Algebraic Patching

Let E be a field, G a finite group, and {G _i |i∈I} a finite set of subgroups of G with G=〈G _i |i∈I〉. For each i∈I we are given a Galois extension F _i of E with Galois group G _i . We suggest a general method how to ‘patch’ the given F _i ’s into a Galois extension F with Galois group G (Lemma 1.1.7). Our method requires extra fields P _i , all contained in a common field Q and satisfying certain conditions making \(\mathcal{E}=(E,F_{i},P_{i},Q;G_{i},G)_{i\in I}\) into ‘patching data’ (Definition 1.1.1). The auxiliary fields P _i in this data substitute, in some sense, analytic fields in rigid patching and fields of formal power series in formal patching.

If in addition to the patching data, E is a Galois extension of a field E ₀ with Galois group Γ and Γ ‘acts properly’ (Definition 1.2.1) on the patching data \(\mathcal{E}\), then we construct F above to be a Galois extension of E ₀ with Galois group isomorphic to Γ⋉G (Proposition 1.2.2).

Norms ‖⋅‖ of associative rings are generalizations of absolute values |⋅| of integral domains, where the inequality ‖xy‖≤‖x‖⋅‖y‖ replaces the standard multiplication rule |xy|=|x|⋅|y|. Starting from a complete normed commutative ring A, we study the ring A{x} of all formal power series with coefficients in A converging to zero. This is again a complete normed ring (Lemma 2.2.1). We prove an analog of the Weierstrass division theorem (Lemma 2.2.4) and the Weierstrass preparation theorem for A{x} (Corollary 2.2.5). If A is a field K and the norm is an absolute value, then K{x} is a principal ideal domain, hence a factorial ring (Proposition 2.3.1). Moreover, Quot(K{x}) is a Hilbertian field (Theorem 2.3.3). It follows that Quot(K{x}) is not a Henselian field (Corollary 2.3.4). In particular, Quot(K{x}) is not separably closed in K((x)). In contrast, the field K((x))₀ of all formal power series over K that converge at some element of K is algebraically closed in K((x)) (Proposition 2.4.5).

Starting from a complete valued field (K,| |), we choose an element r∈K ^×, a finite set I, and for each i∈I an element c _i ∈K such that |r|≤|c _i −c _j | if i≠j. Then we set \(w_{i}={r\over x-c_{i}}\), with an indeterminate x, and consider the ring R=K{w _i |i∈I} of all series with a ₀,a _in ∈K such that for each i the element a _in tends to 0 as n→∞. The ring R is complete under the norm defined by ‖f‖=max  _i,n (|a ₀|,|a _in |) (Lemma 3.2.1). We prove that R is a principal ideal domain (Proposition 3.2.9) and denote its quotient field by P. More generally for each subset J of I, we denote the quotient field of K{w _i |i∈J} by P _J . We deduce (Proposition 3.3.1) that P _J ∩P _J′=P _J∩J′ if J,J′⊆I have a nonempty intersection and P _J ∩P _J′=K(x) if J∩J′=∅. Thus, setting P _i =P _I\{i} for i∈I, we conclude that \(\bigcap_{i\in I}P_{i}=K(x)\). The fields E=K(x) and P _i are the first objects of patching data (Definition 1.1.1) that we start to assemble.

Let K ₀ be a complete field under a discrete ultrametric absolute value and x an indeterminate. We prove that each finite split embedding problem over K ₀ has a rational solution. Thus, given a finite Galois extension K of K ₀ with Galois group Γ that acts on a finite group G, there is a finite Galois extension F of K ₀(x) which contains K(x) with Gal(F/K(x))≅G and Gal(F/K ₀(x))≅Γ⋉G such that res: Gal(F/K ₀(x))→Gal(K/K ₀) corresponds to the projection Γ⋉G→Γ. Moreover, F has a K-rational place unramified over K(x) whose decomposition group over K ₀(x) is Γ.

To construct F we choose finitely many cyclic subgroups C _i , i∈I, of G which generate G. For each i∈I we construct a Galois extension F _i =K(x,z _i ) of K(x) with Galois group C _i in K((x)). Then we consider the ring R=K{w _i |i∈I} as in Section 3.2, where \(w_{i}={r\over x-c_{i}}\), r∈K ₀, c _i ∈K, and |r|≤|c _i −c _j | for all i≠j, and shift F _i into the field P′ _i =Quot(K{w _i }) (Lemma 4.3.5). Choosing the c _i ’s in an appropriate way (Claim A of the proof of Proposition 4.4.2), we establish patching data \(\mathcal{E}\) with a proper action of Γ and apply Proposition 1.2.2 to solve the given embedding problem.

One of the major problems of Field Arithmetic was whether the absolute Galois group of every countable PAC Hilbertian field K is free of countable rank. By Iwasawa, that means that every finite embedding problem of Gal(K) is solvable. The PAC property of K implies that Gal(K) is projective, so it suffices to solve finite split embedding problems over K. Since K is Hilbertian, it suffices to solve finite split constant embedding problems over K(x), where x is transcendental over K. Since K is PAC, it is existentially closed in the field of formal power series \(\hat{K}=K((t))\). By Bertini-Noether, it suffices to solve each finite split constant embedding problem over \(\hat{K}(x)\). Thus, the initial problem of proving that \(\mathrm{Gal}(K)\cong \hat{F}_{\omega}\) is reduced to a problem that Proposition 4.4.2 settles.

The property of being existentially closed in K((t)) that each PAC field K has is shared by all Henselian fields. We call a field K which is existentially closed in K((t)) ample. In that case, the arguments of the preceding paragraph prove that each finite split constant embedding problem over K(x) is solvable (Theorem 5.9.2).

It turns out that ample fields can be characterized in diophantine terms: A field K is ample if and only if every absolutely irreducible curve over K with a simple K-rational point has infinitely many K-rational points (Lemma 5.3.1). Surprisingly enough, each field K such that Gal(K) is a pro-p group for a single prime number p has the latter property and is therefore ample (Theorem 5.8.3). On the other hand, the theorems of Faltings and Grauert-Manin imply that number fields and function fields of several variables are not ample (Proposition 6.2.5).

It is sometimes more difficult to give examples of objects that do not have a certain property P than examples of objects that have that property. A standard method to do that is to prove that each object having the property P has another property P’ and then to look for an object that does not have the property P’. For example, by Corollary 5.3.3, every ample field is infinite. Hence, finite fields are not ample. More sophisticated examples of nonample fields are function fields of several variables over arbitrary fields (Theorem 6.1.8(a)). Likewise we prove that if E/K is a function field of one variable and F is the compositum of a directed family of finite extensions of E of bounded genus, then F is nonample (Theorem 6.1.8(b)). The proof uses elementary methods like the Riemann-Hurwitz genus formula. We have not been able to prove that number fields are nonample by elementary means. We have rather used in Proposition 6.2.5 the deep theorem of Faltings (formerly, Mordell’s conjecture).

Section 6.3 surveys concepts and results on Abelian varieties, Jacobian varieties, and homogeneous spaces (the latter is applied only in 11.5). Likewise, Section 6.4 surveys the very deep Mordell-Lang conjecture proved by Faltings and others. As a consequence we prove that the rational rank of every nonzero Abelian variety over an ample field of characteristic zero is infinite (Theorem 6.5.2). That result combined with a result of Kato-Rohrlich (Example 6.5.5) leads to examples of infinite algebraic extensions of number fields that are nonample. Finally, we prove that for each positive integer d there is a linearly disjoint sequence K ₁,K ₂,K ₃,… of extensions of ℚ of degree d whose compositum is nonample (Example 6.8.9). The proof is based on the concept of the “gonality” of a function field of one variable that we establish in Sections 6.6 and 6.7 as well as on a result of Frey (Lemma 6.8.7) based on the Mordell-Lang conjecture.

Let K ₀ be a complete field with respect to an ultrametric absolute value. In Proposition 4.4.2 we considered a finite Galois extension K of K ₀ with Galois group Γ acting on a finite group G and let x be an indeterminate. We constructed a finite Galois extension F of K ₀(x) that contains K and with Galois group Γ⋉G that solves the constant embedding problem Γ⋉G→Gal(K(x)/K ₀(x)). Using an appropriate specialization we have been then able to prove the same result in the case where K ₀ was an arbitrary ample field (Theorem 5.9.2). This was sufficient for the proof that each Hilbertian PAC field is ω-free (Theorem 5.10.3).

In this chapter we lay the foundation to the proof of the third major result of this book: Giving a function field E of one variable over an ample field K of cardinality m, each finite split embedding problem over E has m linearly disjoint solution fields (Theorem 11.7.1).

Here we let K ₀ be as in the first paragraph, and consider a finite Galois extension E′ of K ₀(x) (where E′ is not necessarily of the form K(x) with K/K ₀ Galois) acting on a finite group H. We prove that the finite split embedding problem Gal(E′/K ₀(x))⋉H→Gal(E′/K ₀(x)) has a solution field F′. Moreover, if H is generated by finitely many cyclic subgroups G _j , then for each j there is a branch point b _j with G _j as an inertia group.

We generalize Theorem 5.9.2 and prove that if K ₀ is an ample field, then not only constant finite split embedding problems over K ₀(x) are solvable but every finite split embedding problem Gal(E/K ₀(x))⋉H→Gal(E/K ₀(x)) has as many linearly disjoint solution fields F _α , with α<card(K) (Proposition 8.6.3). Moreover, let K be the algebraic closure of K ₀ in E. Then each K-rational place φ of E unramified over K ₀(x) with φ(x)∈K ₀∪{∞} extends to a K-rational place of F _α unramified over K ₀(x) (Lemma 8.6.1).

The construction of the solutions for general finite split embedding problems over K ₀(x) in the case where K ₀ is an ample field relies on Proposition 7.3.1, where K ₀ is assumed to be complete under an ultrametric absolute value. For an arbitrary ample field K ₀, we first lift the embedding problem to one over K ₀((t))(x), and apply Proposition 7.3.1 to solve it with additional information on the branch points (in particular they should be algebraically independent over K ₀) and on their inertia groups. Then, we use Bertini-Noether as in Lemma 5.9.1 to reduce that solution to one of the original problem. In order to achieve many linearly disjoint solutions the reduction has to keep track of the branch points and their inertia groups. This can be done once the reduction is normal in the sense of Section 8.1.

Let C be an algebraically closed field of cardinality m, x an indeterminate, E a finite extension of C(x) of genus g, and S a set of prime divisors of E/C. We denote the maximal extension of E ramified at most over S by E _S . If X is a smooth projective model of E/C, then we interpret S as a subset of X(C), call Gal(E _S /E) the fundamental group of X\S, and denote it by π ₁(X\S). Starting from the fundamental group of the corresponding Riemann surface and applying the Riemann existence theorem, one proves that when r=card(S)<∞, Gal(E _S /E) is the free profinite group generated by r+2g elements σ ₁,…,σ _r ,τ ₁,τ′₁,…,τ _g ,τ′ _g with the unique defining relation σ ₁⋅⋅⋅σ _r [τ ₁,τ′₁]⋅⋅⋅[τ _g ,τ′ _g ]=1 (Proposition 9.1.2). Using Grothendieck’s specialization theorem, we generalize that result to an arbitrary algebraically closed field C of characteristic 0 (Proposition 9.1.5). In particular, if r≥1, then \(\mathrm{Gal}(E_{S}/E)\cong \hat{F}_{r+2g-1}\). When m=card(S) is infinite, we take the limit on all finite subsets of S to conclude that \(\mathrm{Gal}(E_{S}/E)\cong \hat{F}_{m}\) (Corollary 9.1.9). In particular, if S is all of the prime divisors of E/C, then card(S)=card(C) and we find that \(\mathrm{Gal}(E)\cong \hat{F}_{m}\) (Corollary 9.1.10). In particular, Gal(E) is projective (Corollary 9.1.11).

The situation is quite different when char(C) is a positive prime number p. We can not use the Riemann existence theorem to determine the structure of Gal(E _S /E). Indeed, if S is nonempty and of cardinality less than that of C, then Gal(E _S /E) is even not a free profinite group (Proposition 9.9.4) as is the case in characteristic 0. What we do know is the structure of the Galois group Gal(E _S,p′/E), where E _S,p′ is the maximal Galois extension of E ramified at most over S and of degree not divisible by p. Using Grothendieck’s lifting to characteristic 0, one proves that the latter group is just the maximal quotient of order not divisible by p of the corresponding group in characteristic 0 (Proposition 9.2.1). But, this does not help us to compute Gal(E). Instead, we prove by algebraic means that Gal(E) is a free profinite group of cardinality m. This proof works over every algebraically closed field and does not use the Riemann existence theorem.

The first step is to prove that Gal(E) is projective (Proposition 9.4.6). Our proof applies some basic properties of the cohomology of profinite groups. Then we use that every finite split embedding problem for Gal(E) has m solutions (Proposition 8.6.3) to conclude that \(\mathrm{Gal}(E)\cong\hat{F}_{m}\) (Corollary 9.4.9).

Interesting enough, the same arguments work if E is a finite extension of K(x), where K is a field of cardinality m of positive characteristic p and Gal(K) is a pro-p group. Thus, even in this case \(\mathrm{Gal}(E)\cong\hat{F}_{m}\) (Theorem 9.4.8).

Next we prove for each nonempty set of prime divisors of E/C that Gal(E _S /E) is projective (Corollary 9.5.8). In addition to the projectivity of Gal(E), the main tool used in the proof is the Jacobian variety of a smooth projective model Γ of E/C. The same tool helps us to prove that Gal(E _S /E) is not projective if S is empty (Proposition 9.6.1). The latter group can be interpreted as the fundamental group of Γ.

Finally we consider the case where E=C(x) and apply algebraic patching to solve each split embedding problems m times in E _S , first in the case that C is complete under an ultrametric absolute value and then when C is an arbitrary algebraically closed field. This proves that \(\mathrm{Gal}(E_{S}/E)\cong\hat{F}_{m}\) if card(S)=m (Theorem 9.8.5). This is an optimal result in characteristic p. In that case, Gal(E _S /E) is not free if card(S)<m (Proposition 9.9.4).

We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a “quasi-free profinite group” (Section 10.6).

A somewhat stronger condition is that of a “semi-free profinite group”. We say that G is semi-free if every finite split embedding problem for G with a nontrivial kernel has m independent solutions (Definition 10.1.5). The advantage of the latter notion on the former one is that the known conditions on a closed subgroup of a free profinite group of rank m to be free of rank m go over to semi-free groups. Indeed, even the method of proof that applies twisted wreath products goes over from free profinite groups to semi-free profinite groups (Section 10.3). Thus, every open subgroup of a semi-free group G is semi-free (Lemma 10.4.1), every normal closed subgroup N of G with G/N Abelian is semi-free, every proper open subgroup of a closed normal subgroup of G is semi-free, and in general every closed subgroup M of G that is “contained in a diamond” is semi-free (Theorem 10.5.3).

An application of the diamond theorem to function fields of one variable over PAC fields appears in the next chapter.

We prove that if K is an ample field of cardinality m and E is a function field of one variable over K, then Gal(E) is semi-free of rank m (Theorem 11.7.1). It follows from Theorem 10.5.4 that if F is a finite extension of E, or an Abelian extension of E, or a proper finite extension of a Galois extension of E, or F is “contained in a diamond” over E, then Gal(F) is semi-free.

We apply the latter results to the case where K is PAC and E=K(x), where x is an indeterminate. We construct a K-radical extension F of E in a diamond over E and conclude that F is Hilbertian and Gal(F) is semi-free and projective (Theorem 11.7.6), so Gal(F) is free. In particular, if K contains all roots of unity of order not divisible by char(K), then Gal(E)_ab is free of rank equal to card(K) (Theorem 11.7.6).

Following (Pop, 2010), we generalize and strengthen Theorem 5.11.3 and prove that the absolute Galois group of a Noetherian domain which is complete with respect to a prime ideal of height at least 2 is semi-free (Theorem 12.4.3).