Free Proalgebraic Groups

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic.


Introduction
Free profinite groups play an important role in the theory of profinite groups and in Galois theory. For example, the Shafarevic conjecture in inverse Galois theory states that the absolute Galois group of the maximal abelian extension Q ab of Q is the free profinite group on a countably infinite set. (See e.g., [Har].) A solvable version of Shafarevic's conjecture was established by Iwasawa in [Iwa53] and a geometric version of Shafarevic's conjecture is known to be true: Theorem 1.1 (Douady, Pop, Harbater). Let k be an algebraically closed field. Then the absolute Galois group of K = k(x), the field of rational functions over k, is the free profinite group on a set of cardinality |K|. This was first proved by Douady in characteristic zero ( [Dou64]) and then later, independently by Pop ([Pop95]) and Harbater ([Har95]) in positive characteristic.
Differential Galois theory ( [vdPS03], [Mag94]) is an extension of classical Galois theory: Instead of polynomials over a field, one considers linear differential equations over a differential field. The differential Galois group of a linear differential equation is a linear algebraic group. Accordingly, the absolute differential Galois group of a differential field is a projective limit of linear algebraic groups, i.e., an affine group scheme or a proalgebraic group.
A basic example of a differential field is the field k(x) with the derivation d dx . No explicit description of the absolute differential Galois group of k(x) is presently known. However, around the turn of the century, Matzat promoted the idea that their should be a differential analog of Theorem 1.1, i.e., a differential version of Shafarevic's conjecture:

Free proalgebraic groups
We introduce free proalgebraic groups through a universal mapping property: The free proalgebraic group on a set X over a field k is a proalgebraic group Γ (X) over k together with a map X → Γ (X)(k) that is universal among all maps satisfying the following property: Every morphism from Γ (X) to an algebraic group maps almost all elements of X to the identity. To establish the existence of free proalgebraic groups we use the tannakian machinery ( [DM82], [Del90]).
As in the theory of profinite groups, where one considers, e.g., pro-solvable groups or pro-p-groups, it is often convenient or interesting to restrict attention to a certain class C of linear algebraic groups, for example the class of all reductive algebraic groups, or the class of all unipotent algebraic groups. As long as the class C satisfies certain closure properties there is a well-behaved notion of a pro-C-group and indeed we introduce free proalgebraic groups in this context, i.e., we define free pro-C-groups.
Our notion of free pro-C-groups encompasses three previously known constructions. Firstly, if C is a class of finite constant algebraic groups corresponding to a class C of finite groups, then a free pro-C-group can be identified with a profinite group, indeed with a free pro-C-group. In this spirit, most of our results contain familiar results about profinite groups as a special case.
Secondly, the free proalgebraic group on a finite set X is the proalgebraic completion of the (abstract) free group on X.
Thirdly, free prounipotent groups have already been introduced in [LM82], where they were studied in connection with the cohomology of unipotent groups and characterized as the prounipotent groups of cohomological dimension one. Their internal structure has further been investigated in [LM83]. Moreover, in [Mag01], a unipotent version of Matzat's conjecture over C has been proved: The differential Galois group of the maximal prounipotent Picard-Vessiot extension of C(x), is a free prounipotent group.
We note that free profinite groups play a distinguished role in the theory of profinite groups ( [RZ10], [FJ08]) and that there are several results about free profinite groups that we do not touch upon in this paper that suggest themselves for a generalization to free proalgebraic groups.

Saturated proalgebraic groups
Roughly speaking, saturated proalgebraic groups behave like universal objects in the category of proalgebraic groups. Our defintion of saturated proalgebraic groups gives a precise meaning to the vague idea of forming the projective limit of all linear algebraic groups, or of forming the projective limit of all linear algebraic groups belonging to a certain class C.
Every proalgebraic group Γ can canonically be written as a projective limit of linear algebraic groups. Indeed, Γ = lim ← − −N Γ /N , where N ranges over all normal closed subgroups of Γ such that Γ /N is algebraic. There is no difficulty in constructing a proalgebraic group Γ such that, up to isomorphism, all linear algebraic groups occur as quotients of Γ . More generally, one may construct a proalgebraic group Γ such that, up to isomorphism, all epimorphisms of linear algebraic groups occur in the canonical projective system for Γ . This property does not determine Γ up to isomorphism. So one may make the stronger requirement, that for every algebraic quotient H of Γ every epimorphism G H of linear algebraic groups occurs, up to isomorphism, in the canonical projective system for Γ , i.e., we ask that the embedding problem Γ 1 1 G / G / G H for Γ has a solution. Even this property does not determine Γ up to isomorphism. However, it turns out that, if k is countable and we require that Γ has countable rank, i.e., Γ is the projective limit of linear algebraic groups, where the projective limit is taken over a countable set, then there exists, up to isomorphism, a unique such Γ . Moreover, if k has characteristic zero, Γ is isomorphic to the free proalgebraic group on a countably infinite set. To summarize, we have: where G is a finite group, is solvable. In fact, we establish a relative version or C-version of Theorem 1.3 so that indeed Iwasawa's result is a special case of ours. Theorem 1.3 is exactly the group theoretic input needed for the proof of the special case of Matzat's conjecture (Theorem 1.2). However, we do provide a version of Theorem 1.3 that applies for base fields of arbitrary cardinality, that we will discuss now. The rank of a proalgebraic group G can be defined as the smallest cardinal κ such that G is the projective limit of linear algebraic groups, taken over a directed set of cardinality κ. A pro-C-group Γ is saturated if for all epimorphisms G H, Γ H of pro-C-groups with rank(H) < rank(Γ ) and rank(G) ≤ rank(Γ ) the embedding problem has a solution. Morally, a proalgebraic group is saturated if it solves as many embedding problems as possible. In Theorem 3.24 we provide several equivalent characterizations of this idea. The general version of Theorem 1.3 states the following: Theorem 1.4 (Theorem 3.42). Let k be a field of characteristic zero and let Γ be a pro-C-group whose rank is such that rank(Γ ) = κ ≥ |k|. Then Γ is the free pro-C-group on a set of cardinality κ if and only if Γ is saturated.
We show by example that the assumption rank(Γ ) ≥ |k| in Theorem 1.4 is necessary.
We conclude this introduction with a brief outline of the article: After introducing some basic terminology and constructions, the existence of free pro-C-groups is established in Section 2 using the tannakian machinery. We also compute some examples of free pro-C-groups. In Section 3 saturated pro-C-groups are introduced and several equivalent characterizations of saturated pro-C-groups are established. We then discuss the questions of existence and uniqueness of saturated pro-C-groups: Saturated pro-C-groups of a fixed rank are unique up to isomorphism. However, their existence can only be guaranteed for a rank at least the cardinality of the base field. We also show that, over a field of characteristic zero, a free pro-C-group Γ on a set of cardinality κ is saturated if rank(Γ ) = κ. Paired with the uniqueness of saturated pro-C-groups, this yields Theorem 1.4.
The author is grateful to Annette Bachmayr, Zoé Chatzidakis, David Harbater, Julia Hartmann, Andy Magid and Anand Pillay for helpful discussions during the preparation of this manuscript.

Free pro-C-groups
The main goal of this section is to establish the existence of free pro-C-groups (Definition 2.18). We first need to introduce some terminology.

Preliminaries and notation
All rings are assumed to be commutative and unital. We work over a fixed base field k throughout. The algebraic closure of k is denoted by k. We assume some familiarity with the theory of linear algebraic groups and affine group schemes. We recall some basic results and notions that will be used throughout the text below. (See e.g., [Mil17], [DG70] or [Wat79].) To avoid endless repetitions of the word affine we use the term algebraic group to mean affine group scheme of finite type over k. In particular, over a field of positive characteristic, an algebraic group need not be reduced. A proalgebraic group is, by definition, an affine group scheme over k. This terminology emphasizes the approach of this paper and is justified by the fact that the proalgebraic groups are precisely the projective limits of algebraic groups. Projective limits are always assumed to be taken over a directed partially ordered set. Projective limits exist in the category of proalgebraic groups and they can indeed be taken pointwise, i.e., (lim for any k-algebra R. A closed subgroup of a proalgebraic group is a closed subgroup scheme. Let G be a proalgebraic group. A closed normal subgroup N of G is called coalgebraic if the quotient G/N is algebraic, i.e., of finite type. The isomorphism theorems from the theory of (abstract) groups hold verbatim for proalgebraic groups.
If X is an affine scheme, we write X red for the underlying reduced scheme. Similarly, if R is a ring, we write R red for the quotient of R by its nilradical.
If X is an affine scheme over k, we write k[X] for the k-algebra of global sections of X, so X = Spec(k[X]). If φ : X → Y is a morphism of affine schemes over k, we denote the scheme-theoretic image of φ by φ(X), i.e., φ(X) is the smallest closed subscheme of Y such that φ factors through the inclusion φ(X) → Y . In terms of ideals, the defining ideal of φ(X) is simply the kernel of the dual map φ * : k[Y ] → k[X]. If φ : G → H is a morphism of proalgebraic groups, then φ(G) is a closed subgroup of H. For a morphism φ : G → H of proalgebraic groups the following conditions are equivalent: • There exists a normal closed subgroup N of G such that H G/N and φ identifies with the canonical map G → G/N .
• For every k-algebra R and h ∈ H(R), there exists a faithfully flat R-algebra S and an element g ∈ G(S) such that g maps to h ∈ H(R) ⊆ G(S) under φ.
If these conditions are satisfied we call φ an epimorphism and indicate this by writing G H. If G H is an epimorphism of algebraic groups, then G(k) → H(k) is a surjective map. (The converse is true if H is smooth.) We will often treat a proalgebraic group G as a functor from the category of k-algebras to the category of groups. For a k-algebra R we then often identify G(R) with Hom(k[G], R), the set of k-algebra homomorphisms from k[G] to R. For a morphism φ : G → H of proalgebraic groups we denote with φ R : G(R) → H(R) the corresponding homomorphism of groups. If confusion is unlikely, we may write φ(g) instead of φ R (g) for g ∈ G(R).
To define the structure of a proalgebraic group on an affine scheme G over k is equivalent to defining the structure of a Hopf algebra on k [G]. The normal closed subgroups of G are in one-to-one correspondence with the Hopf subalgebras of k [G]. In more detail, a normal closed subgroup N of G corresponds to the image of k[G/N ] → k [G], the morphism of Hopf algebras, dual to the canonical map G → G/N . We will usually identify k[G/N ] with this Hopf subalgebra of k [G]. For normal closed subgroups N 1 , N 2 of G, is a finite dimensional vector space over k. The identity component of an algebraic group G is denoted by G o . This notion extends to proalgebraic groups: is exact and fully faithful. Any finite group G can naturally be interpreted as an algebraic group over k. This algebraic group is often called the constant group scheme associated with G, we denote it by G k . The functor G G k from the category of finite groups to the category of algebaic groups is fully faithful and extends to a fully faithful functor from the category of profinite groups to the category of proalgebraic groups: If G = lim ← − −i∈I G i is a projective limit of finite groups G i , then G k = lim ← − −i∈I (G i ) k is a projective limit of finite constant group schemes. The group G k (k) can be identified with G itself. All prime ideals of Spec(k[G k ]) are maximal (and minimal) and the kernel of some morphism k[G k ] → k of k-algebras. Thus Spec(k[G k ]) can also be identified with G. Moreover, the topologies on Spec(k[G k ]) and G agree. There is a one-to-one correspondence between the closed subgroups of G and the closed subgroups of G k . Under this correspondence, the normal open (=closed and of finite index) subgroups of G correspond to the coalgebraic subgroups of G k .

Formations of algebraic groups
To have a well-behaved notion of pro-C-groups, the class C of algebraic groups needs to satisfy certain closure properties that we will now discuss.
We always assume that C is a non-empty class of algebraic groups over k such that any algebraic group isomorphic to an algebraic group in C also belongs to C. A group that belongs to C will also be called a C-group. We call C a formation if it satisfies the following two conditions.
(i) C is closed under taking quotients, i.e., if G H is an epimorphism of algebraic groups and G ∈ C, then H ∈ C.
(ii) C is closed under subdirect products, i.e., if H is a subdirect product of G 1 , G 2 ∈ C, then H ∈ C.
(Recall that a closed subgroup H of G 1 × G 2 is a subdirect product of G 1 and G 2 if the projections H → G i are epimorphisms for i = 1, 2.) Remark 2.1. Condition (ii) above has two equivalent reformulations: (ii)' If G is a proalgebraic group and N 1 , N 2 G are normal closed subgroups such that G/N 1 and G/N 2 are in C, then also G/(N 1 ∩ N 2 ) is in C.
(ii)" If G is an algebraic group and N 1 , N 2 G are normal closed subgroups such that N 1 ∩ N 2 = 1 and G/N 1 , G/N 2 ∈ C, then G ∈ C.
To avoid trivialities we also assume that a formation contains a non-trivial algebraic group.
If C is a formation of (abstract) finite groups ([RZ10, p. 20]), i.e., C is closed under taking quotients and subdirect products, we let C = C k denote the class of all algebraic groups isomorphic to G k for some G in C. Then C is a formation of algebraic groups.
As illustrated by the following example, most of the familiar classes of algebraic groups are formations.
Example 2.2. Note that if C is closed under taking quotients, direct products and closed subgroups, then C is a formation. Therefore the following classes are formations: To be precise, here semisimple and reductive groups are required to be smooth but not necessarily connected. Quotients of semisimple (respectively reductive) algebraic groups are semisimple (respectively reductive). See e.g., [Mil17,Lemma 19.14]. Let us show that in characteristic zero the class of semisimple linear algebraic groups is closed under subdirect products. (The argument for reductive algebraic groups is similar.) Let G be an algebraic group and N 1 , N 2 G closed normal subgroups with N 1 ∩ N 2 = 1 and G/N 1 , G/N 2 semi-simple. Let R(G k ) denote the radical of G k . Since R(G k ) maps into R((G/N 1 ) k ) = 1, we see that R(G k ) is contained in (N 1 ) k . Similarly, R(G k ) ⊆ (N 2 ) k . Thus R(G k ) = 1. Since G is automatically smooth in characteristic zero, it follows that G is semisimple.
The following example shows that over a field of positive characteristic p, the classes of all smooth algebraic groups, all semisimple algebraic groups and all reductive algebraic groups are not formations: The subdirect product H = {(g 1 , g 2 ) ∈ SL 2 × SL 2 | Fr p (g 1 ) = Fr p (g 2 )} of G 1 = SL 2 and G 2 = SL 2 is not smooth. Here, for any 2 × 2 matrix Fr p a b c d = a p b p c p d p The class of connected algebraic groups is also not a formation as shown by the following: the subdirect product {(g 1 , g 2 ) ∈ G 2 m | g 2 1 = g 4 2 } of G 1 = G 2 = G m is not connected. Let C be a class of algebraic groups. A proalgebraic group G is called a pro-C-group if G = lim ← − − G i is a projective limit of C-groups G i , where the transition maps G j → G i (j ≥ i) are epimorphisms. The following lemma shows that, when C is a formation, every pro-C-group can canonically be written as a projective limit of C-groups.   [G]. By (i) the H i 's are C-groups and thus H is a pro-Cgroup.
The following lemma will be useful for constructing free pro-C-groups.
Lemma 2.4. Let G be a proalgebraic group and C a formation of algebraic groups. Then G has a maximal pro-C-quotient, i.e., there exists a pro-C-group π C (G) together with an epimorphism G π C (G) such that for any morphism φ : G → H of proalgebraic groups with φ(G) a pro-C-group, there exists a unique morphism π C (G) → H such that Proof. Let I denote the set of all normal closed subgroups N of G such that G/N ∈ C. We define a partial order on I by N 1 ≤ N 2 if N 1 ⊇ N 2 . Because C is closed under subdirect products (Remark 2.1) we know that N 1 ∩N 2 ∈ C if N 1 , N 2 ∈ C. Thus I is directed. For N 1 ≤ N 2 we have a canonical epimorphism G/N 2 G/N 1 and the collection of all these maps is a projective system. Let π C (G) denote the corresponding projective limit. Then π C (G) is a pro-C-group and k[π C (G)] ⊆ k[G] is the union of all finitely generated Hopf . So φ factors uniquely through G π C (G).
Example 2.5. If G is an algebraic group and C the class of all étale algebraic groups, then π C (G) agrees with π 0 (G) = G/G o , the group of connected components of G ([Mil17, Def. 2.38]).

Coalgebraic subgroups and convergence to one
Let G = lim ← − − G/N be a proalgebraic group, written as the projective limit of the canonical projective system. Then also G(k) = lim ← − − (G/N )(k) and it is possible to topologize G(k) with the limit topology, where the (G/N )(k) are considered with the discrete topology (rather than the Zariski topology). Then {N (k)| N ≤ G is coalgebraic} is a neighborhood basis at 1 for G(k). We have no need to actually use this topology on G(k), but we will introduce some topological language which is reminiscent of this idea.
Definition 2.6. Let G be a proalgebraic group. A set N of coalgebraic subgroups of G is a neighborhood basis at 1 for G, if for every coalgebraic subgroup N of G, there exists N ∈ N with N ≤ N .
Thus a set N of coalgebraic subgroups of G is a neighborhood basis at 1 for G if and only if (i) N is downward directed, i.e., for N 1 , N 2 ∈ N there exists N 3 ∈ N with N 3 ≤ N 1 , N 2 .
(ii) N ∈N N = 1. From a neighborhood basis N at 1 for G we obtain a projective system From Lemma 2.7 we immediately obtain: Proof. As H is algebraic we can chose N = 1 in Lemma 2.7.

Corollary 2.10. Let C be a formation of algebraic groups and G a pro-C-group. If C is closed under taking closed subgroups, then every closed subgroup of G is a pro-C-group.
Proof. Let H be a closed subgroup of G and let N be a neighborhood basis at 1 for G. According to Corollary 2.8 the set {H ∩ N | N ∈ N } is a neighborhood basis at 1 for H.
The free proalgebraic groups that we want to consider are free on a set X such that the map from X to the proalgebraic group has a certain property which is explained in the following definition.
Definition 2.11. Let X be a set, G a proalgebraic group and R a k-algebra. A map ϕ : X → G(R) converges to 1 if for every coalgebraic subgroup N of G almost all elements of X map into N (R), i.e., X ϕ −1 (N (R)) is finite for every coalgebraic subgroup N of G.
For X ⊆ G(R), we say that X converges to 1 if the inclusion map converges to 1. If G is algebraic, it follows that ϕ converges to 1 if and only if almost all elements of X map to 1.

Generating proalgebraic groups
Let G be a proalgebraic group, R a k-algebra and X ⊆ G(R). As the intersection of closed subgroups H of G with X ⊆ H(R) also has this property, there exists a smallest closed subgroup X of G such that X ⊆ X (R). We call it the closed subgroup generated by X. If X = G, we also say that X generates G or that X is a generating set of G.
Lemma 2.14. Let G be a proalgebraic group, R a k-algebra and X ⊆ G(R). If k is perfect and R reduced, then X is reduced (and therefore geometrically reduced).
Proof. Since k is perfect, the underlying reduced subscheme G red of a proalgebraic group G is a closed subgroup of G. (See [Mil17, Cor. 1.39] for the algebraic case or [Gro70, Exposé VI A , Section 0.2] for the general case.) So X red is a closed subgroup of G. It therefore suffices to show that X ⊆ X red (R). If If R is not reduced, X need not be reduced, even when k is perfect. For example, if k is a field of The following example shows that X need not be reduced if k is not perfect, even when R = k.
Example 2.15. Let k be a field of characteristic 2 and a ∈ k such that a There are only four subschemes of H, their defining equations are These are either not subgroups or do not contain X. Thus X = H. Lemma 2.16. Let φ : G → H be a morphism of proalgebraic groups, R a k-algebra and X ⊆ G(R). Then φ( X ) = φ R (X) . In particular, if X = G and φ is an epimorphism, then φ R (X) = H.

Existence of free pro-C-groups
We are now prepared to introduce free proalgebraic groups.
Theorem 2.17. Let X be a set, C a formation and R a k-algebra. There exists a pro-C-group Γ = Γ C R/k (X) and a map ι : X → Γ (R) such that (i) ι converges to 1 and ι(X) is a pro-C-group, (ii) the pair (Γ , ι) is universal, i.e., for every pro-C-group G and every map ϕ : X → G(R) converging to 1 with ϕ(X) a pro-C-group, there exists a unique morphism ψ : Γ → G of proalgebraic groups such that commutes.
Proof. We first show that ι(X) = Γ for any pair (Γ , ι) satisfying (i) and (ii). Let G = ι(X) ≤ Γ and let ϕ : Since ϕ converges to 1 by Lemma 2.13, it follows from commutes. As also id : Γ → Γ has this property, it follows from the uniqueness in (ii) that φ = id. Therefore G = Γ as desired. We will next construct Γ C R/k (X) for the formation C of all algebraic groups. Let F X be the free (abstract) Clearly the cofinite representations of F X are a category. We will show that they indeed are a neutral Tannakian category ( [DM82,Del90]) in a natural way.
For two cofinite representations (V , φ) and (V , φ ), the tensor product V ⊗ k V becomes a cofinite representation by is a morphism of cofinite representations. Therefore the usual associativity constraint on k-vector spaces induces an associativity constraint on the category of cofinite representations. Similarly, we obtain a compatible commutativity constraint. Thus the cofinite representations are a symmetric monoidal category. The identity object 1 is given by the one-dimensional So the cofinite representations of F X are an abelian category.
As k can be identified with the endomorphisms of 1, we conclude that the cofinite representations of F X are a tensor category (in the sense of [Del90, 2.1]).
To conclude the description of our neutral tannakian category we need a fibre functor. We take it to be the forgetful functor ω, that associates the underlying vector space V to a cofinite representations (V , φ).
Let Γ = Aut ⊗ (ω) be the fundamental group of this tannakian category. An element of Γ (R) is, by definition, a family (λ V ), indexed by all cofinite representations, where We have a group homomorphism F X → Γ (R), given by sending g ∈ F X to (φ(g)) (V ,φ) , where (V , φ) ranges over all cofinite representations of F X . We note that Γ is the projective limit of the algebraic groups Aut ⊗ (ω| (V ,φ) ⊗ ), where (V , φ) ⊗ denotes the tensor category generated by the cofinite representation (V , φ), and that Aut ⊗ (ω| (V ,φ) ⊗ ) naturally embeds into GL(V ) as a closed subgroup. The category of cofinite representations of F X is equivalent (via ω) to the category of (finite dimensional) representations of Γ . Explicitly, a cofinite representation (V , φ) corresponds to the canonical projection In particular, the universal element id : Γ → Γ corresponds to the group homomorphism F X → Γ (R).
Let ι : is injective and φ is cofinite, it follows that almost all elements of X map into N (R). Thus ι converges to 1. We next establish the universal property: We claim that for every proalgebraic group G and every map ϕ : X → G(R) converging to 1, there exists a unique morphism ψ : Γ → G such that (2) commutes.
Note that ϕ induces a morphism of groups F X → G(R). Because ϕ converges to 1, every (finite dimensional, k-linear) representation of G induces a cofinite representation of F X . We thus have a tensor functor F : Rep(G) → Rep cf (F X ) = Rep(Γ ), compatible with the fibre functors, where Rep(G) denotes the tannakian category of representations of G and Rep cf (F X ) denotes the category of cofinite representations of F X . By [DM82, Cor. 2.9], the tensor functor F is induced by a unique morphism ψ : Γ → G of proalgebraic groups. By construction, the diagram commutes, then ψ and ψ induce the same tensor functor Rep(G) → Rep cf (F X ) and therefore ψ = ψ (loc. cit.). So the existence of Γ = Γ C R/k (X), is established when C is the formation of all algebraic groups. Let us now assume that C is an arbitrary formation and let (Γ , ι) be as above. We know from Lemma 2.4 that Γ has a maximal pro-C-quotient Γ C R/k (X) = π C (Γ ). Let ι C denote the composition By Lemma 2.12 the map ι C converges to 1. Since ι(X) = Γ (as established in the first paragraph) it follows, using Lemma 2.16, that ι C (X) = π C (Γ ). In particular, ι C (X) is a pro-C-group.
To verify the universal property of (π C (Γ ), ι C ), let G be a pro-C-group and ϕ : X → G(R) a map converging to 1 such that ϕ(X) is a pro-C-group. By the universal property of (Γ , ι), there exists a unique morphism φ : Γ → G such that ϕ = φ R • ι. Using Lemma 2.16, we see that Thus φ(Γ ) is a pro-C-group and by the universal property of (π C (Γ ), π), there exists a unique morphism it follows from the universal property of (Γ , ι) that ψ • π = φ. But then the universal property of (π C (Γ ), π) yields ψ = ψ.
Clearly, the pair (Γ C R/k (X), ι) is uniquely determined (up to isomorphism) by the universal property. Definition 2.18. The proalgebraic group Γ C R/k (X) from Theorem 2.17 is called the free pro-C-group on X/R.
If C is a formation of algebraic groups that is closed under taking subgroups, then the conditions ι(X) is a pro-C-group in (i) and ϕ(X) is a pro-C-group in (ii) of Theorem 2.17 can be dropped (Corollary 2.10).
Remark 2.19. For a pro-C-group to be the free pro-C-group on X/R it suffices that it satisfies the mapping property of Theorem 2.17 for any C-group G.
Proof. Let Γ be a pro-C-group and ι : X → Γ (R) be a map converging to 1 with ι(X) a pro-C-group such that for every C-group G with a map ϕ : X → G(R) converging to 1 with ϕ(X) a C-group, there exists a unique morphism ψ : Let G be a pro-C-group and ϕ : X → G(R) a map converging to 1 with ϕ(X) a pro-C-group. We may write G = lim ← − −i∈I G i as a projective limit of C-groups G i with surjective transition maps G i G j (i ≥ j).
Then the induced maps ϕ i : also converge to 1 (Lemma 2.12) and by Lemma 2.16 ϕ i (X) = π i ( ϕ(X) ) is a C-group. Thus there exist unique morphisms ψ i : Γ → G i with ϕ = (ψ i ) R • ι. By uniqueness the ψ i must be compatible and therefore define a morphism ψ : Γ → G. This is the unique ψ such that ϕ = ψ R • ι.
Remark 2.20. If k is perfect, Γ C (X) is geometrically reduced by Lemma 2.14. If k is not perfect, Γ C (X) need not be reduced. For example, if C is the formation of all algebraic groups over a field of characteristic 2, the non-reduced group H from Example 2.15 would be a quotient of Γ C (X) and therefore Γ C (X) cannot be reduced.
We next consider some examples of free proalgebraic groups. The first three examples explain how our notion of free proalgebraic groups encompasses previously known constructions.
Example 2.21. Let C be a formation of finite groups and let C = C k be the corresponding formation of algebraic groups over k (Section 2.2). Let X be a set and let Γ C (X) be the free pro-C-group on X ([FJ08, Def. 17.4.1]). Then Γ C (X) = (Γ C (X)) k . Since G(k) = G(L) for any field extension L of k and C-group G, one actually has Γ C L/k (X) = (Γ C (X)) k for any field extension L of k. Example 2.22. Let k be an algebraically closed field, X a finite set and C the formation of all algebraic groups. Then Γ C (X) is the proalgebraic completion ([BLMM02, Def. 4]) of the (abstract) free group on X. If k has characteristic zero and X has only one element, one can show that Γ C (X) = G a × D(k × ). See [YZh] or [BLMM02, Example 1].
Example 2.23. Free prounipotent groups over an algebraically closed field of characteristic zero were introduced in [LM82] and further studied in [LM83]. The construction and definition of free prounipotent groups given in [LM82] does not quite agree with ours, however in [LM82, Prop. 2.2] the following mapping property of U (X), the free prounipotent group on a set X (in the sense of [LM82]) is established: for any unipotent algebraic group G, there is a bijection between the set of morphisms U (X) → G with the set of subsets {g x | x ∈ X} of G(k) where g x = 1 for almost all x ∈ X, such that a morphism ψ : U (X) → G corresponds to {ψ(x)| x ∈ X}. It is clear from the construction given in [LM82, Section 2] that the map X → U (X) converges to 1. Thus the mapping property from [LM82, Prop. 2.2] is equivalent to the mapping property of Theorem 2.17 (Remark 2.19). In other words, over an algebraically closed field of characteristic zero, U (X) = Γ C (X), where C is the class of all unipotent algebraic groups.
If G is a unipotent algebraic group over an algebraically closed field k of characteristic zero and if x ∈ G(k) 1, then x G a . (Note that x is abelian and unipotent and therefore isomorphic to G n a for some n, but then necessarily n = 1.) It follows that Γ C (X) = G a if X is a one element set.
If k is not algebraically closed, Γ C (X) is more complicated. Indeed, if k = Q and x = ( Thus G 2 a is a quotient of Γ C (X). Example 2.24. Let k be a field of characteristic zero and C the formation of all abelian unipotent algebraic groups. Thus an algebraic group is in C if and only if it is isomorphic to G n a for some n ([Mil17, Cor. 16.15]). For any set X and k-algebra R we have The kernels of the projections Y G a → Y G a , where Y is a finite subset of Y are a neighborhood basis at 1 for Y G a . Every such kernel contains almost all elements of ι(X). Thus ι converges to one.
Let G be a pro-C-group and ϕ : X → G(R) a map converging to 1. Then G is isomorphic to Z G a for some set Z. For x ∈ X and z ∈ Z we can write ϕ(x) z = b∈B λ(b, x, z)b ∈ R for uniquely determined for any k-algebra S and (s y ) y∈Y ∈ Y G a (S). The above sum is finite because ϕ converges to 1. Then ψ is the unique morphism such that Example 2.25. Let C be the class of all diagonalizable algebraic groups, R a k-algebra and X a set. Then Here ⊕ X R × is the direct sum of |X| copies of the abelian group R × . Proof. Let us define ι : X → D(⊕ X R × )(R) = Hom(⊕ X R × , R × ) as the map that sends x ∈ X to the projection π x : ⊕ X R × → R × that picks out the x-component. We need to check that ι converges to 1.
In general, if M is an abelian group, the inclusion map N = D(M ) → D(M) of a coalgebraic subgroup into D(M), corresponds to a quotient map M M of abelian groups with a finitely generated kernel. So ι converges to 1, because for every finitely generated subgroup A of ⊕ X R × , A ⊆ ker(π x ) for almost all x ∈ X.
Let G = D(M) be a pro-C-group and ϕ : X → G(R) a map converging to 1. We have to find ψ : commutes. This corresponds to finding a morphism ψ : M → ⊕ X R × of abelian groups such that commutes for all x ∈ X. Clearly, the unique solution to this problem is given by ψ(m) = ⊕ x∈X ϕ(x)(m) for m ∈ M. Note that this sum is finite because ϕ converges to 1.
We would like to warn the reader that the formation of free pro-C-groups is not compatible with extension of the base field in the sense that, for example, the free pro-diagonalizable group over k on X can be obtained from the free pro-diagonalizable group over k by base extension from k to k .
We conclude this section with two lemmas that will be needed later. The following lemma is fundamental for showing that certain embedding problems for free pro-C-groups are solvable.
Lemma 2.26. Let k be field of characteristic zero and let G be an algebraic group. Then there exists a finite subset X of G(k) such that G = X .
Proof. A proof, under the additional assumption that k is algebraically closed, can be found in [vdPS03,Lemma 5.13]. But if X generates G k , then X also generates G.
The above lemma fails in positive characteristic, even for smooth groups, for example G a is not finitely generated in positive characteristic.
Lemma 2.27. Let k be a field of characteristic zero, X an infinite set and C a formation. Then Γ C (X) is not an algebraic group.
Proof. By assumption there exists a non-trivial C-group G. Since k has characteristic zero G(k) 1 and we can choose g ∈ G(k) with g 1. Moreover, by Lemma 2.26, we can find g 1 , . . . , g n ∈ G(k) such that G = g 1 , . . . , g n . Pick x 1 , . . . , x n ∈ X and for every finite subset Y of X containing x 1 , . . . , x n define a map . , x n } and ϕ Y (x) = 1 for x ∈ X Y . Then ϕ Y converges to 1 and therefore defines an epimorphism φ Y : Γ C (X) G.
Let {x 1 , . . . , x n } Y 1 Y 2 . . . be a strictly ascending chain of finite subsets of X. Thus N 1 N 2 . . . is a strictly descending chain of closed subgroups of Γ C (X). Hence Γ C (X) cannot be an algebraic group.

Saturated pro-C-groups
The main goal of this section is to characterize free pro-C-groups in terms of embedding problems. To achieve this goal we first study certain pro-C-groups that solve "many" embedding problems, we call these groups saturated.

The rank of a proalgebraic group
As we will show, for any formation C there exists up to isomorphism a unique saturated pro-C-group Γ , once we fix the "size" of Γ and this size is sufficiently big. The goal of this subsection is to make precise this notion of size by introducing the rank of a proalgebraic group.
The rank of a profinite group G can be defined as the smallest cardinal number κ such that G has a set of κ (topological) generators that converges to 1. At first sight, it may seem plausible to define the rank r(G) of a proalgebraic group G, say over an algebraically closed field k, as the smallest cardinal κ such that there exists a subset X of G(k) converging to 1 with X = G. However, this approach has drawbacks and yields pathologies. For example, over a field of positive characteristic, there may not exist such an X. (This is for example the case for G a .) Also, for an infinite index set I, r( i∈I G a ) and r( i∈I G m ) behave quite differently. If k has characteristic zero, then r( i∈I G a ) is infinite. On the other hand, if k has transcendence degree greater or equal to |I|, then r( i∈I G m ) = 1 since (g i ) i∈I is a generator if (g i ) i∈I are algebraically independent over the prime field.
Below we will define the rank rank(G) for any proalgebraic group G. It satisfies the following equalities rank( i∈I G a ) = rank( i∈I G m ) = |I|.
Proposition 3.1. Let G be a proalgebraic group that is not algebraic. Then the following cardinals are equal.

(i) The dimension of k[G] as a k-vector space.
(ii) The smallest cardinal κ such that k[G] can be generated as a k-algebra by κ elements.

(iii) The smallest cardinal κ such that k[G] is the directed union of κ finitely generated Hopf subalgebras.
(iv) The smallest cardinal κ such that there exists a neighborhood basis at 1 for G of cardinality κ.
( Assume that (f i ) i∈I generates k[G] as a k-algebra. For every finite subset J of I let A J be the smallest Hopf subalgebra of k [G] that contains all the f j for j ∈ J. Then A J is finitely generated as a k-algebra ([Wat79, Section 3.3]) and the A J 's form a directed system whose union is k[G]. Thus κ 3 ≤ κ 2 . On the other hand, choosing a finite set of generators for every Hopf subalgebra in a directed system shows that κ 2 ≤ κ 3 . As finitely generated Hopf subalgebras correspond to coalgebraic subgroups we have κ 3 = κ 4 .
If N is a neighborhood basis at 1 for G, then G lim and N i is the kernel of the projection G → G i , then the N i are a neighborhood basis at 1 for G. Thus κ 4 = κ 5 .
We would like to define the rank of a proalgebraic group as the quantity characterized in Proposition 3.1. We note that this notion of rank has nothing to do with the classical notion of rank for algebraic groups defined as the dimension of a Cartan subgroup. For an algebraic group G, the quantities defined in (iii), (iv) and (v) of Proposition 3.1 are all equal to 1, but the quantities defined in (i) and (ii) may be distinct and different from 1. For example, for G = G 2 a , the quantity defined in (i) is |N|, whereas the quantity defined in (ii) is 2. The exact definition of the rank of an algebraic group is largely irrelevant for what follows (as long as it is finite). The notationally most convenient choice is to set it equal to 1 for a non-trivial algebraic group and equal to 0 for the trivial group. We therefore make the following definition.
Definition 3.2. The rank rank(G) of a non-trivial proalgebraic group G is the smallest cardinal κ such that there exists a neighborhood basis at 1 for G of cardinality κ. The rank of the trivial algebraic group is 0.
Thus the rank of a proalgebraic group G is finite if and only if G is algebraic. Let G be an infinite profinite group and let G = G k denote the corresponding proalgebaic group. The cardinality of a neighborhood basis at 1 for G consisting of open normal subgroups does not depend on the neighborhood basis and agrees with the minimal cardinality of any neighborhood basis at 1 for G (not necessarily consisting of groups). This cardinal number is denoted by w 0 (G) in [RZ10]. Since the coalgebraic subgroups of G are in one-to-one correspondence with the open normal subgroups of G, it follows that rank(G) = w 0 (G). In particular, if G is not (topologically) finitely generated, it follows that rank(G) = rank(G). Example 3.3. Let (G i ) i∈I be an infinite family of non-trivial algebraic groups and G = i∈I G i . Then rank(G) = |I|. More generally, for an arbitrary set I let φ i : G i H (i ∈ I) be a family of epimorphisms of proalgebraic groups. Let i∈I (G i H) denote the functor from the category of k-algebras to the category of groups defined by for any k-algebra R. This functor is representable, i.e., a proalgebraic group. Indeed, it is represented by where the direct limit is taken over all finite subsets {i 1 , . . . , i n } of I ordered by inclusion. We call i∈I (G i H) the fibre product of the φ i . If H = 1 this simply reduces to the product. If I = {1, 2} one usually writes G 1 × H G 2 for the corresponding fibre product.
It follows from the above description of the coordinate ring of the fibred product that if κ is an infinite cardinal such that rank(G i ) ≤ κ and |I| ≤ κ, then rank( i∈I (G i H)) ≤ κ.
Example 3.4. If M is a non-trivial abelian group, then rank(D(M)) = |M| unless M is finitely generated, in which case rank(D(M)) = 1.

Lemma 3.5. Let G be a proalgebraic group. If H is a closed subgroup or a quotient of G, then its rank is such that rank(H) ≤ rank(G).
Proof. This is clear if G is algebraic and if G is not algebraic the statement follows from Proposition 3.1, using e.g., characterization (i).

Lemma 3.6. Let φ : G H be an epimorphism of proalgebraic groups with algebraic kernel. Then we have rank(G) = rank(H).
Proof. By Lemma 3.5 it suffices to show that rank(G) ≤ rank(H). By Corollary 2.9 there exists a coalgebraic subgroup M of G such that M ∩ ker(φ) = 1. Let N be a neighborhood basis at 1 for H. We claim that the set N = {φ −1 (N ) ∩ M| N ∈ N } is a neighborhood basis at 1 for G. First of all, note that φ −1 (N ) ∩ M ≤ G is coalgebraic because φ −1 (N ) and M are coalgebraic. Furthermore, N is downward directed because N is and Thus rank(G) ≤ rank(H). The following proposition shows that the rank can also be understood as the minimal length of a subnormal series satisfying certain conditions. Proof. We will show by transfinite induction that rank(G/N λ ) ≤ |λ| for every λ ≤ µ. The initial case λ = 0 is true by definition. If λ = λ + 1 is a successor ordinal, then G/N λ +1 G/N λ has kernel N λ /N λ +1 . Thus rank(G/N λ ) = rank(G/N λ ) ≤ |λ | ≤ |λ| by Lemma 3.6. If λ is a limit ordinal, then rank(G/N λ ) ≤ |λ| by Lemma 3.7.
Our next goal is to determine the rank of a free pro-C-group. The following lemma yields a lower bound.
Lemma 3.9. Let G be a proalgebraic group that is not algebraic and let R be a k-algebra. If X ⊆ G(R) converges to 1, then |X| ≤ rank(G).
Proof. Let N be a neighborhood basis at 1 for G. We have X = N ∈N (X N (R)) (assuming w.l.o.g. 1 X) with X N (R) finite. So |X| ≤ |N |.
The next lemma yields an upper bound for the rank of free pro-C-groups.
Lemma 3.10. Let C be a formation and let X be a set. Then Γ C (X) has at most |k||X| coalgebraic subgroups.
Proof. Let N denote the set of all coalgebraic subgroups of Γ C (X). These all arise as kernels of some epimorphism φ : Γ C (X) H for some C-group H. Such a φ is determined by choosing a finite subset {x 1 , . . . , x n } of X and the images φ k (x 1 ), . . . , φ k (x n ) ∈ H(k). For a fixed H, there are |X||H(k)| choices. There are, up to isomorphism, at most |k| algebraic groups over k. Thus, also allowing H to vary, we see that there are at most |k||X| coalgebraic subgroups.
Combining the previous two lemmas we obtain: Corollary 3.11. Let C be a formation and let X be a set such that Γ C (X) is not algebraic. Then In particular, if |X| ≥ |k|, then rank(Γ C (X)) = |X|.
Both extreme cases of inequality (5) may occur: If X is an infinite set and C is the formation of all abelian unipotent groups over an algebraically closed field of characteristic zero, then rank(Γ C (X)) = |X| by Example 2.24. On the other hand, if C is the formation of all diagonalizable algebraic groups, then rank(Γ C (X)) = |k||X| by Example 2.25.
In some rare cases Γ C (X) can indeed be an algebraic group and then the inequality |X| ≤ rank(Γ C (X)) from Corollary 3.11 may not hold. For example, if k is an algebraically closed field of characteristic zero, X a finite set and C the formation of all abelian unipotent groups, then Γ C (X) = G n a , where n = |X|, by Example 2.24. Also, if k is a field of positive characteristic and C is the formation of all infinitesimal algebraic groups, then Γ C (X) = 1 for any set X (since G(k) = 1 for any infinitesimal algebraic group G). However, if k has characteristic zero and X is infinite, then Γ C (X) is not algebraic by Lemma 2.27. We thus obtain from Corollary 3.11: Corollary 3.12. Let k be a field of characteristic zero, C a formation and X a set with |X| ≥ |k|. Then rank(Γ C (X)) = |X|.
We record one more lemma for later use.
Lemma 3.13. Let C be a formation and let X be an infinite set. If φ : Γ C (X) H is an epimorphism such that rank(H) < |X|, then |X| many elements of X map into ker(φ)(k).

The dimension of a proalgebraic group
For our characterization of saturated pro-C-groups we will also need another measure for the size of a proalgebraic group that is different from the rank. Let G be a proalgebraic group. Then the nilradical a of k[G o ] is a prime ideal and we can define the dimension dim(G) of G as the transcendence degree over k of the field of fractions of k[ If G is an algebraic group this is simply the usual dimension of G. Lemma 3.14. Let G be a proalgebraic group. If H is a closed subgroup or a quotient   Proof. Since I (G H) contains the closed subgroup I ker(α) it suffices to see that dim( I ker(α)) ≥ |I|. But this follows rather directly from the definition.

Embedding problems
We are now prepared to make precise the idea of a proalgebraic group that solves many embedding problems. We also refer to diagram (6) as the embedding problem. If there exists a solution the embedding problem is called solvable. The kernel of (6) is ker(α). If the kernel of (6) is trivial, (6) is called a trivial embedding problem. (In this case, α −1 β is the unique solution.) The embedding problem (6) is algebraic if G (and therefore also H) are algebraic groups. Let C be a formation and assume that Γ is a pro-C-group. Then (6) is a pro-C-embedding problem if G is a pro-C-group. If G is in C, then (6) is a C-embedding problem.
It is instructive to reformulate (6)  Roughly speaking, we are interested in pro-C-groups that solve as many pro-C-embedding problems as possible. There does not exist a pro-C-group Γ such that all pro-C-embedding problems (6) for Γ are solvable. For example, if H = Γ , β is the identity map and (6) is non-trivial, then (6) does not have a solution. Also, if rank(G) > rank(Γ ), there cannot exist an epimorphism Γ → G (Lemma 3.5). It turns out that imposing the restrictions rank(H) < rank(Γ ) and rank(G) ≤ rank(Γ ) is sufficient to allow for the existence of a pro-C-group that solves all such pro-C-embedding problems. Moreover, for every cardinal number κ ≥ |k|, there exists a unique such pro-C-group Γ with rank(Γ ) = κ (Theorems 3.37 and 3.38).
The following theorem summarizes several well-known characterizations of profinite groups that solve many embedding problems. See [RZ10, Section 3.5] or [FJ08, Section 25.1].
Theorem 3.20. Let C be a formation of finite groups and Γ a pro-C-group of infinite rank. We consider nontrivial pro-C-embedding problems The following statements are equivalent: (i) Every pro-C-embedding problem (7) with rank(H) < rank(Γ) and rank(G) ≤ rank(Γ) has a solution.
(iii) Every pro-C-embedding problem (7) with rank(H) < rank(Γ) such that ker(α) is a finite minimal normal subgroup of G has a solution.
To state our generalization of Theorem 3.20 we need two more definitions. , a 1 ⊗ . . . ⊗ a n → φ * i 1 (a 1 ) · · · φ * i n (a n ) is injective. Thus, independence of the φ i , corresponds to linear independence (over k [H] The main goal of this section is to prove the following theorem, which generalizes Theorem 3.20 from finite groups to algebraic groups.

Theorem 3.24. Let C be a formation and Γ a pro-C-group of infinite rank. We consider non-trivial pro-Cembedding problems
The following statements are equivalent: (i) Every pro-C-embedding problem (8) with rank(H) < rank(Γ ) and rank(G) ≤ rank(Γ ) has a solution.
(v) For every C-embedding problem (8)  The proof of Theorem 3.24 will be given at the end of this subsection. It consists of the following implications: In [BHHW,Theorem 3.7] we translate the conditions from Theorem 3.24 to differential embedding problems. The reader may find it instructive to look at the meaning of the above conditions in this context. Definition 3.25. Let C be a formation. A pro-C-group of infinite rank satisfying the equivalent characterizations of Theorem 3.24 is called saturated.

Remark 3.26.
A non-trivial pro-C-group Γ of finite rank, i.e., a non-trivial C-group, cannot satisfy all conditions (i)-(vii) of Theorem 3.24. For example, condition (i) (with H = 1) implies that for any G ∈ C and n ≥ 1, the algebraic group G n is an epimorphic image of Γ . Thus the rank of Γ must be infinite.
It turns out that in characteristic zero, saturated pro-C-groups with rank not less than |k| are free pro-Cgroups in disguise (Theorem 3.42). The following remark explains why we chose the expression "saturated".
Remark 3.27. Let C be a formation of finite groups. There exists a certain first order theory T such that models of T correspond to the class of profinite groups having a certain property (called the Iwasawa property in [Cha98] and [CvdDM82] and the embedding property in [FJ08]). Moreover, a pro-C-group satisfies the equivalent conditions of Theorem 3.20 if and only if it corresponds to a saturated model of T . See [Cha98] and [CvdDM82]. It seems conceivable that there is some first order theory T such that saturated pro-C-groups correspond to saturated models of T . In any case, the abelian groups M, such that D(M) is a saturated pro-diagonalizable group, are exactly the saturated models of a certain first order theory (namely, the theory of divisible abelian groups with infinite p-torsion for every prime p). See Example 3.40.
Remark 3.29. While there is a clear correspondence between (i), (ii) and (iii) of Theorem 3.20 and Theorem 3.24, the other characterizations in Theorem 3.24 (with maybe the exception of (vii)) do not look like (iv) of Theorem 3.20 on the face of it. However, it is not hard to see that (vi) of Theorem 3.24 corresponds to (iv) of Theorem 3.20 in the case when C = C k is a formation of finite constant algebraic groups obtained from a formation C of finite groups.
Indeed, we will show that (vi) of Theorem 3.24 holds for a C k -embedding problem (8) if and only if it has rank(Γ ) solutions: We first note that two solutions of (8) that have the same kernel, only differ by an automorphism of The following three lemmas are a preparation for the proof of (iii)⇒(i) in Theorem 3.24. For a finite affine scheme X over k, let us write |X| for the vector space dimension dim k (k[X]) of the coordinate ring of X, i.e., the number of points of X counted with multiplicities. For clarity of exposition we separated the following lemma from the proof of Lemma 3.31.
Lemma 3.30. Let G be an algebraic group. Then |G/N | is bounded, as N varies over all normal closed subgroups of G with dim(N ) = dim(G).

Proof. In characteristic zero
Here we passed to k to guarantee that the underlying reduced subscheme of a subgroup is a subgroup. Also note that G k /(G o k ) red is a zero dimensional algebraic scheme and therefore is finite. Proof. Assume first that N / N is finite. If there is no normal closed subgroup N of G that strictly contains N and is strictly contained in N , then N / N is an almost minimal normal subgroup of G/ N and we can choose n = 1. So let us assume that such an N exists. In this case, we may choose an N such that |N / N | is maximal (i.e., |N /N | is minimal) and call it N 1 . Then N /N 1 is an almost minimal normal subgroup of G/N 1 . Since |N 1 / N | < |N / N | we may conclude by induction on |N / N |.
Let us now consider the general case. We proceed by induction on dim(N / N ). The base case dim(N / N ) = 0 was just handled above. So we may assume that dim(N / N ) > 0. As above, if there is no normal closed subgroup of G that strictly contains N and is strictly contained in N we may choose n = 1. Otherwise, among all normal closed subgroups N of G that are strictly contained in N and strictly contain N , choose one such that dim(N / N ) is maximal (i.e., dim(N /N ) minimal) and call it N 1 . If dim(N /N 1 ) > 0, then N /N 1 is not finite and an almost minimal normal subgroup of G/N 1 , because a normal closed subgroup of G/N 1 that is strictly contained in N /N 1 must have dimension zero and is therefore a finite subgroup. As dim(N 1 / N ) < dim(N / N ) we conclude by induction on dim(N / N ).
If dim(N /N 1 ) = 0 we proceed as follows: Since |N /N | is bounded, as N varies over all normal closed subgroups of G with N ⊇ N ⊇ N and dim(N /N ) = 0 (Lemma 3.30 applied to G = N / N ), we may choose one such that |N /N | is maximal. Let us call this group N . Since N /N is finite, we know from the first part of this proof that there exists a sequence as in the statement of the lemma. Now if N is a normal closed subgroup of G strictly contained in N and containing N , then dim(N ) < dim(N ) since otherwise the maximality of |N /N | would be contradicted. Considering the normal closed subgroups N ⊇ N of G we may conclude by induction since this case was already treated above.
It is not possible to strengthen "almost minimal" to "minimal" in the above lemma. For example, consider G = G m = N and N = 1. λ is a limit ordinal and (iii) if G has infinite rank and rank(G/N ) < rank(G), then rank(G/N λ ) < rank(G) for λ < µ.

Lemma 3.32. Let G be a proalgebraic group and N a normal closed subgroup. Then there exists an ordinal number µ and a chain
Proof. If N is algebraic, the claim follows from Lemma 3.31. So let us assume that N is not algebraic. Let N denote a neighborhood basis at 1 of G with |N | = rank(G). According to Corollary 2.8 the set N (N ) = {N ∩ N | N ∈ N } is a neighborhood basis at 1 for N . Let µ be the smallest ordinal number such that |µ| = |N (N )|. So we may write N (N ) = {U λ | λ < µ}. We set N 0 = N and for 1 ≤ λ ≤ µ we set N λ = λ <λ U λ . Clearly the N λ 's form a descending chain of normal closed subgroups of G such that N 0 = N and N µ = 1. Moreover N λ /N λ+1 = N λ /(N λ ∩ U λ+1 ) = U λ+1 N λ /U λ+1 is algebraic, because U λ+1 N λ /U λ+1 embeds into N /U λ+1 . If N λ /N λ+1 is not an almost minimal normal closed subgroup of G/N λ+1 , we can use Lemma 3.31 to insert finitely many subgroups between N λ and N λ+1 to achieve (i). On the other hand, (ii) is satisfied by construction.
We are now prepared to prove the implication (iii)⇒(i) in Theorem 3.24.
Proposition 3.33. Let C be a formation and Γ a pro-C-group of infinite rank. We consider pro-C-embedding problems If every pro-C-embedding problem with rank(H) < rank(Γ ) and almost minimal algebraic kernel has a solution, then every pro-C-embedding problem with rank(H) < rank(Γ ) and rank(G) ≤ rank(Γ ) has a solution.
We now work towards a proof of (v)⇒(iii) in Theorem 3.24. Proof. If N 1 , N 2 are normal closed subgroups of a proalgebraic group G, it follows from the universal properties that G/N 1 × G/N 1 N 2 G/N 2 G/ (N 1 ∩ N 2 ). In particular, According to Corollary 2.9, there exists a coalgebraic subgroup N of G with N ∩ ker(α) = 1. So Note that ker(α) is isomorphic to ker(H → H ). If N is a normal closed subgroup of H contained in ker(H → H ). Then N N × H 1 corresponds to a normal closed subgroup of G contained in ker(α).
The following proposition proves (v)⇒(iii) of Theorem 3.24. We next show that (i)⇒(vii): So we assume that a C-embedding problem (8) is given. Let I be a set of cardinality rank(Γ ) and set G = I (G H). Then rank(G ) ≤ rank(Γ ). By assumption the embedding problem For every i ∈ I we can compose φ with the projection π i : G → G that picks out the i-th component, to obtain a solution φ i : Γ G of (8). By construction these solutions are independent. Finally, we show (vii)⇒(vi): So we assume that a C-embedding problem (8) is given together with independent solutions (φ i ) i∈I , where |I| = rank(Γ ). The intersection N of all kernels of all solution to (8) is contained in i∈I ker(φ i ). So the isomorphism Γ / i∈I ker(φ i ) → i∈I (G H) yields an epimorphism Γ /N i∈I (G H). By Lemma 3.5 we have Moreover, if dim(ker(α)) > 0 we have dim(Γ /N ) ≥ dim( i∈I (G H)) ≥ |I| = rank(Γ ) by Lemmas 3.14 and 3.18. But then necessarily dim(Γ /N ) = rank(Γ ), because dim(Γ /N ) ≤ dim(Γ ) ≤ rank(Γ ) (Lemmas 3.14 and 3.16).

Existence of saturated pro-C-groups
We next discuss the question, for which infinite cardinals κ there exists a saturated pro-C-group of rank κ. We will show that the answer is always positive for κ ≥ |k|.
We will need some preparatory results. Two epimorphisms G 1 H, G 2 H of proalgebraic groups are said to be isomorphic if there exists an isomorphism G 1 → G 2 such that commutes. This definition also applies for G 1 = G 2 .
For a formation C we denote by |C| the cardinality of the set of all isomorphism classes of C-groups.
Lemma 3.36. Let κ be an infinite cardinal and let C be a formation such that (ii) every C-group has at most κ normal closed subgroups and (iii) there are, up to isomorphism, at most κ distinct epimorphisms between any two C-groups.
Let H be a pro-C-group with rank(H) ≤ κ. Then the set of isomorphism classes of epimorphisms G H of pro-C-groups with algebraic kernel, has cardinality less or equal to κ.

Such an epimorphism H
H gives rise to an epimorphism H × H H H of pro-C-groups with algebraic kernel. Moreover, by Lemma 3.34, every epimorphism G H of pro-C-groups with algebraic kernel is of this form. Thus there are, up to isomorphism, at most κ · κ = κ epimorphisms G H of pro-C-groups with algebraic kernel.
Theorem 3.37. Let κ be an infinite cardinal. If C is a formation such that (ii) every C-group has at most κ normal closed subgroups and (iii) there are, up to isomorphism, at most κ distinct epimorphisms between any two C-groups, then there exists a saturated pro-C-group of rank κ. In particular, if κ ≥ |k|, there exists a saturated pro-C-group of rank κ for any formation C.
Proof. Let Γ 0 = i G i be a product of C-groups such that every C-isomorphism class occurs exactly κ many times in this product. By (i) and Example 3.3 we have rank(Γ 0 ) = κ. According to Lemma 3.36 there are at most κ isomorphism classes of epimorphisms G Γ 0 of pro-C-groups with algebraic kernel. Let Γ 1 = i (G i Γ 0 ) be a fibred product of epimorphisms of pro-C-groups with algebraic kernel, where every isomorphism class of such epimorphisms occurs exactly κ times. By Lemma 3.36 this fibred product is formed over an index set of cardinality κ. According to Example 3.3 we have rank(Γ 1 ) ≤ κ. Since Γ 0 is a quotient of Γ 1 we actually have rank(Γ 1 ) = κ. We now keep iterating this construction. So Γ 2 would be a fibred product of epimorphisms G i Γ 1 and so on. Let Γ = lim ← − −i∈N Γ i be the projective limit of the Γ i 's. Then Γ is a pro-C-group of rank κ.
We will show that Γ is saturated. To do this we will use condition (vii) of Theorem 3.24. So let ). Then G × H Γ i Γ i is an epimorphism of pro-C-groups with algebraic kernel. By construction of Γ , there is an epimorphism Γ J (G × H Γ i Γ i ), where the product is taken over an index set J of cardinality κ. For j ∈ J let φ j : Γ G be the composition where the map in the middle is the projection onto the j-th factor. Then φ j is a solution to (13) for every j ∈ J. In fact, since the family (φ j ) j∈J of solutions is independent.
Finally, a counting argument shows that (i) Up to isomorphism there are at most |k| algebraic groups.
(ii) An algebraic group has at most |k| normal closed subgroups.
(iii) There are at most |k| epimorphisms between any two algebraic groups.
Thus conditions (i), (ii) and (iii) are satisfied for any C if κ ≥ |k|.
We note that the conditions of Theorem 3.37 are satisfied for any infinite κ if C = C k is a formation of algebraic groups coming from a formation C of finite groups. The conditions are also met for any infinite κ for the class C of all diagonalizable algebraic groups.

Uniqueness of saturated pro-C-groups
The uniqueness of saturated pro-C-groups of a fixed rank follows by a rather standard back and forth argument.
Theorem 3.38. Let C be a formation and let Γ and Γ be saturated pro-C-groups with rank(Γ ) = rank(Γ ). Then Γ and Γ are isomorphic.
So conditions (i), (ii) and (iii) are satisfied for µ.
We continue with two examples of saturated pro-C-groups.
Example 3.39. Let k be a field of characteristic zero and C the formation of all abelian unipotent algebraic groups. (Cf. Example 2.24.) We claim that for any infinite cardinal κ, the proalgebraic group κ G a is the saturated pro-C-group of rank κ.
Proof. It suffices to show that κ G a is saturated. Maybe the clearest way to see this it to observe that the functor V G(V ) with G(V )(R) = Hom k (V , R) for any k-algebra R, is a (contravariant) equivalence of categories from the category of k-vector spaces to the category of pro-C-groups. We note that G(V ) is represented by the symmetric algebra on V and that an epimorphism G(V ) G(W ) of pro-C-groups corresponds to an injection W → V of k-vector spaces. Moreover, for an infinite dimensional k-vector space V , we have rank(G(V )) = dim k (V ). Thus the claim reduces to the following obvious statement: If V is a vector space of dimension κ, V 1 a subspace of V with dim k (V 1 ) < κ and V 1 → V 2 an embedding of vector spaces with dim k (V 2 ) ≤ κ, then V 2 can be embedded into V over V 1 . We next determine the saturated pro-diagonalizable groups.
Example 3.40. Let C be the formation of all diagonalizable algebraic groups and let κ be an infinite cardinal. According to Theorems 3.37 and 3.38 there exist a unique saturated pro-C-group Γ of rank κ. This group Γ is of the form Γ = D(M) for some abelian group M. We would like to determine M. According to Theorem 3.24 and Theorem 3.38 the group M is uniquely characterized by the following properties: