Pro-algebraic and differential algebraic group structures on affine spaces

Piotr Kowalski and Anand Pillay ^{∗}

Abstract We prove the following result, solving a problem raised in [1]: If G is a differential algebraic group whose underlying set is some affine n-space, then G is unipotent. A key result, possibly of independent interest, concerns infinite-dimensional group schemes: a group scheme whose underlying scheme is Spec(K[X_{1}, X_{2}, ....]) (K an algebraically closed field of characteristic 0) is a projective limit of unipotent algebraic groups.

1 Introduction

The work in this paper is motivated by a problem discussed by Buium and Cassidy in [1] concerning the category of differential algebraic groups. Suppose (K, D) is a differentially closed field (of characteristic 0), and G is a group whose underlying set is K^{n} for some n and whose group operation and inverse are given by (sequences of) differential polynomials. (This is what we mean by a differential algebraic group structure on K^{n}.) They ask whether G has to be a linear unipotent differential algebraic group. (A differential algebraic group G is defined there to be unipotent if G has a finite descending normal sequence of differential algebraic subgroups, whose factor groups are D-embeddable in the algebraic group G_{a}. Owing to a result in [9] stating that any differential algebraic group has a D-embedding into some algebraic group, the above definition of unipotence is equivalent to there being a D-embedding of G into some unipotent linear algebraic group.) This is a differential algebraic generalization of what Lazard ([6]) proved for algebraic groups. In any case, Cassidy ([4]) gave a positive answer to the question, in the case that n ≤ 2. Here we give a positive answer in general.

∗Supported by NSF grant DMS-96-96268, and also a grant from the UIUC Research Board

The proof in [9] that any differential algebraic group G D-embeds in some algebraic group, depended on constructing a certain canonical pro-algebraic group G^{∞} from G. If G lives on some affine n-space, then G^{∞} will live on infinite-dimensional affine space. So the main work involves studying such proalgebraic groups. This will be done in section 2. The differential algebraic conclusions will be given in section 3. We give another fast proof of Lazard’s Theorem in section 4.

The reader can look at [2] and [3] for more background on the category of differential algebraic groups. By a “D-homomorphism” we mean a homomorphism between differential algebraic groups in the category of differential algebraic varieties (so given locally by differential rational functions, or equivalently definable in the language of differential fields). Background concerning the model theory of differential fields is provided in [9], and any facts needed concerning linear algebraic groups can be found in [8].

The first author would like to thank Marcin Chalupnik for many inspiring discussions, as well as Jan Dymara and Tadeusz Januszkiewicz for their helpful remarks. The second author would like to thank Alexandru Buium and Phyllis Cassidy for some helpful discussions, and also the MSRI, Berkeley, for their hospitality at the time this work was begun. Finally both authors would like to thank Zoe Chatzidakis and Dave Marker for their helpful comments on a preliminary draft of this paper.

2 Pro-algebraic varieties and groups over C

To begin we work inside a large algebraically closed field K of characteristic 0. The reader can assume this to be C if he or she wishes, and in any case we will assume this later in the section. For the purposes of this paper we will work with rather restrictive notions of pro-algebraic variety and the category of pro-algebraic varieties.

Definition 2.1 (i) By a pro-algebraic variety V we will mean a system (V_{i}, π_{i,j} : i, j ∈ Λ, i ≤ j) where Λ is a countable directed set, V_{i} is a variety for i ∈ Λ, for i ≤ j ∈ Λ, π_{i,j} is a surjective morphism from V_{j} onto V_{i}, and i ≤ j ≤ k implies that π_{j,k}.π_{i,j} = π_{i,k}. (ii) By a point of the pro-algebraic variety V we mean a sequence (a_{i})_{i∈Λ} such that a_{i} ∈ V_{i} for all i and π_{i,j}(a_{j}) = a_{i} whenever i ≤ j. (iii) Let V = (V_{i}, π_{i,j})_{i∈Λ}, W = (W_{i}, ν_{i,j})_{i∈Γ} be pro-algebraic varieties. By a morphism f from V to W we mean a mapping from the set of points of V to the set of points of W which is induced by a family of morphisms f_{j} : V_{g(j)} → W_{j} (j ∈ Γ). What this means is that there should be an ≤- preserving map g from Γ to Λ, and for each j ∈ Γ a morphism f_{j} from V_{g(j)} to W_{j}, such that j ≤ j′ implies ν_{j,j′}.f_{j′} = f_{j}.π_{g(j),g(j′)}, and such that f((a_{i})_{i}) = (f_{j}(a_{g(j)})_{j} (iv) So an isomorphism between pro-algebraic varieties V and W is a bijection f between the sets of points, such that both f and f^{−1} are morphisms. (v). If V = (V_{i}, π_{i,j})_{i∈Λ} and W = (W_{i}, ν_{i,j})_{i∈Γ} are pro-algebraic varieties, then V × W is the pro-algebraic variety (V_{i} × W_{j}, π_{i,i′} × ν_{j,j′})_{(i,j)∈Λ×Γ} where Λ × Γ is given the product ordering. (vi). By a group object in the category of pro-algebraic varieties, we mean a pro-algebraic variety V with morphisms μ : V × V → V (for multiplication) and τ : V → V (for inversion), which equip the set of points of V with a group structure. (vii) By a pro-algebraic group we mean (as usual) a pro-algebraic variety (V_{i}, π_{i,j})_{i} where the V_{i} are algebraic groups and the π_{i,j} are (surjective) homomorphisms.

Remark 2.2 (i) Note the surjectivity (rather than generic surjectivity) requirement in (i). (ii) We are assuming the index set Λ to be countable, so it is easy to see that any proalgebraic variety V will be isomorphic to one whose index set is (ω, ≤) (or a single point). (iii) Note the definitional distinction between a group object in the category of pro-algebraic varieties and a pro-algebraic group. However the latter is clealyr among the former. (iv) We will call a pro-algebraic variety V irreducible if all the V_{i} are irreducible varieties.

(v) We will say that a pro-algebraic variety (V_{i}, π_{i,j}) is defined over k iff all the V_{i} and π_{i,j} are. Similarly for a morphism between pro-algebraic varieties to be defined over k. Assume that (V_{i})_{i} is defined over k. By a generic point of V over k we mean a point (a_{i})_{i} of V such that each a_{i} is a generic point of V_{i} over k. (vi) If the morphism f : V → W is surjective on points, then it follows from the definition that each f_{j} : V_{g(j)} → W_{j} is surjective. (vii) We may omit mention of the surjective morphisms π_{i,j} if the context allows.

The next result was almost proved in [9]. However in that paper we were working birationally, without the formalism of Definition 2.1, which is needed even to state the result.

Proposition 2.3 Any irreducible group object in the category of pro-algebraic varieties is isomorphic (in this category) to an irreducible pro-algebraic group.

Proof. Let V = (V_{i}, π_{i,j})_{i∈Λ}, μ : V × V → V be the group object in the category of pro-algebraic varieties, which we assume to be defined over k. We write a.b for μ(a, b). By a rational map f defined over k from V to a pro-algebraic variety W (also defined over k) we mean data a, b with b ∈ k(a) where a is a generic point in V over k and b some point in W. We write b = f(a) and call f generically surjective if b is a generic point in W over k. We call f birational if k(a) = k(b). It is easy to see that if a, b are generic independent (over k) points of V then so is a.b. By [9], there is an irreducible pro-algebraic group G = (G_{j} : j ∈ Γ) defined over k and a birational map f from V to G such that for generic independent (over k) points a, b ∈ V, f(a.b) = f(a).f(b). For any c ∈ V, let a, b be generic in V over k(c) such that c = a.b, and define f^{∗}(c) = f(a).f(b). It is not difficult to see that f^{∗} is a well-defined abstract group isomorphism between the points of V and those of G, which agrees with f on generic points of V over k. We must show that f^{∗} is an isomorphism in the sense of Definition 2.1 (iv). We first show that f^{∗} is a morphism. Let us fix j ∈ Γ. We must show that for arbitrarily large i ∈ Λ, for all points a of V the value of f_{j}^{∗}(a) depends only on a_{i} and that the induced map from V_{i} onto G_{j} is a morphism of algebraic varieties. Our description of f^{∗} implies that f_{j}^{∗} is type-definable (in the structure (K, +, .)). The compactness theorem yields arbitrarily large i such that f_{j}^{∗}(a) depends only on a_{i} for all points a of V. For any such i we will write the induced map from V_{i} to G_{j} also as f_{j}^{∗}. Let us fix some such i. We can easily find some Zariski-open subset U of V_{i} defined over k such that the restriction of f_{j}^{∗} to U is a morphism. On the other hand, by the definition of V as a group object in the category of pro-algebraic varieties, there is i′ ≥ i, and a morphism P from V_{i′} × V_{i′} → V_{i} defined over k such that for all points a, b of V, (a.b)_{i} = P (a_{i′}, b_{i′}). Choose any c ∈ V_{i′}, and choose d ∈ V_{i′} generic over k(c). Then P (d, c) is a generic point of V_{i} over k so is in U. Thus there is a Zariski open subset U_{1} of V_{i′} containing c such that for all c_{1} ∈ U_{1}, P (d, c_{1}) ∈ U. Let e = f_{j}^{∗}(d). Note now that for any c_{1} ∈ U_{1}, f_{j}^{∗}(c_{1}) = e^{−1}.f_{j}^{∗}(P (d, c_{1}), and thus the restriction of f_{j}^{∗} to U_{1} is a morphism. We have shown that the map f_{j}^{∗} from V_{i′} to G_{j} is a morphism, as required. As j was arbitrary, it follows that f^{∗} : V → G is a morphism. A similar (but easier) argument shows the inverse of f^{∗} to be a morphism. This completes the proof.

Definition 2.4 By A^{∞} we mean the pro-algebraic variety (A^{n} : n = 1, 2, ...) with the natural projections from A^{n} to A^{m} for m ≤ n.

Proposition 2.5 Suppose G = (G_{i})_{i} is an irreducible pro-solvable pro-algebraic group, whose underlying pro-algebraic variety is isomorphic to A^{∞}. Then G is pro-unipotent, namely each G_{i} is linear unipotent.

Proof. Fix i. By Remark 2.2 (vi), there is a surjective morphism from some A^{n} onto G_{i}. The structure of algebraic groups yields a connected linear normal algebraic subgroup H_{1} of G_{i} such that G/H_{1} is an abelian variety. As there are no surjective morphisms from A^{n} onto an abelian variety, G = H_{1} is a (solvable) linear algebraic group. Let H_{2} be the unipotent radical of G (maximal normal unipotent algebraic subgroup). Then G/H_{2} is an algebraic torus (namely isomorphic to G_{m}^{N} for some N, where G_{m} denotes here the multiplicative group of the field). So we obtain a surjective morphism from A^{n} onto G_{m}^{N}, and thus also a surjective morphism onto one factor G_{m}. From the latter we obtain a polynomial map f : K^{n} → K whose image is K \ {0}, which is impossible. Thus G = H_{2} is linear unipotent.

The remainder of this section is devoted to showing:

Proposition 2.6 Suppose G = (G_{i})_{i} is an irreducible pro-algebraic group whose underlying pro-algebraic variety is isomorphic to A^{∞}. Then each G_{i} is solvable (and linear).

We now assume our universal domain K to be the field of complex numbers C. (This can clearly be done as our index sets for pro-algebraic varieties are countable). The advantage is that we can view our varieties (and even pro-algebraic varieties) as topological spaces, and use elementary algebraic topology. The set of complex points of any complex variety V is naturally a topological space. Any morphism f between complex varieties will be a continuous map between the corresponding topological spaces. In particular if V and W are isomorphic as varieties then they are homeomorphic.

We will use some homotopy-theoretic facts about complex algebraic groups to prove Proposition 2.6. We assume familiarity with the homotopy groups π_{n}(X, x) of pointed topological spaces (X, x), for n ≥ 1. We refer the reader to [10]. (Also π_{0}(X, x) is the set of arc-wise connected components of X.) For G a real Lie group with identity element e, π_{n}(G) is by definition π_{n}(G, e) and for n ≥ 1 equals π_{n}(G^{0}) where G^{0} is the connected component of G. We need:

Fact 2.7 (i). Let f : H → G be a surjective continuous homomorphism of Lie groups, then f induces a surjective homomorphism from π_{3}(H) to π_{3}(G). (ii). Let G be a connected complex linear algebraic group (or even a connected complex linear Lie group), with π_{3}(G) = 0. Then G is solvable.

Proof. These are both well-known. We give a brief explanation. For (i): f : H → G is a fibre-bundle, so a fibration, and hence by 7.2.14 of [10], f induces a long-exact sequence ....π_{i+1}(H) → π_{i+1}(G) → π_{i}(L)....π_{0}(H) → π_{0}(G), where L = ker(f). But π_{2} of any Lie group is trivial (this is mentioned on p.82 of [7] for example). So we get surjectivity for π_{3} as required. (ii): Let R be the solvable radical of G (maximal normal solvable connected algebraic subgroup). By (i) π_{3}(G/R) = 0. On the other hand G/R is, if nontrivial, a semisimple complex algebraic group and it is well-known that such a group has nontrivial π_{3}. (For any complex semisimple Lie group H there will be a nontrivial continuous homomorphism of SU(2) (and thus S^{3}) into H, yielding a nontrivial element of π_{3}(H).) Thus G/R is trivial, and G = R is solvable.

Now we can give: Proof of Proposition 2.6. We may assume that G = (G_{i} : i = 1, 2, 3...) with surjective homomorphisms G_{i+1} → G_{i}. As in the proof of 2.5, the G_{i} are linear. We may work over C. Now the underlying pro-algebraic variety of G is isomorphic (in the sense of Definition 2.1) to A^{∞}. Fix i. We can then find a surjective morphism f from some A^{n} onto G_{i} and a surjective morphism g from some G_{j} (j > i) onto A^{n} such that f.g is the given surjective homomorphism from G_{j} onto G_{i}. By Fact 2.7(i) f.g induces a surjective homomorphism (f.g)_{∗} say from π_{3}(G_{j}) onto π_{3}(G_{i}). But (f.g)_{∗} = f_{∗}.g_{∗}, where f_{∗} : π_{3}(G_{j}) → π_{3}(A^{n}) and g_{∗} : π_{3}(A^{n}) → π_{3}(G_{i}) are induced by f, g respectively. But A^{n} = C^{n} has trivial π_{3}. Thus π_{3}(G_{i}) = 0 and by Fact 2.7(ii), G_{i} is solvable.

Corollary 2.8 Let (A^{∞}, .) be a group object in the category of pro-algebraic varieties. Then (A^{∞}, .) is isomorphic, in this category to a projective limit of unipotent algebraic groups.

Proof. By Propositions 2.3, 2.5 and 2.6.

Translated into the language of group schemes this says that a group scheme whose underlying scheme is K[X_{1}, X_{2}, ....] (K an algebraically closed field of characteristic 0) is pro-unipotent.

Remark 2.9 In the proof of 2.5 we concluded pro-unipotence of G = (G_{i})_{i} from the the existence of surjective morphisms from affine spaces to the G_{i}. The assumption that the G_{i} are solvable was necessary, for any semisimple algebraic group H will be the image of some affine space under a morphism: H is generated in finitely many steps by its unipotent subgroups, and this yields a morphism from some finite product of unipotent subgroups onto H.

3 Differential algebraic groups

We will use the results of section 2 to prove:

Theorem 3.1 Let (K, D) be a differentially closed field (of characteristic 0), and let G be a differential algebraic group whose underlying set is K^{n} for some n. Then G is unipotent, namely there is D-embedding of G into some unipotent algebraic group.

Proof. Note that K is an algebraically closed field. We may assume (K, D) to be reasonably saturated, and we treat K also as a universal domain for algeraic geometry.

The meaning of the hypothesis of the theorem is that there are differential polynomials P_{1}, .., P_{n} (over K) in differential indeterminates X_{1}, .., X_{n}, Y_{1}, .., Y_{n} and differential polynomials Q_{1}, .., Q_{n} (over K) in differential indeterminates X_{1}, .., X_{n}, such that for a, b ∈ G, a.b = (P_{1}(a, b), .., P_{n}(a, b)) and a^{−1} = (Q_{1}(a), .., Q_{n}(a)). We can identify any differential polynomial P in differential indeterminates X_{1}, .., X_{r} with an ordinary polynomial P′ over K in indeterminates X_{1}, .., X_{r}, D(X_{1}), .., D(X_{r}), ..., D^{k}(X_{1}), .., D^{k}(X_{r}), ... Note that D operates as a derivation on the ring of such polynomials: by D : K → K as given, and D(D^{s}(X_{i})) = D^{s+1}(X_{i}) and using the Leibniz rule. In particular, each of the given P_{i} “is” an ordinary polynomial P′_{i}, say, over K, in indeterminates X_{1}, ., X_{n}, D(X_{1}), ..., D(X_{n}), ..., D^{m}(X_{1}), .., D^{m}(X_{n}) and Y_{1}, .., Y_{n}, D(Y_{1}), .., D(Y_{n}), ..., D^{m}(Y_{1}), .., D^{m}(Y_{n}), and similarly for the Q_{i}. (We choose m large enough). Notationally, we introduce indeterminates X_{i}^{j} for i, j < ω to represent D^{j}(X_{i}). We construct a canonical group structure, which we call G^{∞} on A^{∞}(K). First we will enumerate the coordinates of a point x ∈ A^{∞} as (x_{1}^{0}, .., x_{n}^{0}, x_{1}^{1}, .., x_{n}^{1}, ..., x_{1}^{r}, .., x_{n}^{r}, ...) . We call x_{i}^{j} the (i, j)th coordinate of x. Let the operation ⋆ on A^{∞} be defined by: (x ⋆ y)_{i}^{j} = D^{j}(P′_{i})(x, y).

Claim 1. ⋆ is a group operation on A^{∞}. Moreover the (i, j)th coordinate of the inverse of x is D^{j}(x_{i}). Proof. We just point out associativity of ⋆, leaving the rest to the reader. Let k be a small differential subfield of K over which the P_{i}, Q_{i} are defined. Let a = (a_{1}, .., a_{n}), b = (b_{1}, .., b_{n}), c = (c_{1}, .., c_{n}) be D-generic D-independent (over k) elements of G. This means that the set {D^{j}(a_{i}) : i = 1, .., n, j = 0, 1, 2, ...} ∪ {D^{j}(b_{i}) : i = 1, .., n, j = 0, 1, 2, ..} ∪ {D^{j}(c_{i}) : i = 1, .., n, j = 0, 1, 2, ...} is algebraically independent over k. As (a · b) · c = a · (b · c), it follows that ((D^{j}(a_{i}))_{i,j}) ⋆ (D^{j}(b_{i}))_{i,j}) ⋆ (D^{j}(c_{i}))_{i,j} = (D^{j}(a_{i}))_{i,j} ⋆ ((D^{j}(b_{i}))_{i,j} ⋆ (D^{j}(c_{i}))_{i,j}). As the latter is expressed by a system of polynomial equations over k, the algebraic independence implies that (x ⋆ y) ⋆ z = x ⋆ (y ⋆ z) for all x, y, z ∈ A^{∞}.

Claim 2. ⋆ is a morphism from A^{∞} × A^{∞} to A^{∞} (in the category of proalgebraic varieties). Similarly for inversion. Proof. By definition. (Note that D^{j}(P′_{i}) is a polynomial in indeterminates X_{1}^{0}, .., X_{n}^{0}, .., X_{1}^{m+j}, .., X_{n}^{m+j}, Y_{1}^{0}, .., Y_{n}^{0}, .., Y_{1}^{m+j}, .., Y_{n}^{m+j}.)

Thus G^{∞} is a group object in the category of pro-algebraic varieties.

Claim 3. The map taking (a_{1}, .., a_{n}) to (a_{1}, ..., a_{n}, D(a_{1}), .., D(a_{n}), .., D^{j}(a_{1}), .., D^{j}(a_{n}), ...) is an (abstract) group embedding of G into G^{∞} Proof. By construction of ⋆.

By Corollary 2.8, G^{∞} is isomorphic to a projective limit (G_{i})_{i} of unipotent linear algebraic groups. The D-embeddability of G in some unipotent group follows as in [9], but we recall the argument. We have (homo)morphisms h_{i} : G^{∞} → G_{i}, such that the intersection of all the ker(h_{i}) is trivial. Composing with the embedding given by Claim 3, gives us homomorphisms f_{i} : G → G_{i}, with ⋂_{i} ker(f_{i}) trivial. Each f_{i} is easily seen to be a D-homomorphism, and so the ker(f_{i}) are differential algebraic subgroups of G. By the DCC on differential algebraic subgroups, some f_{i} is already an embedding, and we finish.

4 Remark’s on Lazard’s Theorem

Lazard showed in [6] that an algebraic group G (in any characteristic) whose underlying variety is some affine n-space is nilpotent, answering a question of Cartier. In [1] Lazard is quoted as proving that G is unipotent. For characteristic 0 this latter result follows from our results in section 2. We will sketch another proof of this, valid in all characteristics, and with a model-theoretic flavour (reminiscent of Ax’s proof that injectivity implies surjectivity for a morphism of a variety into itself). Like Lazard’s proof, it depends on counting points over finite fields. Let us fix our universal domain as K, an algebraically closed field of arbitrary characteristic.

Proposition 4.1 Let G be an algebraic group whose underlying variety is A^{n} for some n. Then G is a unipotent linear algebraic group.

Proof. Note that G is connected and linear. Let R be the solvable radical of G. So we have a surjective homomorphism f : G → H with kernel R, such that H is, if nontrivial, a semisimple linear algebraic group. We show that H is trivial. Suppose not. Then H contains a 1-dimensional algebraic torus T, namely an algebraic subgroup T isomorphic via some h to (K^{∗}, .). We can find a formula ϕ(x, y) in the language of fields, such that for some b ∈ K, ϕ(x, b) defines (G, H, f, T, h), and such that for any algebraically closed field K_{1}, and b_{1} ∈ K_{1}, ϕ(x, b_{1}), if consistent, defines a surjective homomorphism f_{1} : G_{1} → H_{1} between connected linear algebraic groups G_{1}, H_{1}, and an isomorphism h_{1} between the algebraic subgroup T_{1} of G_{1} and K_{1}^{∗} such that ker(f_{1}) is connected and solvable, and the underlying variety of G_{1} is K_{1}^{n}. The sentence ∃x∃yϕ(x, y) is true in K so a consequence of T h(K). Using compactness if necessary, this sentence is true in F̄_{p} for some prime p. So there are G_{1}, H_{1}, f_{1}, T_{1}, h_{1} as above, all defined over a finite field F_{p^{m}} say. For each finite field F_{q} containing F_{p^{m}}, G_{1}(F_{q}) has cardinality q^{n}. On the other hand for each such F_{q} there is q′ > q such that H_{1}(F_{q}) is contained in f_{1}(G_{1}(F_{q′})). It follows that every element of H_{1}(F̄_{p}) has order a power of p. Thus every element of T_{1}(F̄_{p}) has order a power of p. But the latter group is isomorphic (via h_{1}) to the multiplicative group of F̄_{p}, and we have a contradiction. So H is trivial, and G = R. R is the semidirect product of its unipotent radical R_{u} and an algebraic torus T. As above, T is trivial. So G is unipotent. The proof is complete.

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