those of cohomological dimension ones, the dimension of the first and second cohomology groups give cardinalities of generators and defining relations [29], [30]. Definition 11. We say that a map $\mu\colon (\widehat{X}^*,*)\to G(k)$ of a pointed profinite space $(\widehat{X}^*,*)$ into the group of rational points of a prounipotent group $G$ is convergent if there is a decomposition $G(k)\cong \varprojlim G_{\alpha}(k)$ such that $\phi_{\alpha}\mu(x)=1$ for all but finite $x\in \widehat{X}^*$, where $\phi_{\alpha}\colon G\to G_{\alpha}$ is the projection. Remark 8. By construction, free prounipotent groups have the universal property with respect to convergent maps of generators to rational points of prounipotent groups. Let $f\colon F(\widehat{X}^*,*)(k)\to G(k)$ be a homomorphism from a free prounipotent group over a pointed profinite space $(\widehat{X}^*,*)$ onto $k$-points of a prounipotent group $G$, by the proof of Proposition 6, the map $(\widehat{X}^*,1)\to G(k)$ is convergent. 4.5. Simplicial presentations of prounipotent groupsIn [33] and [35], the notion of finite type prounipotent presentation was extended to prounipotent groups over non-algebraically closed fields of characteristic zero. Their simplicial forms also appeared in [33], [35] as free simplicial prounipotent groups of finite type degenerate in dimensions greater than one. In this section, we define the notion of a simplicial prounipotent presentation over punctured profinite spaces. Our exposition is completely analogous in the characteristic zero and pro-$p$-cases, and so we restrict ourselves to the first case. The only difference is that, in positive characteristic, we replace $[R,R]$ with $R^p[R,R]$ and $[R,F]$ with $R^p[R,F]$ in order to work with $\mathbb{F}_p$-vector spaces and their duals in the pro-$p$-case. Remark 9. By Proposition 7, any prounipotent group $G$ can be presented as a quotient of a free pro-unipotent group $G\cong F(X)/R=F(\widehat{X}^*,*)/R$ for some set $X$ of cardinality less than or equal to that of the basis in $Hom(G,k_a)$. If the inflation map $H^1(G,k_a)\to H^1(F,k_a)$ is an isomorphism, then we say that such a prounipotent presentation is proper. In what follows, we will not restrict ourselves only to proper representations, but we consider only those sets of “defining relations” $Y$ for which there is a fixed isomorphism $(R/[R,F](k))^{\vee}\cong k^Y$ or dually $R/[R,F](k)\cong (k^Y)^*\cong k^{(\widehat{Y}^*,*)}$. This isomorphism supposed to be canonical in the sense that it arises from a simplicial presentation, that is, the images of the relators form a pseudobasis (they are linearly independent and topologically generate $R/[R,F](k)$), we will return to this condition in Remark 12 below. We say that $1\to R\to F(\widehat{X}^*,*)\to G\to 1$ is a prounipotent presentation $(X\,|\,R)$ in the form (1) and construct its simplicial form into three steps. Step 1. We choose a basis, say $Y$, in $R/[R,F](k)^{\vee}$. Then $R/[R,F](k)\cong (k^Y)^*\cong k^{(\widehat{Y}^*,*)}$, by duality, and $\tau_F\colon (\widehat{Y}^*,*)\to (\widehat{Y}^*,0)\subset R/[R,F](k)$ is by definition a homeomorphism of punctured profinite spaces, which sends $y\mapsto y$, $y\in \widehat{Y}^*$, and $*\mapsto 0$. Step 2. Let $\psi\colon R/[R,R](k)\to R/[R,F](k)$ be a projection of linearly compact topological vector spaces induced by the inclusion $[R,R]\subset [R,F]$. Then the dual inclusion $\psi^{\vee}\colon R/[R,F](k)^{\vee}\to R/[R,R](k)^{\vee}$ allows us to consider the basis $Y\subset R/[R,F](k)^{\vee}$ as a subset of some basis $T$ of $R/[R,R](k)^{\vee}$, that is, $Y\subset T$. By $\sigma\colon R/[R,R](k)^{\vee}\to R/[R,F](k)^{\vee}$ we denote the projection onto the subspace, which is given on the basis by the rule $\sigma(x)=x$, $x\in Y$, and $\sigma(x)=0$, $x\notin Y$, and, by duality, obtain the closed embedding $\sigma^*\colon R/[R,F](k)\to R/[R,R](k)$, which is a continuous splitting of $\psi$ and $(\widehat{T}^*,0)=(\widehat{Y}^*,0)\cup (\widehat{T\setminus Y }^*,0)$. Step 3. Let $T$ be a basis of $R/[R,R](k)^{\vee}$. By Remark 6, there exists a lifting homeomorphism of pointed profinite spaces $\eta\colon (\widehat{T}^*,0)\to R(k)$ and the image $\eta(\widehat{T}^*,0)$ Zariski generates $R(k)$. We define a pointed profinite space of “defining retations” as the image of
$$
\begin{equation}
\tau=\eta\circ\sigma^*\circ\tau_F\colon (\widehat{Y}^*,*)\to R(k).
\end{equation}
\tag{12}
$$
We claim that $\tau(\widehat{Y}^*)$ Zariski generates $R(k)$ as a normal subgroup of $F(\widehat{X}^*,*)$. The proof of this statement is quite standard (see [29], Theorem 3.11, [42], Proposition 7.8.2) and we recall it. We need to show that $N(k)=\langle\widehat{Y}^*\rangle_F$, that is, the Zariski normal closure of $\widehat{Y}^*$ coincides with $R(k)$. Indeed, the inclusion $N\subseteq R$ gives the restriction homomorphism $\operatorname{Res}\colon H^1(R,k_a)\to H^1(N,k_a)$, which induces a monomorphism (by taking $F$-fixed points) $Hom_F(R,k_a)=H^1(R,k_a)^F\xrightarrow{\operatorname{Res}^F} H^1(N,k_a)^F$ (by construction of $N$, see also Proposition 26, [46]), that is, we have $\operatorname{Ker}(\operatorname{Res})^F=0$ and, therefore, by Lemma 2, $\operatorname{Ker}(\operatorname{Res})=0$. Now suppose that $N\neq R$, then there is a non trivial homomorhism $\xi\colon R/N\to k_a$, and hence $\operatorname{Ker}(\operatorname{Res})^F\neq 0$, a contradiction. Therefore, $N=R$. Definition 12. We say that we are given a simplicial presentation of a prounipotent group $G$ if there exists a punctured profinite space $(\widehat{Y}^*,*)$ such that $G\cong F(\widehat{X},*)/R$, where $R=d_1 \operatorname{Ker} d_0$, is included into the truncated simplicial diagram of prounipotent groups and $d_0(x)=x$, $d_0(y)=1$, $d_1(x)=x$, $d_1(y)=\tau(y)$, $s_0(x)=x$ for $x \in (\widehat{X}^*,*)$, $y \in (\widehat{Y}^*,*)$, where $\tau$ is a homeomorphism (12) of pointed profinite spaces, which identifies $(\widehat{Y}^*,*)$ with the pointed profinite space of normal generators in $R(k)$, that is, $Y^*$ is the dual of some basis $Y$ of $R/[R,F](k)^{\vee}$. Remark 10. As in § 2.1, we can consider a simplicial prounipotent presentation as the second step in the building of the Eilenberg–MacLane complex, we denote it by $F^{(1)}_{\bullet}$, and define the second homotopy group of a prounipotent presentation as $\pi_1F^{(1)}_{\bullet}$. Proposition 8. Let $(X\,|\,R) $ be a prounipotent presentation (1) of a prounipotent group $G$. Then there is a pointed profinite space $(\widehat{Y}^*,*)$ and a convergent map $\tau\colon (\widehat{Y}^*,*)\to R(k), \tau(*)=1$, which identify $(\widehat{Y}^*,*)$ with the pointed profinite subspace of normal generators of $R(k)$. If we assume $d_0(x)=x$, $d_0(y)=1$, $d_1(x)=x$, $s_0(x)=x$, $d_1(y)=\tau(y)$, $y\in (\widehat{Y}^*,*)$, then (13) is a simplicial presentation of $G$, in particular, $G\cong F(\widehat{X}^*,*)/d_1(\operatorname{Ker} d_0)$. Proof. Let $\tau\colon (\widehat{Y}^*,*)\to R(k)\vartriangleleft F(\widehat{X}^*,*)(k)$ be obtained as a restriction to $(\widehat{Y}^*,*)$ of some lifting of the basis $(\widehat{T}^*,0)$ of $R/[R,R](k)$ along the abelianization map $\mu\colon R(k)\to R/[R,R](k)$. It is convergent as mentioned in Remark 8.
Define $d_0(x)=x$, $d_1(x)=x$, $s_0(x)=x$ for $x\in \widehat{X}^*$ and $d_0(y)=1$, $d_1(y)=\tau(y)$ for $y\in \widehat{Y}^*$. The existence of such $d_i$ follows from Lemma 4, since $F(\widehat{X}^*\cup \widehat{Y}^*,*)$ is the direct sum of $F(\widehat{X}^*,*)$ and $F(\widehat{Y}^*,*)$. This proves the proposition. 4.6. SubpresentationsWe now introduce the notion of a subpresentation of a prounipotent simplicial presentation (13). We start with $Y$ accompanied by the isomorphism of $k$-vector spaces $(R/[R,F](k))^{\vee}\cong k^Y$. A subpresentation is constructed from a subset $Y_1\subset Y$ using the embedding $(\widehat{Y_1}^*,*)\hookrightarrow (\widehat{Y}^*,*)$ as the restriction
$$
\begin{equation*}
\tau_1=\tau|_{(\widehat{Y_1}^*,*)}\colon (\widehat{Y_1}^*,*)\to F(\widehat{X}^*,*)
\end{equation*}
\notag
$$
of $\tau$ in Definition 12. We define $R_0(k)$ to be the normal Zariski closure of $\tau_1(\widehat{Y}_1^*)$ in $F(k)$. The following diagram summarizes the above findings: where $\psi$ and $\psi_0$ are induced by the inclusions of the corresponding subgroups $[R,R]\subset[R,F]$ and $[R_0,R_0]\subset[R_0,F]$. $\phi$ is induced by the inclusion $R_0\hookrightarrow R$, and $\chi$ is induced from $\phi$ by the inclusion $[R_0,F]\subset[R,F]$; as shown in Proposition 9, $\xi$ is actually a projection onto a closed subspace. Taking $d'_0$, $d'_1$ as the restrictions of $d_0$, $d_1$ to $(\widehat{Y_1}^*,*)$ in (13), we get a diagram, which is called a subpresentation of the simplicial presentation (13) Proposition 9. If (13) is a given prounipotent presentation, then (15) is a simplicial presentation. Proof. We only need to check that $(R_0/[R_0,F](k))^{\vee}\cong k^{Y_1}$. For this, we only need to note that if $Y_1^*$ are linearly independent in $R/[R,F](k)$, then they are linearly independent in $R_0/[R_0,F](k)$. This proves the proposition. 4.7. Prounipotent (pre-)crossed modulesPro-$p$-crossed modules were extensively studied by Porter [40], and prounipotent crossed modules were introduced and studied in the case of finite type presentations in [33]. The purpose of this section is to extend the previous concepts to prounipotent crossed modules over punctured profinite spaces. We do not divide the presentation into characteristic zero and the pro-$p$-case, since they are completely analogous. A right action of a prounipotent group $G_1$ on a prounipotent group $G_2$ is a natural transformation of the group-valued functors $G_2\times G_1\to G_2$ (see [24], § 2.6). It follows from Yoneda’s lemma that $\mathcal{O}(G_2)$ inherits the left $\mathcal{O}(G_1)$-comodule structure. We always assume, that such an action is prounipotent, that is, there is a system of $G_1$-invariant normal subgroups of finite codimension $\{N_{\lambda}\}$ of $G_2$ and $G_2\cong \varprojlim G_2/N_{\lambda}$, where $G_2/N_{\lambda}$ are unipotent groups. Our definition is equivalent to that of a continuous action (see [40], § 1.3) in the pro-$p$-case, and it follows that, in the pro-$p$-case, the corresponding semidirect product $G_2\leftthreetimes G_1$ is a pro-$p$-group. Lemma 6. Given a prounipotent action of a prounipotent group $G_1$ on a prounipotent group $G_2$, the corresponding semidirect product $G_2\leftthreetimes G_1$ is a prounipotent group. Proof. Suppose that $G_2$ is unipotent. Let $\mu\colon \mathcal{O}(G_2)\to\mathcal{O}(G_1)\otimes\mathcal{O}(G_2)$ define the left $\mathcal{O}(G_1)$-comodule structure on $\mathcal{O}(G_2)$. Since $\mu(\mathcal{O}(G_2))$ is finitely generated as an algebra and any finite set in a Hopf algebra is contained in a finitely generated Hopf subalgebra (see [51], Theorem 3.3), we conclude that $\mu(\mathcal{O}(G_1))\subseteq A\otimes \mathcal{O}(G_2)$, where $A$ is a finitely generated (as algebra) Hopf subalgebra in $\mathcal{O}(G_1)$. Since any Hopf algebra is faithfully flat over its Hopf subalgebra, it follows from Theorem 13.2 in [51] that we have an epimorphism $\operatorname{Spec}(\mathcal{O}(G_1))\to \operatorname{Spec}(A)$, and hence the action is factored via the unipotent group scheme $\operatorname{Spec}(A)$. Therefore, $G_1\leftthreetimes G_2\cong \varprojlim_{W_{\beta} \supseteq A} G_1\leftthreetimes (G_2 / W_{\beta})$. It remains to note that the semidirect product of unipotent groups is a unipotent group (since it contains a finitely generated nilpotent discrete torsion-free dense subgroup Remark 4) and hence $G_1\leftthreetimes G_2$ is a prounipotent group.
In the general case, where $G_2$ is a prounipotent group with a unipotent action of $G_1$, we have the decomposition $G_1\leftthreetimes G_2\cong \varprojlim G_1/N_{\lambda}\leftthreetimes G_2$, where $G_1/N_{\lambda}$ are unipotent, and, therefore, our statement follows from the previous case. This proves the lemma. Example 7. It is worth noting that the action of a prounipotent group on itself by conjugation is always prounipotent. It is also shown in the proof of Proposition 11 below that the natural action of a prounipotent group on its relation module is prounipotent. Definition 13. By a prounipotent pre-crossed module, we mean a triple $(G_2,G_1,\partial)$, where $G_1$, $G_2$ are prounipotent groups, $\partial\colon G_2 \to G_1$ is a homomorphism of prounipotent groups, $G_1$ acts prounipotently on $G_2$ from the right, satisfying, for any $k$-algebra $A$, the identity $\mathrm{CM}\,1)\, \partial(g_2^{g_1}) = g_1^{-1} \partial(g_2) g_1$, where the action is written in the form $(g_2,g_1) \to g_2^{g_1}$, $g_2 \in G_2(A)$, $g_1 \in G_1(A)$. Definition 14. A prounipotent pre-crossed module is called crossed if, for any $k$-algebra $A$, the following additional identity holds for $g_1,g_2 \in G_2(A)$: $\mathrm{CM}\,2)$ $g^{\partial(g_1)}= g_1^{-1} g g_1$. The identity $\mathrm{CM}$ 2) is called the Peiffer identity. Such prounipotent crossed module will be denoted by $(G_2,G_1,\partial)$. Definition 15. A homomorphism $(G_2,G_1, \partial) \to (G'_2,G'_1, \partial')$ of prounipotent crossed modules is a pair of homomorphisms $\varphi\colon G_2 \to G'_2$ and $\psi\colon G_1 \to G'_1$ of prounipotent groups satisfying: a) $\varphi(g_2^{g_1}) =\varphi (g_2)^{\psi(g_1)}$; and b) $\partial' \varphi(g_2) = \psi \partial(g_2)$. Below we will define the notion of a free (pre-)crossed module and show that the prounipotent (pre-)crossed module of (13) is free. Definition 16. A prounipotent (pre-)crossed module $(G_2,G_1,d)$ is called a free prounipotent (pre-)crossed module on a pointed profinite space $(\widehat{Y}^*,1)\in G_2(k)$ if the module $(G_2,G_1,d)$ possesses a universal property with respect to convergent maps $\nu\colon (\widehat{Y}^*,1)\to G'_2(k)$, that is, for any prounipotent (pre-)crossed module $(G'_2,G'_1,d')$ and a convergent map $\nu\colon (\widehat{Y}^*,1)\to G'_2(k)$ and for any homomorphism of prounipotent groups $f\colon G_1\to G'_1$ such that $fd(\widehat{Y}^*,1)=d'(k)\nu(\widehat{Y}^*,1)$, there is a unique homomorphism of prounipotent groups $h\colon G_2\to G'_2$ such that $h(k)(\widehat{Y}^*,1)=\nu(\widehat{Y}^*,1)$ and the pair $(h,f)$ is a homomorphism of (pre-)crossed modules. Proposition 10. Given a simplicial prounipotent presentation (13), $\operatorname{Ker} d_0 \xrightarrow{d_1} F(\widehat{X}^*,*)$ is a free prounipotent pre-crossed module on $(\widehat{Y}^*,1)\subset \operatorname{Ker} d_0(k)\subset F(\widehat{X}^*\cup \widehat{Y}^*,*)(k)$. Proof. Let $A\xrightarrow{\partial} H$ be a prounipotent pre-crossed module. For each convergent map $\nu\colon (\widehat{Y}^*,1) \to A(k)$, we construct a homomorphism $\psi$ in the diagram as follows: where $\widetilde{\partial}(A)=\partial(A)$, $\widetilde{\partial}(H)=\operatorname{id}_H$, $\widetilde{\partial}(a,b)=\partial(a)\cdot b$ for $a\in A(k)$, $b\in H(k)$. The semidirect product $A\leftthreetimes H$ is a prounipotent group, since the action in the pre-crossed module $(A,H,\partial)$ is prounipotent. The fact that $\widetilde{\partial}$ is a homomorphism follows by a direct computation using the fact that, for $\partial$, the property $\mathrm{CM}$ 1) from Definition 13 holds. Indeed, $\widetilde{\partial}((a,b)(a_1,b_1))=\widetilde{\partial}(aa_1,b^{a_1}b_1)=a a_1\cdot\partial( b^{a_1}b_1) =aa_1\, \partial(b^{a_1})\, \partial(b_1)=aa_1a_1^{-1} \, \partial(b)\, a_1\, \partial(b_1)=a\, \partial(b)\, a_1\, \partial(b_1)$.
We define $\psi\colon F(\widehat{X}^*\cup \widehat{Y}^*,*)\to A\leftthreetimes H$ on $k$-points by the rule $\psi(k)(x)=f(k)(x)$ on $x\in \widehat{X}^*$ and $\psi(k)(y)=\nu(y)$ on $y\in \widehat{Y}^*$. Lemma 4 yields a homomorphism of prounipotent groups $\psi\colon F(\widehat{X}^*\cup \widehat{Y}^*,*)\to A\leftthreetimes H$. We define $\varphi\colon \operatorname{Ker}d_0\to A\leftthreetimes H$ by$\varphi=\psi\mid_{\operatorname{Ker} d_0}$, that is, this is the restriction of $\psi$ to $\operatorname{Ker} d_0$. Then the pair $(\phi,f)$ is a pre-crossed module homomorphism. Indeed, let $g_1\in F(\widehat{X}^*,*)$, then there is $\widetilde{g}_1\in F(\widehat{X}^*\cup \widehat{Y}^*,*)$ such that $d_1(\widetilde{g}_1)=g_1$, and hence we have $\phi(g_2^{g_1})= \phi(\widetilde{g}_1^{-1}g_2\widetilde{g}_1)=\phi(\widetilde{g}_1)^{-1}\phi(g_2)\phi(\widetilde{g}_1)$ and, as $A\leftthreetimes H$ is a semidirect product and $\psi|_{F(\widehat{X}^*,*)}=f$, $\phi(\widetilde{g}_1)^{-1}\phi(g_2)\phi(\widetilde{g}_1)=\phi(g_2)^{\phi(\widetilde{g}_1)}=\phi(g_2)^{f(g_1)}$ and a) of Definition 15 holds and it is enough and easy to check b) of Definition 15 for $\widehat{X}^*\cup \widehat{Y}^*$. This proves the proposition. We use the ideas of § 2 to construct prounipotent crossed modules of prounipotent presentations in the form (13). Let $P=\langle a^{d_1(u)}=u^{-1}au\rangle$, where $u,a\in \operatorname{Ker} d_0(k)$, be the normal Zariski (pro-$p$) closure of the Peiffer subgroup and let $C=\operatorname{Ker} d_0/P$. We say that $C \xrightarrow{\overline{d_1}} F$, where $\overline{d_1}$ is induced from $d_1$ and $F$ acts on $\operatorname{Ker} d_0$ by conjugations $a^u=s_0(u)^{-1}as_0(u)$, is a prounipotent crossed module of a prounipotent presentation (13). The prounipotent analog of Brown–Loday lemma (see § 2.1) identifies the commutant $[\operatorname{Ker} d_0$, $\operatorname{Ker} d_1]$ with the image $d^2_2 (NG_2)$ (which is always closed). By Lemma 1, $[\operatorname{Ker} d_0,\operatorname{Ker} d_1]=P$, and so by a crossed module of a prounipotent presentation (13), we also mean the crossed module from Lemma 1 (see also Lemma 3.10 in [33]). For any prounipotent presentation (13), in analogy with Proposition 1, we introduce its second homotopy group, as already suggested in Remark 10, as $u_2(X\,|\,R)=\pi_1(F^{(1)}_{\bullet})$, which is the same as
$$
\begin{equation*}
u_2(X\,|\,R)=\operatorname{Ker}( C(k)\xrightarrow{\overline{d}_1} F(k)).
\end{equation*}
\notag
$$
If $u_2(X\,|\,R)=0$ for a given prounipotent presentation (13), then such presentation is called aspherical, and we are now ready to present the main result of the paper. Theorem 2. Let (13) be an aspherical presentation of a prounipotent group $G \cong F(X)/R$ over the base field $k$ of characteristic zero (or $p \geqslant 2$ is a prime and (13) is a pro-$p$-presentation of a pro-$p$-group $G$) and let (15) be a subpresentation of (13). Then (15) is also aspherical. § 5. Relation modules and aspherical presentationsBy a relation module of a pro-$p$-presentation (1) of a pro-$p$-group $G\cong F/R$, we understand the abelianization $\overline{R}=R/[R,R]$, where $[R,R]$ is the commutant (closed in the pro-$p$-topology), with the conjugation-induced action of $G$ on $\overline{R}$. Relation modules of pro-$p$-groups are known to be $\mathbb{F}_pG$-modules (see [42], § 5.7) and, therefore, they are topological $\mathcal{O}(F)^*$-modules in our terminology. Similar structures were rediscovered in [31] and [33] for prounipotent presentations with finiteness assumptions. Our proof uses a convenient reformulation (Definition 8) of the concept of a topological module in the language of linearly compact vector spaces. We start with the conjugation induced action of the group algebra of a free nilpotent group of finite rank on the corresponding finite-dimensional factor of the relation module, and then complete this action in the $I$-adic topology, obtaining the necessary diagram of linearly compact spaces. Next, it remains to check that the action of the dual algebra of regular functions of a free prounipotent group coincides with that of such a completion. In § 5.1, we prove that relation modules without any assumptions of finiteness are naturally topological modules with conjugation-induced action. In § 5.3, we show that presentation (13) is aspherical if and only if its relation module is free; we also establish a cohomological criterion of asphericity, namely if and only if $\operatorname{cd} G\leqslant 2$. Similar results are known for pro-$p$-groups (see [27], Proposition 7.7) and for proper finite type prounipotent presentations over fields of characteristic zero [30], [35]. In § 5.4, we adapt Tsvetkov’s ideas [49] about pro-$p$-groups of cohomological dimension 2, obtaining a criterion of asphericity for subpresentations; we also replace his suspicious argument in the proof of equivalence (a), (b) and (d), (e) by the spectral sequence argument prepared in § 3.3. The proof of Theorem 2 is given in § 5.5. 5.1. Relation modules of prounipotent presentationsProposition 11. Given $S\lhd F=F(\widehat{X}^*,*)$, a normal subgroup of a free prounipotent group, the rational points $\overline{S}(k)$ of the abelianization $\overline{S}=S/[S,S]$ are naturally endowed with a topological $\mathcal{O}(F)^*$-module structure with conjugation induced action of $F$ on $\overline{S}$. Proof. First, we consider the case when $X$ is finite, then $S(k)$ has a convergent basis [31], say $Z\subset S(k)$, which generates $S(k)$ as a Zariski-normal subgroup and all but finitely many elements of $Z$ lie in the lower series filtration $L_n\leqslant F=F(X)(k)$. In fact, the quotients $L_i/L_{i+1}\cong C_i/C_{i+1}\otimes k$ (see Corollary 3.7, A.3 in [41], where $C_i$ is a lower central filtration of the free discrete group $\Phi(X)$) have finite dimension (see [45], Theorem 1, Chap. I, § 4.6), and hence the quotients $S(k)\cap L_i(k)/S(k)\cap L_{i+1}(k)$ are also of finite dimension and, therefore, taking their lifts in $S(k)$ we get a convergent basis. As a consequence, we see that the quotients $\overline{S}_i(k)=\overline{S(k)/S(k)\cap L_i(k)}$ are in fact finite-dimensional vector spaces over $k$.
We claim that each $\overline{S}_i(k)$ with the conjugation-induced action of $(F/L_i)(k)$ (we take into account that conjugation action of $L_i(k)$ on $\overline{S}_i(k)$ is trivial) is a topological $\mathcal{O}(F/L_i)^*$-module. By Definition 8, this is equivalent to finding a continuous $k$-linear mapping
$$
\begin{equation*}
\mu_i\colon \overline{S}_i(k)\mathbin{\widehat{\otimes}} \mathcal{O}(F/L_i)^*\to \overline{S}_i(k),
\end{equation*}
\notag
$$
arising naturally from the conjugation action and included into the necessary diagrams (10).
By Theorem A.2 and Corollary A.4 in [18], $\Phi(X)/C_i\subset F(k)/L_i(k)$, and hence $\Phi(X)/C_i$ also acts on $S_i(k)=S(k)/S(k)\cap L_i(k)$ by conjugations. We continue this action by linearity to the group ring $k[\Phi(X)/C_i]$:
$$
\begin{equation*}
\widetilde{\mu}_i\colon \overline{S}_i(k)\otimes k[\Phi(X)/C_i]\to \overline{S}_i(k),
\end{equation*}
\notag
$$
given by the rule $r\otimes\sum a_jf_j\mapsto \sum a_jr^{f_j}$, $a_j\in k$, $f_j\in \Phi(X)/C_i$, $r\in \overline{S}_i(k)$.
The required mapping $\mu_i$ will be obtained as a completion of $\widetilde{\mu}_i$. We start from Quillen’s formulas $\mathcal{O}(F/L_i)^*\cong \varprojlim_n k[\Phi(X)/C_i]/I_{\Phi}^n$ and as $\Phi(X)/C_i$ is torsion-free nilpotent, then $\bigcap I_{\Phi}^n=0$, where $I_{\Phi}$ is the augmentation ideal of $k[\Phi(X)/C_i]$, and hence we get the decomposition $\overline{S}_i(k)\cong \varprojlim_n \overline{S}_i(k)/\overline{S}_i(k)I^n_{\Phi}$, where $\overline{S}_i(k)I^n_{\Phi}$ denotes the $k$-linear subspace of $\overline{S}_i(k)$ generated by elements of the form $a\cdot f$, where $a\in \overline{S}_i(k)$, $f\in I^n_{\Phi}$. Observe that $F$ acts trivially on graded quotients $S\cap L_i/S\cap L_{i+1}$ of finite dimension, now we may take the images of their basis under abelianization of $S/S\cap L_i$ to ensure that the quotients $\overline{S/S\cap L_i}$ are of finite dimension. Therefore, since $\overline{S}\cong\varprojlim_i\overline{S/S\cap L_i}$, it is sufficient to check that each of $\overline{S/S\cap L_i}$ can be naturally endowed with the structure of a topological $\mathcal{O}(F/L_i)^*$-module with action induced by conjugation. Let us introduce an auxiliary group $\widetilde{G}=\overline{S}_i(k)\leftthreetimes F(X)/L_i$. By Lemma 6, $\widetilde{G}$ is a unipotent group, and we have well defined, by Quillen’s formulas, the closed subspaces $\overline{S}_i(k)\cdot\widehat{I}^m\subseteq\overline{S}_i(k)$, where $\widehat{I}$ is the augmentation ideal of $\mathcal{O}(F/L_i)^*$. Since $\overline{S}_i(k)$ is of finite dimension, there is $m\in \mathbb{N}$ such that $\overline{S}_i(k)\cdot\mathfrak{I}^m=0$, where $\mathfrak{I}\subset \mathcal{O}(\widetilde{G})^*$ is the augmentation ideal, but $\bigcap\overline{S}_i(k)I^n_{\Phi}\subseteq\bigcap\overline{S}_i(k)\widehat{I}^n\subseteq \bigcap\overline{S}_i(k)\mathfrak{I}^n=0$ and, therefore, $\overline{S}_i(k)I^n_{\Phi}=\overline{S}_i(k)\widehat{I}^n$ for each $n\in \mathbb{N}$. Since $\overline{S}_i(k)/\overline{S}_i(k)I^n_{\Phi}\cong \overline{S}_i(k)/\overline{S}_i(k)\widehat{I}^n$ and $\mathcal{O}(F)^*/\widehat{I}^n\cong k[\Phi(X)/C_i]/I^n_{\Phi}$, for each $n\in \mathbb{N}$ we have the $k$-linear maps of finite dimensional vector spaces
$$
\begin{equation*}
\overline{S}_i(k)/\overline{S}_i(k)\widehat{I}^n\otimes \mathcal{O}(F/L_i)^*/\widehat{I}^n\to \overline{S}_i(k)/\overline{S}_i(k)\widehat{I}^n,
\end{equation*}
\notag
$$
which are completely defined by the conjugation action of $F/L_i$, since by Proposition 3.6 (a) in [41] for $n>i$, the canonical map $\Phi(X)/C_i\to k[\Phi(X)/C_i]/I^n_{\Phi}$ is injective and $\Phi(X)/C_i$ defines this action. Indeed, $(\overline{\widetilde{S}}_i\leftthreetimes \Phi(X)/C_i)^{\wedge}_u(k)\cong \overline{S}_i(k)\leftthreetimes F/L_i(k)$ by Remark 4, where $\overline{\widetilde{S}}(k)$ is a finitely generated dense subgroup generated (as a discrete group) by the same elements as the $k$-vector space $\overline{S}_i(k)$. Observe that the conjugation action of $F/L_i(k)$ on $\overline{S}_i(k)$ coincides with the action defined by completion, and, therefore, the last one, in particular, is independent of the chosen basis $X$ in $F/L_i(k)$. Passing to the inverse limit over $n$, we obtain the required $k$-linear mapping $\mu_i$, which fits into the required diagrams, since the mappings $\widetilde{\mu}_i$ are taken from the associative group ring action with unit and, therefore, $\overline{S}_i(k)$ inherits the structure of a topological $\mathcal{O}(F/L_i)^*$-module. Passing to the inverse limit over $i$, we obtain the desired topological $\mathcal{O}(F)^*$-module structure on $\overline{S}(k)$.
Let $X$ be some chosen basis in $(F/[F,F])^{\vee}$. Then $F=F(\widehat{X^*},*)(k)$. We decompose $S(k)\cong\varprojlim S_{\lambda}(k)$ into the inverse limit of natural projections $S_{\lambda}(k)=\phi_{\lambda}(S)(k)$, where $\phi_{\lambda}\colon F\to F(X_{\lambda}^*,*)$ in order to get the decomposition $\overline{S}=\varprojlim \overline{S}_{\lambda}$. For each $\lambda$, we have $\mathcal{O}(F(X_{\lambda}^*,*))^*$-module structure $\mu_{\lambda}\colon \overline{S}_{\lambda}(k)\mathbin{\widehat{\otimes}} \mathcal{O}(F_{\lambda})^*\to \overline{S}_{\lambda}(k)$ induced by the conjugation action of $F_{\lambda}=F(X_{\lambda}^*,*)(k)$ on $S_{\lambda}(k)$. Since the conjugation action commutes with factorization, we define $\mu=\varprojlim \mu_{\lambda}$, which allows us to define the structure of a $\mathcal{O}(F)^*$-topological module on $\overline{S}(k)$. Proposition 11 is proved. As a byproduct of ideas in the proof, we obtain the following useful corollaries. Corollary 2. Let $M$ and $N$ be topological $\mathcal{O}(G)^*$-modules and let $f\colon M\to N$ be a $G(k)$-homomorphism of linearly compact vector spaces. Then $f$ is a homomorphism of topological $\mathcal{O}(G)^*$-modules. Corollary 3. Let (1) be a prounipotent presentation. Then the linearly compact $k$-vector space $\overline{R}(k) = R/[R,R](k)$ is naturally endowed with a topological $\mathcal{O}(G)^*$-module structure with the action induced by the conjugations of $F(k)$ on $R(k)$. We call such a topological $\mathcal{O}(G)^*$-module the relation module of a prounipotent presentation. Proof. $R\lhd F$ lies in the kernel of the conjugation action of $F$ on $\overline{R}$, and hence the action factors through the action of $G$. Therefore, $\overline{R}(k)$ is a topological $\mathcal{O}(G)^*$-module. This proves the corollary. Proposition 12. Given a prounipotent presentation in the simplicial form (13), $\overline{C}(k)$ inherits the structure of a topological $\mathcal{O}(G)^*$-module. Proof. By Proposition 11, the abelianization $\overline{\operatorname{Ker} d_0}(k)$ of $\operatorname{Ker} d_0$ is an $\mathcal{O}(F)^*$-module. It contains an $\mathcal{O}(F)^*$-submodule $\langle[\operatorname{Ker}d_0,\operatorname{Ker} d_1](k)\rangle$ of $\overline{\operatorname{Ker} d_0}(k)$ generated by the image of $[\operatorname{Ker}d_0,\operatorname{Ker} d_1](k)$ in $\overline{\operatorname{Ker} d_0}(k)$.
Since $\overline{C}(k)\cong \overline{\operatorname{Ker}d_0}(k)/\langle[\operatorname{Ker}d_0,\operatorname{Ker} d_1](k)\rangle$, it follows that $\overline{C}(k)$ is a topological $\mathcal{O}(F)^*$-module. Next, $R(k)$ lies in the kernel of this action. Indeed, let $d_1(x)\in R(k)=d_1(\operatorname{Ker}d_0(k))$, $x,y\in \operatorname{Ker}d_0(k)$. By $\mathrm{CM}$ 2) of Definition 3, we have $x^{-1}yx=y^{d_1(x)}$ and, therefore, $x^{-1}yxy^{-1}=y^{d_1(x)}y^{-1}$, the image of $y^{d_1(x)}y^{-1}$ in $\overline{C}(k)$, is zero. This proves the proposition. Definition 17. Let $G$ be a prounipotent group. We say that a mapping $\mu\colon (\widehat{X}^*,*)\to L$ of a pointed profinite space $(\widehat{X}^*,*)$ into a topological $\mathcal{O}(G)^*$-module $L$ is convergent if there exists a decomposition $L\cong \varprojlim L_{\lambda}$ into the inverse limit of finite dimensional $\mathcal{O}(G)^*$-modules $L_{\lambda}$ such that $\phi_{\alpha}\mu(x)=0$ for all but finite $x\in \widehat{X}^*$, where $\phi_{\lambda}\colon L\to L_{\lambda}$ is the projection. We also call a topological $\mathcal{O}(G)^*$-module $M$ free over the pointed profinite space $(\widehat{X}^*,*)$ ($i\colon (\widehat{X}^*,*)\to M$, $i(*)=0$) and denote it $(\mathcal{O}(G)^*)^{(\widehat{X}^*,*)}$ if it has the following universal property: for any toplogical $\mathcal{O}(G)^*$-module $L$ and a convergent mapping $\mu\colon (\widehat{X}^*,*)\to L$, there exists a homomorphism of topological $\mathcal{O}(G)^*$-modules $\widehat{\mu}\colon M\to L$ such that $\mu=\widehat{\mu}\circ i$. Lemma 7. For any pointed profinite space $(\widehat{X}^*,*)$, there is $M=(\mathcal{O}(G)^*)^{(\widehat{X}^*,*)}$, which is the free topological $\mathcal{O}(G)^*$-module over $(\widehat{X}^*,*)$. Proof. Let $(\widehat{X}^*,*)\,{\cong}\, \varprojlim (\widehat{X_{\lambda}}^*,*)$. It is easy to check that the topological $\mathcal{O}(G)^*$-module $\varprojlim (\mathcal{O}(G)^*)^{(\widehat{X_{\lambda}}^*,*)}$ possesses the required universal properties, its dual $M^{\vee}\cong \mathcal{O}(G)^X$ is an induced $G$-module by Lemma 9. This proves the lemma. Remark 11. By Proposition 15 (below), $H^1(R,k_a)$ is endowed with a structure of $\mathcal{O}(G)$-comodule, and $H^1(R,k_a)$ can be identified with $\overline{R}(k)^{\vee}$ (see Proposition 10 in [35]), and, therefore, by duality, $\overline{R}(k)$ is a topological $\mathcal{O}(G)^*$-module. Such $\mathcal{O}(G)^*$-module structure is the same as $k\langle\langle G\rangle\rangle =\mathrm{End}_G(\mathcal{O}(G)),\mathcal{O}(G))$-module structure on relation modules of [31], [30], since by (5) $k\langle\langle G\rangle\rangle \cong \mathcal{O}(G)^*$ and by Lemma 8(1) in [31], the action of $G(k)$ on $\overline{R}(k)$ is induced by conjugations. It is also proved in Proposition 15 that $H^1(R,k_a)$ is naturally embedded into its injective hull, which is $\mathcal{O}(G)^Y$, and, therefore, by duality as well, we have a natural epimorphism $(\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}\to \overline{R}(k)$ of $\mathcal{O}(G)^*$-modules. Remark 12. Let $C(k)\xrightarrow{\overline{d}_1} F(k)$ be the free prounipotent crossed module of § 4.7. Then $\overline{d}_1$ induces a homomorphism $\partial\colon \overline{C}(k)=(\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}\to \overline{R}(k)$ of topological $\mathcal{O}(G)^*$-modules. The induced mapping on coinvariants $\partial_G\colon \overline{C}(k)_G=k^{(\widehat{Y}^*,*)}\to \overline{R}(k)_G=k^{(\widehat{Y}^*,*)}$ is the “fixed” isomorphism of Remark 9. 5.2. Non-proper presentations and their relation modulesTo reduce non-proper presentations to proper ones, we need the following prounipotent analogue of Propositions 2.2 and 2.4 in [28]. It has a weaker form than [28], since we are not limited to just finitely generated prounipotent groups. Lemma 8. Let $1\to R\to F\xrightarrow{\pi} G\to 1$ be a non-proper prounipotent presentation. Then there is a basis $\widehat{S}^*=\widehat{X}^*\cup \widehat{Z}^*\subseteq F(k)$ such that $\widehat{Z}^*\subseteq R(k)$ and $1\to R_1\to F(X)\xrightarrow{\widehat{\pi}} G\to 1$ is the proper presentation, where $R_1$ is the image of $R$ under the natural epimorphism $F(\widehat{X}^*\cup \widehat{Z}^*)\xrightarrow{\theta} F(\widehat{X}^*)$, which sends $x\mapsto x$ if $x\in \widehat{X}^*$ and $z\mapsto *$ if $z\in \widehat{Z}^*$. Then $\theta$ leads to the exact sequence of topological $\mathcal{O}(G)^*$-modules. Proof. We consider in the case $\operatorname{char}(k)=0$; the arguments in the pro-$p$-case are similar. Since the presentation is not proper, the induced epimorphism of linearly compact topological vector spaces $\widetilde{\pi}\colon F/[F,F](k)\to G/[G,G](k)$ has a non-zero kernel $\operatorname{Ker} \widetilde{\pi}\cong R[F,F]/[F,F](k)\cong R(k)/(R(k)\cap [F,F](k))$ and, therefore, we get the decomposition of topological vector spaces $F/[F,F](k)\cong G/[G,G](k)\oplus R[F,F]/[F,F](k)$. Let $Z$ be a basis of $(R[F,F]/[F,F](k))^{\vee}$ and let $X$ be a basis of $(G/[G,G](k))^{\vee}$. We choose the union uplifts $\widehat{Z}^*$ to $R(k)$ (along the projection $R(k)\to R(k)/(R(k)\cap [F,F](k))$) and $\widehat{X}^*$ in $F(k)$ (in the pro-$p$-case, as always, we use $F^p[F,F]$ and $G^p[G,G]$ instead $[F,F]$ and $[G,G]$ and the uplifts are continuous sections, which exist by Proposition 1 in [46]). By construction, $\widehat{X}^*\cup \widehat{Z}^*$ has the desired properties.
Indeed, for $N=\operatorname{Ker}\theta$ we have an exact sequence $1\to N\to R\to R_1\to 1$ of free prounipotent groups. The corresponding low degree 5-exact sequence in the Lyndon–Hochschild–Serre spectral sequence of $R$-modules takes the form $0\to H^1(R_1,k_a)\to H^1(R,k_a)\to H^1(N,k_a)^R\to 0$, which is equivalent by duality to $0\to \overline{R}_1(k)^{\vee}\to \overline{R}(k)^{\vee}\to (\overline{N}(k)^{\vee})^R\to 0$ and, therefore, by Lemma 3, we get an exact sequence of topological $\mathcal{O}(R)^*$-modules $0\to \overline{N}(k)_{R(k)}\to \overline{R}(k)\to \overline{R_1}(k)\to 0$. By Proposition 10 (with $Y^*=Z^*, d_1(\widehat{Z}^*)=1$ in our current environment), $(N,F(\widehat{X}^*),d_1|_N)$ is a free prounipotent pre-crossed module. Observe that $\operatorname{Ker} d_0=\operatorname{Ker} d_1$ and hence
$$
\begin{equation*}
(C=\operatorname{Ker}d_0/[\operatorname{Ker}d_0,\operatorname{Ker}d_1], F(X), \overline{d}_1)\cong (\overline{N}(k), F(X), \overline{d}_1)
\end{equation*}
\notag
$$
is a free crossed module. Its universal property by Corollary 2 implies that $\overline{N}(k)\cong (\mathcal{O}(F(X))^*)^{(\widehat{Z}^*,*)}$ is a free topological $\mathcal{O}(F(X))^*$-module (cf. Proposition 7 of [9]). Taking coinvariants, we have $\overline{N}(k)_R\cong (\mathcal{O}(G)^*)^{(\widehat{Z}^*,*)}$, inasmuch as $\mathcal{O}(F(X))^{R}\cong\mathcal{O}(F(X))^{R_1}\cong \mathcal{O}(G)$, see [51] for the last isomorphism. This proves the lemma. 5.3. Relation modules and aspherical presentationsIf $G$ is a prounipotent group, we can obtain a precise description of injective hulls. Proposition 13 (see [29], § 1.11). Let $G$ be a prounipotent group, and let $V$ be a $G$-module. Then $V^G\otimes \mathcal{O}(G)$ is an injective $G$-module containing $V$. Each injective $G$-module containing $V$ contains a copy of $V^G\otimes \mathcal{O}(G)$. In view of the previous statement, we can define the injective hull of $V$ by the formula $\mathcal{E}_0(V) = V^G\otimes \mathcal{O}(G)$. Put $\mathcal{E}_{-1}(V)=V$. Let $d_{-1}\colon \mathcal{E}_{-1}(V)\to \mathcal{E}_0(V)$ be the corresponding inclusion. Lemma 9. Let $A$ be a $G$-module over a prounipotent group $G$ and $X$ be a set. Then the following statements are equivalent: A1. $A^*$ is a free topological $\mathcal{O}(G)^*$-module over the space $(\widehat{X}^*,*)$, that is, $A^*\cong (\mathcal{O}(G)^*)^{(\widehat{X}^*,*)}$; A2. $A$ is an induced $G$-module over $X$, that is, $A\cong (\bigoplus_X k_a)\uparrow_1^G$; A3. $H^1(G,A)=0$. Proof. By duality, A1 $\Leftrightarrow$ A2, and also A2 $\Rightarrow$ A3. So, we need to verify A3 $\Rightarrow$ A2.
By Proposition 13, $A$ is a submodule of its injective hull $V=A^G\otimes \mathcal{O}(G)\cong (A^G)\uparrow_1^G$, that is, $A\xrightarrow{j} V= A^G\otimes \mathcal{O}(G)$. Let $C=\operatorname{Coker}(j)$, then we have the exact sequence $0\to A^G\to V^G\to C^G\to H^1(G,A)\to 0$. Since $H^1(G,A)=0$, we have $C^G=0$, and hence $C=0$ and, therefore, $A\cong (A^G)\uparrow_1^G$. This proves the proposition. Proposition 14 (see [29], § 1.12). Let $G$ be a prounipotent group and $V$ be a $G$-module. Define the minimal resolution $\mathcal{E}_i(V)$ and $d_i \colon \mathcal{E}_i(V)\to \mathcal{E}_{i+1}(V)$ inductively, Then $\{\mathcal{E}_i(V), d_i\}$ is an injective resolution of $V$ and $H^i(G, V) = \mathcal{E}_i( V)^G$. Proof. $\mathcal{E}_i(V)=\mathcal{E}_0(\operatorname{Ker} d_i)=(\operatorname{Ker} d_i)^G\otimes \mathcal{O}(G)$. By construction and Proposition 13, $\{\mathcal{E}_i( V), d_i \}$ is an injective resolution, hence $\mathcal{E}_i(V)^G=(\operatorname{Ker} d_i)^G$, and, as $\{\mathcal{E}_i( V)^G\}$ has zero differentials, $H^i(G, V) = \mathcal{E}_i( V)^G$. This proves the proposition. Proposition 15. Given a prounipotent presentation (13) of a pro-unipotent group $G$, then $\operatorname{cd}(G)\leqslant 2$ if and only if there is a non-canonical isomorphism $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$ of right topological $\mathcal{O}(G)^*$-modules. Proof. If $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$, then, clearly, $\operatorname{cd}(G)\leqslant 2$. Conversely, assume that $\operatorname{cd}(G)\leqslant 2$. By continuous duality, the isomorphism $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$ of right topological $\mathcal{O}(G)^*$-modules is equivalent to the isomorphism $Hom_{cts}(\overline{R}(k),k)\cong\mathcal{O}(G)^Y$ of right $\mathcal{O}(G)$-comodules. Proposition 10 in [35] asserts that there is an isomorphism of $\mathcal{O}(G)$-comodules $H^1(R,k_a)\cong Hom(\overline{R},k_a)\cong\overline{R}(k)^{\vee}$. Thus we need to prove the isomorphism of right $\mathcal{O}(G)$-comodules $H^1(R,k_a)\cong \mathcal{O}(G)^Y$.
We first study the case of a minimal presentation; for this, we study the minimal injective $\mathcal{O}(G)$-resolvent of the trivial $\mathcal{O}(G)$-comodule $k_a$. Since $\operatorname{cd}(F)=1$, the minimal $\mathcal{O}(F)$-resolution of $k_a$ takes the form $0\to k_a\to \mathcal{O}(F)\to \mathcal{O}(F)^X\to 0$. We can regard this resolution as the injective $\mathcal{O}(R)$-comodule resolution (see [24], Part I, §§ 4.12, 3.3). Applying the $R$-fixed points functor and taking into account that $R$-fixed points of $\mathcal{O}(F)$ coincide with $\mathcal{O}(G)$ (see [51], § 16.3) we obtain the exact sequence $0\to k_a\to \mathcal{O}(G)\to \mathcal{O}(G)^X\to H^1(R,k_a)\to 0$. It follows from Proposition 13 that the injective hull of $H^1(R,k_a)$ coincides with $ H^1(R,k_a)^G\otimes \mathcal{O}(G)\cong \mathcal{O}(G)^Y$, since $H^2(G,k_a)\cong H^1(R,k_a)^G\cong k^Y$. The composition of the map $\mathcal{O}(G)^X\to H^1(R,k_a)$ with the inclusion of $H^1(R,k_a)$ in its injective hull give rise to minimal injective $\mathcal{O}(G)$-resolution
$$
\begin{equation*}
0\to k_a\to \mathcal{O}(G)\to \mathcal{O}(G)^X\to\mathcal{O}(G)^Y
\end{equation*}
\notag
$$
of the trivial comodule $k_a$. In particular, one has an isomorphism $H^1(R,k_a)\cong \operatorname{Im}(\mathcal{O}(G)^X\to \mathcal{O}(G)^Y)$ of $\mathcal{O}(G)$-comodules. By Proposition 14, $\{\mathcal{E}_i( V)^G\}$ has zero differentials, and hence, since non-trivial modules over prounipotent groups have non-trivial fixed points, the conditions $H^1(R,k_a)\cong\mathcal{O}(G)^Y$ and $\operatorname{cd}(G)= 2$ are equivalent.
If (13) is not proper, then by Lemma 8, we obtain a short exact sequence of topological $\mathcal{O}(G)^*$-modules $0\to (\mathcal{O}(G)^*)^{(\widehat{Z}^*,*)}\to \overline{R}(k)\xrightarrow{\widetilde{\theta}} (\mathcal{O}(G)^*)^{(\widehat{Y\setminus Z}^*,*)}\to 0$, where we identify $Y^*$ with a pseudobasis in $R/[R,F](k)$. Now, the cardinality of $Y$ is the sum of cardinality of a basis of $(R[F,F]/[F,F](k))^{\vee}$ and of the cardinality of a basis of a complement to the embedding $(R[F,F]/[F,F](k))^{\vee}\hookrightarrow (R/[R,F](k))^{\vee}$. This sequence splits along the embedding $(\widehat{Y\setminus Z}^*,*)\hookrightarrow (\widehat{Y}^*,*)$ and, therefore, $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$. It easy to check that $Y\setminus Z$ has the same cardinality as a basis of $(R_1/[R_1,F(X)](k))^{\vee}$, where $R_1$ is obtained as in Lemma 8 from the corresponding proper presentation. Indeed, by construction (in the notation of Lemma 8), $1\to R_1\to F(X)\xrightarrow{\widehat{\pi}} G\to 1$ is a proper presentation and, therefore, $(R_1/[R_1,F(X)](k))^{\vee}\cong H^1(R_1,k_a)^{F(X)}\cong H^2(G,k_a)$. The 5-exact sequence in the Lyndon–Hochschild–Serre spectral sequence of a presentation $1\to R\to F\xrightarrow{\pi} G\to 1$ takes the form $0\to H^1(G,k_a)\xrightarrow{\operatorname{Inf}} H^1(F,k_a)\to H^1(R,k_a)^F\to H^2(G,k_a)\to 0$. By construction, $\operatorname{Coker}(\operatorname{Inf})\cong k^Z$ and, therefore, we have $0\to k^Z\to (R/[R,F](k))^{\vee}\to (R_1/[R_1,F(X)](k))^{\vee}\to 0$, which is the exact sequence of vector spaces. This implies that $k^{Y\setminus Z}\cong (R_1/[R_1,F(X)](k))^{\vee}$ ($Y\setminus Z$ in the sence of cardinality of sets) is the basis of $(R_1/[R_1,F(X)](k))^{\vee}$). Hence (by Lemma 9), $R_1/[R_1,F(X)](k)\cong k^{(\widehat{Y\setminus Z}^*,*)}$. But since $\overline{R_1}(k)$ is free, we obtain $\overline{R_1}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y\setminus Z}^*,*)}$. Proposition 15 is proved. The non-canonicity in the previous Proposition 15 arises in Lemma 8 and is eliminated in the next Proposition 16. Proposition 16. Given a simplicial prounipotent presentation (13) of a prounipotent group $G$, then $u_2(X\,|\,R)=0$ if and only if there is an isomorphism $\overline{R}(k)\cong (O(G)^*)^{(\widehat{Y}^*,*)}$ of topological $\mathcal{O}(G)^*$-modules, that is, $\overline{R}(k)$ is a free topological $\mathcal{O}(G)^*$-module over the pointed profinite space $(\widehat{Y}^*,*)$. Proof. As in § 4.7, we consider the crossed module of a presentation (13). Since $\operatorname{Im}(\overline{d}_1)(k)=R(k)$ is a subgroup of $F(k)$ it is a free prounipotent group (see Corollary 2.10 in [29]). By Lemma 5, $\overline{d}_1(k)$ has a section, say $s$, and, therefore, $C(k)\cong u_2(X\,|\,R)\times sR(k)$. Indeed, $u_2(X\,|\,R)$ is central in $C$, as for $p\in C(k)$, $a\in u_2(X\,|\,R)$ the following equality holds $p^{-1}a^{-1}pa=p^{-1}p^{\partial a} a=1$. But $u_2(X\,|\,R)$ is abelian, and hence $[C(k),C(k)]=[sR(k),sR(k)]$, Therefore, $[C(k),C(k)]\cap u_2(X\,|\,R)=1$, and so, $u_2(X\,|\,R)=\operatorname{Ker}({\overline{C}(k)\to \overline{R}(k)})$. Finally, from Proposition 10 we have $\overline{C}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$ (see also Corollary 3.17 in [33]). If we assume that $u_2(X\,|\,R)=0$, then we have $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$.
Conversely, let $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$ be a free topological module over $(\widehat{Y}^*,*)$. Then its dual $(\overline{R}(k))^{\vee}$ is an injective $G$-module and, therefore, the image of the dual $\overline{d}_1^{\vee}\colon (\overline{R}(k))^{\vee}\to (\overline{C}(k))^{\vee}$ is a direct summand, that is, there is a $G$-module decomposition $(\overline{C}(k))^{\vee}\cong (\overline{R}(k))^{\vee}\oplus B$. By Remark 9, the induced homomorphism on fixed points $(\overline{d}_1^{\vee})^G\colon ((\overline{R}(k))^{\vee})^G\to ((\overline{C}(k))^{\vee})^G$ is an isomorphism, hence $B^G=0$. By Lemma 2, $B=0$ and, therefore, $\overline{d}_1^{\vee}$ is an isomorphism, that is, $u_2(X\,|\,R)=0$. This proves the proposition. 5.4. Tsvetkov’s criterionConsider the diagram (16) of prounipotent groups arising from a prounipotent presentation (13) of a prounipotent group $G$ with $\operatorname{cd}(G)=2$ and its subpresentation (15) of a prounipotent group $G_0$: The inclusion of normal subgroups $R_0\subset R$ of $F=F(X)$ induces a homomorphism of relation modules that is, a homomorphism of topological $\mathcal{O}(G_0)^*$-modules with the module structure of Corollary 3. Since $R_0$ is normal in $F$, $\operatorname{Im}\phi$ is a topological $\mathcal{O}(G)^*$-submodule of $\overline{R}(k)$. We recall that, in the pro-$p$-case, we always assume that $\overline{R}=R/R^p[R,R]$ is a topological $\mathbb{F}_pG$-module (that is, $\mathcal{O}(G)^*= \mathbb{F}_pG$).The short exact sequence of groups $1\to R_0\to R\to H\to 1$ leads to a low degree 5-exact sequence in the Lyndon–Hochschild–Serre spectral sequence
$$
\begin{equation*}
0\to H^1(H,k_a)\xrightarrow{\operatorname{Inf}} H^1(R,k_a)\xrightarrow{\operatorname{\mathrm{Res}}} H^1(R_0,k_a)^{R/R_0}\xrightarrow{\mathrm{Tg}} H^2(H,k_a)\to 0,
\end{equation*}
\notag
$$
where $H^2(R,k_a)=0$ since $R$ is free. The image of the restriction map $\operatorname{Res}\colon H^1(R,k_a)\to H^1(R_0,k_a)$ in the 5-exact sequence coincide with $\operatorname{Im}\phi^{\vee}$ and in particular, since $\operatorname{Im}(\operatorname{Res})\subseteq H^1(R_0,k_a)^{R/R_0}$, $\operatorname{Im}(\operatorname{Res})$ is a $G$-module. Consider two exact sequences of $G$-modules
$$
\begin{equation}
0\to H^1(H,k_a)\xrightarrow{\operatorname{Inf}} H^1(R,k_a)\to \operatorname{Im}\phi^{\vee}\to 0
\end{equation}
\tag{17}
$$
and
$$
\begin{equation}
0\to \operatorname{Im}\phi^{\vee}\to H^1(R_0,k_a)^{R/R_0}\xrightarrow{\mathrm{Tg}} H^2(H,k_a)\to 0.
\end{equation}
\tag{18}
$$
Proposition 17. Given a diagram (16) of prounipotent groups, the following conditions are equivalent: (i) $\operatorname{cd}G_0\leqslant2$, (ii) $\operatorname{cd}H\leqslant1$ and (I) $\operatorname{Im}\phi$ is a free topological $\mathcal{O}(G)^*$-module, (II) $H^0(G,\operatorname{Im}\phi^{\vee})\cong H^0(G_0,H^1(R_0,k_a))$, the map that arises from applying the fixed points functor to (18) is an isomorphism. Proof. First, we show that conditions (I) and (II) are equivalent to:
$(*)$ $H^1(G,\operatorname{Im}\phi^{\vee})=0$; $(**)$ $H^0(G,\operatorname{Im}\phi^{\vee})\cong H^0(G,H^1(R_0,k_a)^{R/R_0})$, the map that arises when applying the fixed points functor to (18) is an isomorphism. For any $G$-module $A$ and a normal subgroup $H\lhd G$, we have $(A^H)^{G/H}=A^G$, taking $R/R_0\lhd G_0$, we obtain $H^0(G_0,H^1(R_0,k_a))\cong H^0(G,H^1(R_0,k_a)^{R/R_0})$ and hence $(**)$ is equivalent to (II). By Lemma 9, (I) is equivalent to $H^1(G,(\operatorname{Im}\phi)^{\vee})= 0$ and, since $\operatorname{Im}\phi^{\vee}\cong (\operatorname{Im}\phi)^{\vee}$, (I) is equivalent to $(*)$. Now we prove that $(*)$ and $(**)$ are equivalent to: (a) $H^1(G,H^1(H,k_a))\to H^1(G,H^1(R,k_a))$ is an epimorphism; (b) $H^2(G,H^1(H,k_a))\to H^2(G,H^1(R,k_a))$ is a monomorphism; (c) $H^0(G,H^2(H,k_a))=0$. This follows from the long exact sequences of cohomology of (17) and (18):
$$
\begin{equation*}
\begin{aligned} \, 0 &\to H^0(G,H^1(H,k_a))\to H^0(G,H^1(R,k_a))\to H^0(G,\operatorname{Im}\phi^{\vee}) \\ &\to H^1(G,H^1(H,k_a))\to H^1(G,H^1(R,k_a))\to \mathbf{H^1(G,\operatorname{Im}\phi^{\vee})} \\ &\to H^2(G,H^1(H,k_a))\to H^2(G,H^1(R,k_a))\to H^2(G,\operatorname{Im}\phi^{\vee})\to \cdots \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, 0 &\to \mathbf{H^0(G,\operatorname{Im}\phi^{\vee})}\to \mathbf{H^0(G,H^1(R_0,k_a)^{R/R_0})}\to H^0(G,H^2(H,k_a)) \\ &\to \mathbf{H^1(G,\operatorname{Im}\phi^{\vee})}\to H^1(G,H^1(R_0,k_a)^{R/R_0})\to H^1(G,H^2(H,k_a))^{R/R_0}\to \cdots. \end{aligned}
\end{equation*}
\notag
$$
The next step is to show that conditions (a), (b), (c) are equivalent to:
(d) $d_2^{1,1}\colon H^1(G,H^1(R,k_a))\to H^3(G,k_a)$ is an epimorphism; (e) $d_2^{2,1}\colon H^2(G,H^1(R,k_a))\to H^4(G,k_a)$ is a monomorphism; (f) $\operatorname{cd}(H)\leqslant 1$. By Lemma 2, (c) is equivalent to $H^2(H,k_a)=0$, and hence, to $\operatorname{cd}H\leqslant1$ (that is, $H$ is free), therefore (c) is equivalent to (f). We arrange (a) and (b) into the commutative diagram where $\operatorname{Inf}^*$ is induced by the inflation map $\operatorname{Inf}\colon H^1(H,k_a)\to H^1(R,k_a)$, $\operatorname{Inf}^*$ coincides with (a), when $p=1$, and with (b), when $p=2$. The maps $\widetilde{d}_2^{p,1}$ and $d_2^{p,1}$ are the differentials in the spectral sequences $\widetilde{E}_2^{p,q}$, $E_2^{p,q}$ of group extensions $1\to H\to G_0\to G\to 1$ and $1\to R\to F\to G\to 1$, respectively. Let us take $G_0$-module resolution (9) of the trivial module $k_a$ and $F$-module resolution (9) of the trivial module $k_a$; these resolutions are also injective $H$ and $R$-module resolutions of the trivial module $k_a$. Hence $\operatorname{Inf}$ is induced on the cohomology by the inclusion of the complexes (see [24], § 6.10)
$$
\begin{equation*}
(k_a\otimes \mathcal{O}(G_0)^{\otimes(q+1)})^H\to (k_a\otimes \mathcal{O}(F)^{\otimes(q+1)})^R.
\end{equation*}
\notag
$$
Let $\gamma_0\colon \widetilde{E}_0^{p,q}\to E_0^{p,q}$ be the induced inclusion of the double complexes
$$
\begin{equation*}
(k_a\otimes \mathcal{O}(G_0)^{\otimes(q+1)})^H\otimes \mathcal{O}(G)^{\otimes p}\to (k_a\otimes \mathcal{O}(F)^{\otimes(q+2)})^R\otimes \mathcal{O}(G)^{\otimes p},
\end{equation*}
\notag
$$
then $\gamma_0$ commute with $d_0^{p,q}$, $\widetilde{d}_0^{p,q}$, $d_1^{p,q}$, $\widetilde{d}_1^{p,q}$ and, therefore, by [23], Chap. VIII, § 9, $\gamma_0$ induces a homomorphism of spectral sequences $\gamma_n\colon \widetilde{E}_n^{p,q}\to E_n^{p,q}$ and, in particular, $\gamma_2\colon H^p(G,H^1(H,k_a))\to H^p(G,H^1(R,k_a))$. By construction, $\gamma_2$ coincides with $\operatorname{Inf}^*$ (an explicit mapping is given in Remark 2) and, therefore, $\widetilde{d}^{p,1}_2=d_2^{p,1}\circ \operatorname{Inf}^*$.
The Lyndon–Hochschild–Serre spectral sequence of the group extension $1\to R\to F\to G\to 1$ degenerates (since $R$, $F$ are free). Hence $d_2^{p,1}$ is an isomorphism and, therefore, conditions (a), (b) are equivalent to (d), (e). Since $H$ is free, the Lyndon–Hochschild–Serre spectral sequence $E_2^{p,q}=H^p(G,H^q(H,k_a))$ of the group extension $1\to H\to G_0\to G\to 1$ also degenerates for $q>1$, and, therefore, $E_3^{p,q}\cong E_{\infty}^{p,q}$. The filtration $E_{\infty}^{p,q}$ of $H^{p+q}(G_0,k_a)$ is as follows:
$$
\begin{equation*}
E_{\infty}^{p,q}\cong gr_pH^{p+q}(G,k_a)=F^pH^{p+q}(G,k_a)/F^{p+1}H^{p+q}(G,k_a)
\end{equation*}
\notag
$$
and, therefore, (d)–(f) are equivalent to $H^3(G_0,k_a)=0$, $\operatorname{cd}H\leqslant 1$, that is, to conditions (i), (ii) of Proposition 17. Proposition 17 is proved. 5.5. Proof of Theorem 2We use the notation of §§ 4.6, 5.4, that is, $R_0(k)=\langle \tau(\widehat{Y}_1^*)\rangle_{F(k)}$, where $Y_1\subset Y$. Proof of Theorem 2. The inclusion of normal subgroups $R_0\subset R$ induces the abelianization homomorphism $\phi\colon \overline{R_0}(k)\to \overline{R}(k)$. Note that $\operatorname{Im}\phi\cong (\mathcal{O}(G)^*)^{(\widehat{Y_1}^*,*)}$ is a free topological module. By Proposition 16, $\overline{R}(k)\cong (\mathcal{O}(G)^*)^{(\widehat{Y}^*,*)}$, and $\operatorname{Im} \phi$ is a closed submodule of $\overline{R}(k)$ generated as a topological $\mathcal{O}(G)^*$-submodule by the image of $(\widehat{Y_1}^*,*)$ in the notation of § 4.6.
We next claim that the isomorphism $H^0(G,\operatorname{Im}\phi^{\vee})\cong H^0(G_0,H^1(R_0,k_a))$ is induced by the exact sequence (18). The last isomorphism is equivalent to the isomorphism $(\operatorname{Im}\phi^{\vee})^{G}\cong H^1(R_0,k_a)^{G_0}$ of $k$-spaces arising from the embedding of (18), and, by continuous duality, to $((\operatorname{Im}\phi^{\vee})^G)^*\cong ((H^1(R_0,k_a))^{G_0})^*$. By Lemma 3 that is equivalent to the isomorphism $(\operatorname{Im}\phi^{\vee})^*_G\cong (H^1(R_0,k_a)^*)_{G_0}$. But $\operatorname{Im}\phi^{\vee}\cong (\operatorname{Im}\phi)^{\vee}$ and so we must check that taking $G$-coinvariants of $\phi$ imply the isomorphism $(\operatorname{Im}\phi)_G\cong R_0/[R_0,F]$. Indeed, $(\operatorname{Im}\phi)_G\cong ((\mathcal{O}(G)^*)^{(\widehat{Y_1}^*,*)}_G)\cong k^{(\widehat{Y_1}^*,*)}$, but $R_0/[R_0,F]\cong k^{(\widehat{Y_1}^*,*)}$ by Proposition 9. We are now ready to apply Proposition 17 (to the effect that $\operatorname{cd} G_0\leqslant 2$) and Propositions 15, 16 in order to show that $(X\,|\,R_0)$ is aspherical. Theorem is proved. § 6. On J.-P. Serre philosophy and subcontractible presentationsTo reveal the reason for the parallelism of discrete and pro-$p$-results, we formulate Proposition 2 in [36], which shows how pro-$p$-groups, together with pro-unipotent groups of characteristic zero, using the arithmetic square, completely determine the homotopy type of the integral Bousfield–Kan completion $\mathbb{Z}_{\infty}X := \mathbb{Z}_{\infty}S_1X$ of a connected $\mathrm{CW}$-complex $X$ of finite type, where $S_1$ is the first Eilenberg subcomplex. Proposition 18. Let $X$ be a connected $\mathrm{CW}$-complex with finite number of cells in each dimension and $\pi$ be a set of prime numbers. Then $\mathbb{Z}_{\infty} X$ is up to homotopy a homotopy pull-back of the arithmetic square as follows: Assume we have a problem regarding some properties on the homotopy type of a finite presentation of a discrete group, and it is known a priori (from some considerations) that these homotopy properties depend only on homological data with $\pi_1$-trivial local systems of coefficients. Note that the homology $H_*(X,R)$ of the original space $X$ is distinguished by the direct summand in the homology $H_*(\mathbb{Z}_{\infty}X,R)$ of $\mathbb{Z}_{\infty}X$ (see [8], Chap. I, Lemma 5.4). By Proposition 18, the homotopy type of $\mathbb{Z}_{\infty}X$ is a homotopy pull-back of the arithmetic square. It is determined by the properties of pro-$p$-presentations and prounipotent presentation over $\mathbb{Q}$ and $\mathbb{Q}_p$. Schematic techniques of this paper demonstrate that if some result is true for pro-$p$-presentations, then it turns out to be true for prounipotent presentations over a field of characteristic zero, and thus the homotopy properties of $\mathbb{Z}_{\infty }X$ are determined by the $p$-adic part. Then J.-P. Serre’s reference is specifically to pro-p-presentations, as defining (co)homological behavior, acquires a rigorous mathematical meaning (interpretation). Note that Proposition 18 could be formulated in an equivalent form using free simplicial groups (for example, simplicial presentations in the 2-dimensional case). Namely, for each connected $\mathrm{CW}$-complex $X$, Kan (see Theorem 6.2 in [25]) constructs a free simplicial group $B_X$ (mentioned in § 2.2) whose non-degenerate free group $(B_X)_n$ generators correspond to $(n+1)$-cells of $X$ and there is a homotopy equivalence $B_X\simeq GS_1X$. This is where the chain of homotopy equivalences $\overline{W}B_X \simeq \overline{W}GS_1X\simeq S_1X$ of connected simplicial sets arises. But the functors $R_{\infty}$ are homotopy invariant, and so in the formulation of Proposition 18 we can always replace $\mathbb{Z}_{\infty}X := \mathbb{Z}_{\infty}S_1X$ with $ \mathbb{Z}_{\infty}\overline{W}B_X$, and we assume this remark in the formulation of the following Theorem 3, proved in Theorem 4.1 of [36] using the methods of this work. The following proposition (see [5], Corollary 4.8) clarifies the importance of finite subcomplexes of contractible 2-complexes and becomes the starting point for [36]. Proposition 19. The Whitehead conjecture is equivalent to the conjecture that the fundamental group of a finite subcomplex of a contractible $2$-complex has rational cohomological dimension at most $2$. We conclude this work with the following result, where $(X\,|\,Y)$ is nothing other than $B_{K(X\,|\,R)}$, and $K(X\,|\,R)$ is the geometric realization of the presentation $(X\,|\,R)$ of the discrete group in the notation of § 2.1. Theorem 3. Let $(X_1\,|\,Y_1)$ be a finite simplicial subpresentation of a simplicial contractible presentation $(X\,|\,Y)$ and $R=k$ be a field of characteristic zero, or $R\,{=}\,\mathbb{F}_p$, or $R=\mathbb{Z}$. Then the Bousfield–Kan completions $R_{\infty}\overline{W}(X_1\,|\,Y_1)$ are aspherical. The author would like to thank Jean-Claude Haussmann for helpful conversations and Nick Gilbert, who recommended reading the advanced article [22]. I would also like to express my deep gratitude to Vladimir Baranovsky for identifying and correcting a number of English usage errors.
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